Information, Optimizations and Systems Controls in Engineering

Advances in Information and Systems Sciences Volume 2

Information, Optimizations and Systems Controls in Engineering

Edited by

Jinkuan Wang, Qingling Zhang and Shuhua Zhang

ISSN 1916-176X ISBN 978-1-55195-221-5

December 2007 ©ISCI

Advances in Information and Systems Sciences Volume 2

Information, Optimizations and

Systems Controls in Engineering

———————————————— AiSS ————————————————

Advances in Information and Systems Sciences Volume 2ISSN 1916-176X

Institute for Scientific Computing and InformationPO Box 60632, UofA Postal OutletUniversity of AlbertaEdmonton, Alberta T6G 2S8Canada

Managing Editors:Yanping Lin and Qingling Zhang

J. Wang, Q. Zhang and S. Zhang

Information, Optimizations andSystems Controls in Engineering

2007 c©ISCIEdmonton, Alberta

Edited by

Jinkuan WangNortheastern University at QinghuandaoQinhuangdao, China

Qingling ZhangInstitute of System SciencesNortheastern UniversityShenyang, China

Shuhua ZhangDepartment of MathematicsTianjin University of Finance and EconomicsTianjin, China

ISSN 1916-176XISBN 978-1-55195-221-5

c©2007 ISCI All right reserved.ISCI, PO Box 60632, UofA Postal Outlet, University of AlbertaEdmonton, Alberta T6G 2S8 Canada

Preface

We are pleased to present this special volume, which is the proceedings of theConference on Information and Systems Sciences held in the Sub-Campus of North-eastern University at Qinhuangdao from 13 August to 16 August of 2006. This is thefirst of a series of conferences on information and systems sciences to be held in thePacific Rim region, which have the goal of bringing together mathematicians, scien-tists and engineers working in the field of information and systems sciences to solvescientific and industrially oriented problems and to provide a forum for the partic-ipants to meet and exchange ideas of common interest in an informal atmosphere.The focus of this particular conference was on the problems and methods relatedto financial applications, dynamical systems and scientific computing, decision andestimation, finance, economics and managements, complex systems, control theoryand systems, signal and image processing, scheduling, data mining, discrete eventsystems, and related areas in engineering and scientific applications. There weremore than 80 people from all over the world to attend the conference. This specialissue contains 27 representative papers selected from many high quality submissionsthrough a thorough peer-review process in accordance with the guidelines of theproceedings.

Jinkuan Wang, Qingling Zhang and Shuhua ZhangQinhuangdao, Hebei, P. R. China 2007

About the Editors

Dr. Jinkuan Wang, graduated in 1981 from Northeastern Uni-

versity (NEU), majoring in automation, got the master’s degree of au-

tomation in NEU in 1985, and the Ph.D from University of Electrical

Communication in Japan in 1993. In the same year he participated in

the research of laser radar, adaptive array antenna of mobile telecom-

munication in the Space Science Research Institute of the Ministry of

Education. Returning back to China in 1998, he served as the vice-

president of Northeastern University at Qinhuangdao (NEUQ) and

then in 1999, the president of NEUQ. As a professor and doctoral

supervisor, he is also a director of China Instrument and Meter Association. Research

interests include adaptive array signal processing, sensor network and intelligent control.

He has published more than 80 academic papers and 6 books.

Dr. Qingling Zhang, received the B.Sc. and M. Sc. de-

grees from the Mathematics Department and the Ph.D. degree

from the Automatic Control Department of Northeastern Univer-

sity, Shenyang, China, in 1982, 1986, and 1995, respectively. He

finished his two-year Postdoctoral work in Automatic Control De-

partment of Northwestern Polytechnical University, Xi’an, China,

in 1997. Since then, he has been a Professor and Dean of the Col-

lege of Science at Northeastern University. He is also a member

of the University Teaching Advisory Committee of National Ministry of Education. He

has published six books and more than 230 papers on control theory and applications.

Dr. Zhang received 14 prizes from central and local governments for his research. He

has also received the Gelden Scholarship from Australia in 2000. During these periods,

he visited Hong Kong University, Sydney University, Western Australia University and

Niigata University, Pohan University of Science and Technology, National Seoul Univer-

sity, University of Alberta, Lakehead University and University of Windsor as a Research

Associate, Research Fellow, Senior Research Fellow and visiting professor, respectively.

Dr. Shuhua Zhang, received the B.S., M.S., and Ph.D.

in computational mathematics from Tianjin Normal University,

Tianjin University, and Institute of Systems Science of Chinese

Academy of Sciences in 1982, 1989, and 1996, respectively. He is

now a professor of Tianjin University of Finance and Economics.

His research interest includes numerical methods for PDEs, pric-

ing financial derivatives, and optimal control problems. He has

published more than 50 papers, and won the second-class Tianjin

Natural Science Prize and the third-class Tianjin Natural Science

Prize in 2000 and 2006, respectively.

Contents

Preface

H. Liang, R. Chang, P. Yu, S. Li and Y. Gao, An overview of BGF/MPLSIP VPNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1F. Li and W. Jin, A new method of high accurate measurementfor superfine scale with virtual instrument technology . . . . . . . . . . . . . . . . . . . . . . . . 11F. Zhao, X. Li, P. Wu and L. Kong , Design of vision system for robotsoccer system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

H. Wang, D. Yang, C. Wang, Y. Zhao and Y. Gao, An optimizingalgorithm for automated service composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27J. Zhang, J. Gao, Y. Liu and J. Gai, The realization of constantpressure water supply drive system based on the fuzzy controller . . . . . . . . . . . . . 34J. Chen, L. Kong and J. Feng, Research on performance of replicationin grid system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40J. Yuan, X. Xu and Y. Gao, In integrated network managementsystem Q3 interface research and application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47S. Chen, L. Hou, L. Wang and F. Liu, Super-resolution imagereconstruction with edge-directed interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53M. Zhang, L. Chai and S. Fei, A H∞ control method for Lure’scontrol systems with time-delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59T. Ren and W. Jin, Development of distributed architecture withmulti-client singe server based on pxi for cold-rolled aluminum foil . . . . . . . . . . . 67W. Jin, X. Wang and F. Li, Measurement of diameters of electricalwires using laser with virtual instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73X. Wang, W. Jin and F. Li, The accurate comparison of calculatedmethod about light distribution in Fraunhofer diraction . . . . . . . . . . . . . . . . . . . . . . 81X. Wang and L. Kong, Resource discovery mechanism based onpeer-to-peer protocols in grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Y. Meng, J. wang, J. Zhu and X. Song, Semi-blind multiuser detectionbased on improved PASTD subspace tracking for MC-CDMA uplink . . . . . . . . . 91Y. Zhao, Y. An, D. Yang, H. Wang and Y. Gao, An adaptive modelfor live media service grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98Y. Han, J. Wang, X. Song and Y. Zhang, A fast and robustalgorithm for parameter estimation of distributed source . . . . . . . . . . . . . . . . . . . . 105F. Liu, J. Wang, X. Zhou and C. Li, Space-time smoothingtechnology for joint angle and frequency estimation . . . . . . . . . . . . . . . . . . . . . . . . . 114L. Yang. Q. Zhang, Z. Qiu and X. Yang, Dissipative control fora class of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Q. Wang, Y. Yuan and Z Qiu, Research on coal pulverizing system ofthe power plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128S. Tong, Q. Zhang, Y. Ma and C. Liu, Decentralized robust controlfor the uncertain nonlinear systems with input saturation . . . . . . . . . . . . . . . . . . . 138J. Liu, D. Zhang and S. Yang, Bifurcation analysis of a food chainmodel with B-D functional response in varying environment . . . . . . . . . . . . . . . . . 144S. Yang, Z. Yang and W. Liu, Decision of pricing for multi-portin the same hinterland based on game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152X. Yang, Q. Zhang, L. Li, X. Liu and S. Yang, Relaxed stabilityconditions for nonlinear fuzzy time delay systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 160X. Feng, X. Xu and X. Liu, Fuzzy rule extraction from fuzzy decisiontrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168Z. Feng, Y. Zhang and Y. Liu, Image encryption based on reversiblecellular automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178L. Li, W. Liu and X. Liu, H∞ control for parameter uncertain T-S fuzzysystems with time-delay in space and control input . . . . . . . . . . . . . . . . . . . . . . . . . .187N. Chen and Y. Zhang, Fuzzy clustering study on the driftingsea-ice in the north of Liaoning gulf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

Received by the editors June 2, 2006 and, in revised form, December 21, 2006 1

INTERNATIONAL JOURNAL OF

ages 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, P

ADVANCES IN INFORMATION AND SYSTEMS SCIENCES Volume 2, Pages 1-10

©2007 Institute for Scientific Computing and Information

AN OVERVIEW OF BGF/MPLS IP VPNS

HAIYING LIANG1,2, RUI CHANG2, PING YU2, SHUMEI LI2 AND YUAN GAO1

Abstract There are a number of types of the Virtual Private Network (VPN). One of the latest methods to build VPN has been defined in RFC4364 (i.e. BGP/MPLS IP VPNs). This approach allows providers to use a Server Provider (SP) backbone network to provide VPN services to their customers. In this paper, we firstly discuss the features of the two types of VPN, the Layer 2 VPNs (L2VPNs) and the Layer 3 VPNs (L3VPNs), then describe in detail that how Border Gateway Protocol (BGP) is used to exchange VPN routing information and how Multiprotocol Label Switching (MPLS) is used to forward VPN data traffic in BGP/MPLS IP VPN, finally enumerate advantages of BGP/MPLS IP VPNs by comparing with other VPN types, and conclude that the BGP/MPLS IP VPN is a better solution for both customers that seek VPN services and providers that offer VPN services due to its scalability, flexibility, availability, economy and privacy.. Key Words, BGP, MPLS, VPN, RT, VRF

1. Introduction

A VPN is a network where customers have connectivity across a shared infrastructure using the same access and security policies as a private network. In VPNs, there are two entities, the provider and the customer. The difference between the two entities is that the customer only operates one VPN whereas the provider potentially operates a vast number of VPNs. So, a good VPN solution minimizes the operational complexity and cost for both the network provider and its customers. Detailed requirements will be expressed from both the provider's and the customer's perspective.

2. The Classification of VPNs

VPNs come in two fundamentally different types, L2VPNs and the L3VPNs, by Internet Engineering Task Force (IETF). Fig.1 shows the classification of various VPN technologies.

2.1. L2VPNs

In L2VPNs [1], the provider extends L2 services to the customer sites but is unaware of L3 VPN information. The customer and the provider do not exchange any routing information with each other. Forwarding decisions in the SP network are based solely on L2 information such as MAC address, ATM VC identifier, and so on.

Two different approaches to L2VPNs are described in IETF, the Virtual Private Wire

Service (VPWS) and the Virtual Private LAN Service (VPLS). The VPWS approach can

2 H. LIANG, R. CHANG, P. YU, S. LI AND Y. GAO

be regarded as a generalized version of the traditional leased line service, in which the sites are connected in a partial or full mesh. Provider Edge (PE) devices provide a logical interconnect such that a pair of Customer Edge (CE) devices can be connected by a single logical L2 circuit. PE devices act as L2 circuit switches. L2 circuits are then mapped onto tunnels in the SP network. These tunnels can either be specific to a particular VPWS, or shared among several services. VPWS applies for all services including Ethernet, ATM, Frame Relay (FR), etc. In VPLS, PE devices provide logical interconnect such that CE devices belonging to a specific VPLS can be connected by a single LAN. The VPLS approach emulates a LAN environment where a site automatically gains connectivity to all the other sites attached to the same emulated LAN. In a VPLS, a customer site receives L2 service from the SP. The PE is attached via an access connection to one or more CEs. The PE performs forwarding of user data packets based on information in the L2 header, such as a MAC destination address.

2.2. L3VPNs In L3VPNs[2], according that the point where tunnels terminate and the functions are

implemented is the CE or PE devices, L3VPNs are differentiated into the two major categories, namely PE-based and CE-based L3VPNs.

In the CE-based L3VPN [3], typically a single level of tunnel terminates at pairs of CE

devices. Usually, a CE device serves a single customer site and therefore the forwarding and routing is physically separate from all other customers. Furthermore, the PE device is not aware of the membership of specific CE devices to a particular VPN. Hence, the VPN functions are implemented using provisioned configurations on the CE devices whereas the shared PE and Provider (P) network is used to only provide the routing and forwarding that supports the tunnel endpoints on between CE devices.

In a PE-based L3VPN, a customer site receives L3 service from the SP. The PE device

is attached through an access connection to one or more CE devices. The PE device performs forwarding of user data packets based on information in the IP layer header, such as an IPv4 destination address. The CE device sees the PE device as a L3 device such as an IPv4 router.

There are two dominating PE-based L3VPN approach, the BGP/MPLS IP VPNs and

the Virtual Router (VR). Both approaches concentrate the VPN function at the edge of the SP network and prevent VPN information from the provider core devices. A VR [4]

OVERVIEW OF BGP/MPLS IP VPNS 3

has exactly the same mechanisms as a physical router, and therefore can inherit all existing mechanisms for configuration, deployment, and operation. Multiple VRs can exist in a single physical device. VRs can be deployed in various VPN configurations. Direct VR to VR connectivity may be configured through L2 links or through a variety of tunnel mechanisms. VPN reachability information to and from customer sites can be dynamically learned from the CE device using standard routing protocols, or it can be statically provisioned on the VR. Since the VR-to-VR connectivity can use tunnels, the inter-VR routing protocol can be independent of the routing protocol used in the backbone network.

3. BGP/MPLS IP VPNs

In BGP/MPLS IP VPNs [5,6,7], CE routers send their routes to PE routers. BGP is then used by the service provider to exchange routes of a particular VPN among PE routers that are attached to that VPN. This ensures that routes from different VPNs remain distinct and separate. PE routers distribute, to CE routers in a particular VPN, routes learned from other CE routers in that VPN. CE routers do not peer with each other. Backbone core routers do not need to know VPN routes.

Two fundamentals flows occur in BGP/MPLS IP VPNs. A control flow is used for

VPN route distribution and Label Switched Path (LSP) establishment, and a data flow is used to forward customer data traffic. The control flow consists of two sub-flows. The first one is responsible for the exchange of routing information between CE and PE routers at the edges of the SP backbone, as well as the information distribution between PE routers across the SP backbone. The second control sub-flow is responsible for establishment of LSPs. BGP is used to exchange labelled VPN routes and MPLS is used to switch labelled VPN data traffic across the SP network. To illustrate the operation of BGP/MPLS IP VPNs, we consider the example showed in Fig. 2.

Fig.2 an Example of BGP/MPLS VPNs

In this figure, we show two VPNs, VPN A and VPN B, that consists of three customer

sites respectively. As an example, we assumed that CEB1 want to communicate with CEB3. Before we describe in detail how BGP is used to exchange VPN routing information and how MPLS is used to forward VPN data traffic in BGP/MPLS IP VPNs in following sub-sections of this section, we firstly discuss some technical terminologies about


BGP/MPLS IP VPNs.

3.1. BGP/MPLS IP VPNs Technical Terminologies 3.1.1. VPN Routing and Forwarding (VRF) Tables

A VRF table defines the VPN membership of a customer site attached to a PE router. Each PE router maintains per-VPN VRF tables and the IP addresses only need to be unique within each VRF table. The PE router is only required to maintain VPN routes for those VPNs to which it is directly attached. Only sites belonging to a VPN should be able to communicate with each other.

3.1.2. VPN-IPv4 Address Family

VPN customers often use the private address space [8] to manage their networks. If two VPNs have no sites in common, then they may have overlapping address spaces. Of course, within each VPN, each address must be unambiguous. BGP/MPLS IP VPNs support a mechanism that converts nonunique IP address into globally unique address by combining the use of the VPN-IPv4 address family with the deployment of Multiprotocol Extensions for BGP-4 (MP-BGP) [9]. A VPN-IPv4 address contains 12 bytes, of which 8 bytes represents a Route Distinguisher (RD) followed by a 4-byte IPv4 address prefix. A RD is an identifier configured on PE devices so that every SP can administer its own numbering space, without conflicting with the RD assignments made by any other service provider.

3.1.3. Route Target (RT) Attribute

Associating a particular RT attribute with a route, allows that route to be placed in all VRF tables that are used for routing traffic received from the corresponding sites. The RT attribute is BGP extended community attribute [10].

3.1.4. CE Routers

A CE router is a customer border device that connects the customer site via a data link, such as FR, ATM or leased line, to one or more PE routers. The CE device is a host or a router. Typically, the CE device is a router that establishes an adjacency with its directly connected PE routers. In this case, CE routers exchange routing information for network reachability of a customer VPN with PE routers using static routing, Interior Gateway Protocol (IGP) or External BGP (EBGP) between Autonomous Systems (ASs).

3.1.5. PE Routers

PE routers are border routers in the SP backbone network that attach directly to CE devices. The PE router first learns local VPN routes from CE devices, and then exchanges VPN routing information with other PE routers using Internal BGP (IBGP) within AS. 3.1.6. P Routers

P routers are core routers in the SP backbone network that does not attach to CE devices. P routers function as MPLS transit Label Switch Routers (LSRs) and will switch the VPN data traffic between PE routers based on the MPLS label. They need only maintain routing information to PE routers and not required to maintain the specific customer VPN routing information.


3.2. VPN Route Distribution via BGP 3.2.1. CE-PE Route Distribution

To enable inbound traffic, the CE router advertises local routes to the PE router. These routes are then converted to VPN-IPv4 routes, and exported to BGP. If there is more than one route to a particular VPN-IPv4 address prefix, BGP chooses the best one, using the BGP decision process.

For outbound traffic, the CE router may have a default route to the PE router. At egress

PE routers, BGP will again choose the best route for a particular VPN-IPv4 address prefix. Then the chosen VPN-IPv4 routes are converted back into IP routes, and imported into one or more VRF tables. Any route installed in a VRF table may be distributed to the associated CE routers.

3.2.2. PE-PE Route Distribution

If two sites of a VPN attach to PE devices that are in the same AS, PE devices can exchange these VPN-IPv4 routes with other PE devices by MP-BGP. Alternatively, each can have an MP-BGP connection to a Route Reflector (RR). Since VPNs may use overlapping address spaces, MP-BGP may learn routes to destinations with the same IPv4 address prefix by using VPN-IPv4 address family. Once PE devices have learned VPN routes from other PE devices, they must install only those routes for the VPNs to which PE devices are attached. This is done by tagging routes that are advertised by PE devices with the RT attribute. When MP-BGP distributes VPN-IPv4 route, it tags the route with its BGP next hop associated RT attribute and MPLS labels. BGP distributes labelled VPN routes [11]. The labels are carried by MP-BGP in the MP-REACH-NLRI and MP-UNREACH-NLRI attributes.

If two sites of a VPN are connected to different ASs, the PE routers attached to that

VPN use MP-BGP to distribute VPN-IPv4 routes. The PE routers use MP-BGP to redistribute labelled VPN-IPv4 routes either to an Autonomous System Border Router (ASBR), or to a RR of which an ASBR is a client. The ASBR then uses MP-BGP to redistribute those labelled VPN-IPv4 routes to an ASBR in another AS, which in turn distributes them to the PE routers in that AS.

Fig.3 Route Distribution


Fig.3 shows how to distribute route via BGP in BGP/MPLS IP VPNs. In the first step, the route for VPN A is distributed from CEB3 at site 3 to the PE router that this site is connected to, i.e. PE3, by BGP, static routing, OSPF, or RIPv2. In the second step, this route is exported into the provider’s BGP. The ingress router PE3 attaches the appropriate RT to the route. Further on, in step 3, this route is distributed to other PE routers using MP-BGP procedures. In step 4, at the egress router PE1, the route filtered by PE1 based on the RT carried by the route information is imported from the provider’s BGP. Finally in step 5, the route is distributed from PE1 to router CEB1, at site 1 of VPN A by BGP, or other protocol.

3.3. LSP Establishment via MPLS

One of the advantages of using BGP/MPLS is that P routers are not aware of VPN. The VPN routes are exchanged between PE devices and P routers are blissfully unaware of these routes. In order to use MPLS to forward VPN traffic across the SP backbone, a LSP must be established between the ingress PE router and the egress PE router. LSPs can be established and maintained across the SP network by using the Label Distribution Protocol (LDP), the CR-LDP, or the RSVP-TE. A provider uses LDP to establish a best-effort LSP between two PE routers. When a provider wants to either assign bandwidth to the LSP or use traffic engineering for selecting an explicit path, RSVP-TE or CR-LDP is used. Fig.4 shows a selected LSP via MPLS.

Fig.4 LSP Establishment

3.4. VPN Data Traffic Forwarding via MPLS To implement the hierarchy of routing knowledge, two levels of labels are used. The

outer label is associated with a route to an egress PE router, and therefore provides forwarding from an ingress PE router to the egress PE router. These outer labels could be distributed either via LDP, CR-LDP, or RSVP-TE. The inter label controls forwarding at the egress PE router. The inter label is distributed via MP-BGP, together with VPN-IPv4 routes.

The ingress PE router performs a destination address lookup in the VRF table to


identify the packet’s next hop. If the route does not exist in the VRF table, then the default forwarding table is consulted. If a match is found, IP packet that corresponds to the route that is the best match to the packet's destination address is encapsulated with the MPLS label. Two labels are pushed into the VPN packet, so that it is tunnelled across the SP backbone to the proper egress PE router. The P routers simply switch the labelled VPN packet based on the outer label. The penultimate LSR pops the outer label before handing over the labelled VPN packet to the destination PE router. The egress PE receives a labelled packet with the inter label, removes the label from the packet, and then forwards the packet to the appropriated CE router.

In Fig.5, the IP packet at site 1 of VPN A goes through CEB1, and then it arrives at PE1.

PE1 performs lookup in the appropriate table and finds the BGP route with Next-hop (i.e. PE3), inter label (the label that PE3 distributed to PE1 via BGP, such as 8) and outer label (the label that P1 distributed to PE1 via LDP, CR-LDP, or RSVP-TE, such as 9), then attaches two labels to the packet and sends the packet to P1 based on outer label, which in turn, uses the switched outer label (the label that P2 distributed to P1 via LDP, etc., such as 6), to make its forwarding decision and to forward the packet to P2. Since P2 is the penultimate hop, P2 pops the switched outer label before sending the packet to PE3. When PE3 receives the packet, it uses the inter label carried by the packet to make its forwarding decision. It then removes the label and sends the packet to CEB3.

Fig.5 VPN Data Traffic Forwarding

4. ADVANTAGES OF BGP/MPLS IP VPNS Since BGP/MPLS IP VPNs belongs to the PE-based L3VPNs, so in one hand, it has

several advantages that are different from L2VPNs and CE-based L3VPNs, in the other hand, the technology of BGP/MPLS IP VPNs is not similar to one of VRs that also belongs to PE-based L3VPNs. In following section, we enumerate the advantages of BGP/MPLS IP VPNs by comparing with other VPN types.

4.1. Advantages of BGP/MPLS IP VPNs by Comparing with L2VPNs

In L2VPNs, the provider determines what L2 technology to be used. If a customer


possesses sufficient knowledge, the required L2 equipment can be supplied and managed by the customer to set up the CE device. If not, the customer must buy additional equipment and management services from a third party, typically the provider. Besides the additional cost it will make the solution less flexible.

In L3VPNs, the provider offers L3 connectivity between customer sites. The customer

can optionally specify more advanced L3 topologies than simple partial of full mesh, such as intranet and extranet integration or hub-and-spoke. PE devices and CE devices exchange routing information. PE devices maintain separate routing contexts, one of each directly attached VPN. This increases the flexibility.

BGP/MPLS IP VPNs do not impose any protocol software upgrades. Therefore,

migration from an older VPN technology to BGP/MPLS can be done easily and quickly.

4.2. Advantages of BGP/MPLS IP VPNs by Comparing with CE-based L3VPNs A CE-based L3VPN is built by connecting customer sites through leased lines or

tunnel technology run on Customer Premises Equipment (CPE). The tunnel topology connecting the CE devices may be a full or partial mesh, depending upon VPN customer requirements and traffic patterns. These technologies are not cost-effective types for small enterprises and are not scalable for large enterprises.

The scalability in BGP/MPLS IP VPNs is obtained easily than CE-based L3VPNs. In

one hand, the SP backbone network consists of PE routers, BGP RRs, P routers, and ASBRs. P routers do not maintain any VPN routes. In order to properly forward VPN traffic, the P routers need only maintain routes to the PE routers and the ASBRs. The use of two levels of labels keeps the VPN routes out of the P routers. PE router maintains VPN routes, but only for those VPNs to which it is directly attached. RRs can be partitioned among VPNs so that each partition carries routes for only a subset of the VPNs supported by the service provider. Thus no single RR is required to maintain routes for all VPNs. For inter-AS VPNs, the ASBRs can be partitioned among VPNs in a similar manner, with the result that no single ASBR is required to maintain routes for all the inter-provider VPNs. As a result, no single component within the SP network has to maintain all the routes for all the VPNs. So the total capacity of the network to support increasing numbers of VPNs is not limited by the capacity of any individual component. This increases the scalability of the providers. In the other hand, a RD is an identifier used in BGP/MPLS IP VPNs to ensure uniqueness of address prefix among VPNs when multiple VPNs use the same address space. Thus it is allowing customers to efficiently use the private address space. A customer site only peers with one PE router attached to all other CE routers that are members of the VPN. This increases the scalability of customers.

4.3. Advantages of BGP/MPLS IP VPNs by Comparing with VR

Although BGP/MPLS IP VPNs and VRs are belong to PE-based L3VPNs, the main difference between VRs and BGP/MPLS IP VPNs residents in the model used to achieve VPN reachability and membership functions. In the VR approach, each VR has its own Routing Information Base (RIB) and separate MPLS data forwarding engine with its own device address space, which prevents one VR from affecting other VRs, hence consuming more memory resources.


In BGP/MPLS IP VPNs, the VPN network layer is terminated at the edge of the

backbone, and a backbone routing protocol (i.e., MP-BGP) is responsible for disseminating the VPN membership and reachability information between PE routers for all the VPNs configured on the PE routers. This decreases the cost of the resources.

5. CONCLUSION

BGP/MPLS IP VPNs is emerging as the popular choice by service providers to build VPN due to its scalability, flexibility, cost and the ability to provide VPN services across the SP network. They are more flexible, available and cost-effective than L2VPNs, more scalable than CE-based L3VPNs and more economical than VRs that also belong to PE-based L3VPNs. So BGP/MPLS IP VPN is a better solution for customers who seek VPN services and providers who offer VPN services. As the underlying and very promising MPLS technology and BGP technology used in VPNs are new and challenging topics, there are many open issues that still need to be researched.

Acknowledgments This work was supported by the National Science Foundation of China (NSFC) under the grant No. 60073059 and No. 60273078. References [1] W. Augustyn, and Y. Serbest, “Service Requirements for Layer-2 Provider Provisioned Virtual Private”,

Internet Draft, draft-ietf-l2vpn-requirements-07.txt, June 2006.

[2] M. Carugi, and D. McDysan, “Service requirements for Layer 3 Provider Provisioned Virtual Private

Networks (PPVPNs)”, RFC4031, April 2005.

[3] Jeremy De Clercq, Olivier Paridaens, “An Architecture for Provider Provisioned CE-based Virtual Private

Networks using IPsec”, draft-ietf-l3vpn-ce-based-03.txt, December 2005

[4] Paul Knight, Hamid Ould-Brahim,Bryan Gleeson, “Network based IP VPN Architecture Using Virtual

Routers”, draft-ietf-l3vpn-vpn-vr-03.txt, March 6, 2006

[5] E. Rosen and Y. Rekhter, “BGP/MPLS IP Virtual Private Networks (VPNs")”,RFC4364, February 2006

[6]E. Rosen, P. Psenak, and P. Pillay-Esnault, “OSPF as the Provider/Customer Edge Protocol for BGP/MPLS

IP Virtual Private Networks (VPNs)”，RFC4577, June 2006.

[7] E. Rosen, “Applicability Statement for BGP/MPLS IP Virtual Private Networks (VPNs")”, RFC4365,

February 2006

[8] Y. Rekhter, and B. Moskowitz, “Address Allocation for Private Internets”, RFC1918, February 1996.

[9] T. Bates, Y. Rekhter, R. Chandra, and D. Katz, “Multiprotocol Extension for BGP-4”,

draft-ietf-idr-rfc2858bis-10.txt, March 2006.

[10] Srihari R. Sangli, and Daniel Tappan, “BGP Extended Communities Attribute”, RFC4360, February 2006.

[11] Y. Rekhter, and E. Rosen, “Carrying Label Information in BGP-4”, RFC3107, May 2001.


d several papers in these areas.

Haiying Liang, School of Information Science and Engineering, Northeastern

University, China. Email: [email protected]. Ms. Liang is an associate

professor at School of Computer, Jilin Normal University, Siping, China. She received

her BSc degree in Computer Application Technology from Yanshan University, China

in 1991, MSc degree in Computer Application Technology from Northeastern

University, China in 2003. Now she is a doctor student in Computer Application

Technology at Northeastern University, China. She research interests are in the areas of

Network management. She has publishe

1School of Information Science and Engineering, Northeastern University, Shenyang, 110004, China 2School of Computer, Jilin Normal University, Siping, 136000, China Email: [email protected]



ges 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, PaVolume 2, Pages 11-19 INFORMATION AND SYSTEMS SCIENCES ADVANCES IN

Computing and Information ©2007 Institute for Scientific

A NEW METHOD OF HIGH ACCURATE MEASUREMENT FOR SUPERFINE SCALE

WITH VIRTUAL INSTRUMENT TECHNOLOGY

FEN LI AND WEI JIN

Abstract Based on diffraction theory, by means of the curve of luminance distribution, the calculation method and equipment with indirect measurement for filament and narrow-slit has been invented long ago. After using laser as a homochromy lamp-house, the measurement is more precise and becomes applications in superfine scale and grating reticle. Measuring the location of the count k level dark strip and calculating the superfine scale by formula is the tradition method. Taking into account of the measure characteristic, a new method is given to calculate the value of superfine scale by means of searching the least of standard difference of the curve of luminance distribution measured and the curve calculated on theory with setting value of superfine scale. In this method, the computer and the graph program technology with virtual instrument is used to search the best match between both curves in setting value subdivision automatically. So the value of superfine scale is calculated and determined correspondingly with best match, thus the accuracy of measurement has been improved.. Key Words, diffraction, luminance distribution, virtual instrument, precision measurement

1. Introduction

There are many optical methods known for measuring the superfine scale with high accuracy [1]. These methods are based on physical principle, including diffraction theory [1,2], scattering [3], interference [4,5] and shadow techniques [6].

According to diffraction theory, calculation method and equipment with indirect measurement for filament and narrow-slit has been invented long ago. After using laser as a homochromy lamp-house, the measurement is more precise and becomes applications in superfine scale and grating reticle. A new method improved was used for the measurement of diameter and form of cylinders, sphere and cubes under clean room or laboratory conditions and the uncertainty of outer diameter measurements was reached to U =10 nm per 100 mm [7]. By means of observing the curve of luminance distribution and measuring the location of the count k level dark strip and calculating the superfine scale by formula is the tradition method in which a laser was used as the source of light.

The customary model in the formula for calibrating the superfine scale, such as slots and thin wires, by optical diffraction is expressed as

12 F. LI AND W. JIN

(1) d sin ϕ = k λ where d is the slot width (or diameter of the wire), ϕ is the diffraction angle, k is the order of diffraction and λ is the wavelength of the radiation.

When dealing with laser beam, the source and observation planes are effectively at large distance from the strip object that is located at the diffraction plane, and so the Fraunhofer assumptions representation will often be adequate. The analysis detailed on diffraction pattern were studied in Ref. [1∼5].

A laser scanner and diffraction pattern detection were used in Ref. [4] for tracing a moving wire and gauging its diameter. Fiber curvature was examined for its effect on apparent measured fiber diameter in a double-diffraction based instrument that are in widespread use in the wool industry [2].

The Fraunhofer diffraction was compared with a numerical simulation based on the Lorenz–Mie theory to improve the performance of a laser diffraction instrument [3]. A new method of high accurate measurement for superfine scale is bring forward in this paper and with virtual instrument technology the method has been completed. The accuracy of measurement is improved.

Taking into account of the measure characteristic, this new method is given to calculate the superfine scale by means of searching the least value of standard difference of the curve measured first on luminance distribution on Fraunhofer diffraction and the curve calculated lately on theory with setting value of superfine scale. In this method, the computer and the graph program technology of virtual instrument were used to search the best match on both curves in subdivision automatically. So the measuring value of superfine scale was calculated when the best match or the least value of standard difference of both curves is reached, thus the measure accuracy was improved. 2. Theory

An recurred experiment system of Fraunhofer diffraction as depicted in Fig.1, consists of a 2 mW stabilized He–Ne laser (MELES QTA−260, CHINA) of nominal wavelength 632.8 nm. Fig. 1 Experiment of Fraunhofer diffraction

Laser lamp-house

Electrical

CCD

D

Screen

Camera To PC

HIGH ACCURATE MEASUREMENT 13

The electrical wire is perpendicular to the propagation axis, thus setting up a diffraction pattern, which is displayed on a projection screen. The receiver is a CCD camera. The collecting lens of the camera acquires the whole diffraction pattern. The camera converts the diffraction pattern to standard video signal with horizontal resolution of 460 TV lines (450 PAL) and directed it to the PC through frame grabber video card. The image of the pattern is printed, as shown in Fig.2. Intensity of light, the location of stripe, is the most at the center and there are the bright or the dark stripes along the bi-axes alternately,

The intensity I of light is defined as the time average of the amount of energy which crosses in unit time a unit area perpendicular to the direction of the energy flow. For a plane wave, according to [8]

(2) 22/1]/)[4/()( EcvI μεπω ==

where ν is the frequency of the incident radiation, ε is the dielectric constant, μ is the magnetic permeability, E is the electric vector and the quantity <E2> is a measure of the intensity.

(a) Photograph of Fraunhofer diffraction

(b) Location of stripe

Fig. 2 Photograph of Fraunhofer diffraction and Location of stripe

Since we are mainly concerned with monochromatic fields, the electric vector E is represented as

(3) ])()()[2/1()(),( * tjtjtj erAerAerAtrE ωωω −−− +=ℜ=

where A is a complex vector, (r) = k.r − δj; (j = 1,2,3), k is the propagation vector, δj’s

I

Location of bright

Location of dark

14 F. LI AND W. JIN

are phase constants and ω is the angular frequency. If two monochromatic waves E1 & E2 are superposed at some point P, then the total electric field at P is

(4) E = E1 + E2 So that (5) E2 = E1

2 + E22 +2 E1 E2

Hence the total intensity at P is (6) I = I1 + I2 + J12

where I1 = <E12> , I2=<E2

2> are the intensities of the two waves, and J12 =2 <E1. E2> =2[I1 I2]1/2cos δ is the interference term. Evidently, there will be maximum and minimum intensities given by

(7) Imax = I1 + I2 + 2[I1 I2]1/2; δ = 0, 2π, 4π,… and

(8) Imin = I1 + I2 + 2[I1 I2]1/2; δ = 0, 3π, 5π,… In the special case when I1 = I2, Eq. (6) reduces to (9) I = 2I1 (1+ cos δ)=4 I1 cos2 (δ/2)

and the intensity varies between a maximum value Imax=4 I1 , and a minimum value Imin = 0. A combined interference and diffraction pattern resulting from illuminating a vertical thin cylindrical wire by a collimated laser beam of known cross-section gives a system of equidistant interference fringes within the profile of diffraction pattern of the wire. Its central region closely resembles the case of non-localized fringes formed by double slits as in the case of Young’s experiment. From Babinet’s principle [8], the form of Fraunhofer diffraction pattern of the wire is the same as that of a slot. The wire is considered as a negative slot and the intensity at an angle θ from the axis is given as

(10) I1 = C2[sin B/B]2 where C is the amplitude, B=πd sin θ/λ and d is the diameter of the wire. It follows from Eqs. (9) and (10) that

(11) I = 4C2[sin B/B]2 cos2 (δ/2) Eq. (11) is the intensity of combined interference and diffraction effects, considering

Eqs. (10), B=πd sin θ/λ , when θ=0, I= I0 = Imax and when dsin θ=kλ ( k=±1, ±2, ±3,…), I=0.

It is to be noticed that the profile of the diffraction pattern of an opaque wire is characterized by symmetrically spread minima on both sides of the central maximum. When the diffraction pattern is displayed on a screen at a distance D from the wire, the point corresponding to an angle θ is at a distance X=D tanθ from the axis. But, considering first-order diffraction, we assume the following approximation for small angles [1]:

(12) sin θ ≈ tan θ ≈ θ = X/D


Thus, for a given wavelength, the intensity distribution in the diffraction pattern is (13) I1 = I0[sin2 (πdX/Dλ)]/(πdX/Dλ)2] The analysis confirms the experimental results of Young’s interference fringe pattern.

It remains to investigate light from a real source and understand the role it plays in interference effects. The distances of the various minima from the center of the pattern are then given by

(14) B = ±π,±2π,±3π,… or

(15) X = ±λD/d,±2λD/d,3λD/d,… where d is the diameter of the wire, λ is the wavelength, D is the distance between the wire and the screen and X is the mean fringe spacing. By X, the location of dark stripe, for example first location of dark stripe X1, the d is calculated as follows:

(16) d = λD/ X1 3. Experiment of data acquirement

In experiment of data acquisition which is converting the intensity of light into electrical analog signal, is shown in Fig.3, we adopt module components such as small drift amplifier and A/D transducer of high precision …etc. Take the displacement control as an example, displacement control block adopt a chip microcomputer to receive the control signal of computer via serial port, then transfer the signal into power signal to control the magnetic stepping motor, drive the sensor of photoelectric element moving on line.

The displacement control is on moving by 0.1 mm within each step, then collecting the signal of photoelectric sensor, acquiring the data of intensity distribute, and depositing into computer memory automatically.

Current

Fig. 3 Experiment of data acquirement

The next step is completing the data by computer software, including the calculation of absolute value, average value and the unitary value, the location of central, drawing

Slot

Laser

Sensor

Line shift

To PC

D

16 F. LI AND W. JIN

the curve of intensity distribute and outputting the calculation result of the value of superfine scale. Some of data are showed in Table 1.

Measuring the location of the count k level dark strip and calculating the superfine scale by formula, i.e. Eq. (16), is the tradition method. But there are some factors affecting the veracity of measuring the location and limiting the precision for the result of measuring superfine scale.

Table 1 Data Measured sequence 1 2 3 … n

distance 0 0.1 0.2 … X

1 150.3 … ―

2 150.6 … ― Measure

time 3 150.1 … ―

evenness 150.33 … ―

unitary 1.00 … ―

4. New arithmetic Taking into account of comparing the data of Table 1 and the data calculated on Eq (13), an ideal has been put forward to search the best match between them. The data calculated on Eq (13) are calculated upon setting d subdivision and other parameters are known before experimentation and calculation. The more precision the d settled each and every, the more precision in comparing between both the data it is. When the d selected on Eq (13) is more and more near real superfine scale, match of both data or curves is becoming close and close. We use the standard difference as the method to calculate the difference between the data of Table 1 and the data calculated on Eq (13), so the new arithmetic is as follows:

(17) ∑=

−=n

iimi II

1

21 )(

1-n1σ

where σ is the value of standard difference, Imi is the value measured, i.e. the data of Table 1, I1i is the value calculated on Eq (13), n is the number of samples. In order to find the real superfine scale in anticipant precision, we should selected in subdivision and let d increase step by step automatically from the value less than real value to the value great than real one. First the value of standard difference is bigger, i.e. biggest if at the jumping-off point to search, because d settled is far away from real value or more less than real value and both curves is more different. As the d increase in subdivision step by step, the value of standard difference is becoming littler and littler. At


the best match point, the value of standard difference is the most little. After best match, both curves will become more and more fall away and the value of standard difference will become bigger and bigger, as the d increase in subdivision step by step. Because we can celect d in subdivision or in more precision step, the result, d value, calculated is to be more precision. At the best match point, the d value settled is the best match with the real value. So the superfine scale is to be made sure. In this way, superfine scale by means of searching the least of standard difference of the curve of luminance distribution measured and the curve calculated by theory, the value of superfine scale is calculated correspondingly, thus the accuracy of measurement has been improved. In this method, the computer and the graph program technology of virtual instrument is used to search the best of value in subdivision automatically. An example, as shown in Fig.4, describes the process of the new calculation in which the curve on screen is the standard difference between the curve of luminance distribution measured and the curve calculated by theory. The least value, i.e. the bottom of the curve, is the searching result about d value with regard to Eq. (12). Subdivision of searching step is selected as 0.001mm.

Fig. 4 The experiment front panel of Virtual Instrument

5. Conclusion

A laser diffraction pattern is suitable for superfine scale, such as diameter of wire and width of slot, measurements and provides a simple and fairly accurate technique, but is

18 F. LI AND W. JIN

sensitive to environment. There are some factors affecting the veracity of measuring the location and limiting the precision for the result of measuring superfine scale. In comparison with most of the similar works on measuring superfine scale, it is the new method which is presented the first time on this paper that superfine scale is calculated by means of searching the least value of standard difference of the curve of luminance distribution measured and the curve of theory. The results obtained in this work are more accurate, the estimated uncertainty in our measurements could be decreased by improving the calculating. Using automatic calculating and analysis would improve the results and decrease the uncertainty of the measured values. References: [1] Tang W, Zhou Y, Zhang J. Improvement on theoretical model for thin-wire and slot measurement by optical diffraction. Meas Sci Technol 1999;10:N119–23. [2] Lebrun D, Belaid S, Ozkul C, Kuan Fang R, Grehan G. Enhancement of wire diameter measurements: comparison between Fraunhofer diffraction and Lorenz–Mie theory. Opt Eng 1996;35(4):946–50. [3] Zimmermann E, Dandliker R, Souli N. Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach. J Opt Soc Am A 1995;12(2): 398–403. [4] Butler DJ, Forbes GW. Fiber-diameter measurement by occlusion of a Gaussian beam. Appl Opt 1998;37:2598–606. [5] Krattiger B, Bruno AE, Widmer HM, Geiser M, Dandliker R. Laser-based refractive-index detection for capillary electrophoresis: ray-tracing interference theory. Appl Opt 1993;32:956–65. [6] Dobosz M. Measurement of fiber diameter using an edge diode beam of light. Opt Commun 1986;58(3):172–6. [7] Neugebauer M, Ludicke F, Bastam D, Bosse H, Reimann H, Topperwien C. A new comparator for measurement of diameter and form of cylinders, spheres and cubes under clean-room conditions. Meas Sci Technol 1997;8(8):849–56. [8] Born M, Wolf E. Principles of optics. New York: Pergamon; 1980.p. 256–381. [9] Krattiger B, Bruno AE, Widmer HM, Geiser M, Dandliker R.Laser-based refractive-index detection for capillary electrophoresis: ray-tracing interference theory. Appl Opt 1993;32:956–65. [10] Dobosz M. Measurement of fiber diameter using an edge diode beam of light. Opt Commun 1986;58(3):172–6. [11] Baxtar BP, Brims MA, Teasdale DC. The optical fiber diameter analyser (OFDA) new technology for the wool industry. Wool Conference, University of New South Wales,


Australia 1–2 October 1992, p. 131–4. Fen LI received the B. Eng degree in electronics from University on Broadcast and

TV, Liaoning. She is currently a senior teacher in experiment in Northeastern

University at Qinhuangdao, China. Her research interests include Photics and Control

& Measurement.

Center of Laboratory, Northeastern University at Qinhuangdao, Qinhuangdao, Hebei Province, P.R. China, 066004 Email: [email protected]



ges 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, Pa



DESIGN OF VISION SYSTEM FOR ROBOT SOCCER SYSTEM1

FENGDA ZHAO, XIANSHAN LI, PEILIANG WU AND LINGFU KONG

Abstract In recent years, robot soccer has become a popular research topic and been taken as a standard test bed for robotics and artificial intelligence. Vision system is one of the modules of robot soccer, and the tasks is to identity the robots and ball on the playground, and then provide the vision information to the decision-making system at every sampling time. With the ever increase in number of robots a bigger technical challenge is faced to the vision system. In this paper, two designs of color patch and searching algorithms for vision-based robot soccer system are proposed to better resolve the problems of accuracy and speed of identifying robots and conglutinations. Each type of design is used in a certain circumstance, so the two designs are compared after analyzing the experimental results. Then a selection approach is given and the performance of the designs are discussed. The practicability and efficiency has been proved in real competitions. Our team, YSU team, has won the second place three times in China Robot-Soccer Championship from 2003 to 2005. Key Words, Robot soccer, color patch, Searching algorithms

1. Introduction

In recent years, robot soccer has become a popular research topic and been taken as a standard test bed for robotics and artificial intelligence [1]. Vision system is one of the modules of robot soccer, and the task is to identity the robots and ball on the playground, and then provide the vision information to the decision-making system at every sampling time. The vision information includes the location and angle of heading of the team robots, the location of the opponent team robots, and the location of the ball. Whether the vision system can get the vision information accurately and fast in a real-time circumstance is crucial for a team to win a competition. The team of the Newton Labs, which won first place in the MiroSot’96 Miro-Robot World Cup Soccer Tournament, owed the success to the speed of their vision system [2].

According to the rules of MiroSot [3], two pieces of color patch are used to identify a

robot. One is called team color patch and only yellow or blue can be used to identify the robots in a team. The other is called robot color patch and many colors can be used, other than yellow, blue and orange (the color of the golf ball), used to identify the member in a robot team. In a vision system, the algorithms of tracking and identify an object rely largely upon the designs of color patch. At present, the literatures mainly relate to color modules and image segmentation [4][5][6]. In this paper, we first give the design

1 This work is partially supported by YDJJ Grant #2004018

DESIGN OF VISION SYSTEM FOR ROBOT SOCCER SYSTEM 21

approaches of two types color patch and the algorithms of computing the location and angle of heading in term of the shapes and ubiety of the two pieces of color patch. Then we analyze the searching approach based on the proposed color patch scheme. Finally, the experimental results and performance in our work are summarized and future work is presented.

2. Symmetrical color patch scheme

A. Design approach of symmetrical color patch scheme The symmetrical color patch scheme has been used widely for it is simple. The

character of this scheme is that both the team color patch and the robot color patch are regular geometrical shapes, such as square, rectangle and roundness. Further more, they are attached to the topside of a robot symmetrically. The center of a robot is not the center of either color patch, but the midpoint of the line, which linked by the two centers of the color patches. This kind of design is shown in Fig. 1.

Fig. 1 Symmetrical color patch scheme

B. Computation of location and angle of heading

For the center of a robot is the midpoint of the line, which linked by the two centers of the color patches, so the location (Xc, Yc) can be worked out by the following formulas:

2XtXrXc +

= (1)

2YtYrYc +

= (2)

where Xr and Xt represent the X coordinates of the robot color patch and the team color patch respectively, and Yr and Yt represent the Y coordinates of the robot color patch and the team color patch respectively. For a binary image (size is m×n), the size of a color patch can be figured out by a formula, as given in formula (3):

[ ]∑∑= =

=n m

BA1i 1j

ji, (3)

where A is the size of a color patch, when the pixel is in the color patch, otherwise is 0. Then the center coordinate (X, Y) of a color patch are given as,

22 F. ZHAO, X. LI, P. WU AND L. KONG

[ ]

A

jijBX

n

i

m

j∑∑= == 1 1

, (4)

[ ]

A

jiiBY

n

i

m

j∑∑= == 1 1

, (5)

From the above formulas, we can see that the center coordinate of a team color patch

or a robot color patch is the average of the coordinate of all pixels in the color patch. As shown in Fig. 1, the angle of heading of a robot is the direction of a radial from the

team color patch to the robot color patch after contrarotating 45°, and given as,

4tan 1 πθ −⎟

⎠⎞

⎜⎝⎛

−−

XtXrYtYr－＝ (6)

Where denotes angle of heading of a robot, Yr, Yt, Xr and Xt are the same as in formula (1) and (2). 3. Unsymmetrical color patch scheme A. Design approach of unsymmetrical color patch scheme

The character of unsymmetrical color patch scheme is that the center of a robot is just the center of a team color patch, and the purpose is to avoid the mixture of two different colors, unlike the symmetrical color patch scheme. Fig. 2 shows the design approach.

From Fig. 2, we can see there is a little interspace between the team color patch and the robot color patch, and the purpose is to avoid the mixture of two different colors.

Fig. 2 Unsymmetrical color patch scheme

B. Computation of location and angle of heading To calculate the center of the team color patch, the statistical approach is used here.

First, partitioning the patch into eight regions, as shown in Fig 3. Then, Calculating the sum of X coordinate and the sum of Y coordinate of all pixels in each region, at the same time, calculating the number of all pixels in the team color patch. At last, the center coordinates (Xt, Yt) are given as,


821

821

AAA

AAA

nnnxxx

Xt+++

+++= ∑ ∑∑

L

L

(7)

821

821

AAA

AAA

nnnyyy

Yt+++

+++= ∑ ∑∑

L

L

(8)

where ∑ Aixand ∑ Aiy

denote the sum of X coordinate and the sum of Y

coordinate of all pixels in region Ai（i＝1, 2, …, 8）respectively, and is the number of pixels in region Ai. Thus, the location of a robot is got, because (Xc, Yc) = (Xr, Yr) according to the unsymmetrical color patch scheme.

Ain

The calculation of angle of heading is much more complex than that of symmetrical color patch scheme. One of the algorithms is angle-compensating algorithm, which is too complicated in computation to satisfy the requirement of real time. A method of least squares is proposed in this paper to calculation the angle of heading. Compared with the angle-compensating algorithm, this algorithm can reduce the computation to ensure the real-time character of vision system, and the accuracy can be got at the same time. The experimental results will prove this. The main idea of this algorithm is making use of the symmetry character of every two regions of the partitioned team color patch. From Fig. 3, we can see A1 and A7, A3 and A5, A4 and A6, A2 and A8 are symmetrical respectively. The steps of this algorithm is listed simply as following:

Fig. 3 Division of a team color patch

Step 1: Calculating the center of the each region according to the method mentioned former, and Aci (i＝1, 2, …, 8 )representing the center coordinates of each region is got.

Step 2: Calculating the midpoints of the lines of Ac1Ac7, Ac3Ac5, Ac4Ac6 and Ac2Ac8.

Step 3: Using the method of least squares to get the slope based on the four midpoints, and then the angle of heading is got.

Since the method of least squares is widely used, this paper will not give the detailed descriptions. 4. The algorithms of searching color patch in vision system In order to identify the robots, the vision system will search the color patches in the grabbed image. For the color patches are made up of team color patch and robot color


patch, the vision system has two choices to search the color patches: team color patch first or robot color patch first.

The function of searching the team color patches first is the vision system is capable of finding all the robots on the playground without losing any robot for the team colors, yellow and blue, are easy to identify. However, the disadvantage of this approach is the vision system does not enable to make certain the ID of home robots only by the team color patches. So the additional work for the vision system is to make sure the ID of home robots after several times pixels matching when searching the team color patches, and this will prolong the vision processing time. Furthermore, the window tracking method, which can speed the vision system by searching the objects in a dynamic window not in the whole image cannot be used, this will restrict the speed of the vision system much more.

The advantage of searching the robot color patches first is the vision system has make sure which one of home robots will be searched, so re-matching is not needed. At the same time, the window tracking method can be used in this searching approach. However, a problem is unavoidable when using this approach, which is there is more than one robot color as a searching result when the opponent team uses the same robot colors with ours or the similar robot colors to ours. The resolving method is to identify the home team color near the result colors, if found, then the vision system can be sure that the result color is home robot color. 5. Experiments A. Hardware and software configurations

A conventional NTSC color camera is used in our platform., and the image size is 640×480.

A MyVision frame grabber is used and it is capable of grabbing at a full 30Hz frame rate.

The host CPU is Pentium4 1.7GHz. Operating software is Windows2000.

B. The experimental results

The effectiveness of design of the color patch can be quantitatively determined with respect to the criteria, namely accuracy, rate and reliability.

To test the accuracy and rate, we placed five robots at three specific positions with specific headings on a playground used in FIRA Middle League MiroSot. Then we recorded the data in 1000 vision samplings, and the data showed in Table 1 are the averages.

Table 1 The Accuracy and Rate Results

(X1, Y1, θ1) (X2, Y2, θ2) (X3, Y3, θ3) Rate (ms)

Real Data (55, 30, 0) (110, 90, 60) (165, 150, 90)

Symmetrical color patch (53, 29, 3) (113, 92, 58) (164, 147, 92) 7ms

Unsymmetrical color patch (54, 30, 2) (111, 89, 61) (167, 151, 89) 11ms


To test the reliability, we take the times of conglutination of the robots as a

criterion. We recorded the times of conglutination of the robots in a real competition in five minutes, and Table 2 shows the results.

Table 2 The Reliability Results

Total Times Average (times/minute)

Symmetrical color patch 18 3.8

Unsymmetrical color patch 10 2

C. The analyses of the experimental results From the above experimental results, we can draw some conclusions:

1. The rate of symmetrical color patch is faster than the unsymmetrical color patch. The first reason is the computation in the symmetrical color patch is simpler than the other, and second is the window tracking method is used in the symmetrical color patch.

2. The unsymmetrical color patch has better accuracy than the symmetrical color

patch. The reason is the center of the robot is just the center of the team color patch, so the recognizing results only depend on the single color patch.

3. There are more conglutinations in the symmetrical color patch than in the

unsymmetrical color patch. The reason is they are symmetrical between the team color patch and the robot color patch, so it is easy to conglutinate when two or more robots meet. But conglutinations are inevitable in robot soccer game.

D. The selection of the two color patch designs

In general, we select the color patch design according to the lighting conditions and the opponent color patch design. If the lighting conditions are well and the opponent robot colors are not same with us, especially the opponent uses the unsymmetrical color patch, we can use the symmetrical color patch design. The purpose is to get a fast vision processing speed. Otherwise, we should choose the unsymmetrical color patch design to decrease the conglutinations. Despite this will effect the speed of the vision system, the accuracy and reliability can be got.

So the proposed two designs of color patch are used in different circumstance. In fact, our team uses both of them in our vision system. We will select an appropriate one in a real competition. 6. Conclusion

With the ever increase in number of robots a bigger technical challenge is faced to the vision system. In this paper, two designs of color patch, symmetrical color patch scheme and unsymmetrical color patch scheme, are proposed. After analyzing the experimental results, the characters and the selection of the two designs are given. With the proposed designs of color patch, our MiroSot team, YSU team has won the second place three times, one is in Master Cup Chinese Robot Competition held in Beijing, in


August 2003, another two are in the 5th and 6th China Robot-Soccer Championship, held in Wuhan, in July, 2004, and in Chengdu, in July, 2005 respectively. However, vision system is a complex system, further research will be continued.

References [1] Kyung-Hoon Kim, Kut-Won Ko, The Development of a Micro Robot System for Robot Soccer Game, Proceedings of the 1997 IEEE International Conference on Robotics and Automation, 1997. [2] R. Sargent, B. Bailey, C. Witty and A. Wright, Importance of fast vision in winning the First Micro-Robot World Cup Soccer Tournament, Robotics and Autonomous Systems v21 n2 Sep 10 1997, pp139-147. [3] http://www.fira.net [EB/OL]. [4] Gordon Wyeth and Ben Brown, Robust adaptive vision for robot soccer, Compute Scinece and Electrical Engineering, University of Queensland, Brisbane, Australia 4072. [5] C. S. Hong, S. M. Chun, J. S. Lee and K. S. Hong, A vision-guided object tracking and prediction algorithm for soccer robots, Proc. IEEE Int. Conference on Robotics and Automation, 1997, pp346-351. [6] James Bruce, Tucker Balch and Manuela Veloso, Fast and cheap image segmentation for interactive robots, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 15213

Northeastern University at Qinhuangdao, Qinhuangdao, Hebei Province, P. R. China, 066004




ADVANCES IN INFORMATION AND SYSTEMS SCIENCES Volume 2, Pages 27-33.


AN OPTIMIZING ALGORITHM FOR AUTOMATED SERVICE COMPOSITION

HUI WANG, DEGUO YANG, CUIRONG WANG, YUHUI ZHAO AND YUAN GAO

Abstract We present an architecture to facilitate automated discovery, selection, composition and execution of multimedia Web services. Then, we describe an algorithm optimizing the discovery process for composed semantic web services. The algorithm can be used to improve discovery of appropriate component services at invocation time. It performs semantic matchmaking of goals of a composed service to appropriate component services at publishing time. The semantic discovery problem at invocation time is therefore reduced to a selection problem from a list of available (already discovered) component services matching a goal of the composed service. Key Words Software Agents, Web Services, Services Composition

1. Introduction

Digital multimedia is permeating and enriching every aspect of our life. Webcams, surveillance cameras, on-line radio/TV portals, and media-enabled personal communication devices are being widely adopted. Meanwhile, modern computing platforms, ranging from desktops, PDAs, to digital appliances, are ready to process and display multimedia data of varying format and quality. Much recent networking research has been based on the assumption that a large number of heterogeneous and likely mobile network-enabled devices may be used to access both static and streaming media content over the Internet. Such devices may be connected to the network via one or more instances of an ever-growing range of last-hop connection technologies (e.g. GPRS, Wi-Fi, ADSL), and may host any number of media display applications (e.g. realplayer, mediaplayer). Users are also a source of heterogeneity, for example in terms of languages, monetary budget, trust-levels, and other preferences. Given the broad heterogeneity of users, applications, devices and their network interfaces, certain items of multimedia content may be required to be adapted, filtered, or transformed in some way before they can be delivered according to cost and/or QoS constraints, or properly displayed to the user. It is unrealistic to expect that a single adaptation service is able to perform all of the required media processing operations. In general, composing multiple multimedia services (also called services in the rest of this paper), rather than accessing a single service, is essential and provides more benefits to users. Therefore, there should be a program or a software agent which can automatically combine existing services together in order to fulfill the request.

This need strongly calls for a modular design approach, which the Service-Oriented Architecture (SOA) paradigm tends to provide. In this approach, mostly illustrated by the Web service architecture, applications are built using independent, loosely-coupled pieces of software (called services) that achieve a specific, coarse-grained functionality.

28 H. WANG, D. YANG, C. WANG., Y. ZHAO AND Y. GAO

Each service is developed independently, and can be invoked by other services or clients using a declarative description of access points and accepted messages. Using a service-oriented approach, a composite service obtained by combining available services might be used. Discovering the component services, adding the component services to a composite service, triggering the composite service for execution, and monitoring the execution in case of exception handling are among the operations that users will have to be responsible for. Within this big picture, the paper describes an optimization of semantic discovery in the context of composed web services.

The optimization service discovery algorithm we describe in this paper is a part of our overall distributed service discovery approach [1]. During the service discovery phase, taking a user’s service request to perform a composed service which is compared by a set of component services. If a composed service identifies its component services using time consuming semantic discovery techniques each time it is invoked, discovery may become a bottleneck. In this paper, we focus on how to resolve overcome this problem. We propose to make the goals of the composed service used to discover component services part of its public description. This would allow the application of a new discovery algorithm, which reduces the efforts to discover component services at invocation time significantly.

The paper is organized as follows. In section 2, we review related work. Section 3 overviews the system framework. In section 4, we describe in detail an algorithm optimizing the discovery process for composed semantic web services. In section 5, we evaluate our framework and compare our algorithm with a simple automated back-tracking composition algorithm. Finally, we present conclusions and future work.

2. Related Work

Workflow composition in Service-Oriented Architectures has been the focus of much research recently, although the composition problem has been actively investigated in the database and agent communities [2-5]. Workflow composition is an active field of research and are most closely related to our research. Workflow composition can be broadly classified into three categories: manual, semi-automated and automated composition. Manual composition frameworks [6, 7, 8] expect the user to generate workflow scripts either graphically or through a text editor, which are then submitted to a workflow execution engine.

Semi-automated composition techniques [9, 10, 11, 12] are a step forward in the sense that they make 'semantic suggestions' for service selection during the composition process; the user still needs to select the service required from a shortlist of the appropriate services and link them up in the order desired. Automated composition techniques [13-16, 17] automate the entire composition process by using AI planning or similar technology.

An important component of automated composition is the discovery of the services required. Research in this area has focused on the use of DAML-S to describe service capabilities. A matchmaker engine then compares the DAML-S description of the service requested with those of the services advertised [18, 19]. This work is extended by Sycara et. al. [20] to propose a preliminary discovery and execution framework based on AI planning technology, the DAML-S Matchmaker and the DAML-S Virtual Machine. The framework assumes a composition plan is available and hence concentrates on identification of suitable services and execution of the plan. Additionally, the framework does not incorporate the use of a workflow repository. Hence, a workflow has to be

AN OPTIMIZING ALGORITHM 29

recomputed each time a request is received. All workflows are based on implemented services available. As a result, the workflows are not reusable or cannot be shared as there is no guarantee that any component service will be available in the future. 3. Framework Overview

In this section, we outline the general requirements for the framework and present the proposed architecture.

Fig. 1 The distributed service discovery framework.

The framework architecture is shown in Figure 1. Everything is a service. Services are described and invoked based on their descriptions. The framework consists of two repository and five supporting service agents: Services Repository (SR), Process Repository (PR) , Plan Agent, Composition Agent, Management Agent, Service Agent.

The Services Repository stores Web services information that are registered by Web service provides using publish service. Process Repository store process plans (composite service) that are compose of a set service. The process plans generated during service discovery process or at service publishing time. The Plan Agent accepts an incoming user-specified high-level goal. At this level, all tasks and their inputs and outputs are referred to by their logical names. The Plan Agent will typically query the Process Repository to ascertain if the same request has been processed previously. If so, the specification of process plan will be transmitted to the Composition Agent. If not, a request will be made to the reasoning service to retrieve a process template from the rule base which can satisfy the request. If a process template is not available, an attempt will be made to retrieve a combination of tasks that provide the same functionality based on the inputs, outputs, preconditions and effects. The same process will apply to all constituent services. An process plan is generated and store at Process repository(PR) and the specification of the process plan are transmitted to the Composition Agent. Then Composition Agent, Management Agent and service Agent cooperate with each other to execute the composition service instance according to the specification of the process plan. 4. The Optimization Algorithm

Semantic discovery of Web Services means semantic reasoning over a knowledge base where a goal describes the required web service capability as input. Semantic discovery adds accuracy to the search results in comparison to traditional Web Service discovery


techniques, which are based on syntactical searches over keywords contained in the web service descriptions [8], [9]. The additional accuracy of a match is expensive in terms of required computational power. The expensiveness of semantic matchmaking has several aspects influencing the design of the Semantic Web Services infrastructure in different ways. Besides the quality of the semantic match, we consider the following two aspects of outmost importance: Response time Scalability

Response time specifies how long semantic discovery takes to find a set of web services matching a goal. This criterion can restrict or even prohibit service usage in certain scenarios, where a Web Service has to be discovered and directly invoked and the total time for discovery and invocation has to be short. For instance, a composed service may semantically discover required component services at runtime. If potential service requesters of the composed service expect short response times this service becomes unusable to them even if the functionality offered by the service meets their needs.

Scalability specifies how many semantic discovery requests can be processed within a given unit of time. If a composed service identifies its component services using time consuming semantic discovery techniques each time it is invoked, discovery may become a bottleneck.

A practical approach to the above problems is to execute semantic discovery only if necessary, e.g. by saving results from former searches in the Process Repository in our framework. We propose to make the goals of the composed service used to discover component services part of its public description. This would allow the application of a new discovery algorithm, which reduces the efforts to discover component services at invocation time significantly. Using the algorithm described below, the discovery process during the service invocation will take a constant time in order to find a set of services matching the goal, so the overall process identifying component services takes a constant time for semantic discovery plus the time for the selection from a list of services matching the goal.

If the goals are known, the discovery process can be executed by a registry at publishing time and the list of the available services matching the goal can be stored (linked in some way to the goal) in the plan repository. This means that, if the service sends a discovery query with the goal to a plan agent, the plan agent first tries to find the goal and returns in case of success the list of process plans linked to it. Only if the goal could not be found, the plan agent starts the semantic discovery in the “traditional way”. This does not yet solve the problem of detecting new, changed and removed services possibly impacting the functionality of the process plan, since the service could make the discovery once and store it for the further use internally. So what is needed is a way to update the list of available process plans linked to the goal if something changes (e.g., a new service is added to the registry, or an existing service is changed or removed).

We propose using the process repository (SR) to store associations (links) between explicitly described goals in the Web Service Description(WSD) of a process plan and the WSDs of potential component services (containing matching service capabilities).

The following algorithm performs semantic discovery and keeps the goal-service association storage consistent for any changes of goals or service capabilities published to the registry.

publishService(wsd)


choose (wsd) newPlan: perform InsertNewPlan, InsertNewService; updatePlan: perform UpdatePlan, UpdateService; deletePlan: perform DeletePlan, DeleteService; InsertNewPlan: Extract goal descriptions from the service capability description of the process plan and use the PlanAgent to find matching service capabilities. Store the lists of found service capabilities (together with the goals) to the PR i.e. associate them with the matching goals. UpdatePlan: Remove old associations (between goals and service capabilities) from the PR, find and store new associations in the PR. Further optimization possible, e.g. compare the new and old WSD version to find differences between old and new goals. If a goal has not changed no action is required, i.e. no reasoning is required and the already existing association remains valid. DeletePlan: Remove associations for the goals contained in the WSD from the PR. 5. Implementation and Evaluation

In this section, we first describe the implementation of our framework and provide some preliminary results of performance evaluation experiments. We use existing Semantic Web tools, like RDF, OWLS, and reasoning mechanisms to provide the basic representation and reasoning technology. Services are described using OWL-S. The repositories are currently implemented as flat files containing the OWL-S profiles and process models.

The purpose of our experiments is two-fold: to assess plan generation times and success rates. We first evaluate the time for the discovery process during the service invocation (see Figure 2). There is an exponential relationship between the number of services in a process plan and the time required to match and compose an executable graph. We also looked at the process plan generation success rates (see Figure 3). We compared our framework with a simple automated back-tracking composition algorithm. We used the same process ontology and composition requests for both of these techniques. Based on 50 requests, the results show that our algorithm had a success rate of 80% while the back-tracking algorithm had a success rate of about 60%. This confirms our expectation that our optimizing algorithm has a higher probability of successfully generating composition services. Further experiments are currently being done to ascertain the range over which this results apply.

0

20

40

60

80

100

120

10 20 30 40 50 60

Number of services

Plan generation time

(ms)

Fig. 2. Plan Generation Times.


0

0.2

0.4

0.6

0.8

1

optimizing automatedPlan generation rate hhhh

Fig. 3. Plan Generation Success Rates.

6. Conclusion The introduced approach demonstrates for composed services the semantic

matchmaking between goals and service capabilities can be executed at the publishing time instead of invocation time. This approach reduces also the total number of semantic queries due to buffering of existing matches in the process repository. The introduced algorithm ensures that the goal-service association storage can be kept consistent for any changes of goals or service capabilities. The work of this paper is a part of our ongoing research work, which aims to provide an open reusable infrastructure for essential service composition mechanisms. In the future, we plan to put more effort on improving our service discovery model and to deploy this model in a real web service integration environment to observe its practicality. References [1] Hui Wang, Deguo Yang ,Yuhui Zhao, and Yuan Gao. “Ubiquitous Media Agents for Dynamic Services

Composition”. The First International Symposium on Pervasive Computing and Applications”(SPCA2006). [2] Zakaria Maamar, Soraya Kouadri Moste′ faoui, and Hamdi Yahyaoui. “Toward an Agent-Based and

Context-Oriented Approach for Web Services Composition”. In IEEE Transactions on Knowledge and Data

Engineering,, VOL. 17, NO. 5, MAY 2005. [3] Kashyap, V. and A. Sheth. “Semantic Heterogeneity in Global Information Systems: The Role of Metadata,

Context and Ontologies”, Academic Press.

[4] Preece, A.D., K.Y. Hui, Gray, W.A., Marti, P., Bench-Capon, T.J.M., Jones, D.M., and Cui, Z. “The KRAFT

Architecture for Knowledge Fusion and Transformation”. 19th SGES International Conference on

Knowledge-based Systesm and Applied Artificial Intelligence (ES’99) , Springer, Berlin

[5] Bayardo, R.J., W. Bohrer, R. Brice, A. Cichocki, J. Fowler, A. Helal, V. Kashyap, T. Ksiezyk, G. Martin, M.

Nodine, M. Rashid, M. Rusinkiewicz, R. Shea, C. Unnikrishnan, A. Unruh and D. Woelk: InfoSleuth.

“Agent-Based Semantic Integration of Information in Open and Dynamic Environments”. Proceedings of the

ACM SIGMOD International Conference on Management of Data, ACM Press, New York. pp. 195-206.

[6] Taylor, I., Shields, M., Wang, I., and Philp, R. “Distributed P2P Computing within Triana: A Galaxy


Visualization Test Case.”. in the IPDPS 2003 Conference, April 2003

[7] IBM Alphaworks, BPWS4J, http://www.alphaworks.ibm.com/tech/bpws4j

[8] Benatallah, B., Sheng, Q.Z., and Dumas, M. “The Self-Serv Environment for Web Services Composition”,

Jan/Feb, 2003, IEEE Internet Computing. Vol 7 No 1. pp 40-48.

[9] Cardoso, J. and Sheth, A. 2002. “Semantic e-Workflow Composition”. Technical Report, LSDIS Lab,

Computer Science, University of Georgia.

[10] Chen, L, Shadbolt, N.R, Goble, C, Tao, F., Cox, S.J., Puleston, C., and Smart, P. 2003. “Towards a

Knowledge-based Approach to Semantic Service Composition”. 2nd International Semantic Web Conference

[11] Sirin, E., Hendler, J., and Parsia, B. “Semiautomatic composition of web services using semantic

descriptions”. In Web Services: Modeling, Architecture and Infrastructure Workshop in conjunction, ICEIS

2003

[12] Stevens, R.D., Robinson, A.J., and Goble, C.A “myGrid: Personalised Bioinformatics on the Information

Grid”. Bioinformatics Vol. 19 Suppl. 1 2003, (Eleventh International Conference on Intelligent Systems for

Molecular Biology)

[13] Wu, D., Sirin, E., Hendler, J., Nau, D., and Parsia, B. 2003. “AutomaticWeb Services Composition Using

SHOP2”. Twelfth World Wide Web Conference.

[14] Sheshagiri, M.,desJardins, M., and Finin, T. 2003. “A Planner for Composing Service Described in

DAML-S”. Workshop on Planning for Web Services, International Conference on Automated Planning and

Scheduling.

[15] Deelman, E., J. Blythe, Y. Gil, C. Kesselman, G. Mehta, K. Vahi, K. Blackburn, A. Lazzarini, A. Arbree,

Cavanaugh, R. and Koranda, S. “Mapping Abstract Complex Workflows onto Grid Environments”. Journal of

Grid Computing.Vol. 1, 2003

[16] McIlraith, S. and Son, T.C. 2002. “Adapting golog for composition of semantic web services”. In Proc. of

the 8th International Conference on Knowledge Representation and Reasoning (KR ’02), Toulouse, France.

[17] Motta, E., Domingue, J., Cabral, L. and Gaspari, M. “IRS-II: A Framework and Infrastructure for

Semantic Web Services”. 2nd International Semantic Web Conference (ISWC2003) 20-23 October 2003,

Sundial Resort, Sanibel Island,Florida, USA

[18] Paolucci, M., Kawamura, T., Payne, T., Sycara, K. “Semantic Matching of Web Services Capabilities”.

Proceedings of the 1st International SemanticWeb Conference (ISWC), pp. 333-347, 2002.

[19] Lei Li and Ian Horrocks. “A software framework for matchmaking based on semantic web technology”. In

Proc. of the Twelfth International World Wide Web Conference (WWW 2003), pages 331-339. ACM, 2003.

[20] Sycara, K., Paolucci, M., Ankolekar, A., and Srinivasan N.. “Automated Discovery, Interaction and

Composition of Semantic Web Services”, Journal of Web Semantics, Volume 1, Issue 1, December 2003.



INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, Pages 1-22

ADVANCES IN NI FORMATION AND SYSTEMS SCIENCES Volume 2, Pages 1-10


©2007 Institute for Scientific Computing and Information ©2007 Institute for Scientific Computing and Information

THE REALIZATION OF CONSTANT PRESSURE WATER SUPPLY DRIVE SYSTEM BASRD ON THE FUZZY CONTROLLER

JIASHENG ZHANG, JUN GAO，YANG LIU AND JUNXUE GAI1

Abstract The constant pressure water supply method is realized by the adopted fuzzy control technique. The constant pressure water supply system based on the fuzzy controller is made of the fuzzy controller, pressure transmitter, frequency transformer, single-chip microcomputer, the system software and so on. This system is obvious effect than old system in saving energy. The practical running results are given to demonstrate the effectiveness of the proposed method.

Key Words, Fuzzy control; rules; water system; frequency transformer; constant pressure

1. Introduction

The tradition constant speed pump system of supply water in manner of converting pressure shows unstable water pressure and wastes electric energy. The electric energy consumption of water pump is about 20% of the total electric energy consumption in China annually. The electric energy consumption is about 60% of the water costs. Optimizing control pump is very significant. The fuzzy control is a good kind of the optimizing control.

A block diagram of a general fuzzy control system is shown in Figure 1. You should view the fuzzy controller as an artificial decision maker that operates in a closed-loop system in real time. It gathers plant output data , compares it to the reference input , and then decides what the plant input should be to ensure that the performance objectives will be met.

)(ty)(tu)(tr

The fuzzy controller is composed of the following four elements: A rule-base ( a set of If-Then rules ), which contains a fuzzy logic quantification of the

expert’s linguistic description of to achieve good control. An inference mechanism (also called an “inference engine” or “fuzzy inference”

module), which emulates the expert’s decision making in interpreting and applying knowledge about how best to control the plant.

A fuzzification interface, which converts controller inputs into information that the inference mechanism can easily use to activate and apply rule

CONSTANT PRESSURE WATER SUPPLY DRIVE SYSTEMS 35

Figure 1. Fuzzy control architecture

A defuzzification interface, which converts the conclusions of the inference

mechanism into actual inputs for the process.

2． The design of the water supply system When the electromotor is directly connected to the water pump, electromotor power is

310)1( −=ηρθHS

where ρ is fluid density. θ is the water pump flux. H is the water pump delivery lift. η is water pump efficiency. The water pump fluid density is in direct proportion to its rotating speed. The water pump delivery lift is in direct proportion to its rotating speed square. The power of water pump axis is in direct proportion to its rotating speed cube.

The factory water discharge changes largely. According to the fact condition the system is designed. The block diagram of fuzzy control water supply system is shown in Figure 2. The water outlet contains the fire fighting water supply and produce supply and live water supply. Constant pressure water supply system main circuit diagram is shown in Figure 3.

Figure 2. Fuzzy control water supply system Figure 3. Constant pressure water supply

system main circuit diagram 3. The realization of constant pressure water supply system

The fuzzy control water supply system is made of three 7.5W water pump sets, a MicroMaster 430 pump frequency transformer, a S3C8475 single-chip microcomputer fuzzy controller. The range ability of the pressure transmitter is 0 ~ 2 Mpa. The precision

36 J. ZHANG, J. GAO，Y. LIU AND J. GAI

is 1.5 grade. The output voltage range ability is 0 ~ 10 V. When the pressure transmitter detects the water pressure pm of the municipal water supply is more than the water pressure desired value ps and the duration is more than 10 min, the constant pressure water supply system stops working and the municipal water supply system supplies directly the water. When the water pressure pm is less than the water pressure desired value ps and the duration is more than 10 min, the constant pressure water supply system works.

According to the basic equation of the water pump and pipe line, the parameter equation: 2

0)2( iHi QSPP −= where p0 is the out water pressure, SH is the pipe line impedance of the parallel branch, pi and Qi are respectively the water pressure and flux of the one branch. The total output water flux of the water pump set:

∑=

=3

10)3(

iiQQ

When one pump is not Qi work is equal to zero. According to (2) and (3), when the

output total water flux changes the out water pressure p0 changes. The collected water pressure signal through the water outlet is compared with the water pressure desired value. The error e and error change rate ec are achieved. The one pump is driven by the frequency conversion speed motor. The PWM duty ratio is controlled by the fuzzy control rules. The out water pressure p0 is kept on constant by changing the out frequency of the frequency transformer.

The water level detector can detect the water level of the cistern. When the water level less than the lower limit water level the controller opens the electrovalve and the water is injected into the cistern. When the water level more than the upper limit water level the controller shuts the electrovalve and the water is not injected into the cistern. 4. The controller algorithmic design

Because there are many pipeline network coupling variables in the water supply system and the nonlinear variables in water pumps the accurate mathematical models are hardly built. The PID parameters is hardly determined by the engineering algorithmic. The fuzzy control algorithmic is a kind of useful method.

The desired value ps of water pressure , the error e=( p0 - ps ) and the error change rate ec are the input variables of the fuzzy controller. The input variables are converted to the fuzzy sets of the fuzzy control language. The fuzzy control rules are built between the input and output. According to fuzzy control rules the fuzzy control rule table is reasoned off-line process. The fuzzy control rule table is stored in the single-chip microcomputer. The water pressure error e, the error change rate ec and the control variable are described by the following values:

“NL” to represent “neglarge” “NS” to represent “negsmall” “Z” to represent “zero” “PS” to represent “possmall” “PL ” to represent “poslarge”


They are quantified into 9 grades ( -4 –3 –2 –1 0 +1 +2+3 +4 ). The membership function is shown in Figure 4.

Figure 4. Membership function for The water pressure error, the error change rate

and the control variable The fuzzy control rule table 1 represents abstract knowledge that the expert has

about how to control the water pump. We have the follow rule relationship: if E and EC then U

According to fuzzy control rule the fuzzy control table is calculated by min – max barycentric method. The fuzzy control table is shown in table 2. The fuzzy control table 2 is stored in the single-chip microcomputer. During the real-time pressure control the p0 water pressure

Table 1 Control rule Table 2 Fuzzy control

38 J. ZHANG, J. GAO，Y. LIU AND J. GAI

collected by the single-chip computer is compared with the water pressure desired value . psThe error e and the error change rate ec are gained. e and ec multiply respectively by the quantized factor. The control variable value is gained by looking up the control table 2. The duty ratio of the PWM wave is controlled by the control variable value multiplying proportion factor. The water pressure can controlled by the frequency transformer changing the motor rotating speed. 5. The fuzzy controller hardware and software realization

The system software is programmed by the assembly language. The software system is constructed of the main program, the constant pressure control subprogram, the fire control subprogram, the fuzzy control subprogram, the show subprogram, the keyboard subprogram and so on. The constant pressure control subprogram flow chart is shown in Figure 5. There are 16KROM, 272bytes RAM, 10×8 ADC，two PWM outputs in the

Figure 5. The constant pressure control subprogram flow chart

single-chip microcomputer S3C8475. The fuzzy controller hardware block diagram is shown in Figure 6.

Figure 6. Fuzzy controller hardware system

6. Conclusions

The constant pressure water supply system based on the single-chip microcomputer fuzzy controller is realized in small investment and small development time. The constant pressure water supply system raises the water supply quality. When the system works in constant pressure the desired value is set from 0.02 to 0.65Mpa. The error is less than ±0.01Mpa. The response time of pressure from 0 to 0.65 is 2min 36s. The system is in good working order in two years. The supply system is reliable. The adopted


fuzzy control technique keeps away from adjusting the complex tradition PID parameter. References [1] J. M. Mendel. Fuzzy logic system for engineering: A tutorial. Proc. of the IEEE, Special Issue on Fuzzy

Logic in Engineering Applications, 83(3): 345-377, March 1995. [2] G.C. Mouzouris and J. M. Mendel. Nonsingleton fuzzy logic systems: Theory and application. IEEE

Trans. on Fuzzy systems, 5(1): 56-71, February 1997. [3] W. Pedrycaz. Fuzzy control and Fuzzy systems. Wiley, New York, second edition, 1993. [4] T. Takagi and M. Sugeno. Fuzzy identification of systems and its applications to modeling and control.

IEEE Trans. on Systems, Man, and Cybernetics, 15(1): 116-132,January 1985. [5] Liu Suqin, and Liu Xinping, Comparison and Improvement of the PID-control Algorithm and the Fuzzy

Control Algorithm. Control Engineering of China.Vol.10 No.1, pp. 51-53, January 2003. [6] Xie Mujun, Fu Hong and Wang Zhiqian, Application of a Novel Fuzzy Control Algorithm in Water

Temperature Control System. Vol.10 S, pp.78-80, July 2003.

Jiasheng, Zhang, received a B.S. degree in automation in 1982, and M.S. degree

in system engineering in 1990 from Northeastern University. Zhang Jiasheng is

currently a professor in Northeastern University at Qinhuangdao. He is a director

of the test and education center and a direct teacher of M.S. graduate student. His

research interests are automatic control and system engineering.




RESEARCH ON PERFORMANCE OF REPLICATION IN GRID SYSTEM

JING CHEN, LINGFU KONG AND JIANZHOU FENG

Abstract Regarded as the third generation Internet, Grid is a new technology built on the network service. It should support the resources sharing and cooperating work in virtual organization. But network delay is the main obstacle to resource sharing in the WAN. Replication technology becomes the key technology improves the performance in grid system. In this paper, replication is looked as a kind of service, we analyze the factors of effecting grid performance, put forward the strategies of improving, design the frame of replication, and discuss performance factors by mathematical ways. Key Words, Grid system, Replication, Performance factors

1. Introduction

Internet provides multiple services of e-mail and browses Internet and communication for people, but the function of grid is much better, it is a new technology that base on the network service. It connects with high speed internet and HPC (high performance computer) and large scale database and sensors and long-range devices, and provides functions and resources and interactive that make people use computer and other resources transparent, in order to support the resources sharing and cooperating work of virtual organization [1]. Grid is also defined as a resource pool for providing computing [2]. The kernel problem of grid system is how to share resources and work cooperative in virtual organization of distributing section from the definition of grid. The requirement and supply of resource variety dynamically and distribute widely, if you want to finish a requisite service of user will involve in devices and servers and software resource of different position, the factors like response time and the bandwidth of network and the complexity and flexibility of grid management software and interactive devices affect the performance of grid system.

In the present, the major barrier of supporting fast access in a grid is the high latencies of Wide Area Networks and the Internet. To overcome this barrier, large amounts of resources need to be replicated in multiple copies at several world wide distributed sites. Replication involves creating copies at different sites in the Grid. Replication is a technique of fast access, and is also critical for maximizing overall job throughput [3~4]. In this paper, we look replication as a kind of service, and connect the technique of OpenLDAP with Kerberos to solve the relation of mapping information and duplication information and the problems of accessing information in different areas. Designing the architecture of replication based on the relation and technique, and realizing the trans-areas service by single sign-on. It is a base of improving the performance of replication, and we point out the factors of effecting performance.

40

This work is supported by Natural Science Foundation of Hebei Province (Grant No. F2006000281)

RESEARCH ON PERFORMANCE OF REPLICATION IN GRID SYSTEM 41

2. The model design of replication service

In a grid, a user typically submits a job to the Grid. In order for a job to be executed, three kinds of resources are required: computing, storage and network. The Grid must make scheduling decisions for each job based on the current state of the Grid resources (workload and features of Computing Elements, location of data, network load).

Grid is a supporting platform of distributing and parallel system [5~6], and the nodes of grid are the suppliers of grid service. To descript the process of replication thoroughly, we abstract the process of replication as the process that customers request service in grid, so we extend the definition of customers and services.

Definition 1: Service is a function that networking device and application programs and system resources supply.

Definition 2: Customers is a functional user or computer system that use grid resource. 2.1. The service model that customers access trans-area

Because the most replication is file level, we discuss file replication. It involves located files position by replication directory and multiple copies of a file spread across the Grid. When files enter a site or replicates between sites, it should register in file replication directory, because they are available and perceptive for grid. To facilitate the management of replication process, two types of site are used: a node stored physical files (PFS: Physical Files Site) and nodes stored replication files (RFS: Replication Files Sites). The files stored in RFS are an abstract reference to files in PFS, which is coherence replication of files in PFS. A PFS refers to a specific replica of some RFS, located at a definite site. Each RFS will have one PFS for each replica in the Grid. We resolve the problems that customers access services, furthermore, realize the process of replication services trans-area, from the point of view of replication services.

In the network, when we use resources in different areas, the authentication is implemented, the same problems will occur in the grid, because networks provide resources for grid. Therefore, we design the architecture of replication process in grid, according to technique of Kerberos that we realize the access trans-area. The model of replication is illustrated in figure 1.

S S S

Cus t omer s

Ar ea 1

1

2

3

Aut hent i c at i onCent er ( AC)

A rea 2 AC

A rea 3 AC

A rea n AC

4

Figure 1. The architecture of replication process We summarize the flow of replication as four steps from figure 1. (1) When the customers success registering in AC (Authentication Center), AC will

return UT (User Tickets) and ST (Service Tickets) to customers according to the services that customers registered; (2) AC map the registered services of customers to correlative server nodes;

(3) Customers access registered services according to ST; (4) Customers access services do not exist in this area, then send UT to other ACs. If

UT exists in other ACs, then customers will use services in back-fence ACs. The process realizes to access service by single sign-on really. In additional, it avoids transferring customers password in network and improving security problems, according to

42 J. CHEN, L. KONG AND J. FENG

transferring UT information. Each AC cooperates to maintain the whole grid. 2.2. The realization of every step and performance

We design and realize the process that customers access services trans-area, then discuss the implement of replication services. We choose LDAP (Light Directory Access Protocol) database to store information, there are a lot of management software of LDAP. Designing the hiberarchy architecture of replication service. It is illustrated in figure 2.

RFS RFS RFS

Cust omer s

Area1PFS

ReplicationService--S

Figure 2. The hierarchy architecture of replication service

AC map replication services to correlative nodes, we design master/slave structure of LDAP database. Choosing a PFS and some RFS in a area, customers submit files into PFS and map files from PFS to RFS. The PFS and RFS stored by OpenLDAP database, we obtain coherence replica in RFS. From hierarchy architecture, we know that the process that customers access files is the process of requesting a RFS, typically, optimization proceeds through resolving a RFS reference by finding the PFS that can be accessed with the minimal time. Thus, the main optimization criteria are to minimize the access latencies. It is an effective method to improve the rate of accessing. When customers modify files in RFS, the modified results will send messages from RFS to PFS, and PFS will inform all other RFS in this area.

3. The performance of replication service

There are a lot of factors that affect the performance of replication. We know that the performance of replication lie on the service of nodes of gird to arrival customers mostly [7]. Sometime the node is busy with serving customers, then new customers must wait until the node is unoccupied to accept service and participate in computing, but as the same time, the statue of other nodes is maybe unoccupied, which result in allotting unbalance of resources and services. To resolve this problem, we study the characteristic of customer arrival. In fact, the probability every node’s arrival is stochastic. The arrival of customer has special characters, and according to certain stochastic distributing. [8,9]. 3.1. Response time of replication service

We discuss the response time by two kinds of instance. A. Access service in this area

If customers register and access files of existing in RFS in this area (Area 1 in figure 1), the response time which customers access files from RFS is composed of two parts. Namely, serviceregisterresponse TTT += , represents customers’ register time in area one, we computes it by the following way.

registerT

Ser ver

Cust omer s ar r i val

Depar t

Figure 3. The queue model that customers submit files into PFS node Wai t i ng queue l engt h


We simulate register time with this kind of topology in figure 3. In area 1, customers who locate different nodes refer requests to PFS, so we place the requests into a waiting queue, and look PFS as server.

m

Supposing the capacity of the queue is unlimited, namely service may accepted requests anytime. We described register time by using of the model of M/M/1.

Supposing that the arrival rate of customers and service rate are stochastic

variables λ and μ respectively, and the parameters of performance are same in this

model, then the average arrival rate of customers is

m

mλ , the average service rate is μ ,

and sojourn times (including waiting time and register service time) isμm

λμ

register−

=1

1T ,

μλρ

m= is the utilization rate of server.

serviceT ： is service time，which finish the process of replication service, after customers register. It is composed three parts: access time, writing time and delay time.

serviceT

Given: Each node in the overall system and a data object , the ratio of the write cost for file is

w iiv,λ and the read node is iu ,λ , they accord certain stochastic distributing.

Then the response time for the accessing and writing is:

(1) ),(,, )(* xwiviv tisizett += λ

)(* tisizet

(2) ),(,, xwiuiut += η λ

Where is delay time between nodes and , it is a function of available bandwidth, is a consumptive time of accessing files, is a consumptive time of writing files,

),( xwtiv

t,λ

w iiu

t,λ

η is the ratio of the write for node, and is defined as an experiential constant. So the response time is:

(3) serviceregisterresponse TTT += ＝

μλ

μ

mT

−=

1

1

留驻

+ + ivt , iut ,

B. Access service outside this area The files that customers access do not exist in RFS in this area, the response time is

composed of visiting time during areas and service time. Namely, serviceVisitresponse TTT += . is defined same as case 1 (realizing trans-areas service by single sign-on

according to Kerberos protocol, if UT exist in other area, then customers use service is same in this area and outside this area). Now defining by the following way. Customers send UT to -1 different ACs, we describe the case in figure 4.

serviceT

VisitTn


Us er

1

2

n -1

...

Figure 4. Customer sends UT to

is composed of acce ，

n -1 areas

VisitT =γ ivt , ＋ answert ，γVisitT ssing time and answer time namely is

delay coefficient of accessing trans-area. In this case, responseT serviceVisitT T+= ＝(1+γ )

k bandw

ivt ,

＋ answert ＋ iut , W now t te k tha he effecting factors of replication

th

during the replication service, which is a

service include networ idth, e placement of replication node and network delay. So we put forward some methods

to improve performance to solve these factors. 3.2. Storing files in PFS by using of LDAP

We manage the files in PFS by LDAP databasekind of operation of mainly accessing, and LDAP is preponderant. LDAP may manage

resources of every node and visit the information of storage directory, it run on the TCP/IP. The information of LDAP directory organizes and stores according to tree architecture, and material information store in the data structure of entry. The most important aspect is directory design, because directory service is a database system and optimizes specifically access and browse and search operations. Directory includes descriptive information and support refined and complex filtration ability based on attribute information. The file information is static and stores the descriptive information of file, such as ID and name and the date created. So we design the database format in figure 5.

Vi r t ual Root node

Ar ea 1 Ar ea 2 Ar ea n

Host 1 Host 2 Host n

Fi l e 1 Fi l e n

I D Name Si ze Ti me

…

…

…

…

I P

OS

Figure 5. The format of accessing files in LDAP

3.3. Visiting ACs each othem which we design is implemented by ACs in different

ar

er The maintenance of grid systea. Now we explain the principle cooperated work in ACs. There are two cases:

customers find the file on the node inside and outside area. The latter need cooperate in ACs to finish the process of replication. The process is illustrated in figure 6.


St ar t

Cust omer s accessr epl i cat i on ser vi ce

Sear ch RFS i n t hi s ar ea

Fi l es exi se

Y

Usi ng st r at egi es

Cust omer s vi s i t RFS

End

Cooper at i ng wi t h AC

Send massages t oadj acent ACs

UT exi st

Y

N

Ret ur n f ai l messagesFi l es exi st

Ser vi ceexi st

Y

N

N

N

Figure 6. The executive process of different ACs during replication

4. Strategy of replication service

The superior or inferior of replication strategies will affect the performance of grid. When accessing files have two cases.

(1) The attribute of files is known: When accessing files, we search nodes that are the most nearest geographically to requesting node, according to layout arithmetic;

(2) The attribute of files is unknown: First, we visit the most highest frequency nodes whether the file exists. If existing, then search nodes that are the most nearest geographically to requesting node according to browsing the layout of the file and carrying our DFS (deepness precedence search). If it is not response in a restrictive time, then send messages to requesting node, and search the nodes that are adjacent and ancestor nodes of highest frequency. We do not think the file exists in this area, if overtime. In this condition, cooperating with ACs, sending and verifying UT and ST to other area, searching files in a restrictive time, and the process of operation is same as the operation of this area. In additional, we will adopt two strategies to apply in the process of replication service.

The choices of replication service strategies are based on replication topology. The first mapping from the files in PFS to RFS, and we finish mapping to the first level to other levels according to files attribute information, the strategies we choose can realize these function.

(1) LRU (Least Recently Used): In this methods, we browser the attributes and frequency of file that monitor software display, and map high frequency files into a node or nodes group according to files grouping information in first level and replace the files of low frequency. Pay attention to modify directory by sending messages to correlative nodes when the mapping from the first level to next levels finished.

(2) SSTF (Short Files Size Time First): IF customers research a group files that exist in this area, then send files to customers, we adapt to this kind of strategy to avoid the time of accessing or transferring files too long. Because if requesting node do not get response in a restrict time, then it maybe think the files do not exist in this area, and it will request to the adjacent ACs, if we send SSTF files in this group to customers, then the price of visit process will be reduced.


5. Conclusion

Replication is an important technique of optimizing grid performance. In this paper, we look replication as a kind of service. First, resolving the services access trans-area to realize the process of replication services in grid, referencing two types of sites that place different files are PFS and RFS, because most replication is files level, expatiating the whole process of replication services, and design replication strategies, analyzing the effecting factors of performance in grid. We model the factors of effecting performance, and solve the store in LDAP dynamically and the access of multi-levels, and optimize layout arithmetic and replication strategy, which achieve the ultimate goal that the highly dynamic environment of grid is scalability and adaptability. They are our future work.

References

[1] Liu Peng. “The Boundary of Grid Concept”. http://www.chinagrid.net [EB/OL]

[2] Du Zhi-Hui, ChenYu, Liu Peng. “Grid Computing “.The Publishing House of TsingHua University

10(2002), 1-10

[3] “Evaluating Scheduling and Replica Optimisation Strategies in OptorSim”. http:// www.gridforum.org

[EB/OL]

[4] Houda Lamehamedi, Zujun Shentu, Boleslaw Szymanski. “Simulation of Dynamic Data Replication

Strategies In Data Grids”. Proceedings of ICAP’03, Beijing, China October 2002,IEEE Computer Science Press,

Los Alam-itos,CA.

[5] Gu Xiao-LIn,Qian De-Pei. “Study on Grid Computing Model Based on Internet”. Journal of Xi-an

JiaoTong University, 35(10)(2001),1008-1011

[6] Zhang-Yan, Sun Shi-Xin, Peng Wen-Qin. “An Improved Allot Arithmetic of Sub-network in Grid

Multi-processor”. Journal of Software, 12(8)(2001),1250-1257

[7] Lu Zhao-Yi, Wang Si-Ming. “Information Theory of Computer Communication Network”. Publishing

House of Electronics Industry, 1997.

[8] Zhang Q, Yuan Z, Xiao H. “Analysis of distributed computing strategy based on grid”. Proceedings of the

International Workshop on Grid and Cooperative Computing . Beijing: Publishing House of Electrical

Industry.2002

[9] Niu Zhi-Sheng. “The Basic of Communication Network”. Electron Engineering of TsingHua University.

2(2004),26-30 Northeastern University at Qinhuangdao, Qinhuangdao, Hebei Province, P. R. China, 066004

http://www.gridforum.org/






IN INTEGRATED NETWORK MANAGEMENT SYSTEM Q3 INTERFACE RESEARCH AND APPLICATION

JINGYUAN, XIN XU AND YUAN GAO

Abstract This article is based on the point of view of current network management. The purpose of using Q3 Interface is to improve the quality of service、to cut down the cost of circulation and to use reasonable management means to preserve network. Because Q3 Interface could make the development of network management system get rid of the formerly dependence of special interface which is provided by many net base element factories, could simplify and accelerate development speed, could diminish the capitalized cost of circulating and preserving net management system, could improve the stability of net management system.. Key Words, Q3 Interface; Network Management; Cost

1 INTRODUCTION

With the high speed developing of communication technology, the constantly expanding of communication network, equipments of diverse factories were introduced in network. When we are getting more and more benefit, the network complexity is increasing day after day. The requirements for NMS are more and more, higher and higher. So reasonable management means to maintain the network, which is complex day after day is a problem needed resolved right now.

As a stander interface of ITU-T, Its norms are more mature. So it’s a effective way

of resolving diverse factories problem, and supporting system reciprocal operation. Q3 Interface could make the development of network management system get rid of the formerly dependence of special interface which is provided by many net base element factories, could simplify and accelerate development speed, Meanwhile Q3 Interface could shield different factories’ special interface, could make Netware management system software basically unchanged, could diminish the capitalized cost of circulating and preserving net management system, could improve the stability of net management system.

2 Development of Net Management Technology In 1988, ITU formed M.30 series proposal, which defined TMN structural framework,

48 J. YUAN, X. XU AND Y. GAO

functional modules, information model, and physical model. It is used to effectively manage dissimilar structural internet、facilities and business. It is a solution of network centralized management. Its basic idea is to use Q3 interface as system management interface. In early 1980s, the appearance and development of internet make R & D persons bring forward many net management solutions. When ISO constantly modify CMIP/CMIS and make it mature, SNMP was tested and developed in practical application environment. Now SNMP has become a indeed industry standard in the area of computer network management, and it is widely supported and applied.

Integrated NMS could manage and control different interlinked network facilities by using a network management workstation. Then it makes the whole net integrated management become true, which involve the functions of whole network malfunction analysis and orientation and so on. In this way it could make maintenance、usage convenient and also could improve system utilization ratio. 3 Q3 Interface 3.1 Basic Concept of Q3 Interface The main way to solve mutual operation is to adopt a series advanced interface technology, such as separating grammar form lexeme, separating operation from communication, lexeme description being independent of facility, object-orientation technology, etc. So a standard interface: Q3 interface was formed, which supports interconnection、intercommunication、mutual operation. Q3 interface is the physical mapping of q3. In physical structure, Q3 interface lies between OS and NE. Q3 interface is an asymmetry interface. The entities at two ends of interface are called Manager and Agent. Therein, Manager is initiative side and Agent is passivity side shown in figure below.

INTEGRATED NETWORK MANAGEMENT SYSTEM 49

management and application program

manager

Communication entity

Agent

communication

identity

management information

net management proposal

communication proposal

Q3

figure 3-1 Q3 interface structure

Manager provides service to management application program. In most cases, after Manager receive management orders from management application program, Manager will send management orders to Agent, then Agent execute these orders. In the other cases, Agent also initiative reports some information. Manager will send information to management application program. 3.11 Communication Protocal Stack Communication Protocal Stack defined using communication protocal of Q3 interface，From complicated OSI stack to simple CMIP ITU-T defined eight kinds of communication protocal stack based on TCP/IP、ATM-ALL5-BASED etc. Shown in figure 3-2 below：


X710,X711(CM SE)X217,X227(ACSE)X219,X225(FCSE)

7

X.216,X.226(Presentation)x.209(BER)6

5 X.215,x.225(sessi on)

1

2

3

4 Any OSI

Lower Layer

Defined In

Q.811/Q.821

RFC1006(TP0)4

RFC793(TCP)

RFC791(IP)

4

3

RFC1557/AAL5/ATM2

Any Defined PHYSupporting AAL5/ATM

1

Any DL/PHY

SupportingIP

1

2

figure 3-2 3 kinds of communication protocal stack

3.2 Management Information Mode

Management information mode is part of lexeme of Q3 interface. Management information mode is based on ISO system management mode. It adopts Object-Orientation methods, using abstract methods to make lexeme description independent of facilities.

The basic element of management information mode is MO. MO is a abstract description to manifold resource on the communication net. MO is made up of physical resource and logical resource. Physical resource comprises switch、transfer equipment、rack、circuit board, etc; Logical resource is software、number、log、warning threshold, etc. Each MO has some parameters, such as abstract、notify、action, etc. 3.3 Description of Management Object

MO(Management Object) is the kernel of net management information mode. Applying standard and OO method to descript MO is a important step of implementing Q3 interface.

In GDMO, memory tool defined by the managed object is template. Template

which is a standard description format, express all aspects of target definition and related naming structure of defined by the managed object. Each templates of GDMO is provided they should be covered and include various components, and provides a variety of templates for each component should appear in the order.

What needs to be pointed out is that GDMO template only provides half a standardized way to describe management object, Because it is not provided by the

INTEGRATED NETWORK MANAGEMENT SYSTEM 51

language standards asn.1, The files of the managed object written by templates format also require asn.1 coding rules of data. Nevertheless, GDMO template has provided a convenient and effective tool to make a unified description for the managed object. 4. Designation of Q3 Interface in Integrated Network System

Integrated NMS mainly achieve the following three management functions 4.1 Integrated Network Management

Network failures management use the interfaces of manufacturer management system to position network failures, to access associated with the related operational, to export business affected table, to activate function for noticing customer service centers and departments, and to provide rapid resumption of business. Functions of failures management include: collecting, shining upon, storing, shielding, filtering and displaying of alarms information; positioning alarms, and linking with gateway of business; recovering business. 4.2 Network Resource Management

Including network resource management, resources activation management and systems operation management three functional system.

Which mainly consists of resource management are physical plant resources

management, pipeline network resource management, cable network resource management, transmission network management, exchange network resource management and digital dispatch resources management. 4.3 Business Operations Management based on Workflow and Sending Workers Table

Based on workflow management platform, we can detailed record information of staff and time in each alarms handling link, and achieve visualized function of business process management. When conditions to meet, in accordance with procedures defined automatic implementation of the operation. 5 Summary

Integrated NMS makes unified, integrated management, it is a Integrated management system which collects, transmits, processes and storages of the information of network maintenance, operation and management .The purpose of establishing a comprehensive business system is making maximize the use of network resources, improving the quality and efficiency of network operations, simplifying management in many manufacturers mixed network environment, controlling costs of network operating, enhancing means of Maintenance, and providing long-term network planning.


Reference: [1 ] ILOG J Rules White Paper 1.0 2003.

[2 ] BradleyN1The XML Companion /Neil Bradley Addison2Wesley,Harlow, UK, 2000.

[3 ] Vale ZA, FerandesM F, et al., Better KBS for Real2time Applications in Power System Control Centers:

The Experience of the SPARSE Pro2 Ject Computers in Industry, 1998, 37: 972111

[4 ] Cheon SW, Chang S H, Chung H Y, et al., Application of NeuralNet2 works to Multiple Alarm Processing

and Diagnosis in Nuclear Power Plants, IEEE Trans1Nucl1Sci1, 1993, 40: 11220.

[ 5 ] Tidwell D. XSLT, I. OReilly, Cambridge, MA, USA, 2001.

[6 ] ILOG JTGO 3. 5 Users Manual, 1 December 2002

Jing Yuan His research interests are in the areas of VOD, and Network Management

Information Systems. He has published several papers in these areas.

Liaoning Technical University, Network Management Center, Fuxin, Laioning, P. R. China Email: [email protected]..

mailto:[email protected]



SUPER-RESOLUTION IMAGE RECONSTRUCTION WITH EDGE-DIRECTED INTERPOLATION

SHUZHAN CHEN, LI HOU, LIN WANG AND FENDI LIU

Abstract A super-resolution image reconstruction algorithm based on edge-directed interpolation is proposed. Before interpolation, a frequency domain algorithm is employed to register a set of aliased low-resolution images precisely. A high-resolution image is then reconstructed using a new edge-directed interpolation. Several experiments are presented to demonstrate the effectiveness of the algorithm. Key Words, Super-Resolution, Edge-Directed interpolation, Frequency domain image registration

1. Introduction Super-Resolution (SR) image reconstruction is one of the terms applied to the problem

of transcending the limitations of optical imaging systems. It increases image resolution through the use of image processing algorithms, which presumably are relatively inexpensive to implement. The basic idea behind SR is the fusion of a sequence of low-resolution noisy blurred images to produce a higher resolution image or sequence.

SR reconstruction methods are categorized into two main divisions – frequency

domain and spatial domain, depending on the domain in which the restoration method is affected. The idea of SR was first introduced in 1984 by Tsai and Huang [1] for multiframe image restoration of bandlimited signals. They considered the problem subject to the assumption that the low-resolution frames have neither been corrupted by noise nor degraded by a blurring phenomenon. Kim et al extended the formulation to consider observation noise as well as the effects of spatial blurring [2].

Spatial domain methods generally allow for more general motion models compared

with frequency domain ones. They can be based on the whole image or on a set of selected corresponding feature vectors. Keren et al. [3] developed an iterative planar motion estimation algorithm based on Taylor expansions. A pyramidal scheme is used to increase the precision for large motion parameters. Schultz et al. [4] use a maximum a posteriori (MAP) statistical method to build the high-resolution image. Patti et al. [5] propose a POCS algorithm, where the estimated reconstruction is successively projected on different convex sets. Each set represents constraints to the reconstructed image that are based on the given measurements and assumptions about the signal. Elad and Feuer [6] present a SR framework that combines a maximum likelihood/MAP approach with a POCS approach to define a new convex optimization problem. Farsiu et al. [7] propose a new and robust super-resolution algorithm. Instead of the more common L2 minimization, they use the L1 norm, which removes outliers efficiently, resulting in high-resolution images with sharp edges.

In this paper, we propose a SR image reconstruction algorithm based on edge-directed

Received by the editors August 10, 2006 53

54 S. CHEN, L. HOU, L. WANG AND F. LIU

interpolation. A set of aliased low-resolution images is registered through a frequency domain algorithm. When the low-resolution images are accurately registered, the samples of the different images can be combined to fit a synthetical low-resolution image. A high-resolution image is then reconstructed using a new edge-directed interpolation [8]. Experiments results show, both visual and in PSNR, the reconstruction effects are improved compared to the algorithm by Vandewalle et al [9].

2. Edge-Directed Interpolation

Image interpolation addresses the problem of generating a high-resolution image from its low-resolution version. The model employed to describe the relationship between high-resolution pixels and low-resolution pixels plays the critical role in the performance of an interpolation algorithm. Conventional linear interpolation schemes (e.g., bilinear and bicubic) based on space-invariant models fail to capture the fast evolving statistics around edges and consequently produce interpolated images with blurred edges and annoying artifacts. Linear interpolation is generally preferred not for the performance but for computational simplicity.

In order to improve the subjective quality of the interpolated images, many algorithms

have been proposed, one of which is edge-directed interpolation [8]. It is a novel noniterative orientation-adaptive interpolation scheme for natural-image sources, proposed by Xin Li and M. Orchard. The basic idea is to first estimate local covariance coefficients from a low-resolution image and then use these covariance estimates to adapt the interpolation at a higher resolution based on the geometric duality between the low-resolution covariance and the high-resolution covariance. The edge-directed property of covariance-based adaptation attributes to its capability of tuning the interpolation coefficients to match an arbitrarily oriented step edge, which is demonstrated detailed in [10].

A hybrid approach of switching between bilinear interpolation and covariance-based

adaptive interpolation is proposed in [8] to reduce the overall computational complexity. Covariance-based adaptive interpolation is only employed for the pixels around edges called “edge pixels”. For the pixels in the smooth regions called “nonedge pixels”, bilinear interpolation is still used due to its simplicity. Since edge pixels often consist of only a small fraction of pixels in the image, the hybrid approach effectively alleviates the burden of the computational complexity without sacrificing the performance [8].

The computational process is introduced briefly below. Assume the low-resolution

image of sizeji,X WH × , came from the high-resolution image ji, of size , i.e. i2 . Then the interlacing lattice 12,12 ++ ji can be interpolated from the lattice , constrained the approach to the fourth-order linear interpolation

Y WH 22 ×ji,YX=

j2, X=ji 2,2Y

Yji,

(1) ∑∑= =

+++++ =1

0

1

0)(2),(2212,12

ˆk l

ljkilkji YY α

where the interpolation includes the four nearest neighbors along the diagonal directions. The interlacing lattice ji, （Y =+ ji

02 =j Yα

odd）can be interpolated from the lattice ji,Y（ even），using the four nearest neighbors along the diagonal directions in the high-resolution lattice, e.g.,Y .

=+ ji1+j2,1232,22212,1212,2,1 ++−+ +++ ijijiji YYY ααα2

ˆ+i

SUPER-RESOLUTION IMAGE RECONSTRUCTION WITH EDGE-DIRECTED INTERPOLATION 55

Analyzed the rectangular lattice structure, the geometric duality when interpolating the

interlacing lattice ji, （Y =+ ji odd） from the lattice ji,Y （ =+ ji even）and the one between the high-resolution covariance and the low-resolution covariance when interpolating the interlacing lattice 12, from the lattice jij are isomorphic up to a scaling factor of and a rotation factor of

12 ++ jiY2/12

i ,2,2 XY =4/π .Therefore the

lattice （ji,Y =+ ji odd）can be estimated through little change to (1). The factor in (1) can be formulated using the classical covariance method [11], α

(2) )()( 1 xCCCα TT −=

where Mk is the data vector containing the Txxx ]......[ 21=x MM × pixels inside the local window and C is a 24 M× data matrix whose th column vector is the four nearest neighbors of along the diagonal direction.

kkx

3. SR Image Reconstruction Algorithm

In this paper, we use the edge-directed interpolation for SR image sequence reconstruction. First, we employ a frequency domain algorithm to register a set of aliased low-resolution images and then with bicubic interpolation produce a synthetical low-resolution image, which is interpolated through the edge-directed interpolation to get a high-resolution image.

At the registering step, we introduce a frequency domain algorithm by Vandewalle et

al. In [9], a frequency domain technique is proposed to precisely register a set of aliased images, based on their low-frequency, aliasing-free part. This is the part of the signals with the highest signal-to-noise ratio, and in Vandewalle’s setup, the aliasing-free part of the images. The motion parameters are estimated between the reference image and each of the other images. Only planar motion parallel to the image plane is allowed. The motion can be described as a function of three parameters: horizontal and vertical shifts and a planar rotation angle. So, the registration algorithm includes a precise rotation estimation algorithm, a subpixel shift estimation algorithm and an adaptation of this method to estimate motion accurately in aliased images. Compared this approach in a simulation to other spatial domain and frequency domain registration algorithms, such as the spatial domain method by Keren et al [12] and the frequency domain algorithms by Marcel et al. [13] and by Lucchese and Cortelazzo [14], the algorithm can better estimate shift and rotation parameters than the other methods, in particular when some strong directionality is present in the image.

4. Simulation Results

In the simulation, we use the standard high-resolution grayscale images Lena and Baboon (with the size of 512×512) to test our SR algorithm.

First, three shifted and rotated copies are created from the high-resolution test image.

The different images are then low-pass filtered and the first of these images (not-moved reference image) will be the reconstruction target for the super-resolution algorithm. Finally, the four images are downsampled by a factor four. This results in four low-resolution, shifted and rotated images (with the size of 128×128) that can be used as input for the super-resolution algorithm. Then, the low-resolution image sequence is


registered in frequency domain from which a synthetical low-resolution image(with the size of 128×128) is produced with bicubic interpolation. Finally, a high-resolution image (with the size of 256 × 256) is reconstructed through the edge-directed interpolation.

Our SR algorithm is compared with the Vandewalle et al. algorithm in which bicubic

interpolation is implemented after the frequency-domain registration. Table 1 and Fig.1 and 2 show the comparisons. It can be observed that the effect of our method is better both in objective and subjective terms.

Table 1 Comparison of the PSNR results

Method Image

Lina Baboon

Our algorithm 29.5269 25.1971

Vandewalle et al 23.5636 20.7406

(a) (b) (c)

Fig. 1. (a) original Lina image, (b) reconstructed image by our algorithm, (c) reconstructed image by Vandewalle et al. algorithm

(a) (b) (c)

Fig. 2. (a) original Baboon image, (b) reconstructed image by our algorithm, (c) reconstructed image by Vandewalle et al. algorithm

5. Conclusions A SR image reconstruction algorithm is presented. The main elements of this

algorithm are edge-directed interpolating and frequency-domain registering. Regarding edge-directed interpolation, adaptive interpolation is employed for the edge pixels covariance-based and bilinear interpolation is still used for the nonedge pixels, which

SUPER-RESOLUTION IMAGE RECONSTRUCTION WITH EDGE-DIRECTED INTERPOLATION 57

reduces the overall computational complexity and keeps the interpolation performance. The implementation of registration method in frequency domain achieves precise estimation of shift and rotation parameters for the aliased low-resolution image sequence. Our results show that the proposed method improves the reconstruction performance effectively. How to interpolate more accurately with computational simplicity and how to fuse multiple SR reconstructed images both deserve further study.

References [1] Huang T. S., Tsai R. Y., “Multi-frame image restoration and registration.”, Adv. Comput. Vis. Image

Process., 1(1984), 317-339

[2] Kim S. P., Bose N. K., and H. M. Valenzuela., “Recursive reconstruction of high resolution image from

noisy undersampled multiframes.”, IEEE Trans Acoust., Speech, Signal Processing, 38(6)(1990), 1013-1027

[3] Keren D., Peleg S., and Brada R., “Image sequence enhancement using sub-pixel displacement.”,

Proceedings IEEE Conference on Computer Vision and Pattern Recognition, (1988), 742-746

[4] Schultz R. R., Meng L., and Stevenson R. L., “Subpixel motion estimation for super-resolution image

sequence enhancement.”, Journl of Visual Communication and Image Representation, 9(1)(1998), 38-50

[5] Patti A. J., Sezan M. I., and Tekalp A. M.. “Superresolution video reconstruction with arbitrary sampling

lattices and nonzero aperture time.”, IEEE Transactions on Image Processing, 6(8)(1997), 1064-1076

[6] Elad M., Feuer A., “Restoration of a single superresolution image from several blurred, noisy, and

undersampled measured images.” IEEE Trans-actions on Image Processing, 6(12)(1997), 1646-1658

[7] Farsiu S, Robinson D, Elad M., and Milanfar P., “Fast and robust multiframe super resolution.”, Image

Processing, IEEE Transactions, 13(10)(2004), 1327-1344

[8] Li X., Orchard M.T. “New edge-directed interpolation.”, Image Processing, IEEE Transactions,

10(6)(2001), 813-817

[9] Vandewalle P., Süsstrunk S., and Vetterli M. “A frequency domain approach to registration of aliased

images with application to super-resolution.”, EURASIP Journal on Applied Signal Processing (special issue

on Super-resolution), 2005

[10] Li X., M. Orchard. “Edge directed prediction for lossless compression of natural images.”, IEEE Trans.

Image Processing, 10(2001), 813-817

[11] Jayant N., Noll P., “Digital coding of waveforms: principles and applications to speech and video”,

Englewood Cliffs, NJ: Prentice-Hall, 1984

[12] Keren D., Peleg S. and Brada R., “Image sequence enhancement using sub-pixel displacement.”,

Proceedings IEEE Conference on Computer Vision and Pattern Recognition, (1988), 742–746

[13] Marce B., Briot M. and Murrieta R., “Calcul de translation et rotation par la transformation de Fourier.”,

Traitement du Signal, 14(2)(1997), 135-149

[14] Lucchese L., Cortelazzo G. M.,“A noise-robust frequency domain technique for estimating planar

roto-translations.”, IEEE Transactions on Signal Processing, 48(6)(2000), 1769-1786


Shuzhen Chen, College of Information Science and Engineering, YanShan

University. Email: [email protected] received her B. Eng. degree in

Wireless Technique from Yanshan University in 1991, M. Eng. degree in Electric

Circuits & Systems from Yanshan University in 2004. Her research interests are in

the areas of digital image processing.

Li Hou, College of Information Science and Engineering, YanShan University.

Email: [email protected]. She received her B. Eng. degree in Electronic

Information Enginerring from Yanshan University in 2004, and will receive M. Eng.

degree in Electric Circuits & Systems from Yanshan University in 2007. Her

research interests are in the areas of digital image processing and Super-Resolution

image reconstruction.

Lin Wang, College of Information Science and Engineering, YanShan University.

Email: [email protected]. He received his B. Eng. degree in Electronic

Information Enginerring from Yanshan University in 2004, and will receive M. Eng.

degree in Electric Circuits & Systems from Yanshan University in 2007. His

research interests are in the areas of digital image processing and iris recognition.

Fendi Liu, Department of Basic Courses, Northeastern University at Qinhuangdao.

Email: [email protected]. She received her B. Eng. degree in Electronic

Enginerring from Northeastern University in 1996, M. Eng. degree in Electronic

Enginerring from Northeastern University in 1999. Her research interest is in the

areas of single processing.







A H ∞ CONTROL METHOD FOR LUR’E CONTROL SYSTEMS WITH TIME-DELAY

MIN ZHENG LIN CHAI AND SHUMIN FEI

Abstract For a class of Lur’e systems with time-delay, delay-dependent H ∞control

problem is investigated in this paper. By using an improved method, which avoids bounding cross terms, a new type of Lyapunov-Krasovskii functional for such systems is constructed. Based on the new functional, the sufficient conditions for H ∞

control are derived in terms of LMIs derived upon a come complementary approach proposed by L. El Chaoui. Numerical example is presented which illustrates the effectiveness of the new theory. Key Words. Time-delay, Lur’e systems, absolute stability, LMI, Lyapunov-Krasovskii functional

1. Introduction

In recent years, much progress has been made in the problem of delay-dependent stability of Lur’e systems with time-delay[1-3,6,7,9]. [1-3, 6] mainly discuss the problem of stability criteria for Lur’e systems, whereas in [7, 9] the problem of H ∞ output feedback control have been studied for a class of uncertain Lur’e systems with time-delays. To obtain less conservative results, the Lyapunov-Krasovskii, which is based on “descriptor form” in [4], has been applied in many time-delay systems [2,7], for this type of functional which can significantly reduce the design over the traditional methods due to the facts that it is based on a transformed model that is equivalent to the original system and fewer terms need to be bounded in the derivation. However, there still exists an upper bounding on the weighted cross products of the state and the delayed state, so conservativeness can’t be avoided.

In this paper, inspired by the idea of the conclusion in [5], we introduce a new type of Lyapunov-Krasovskii functional to study the problem of H ∞ control for Lur’e system with time-delay, and no inequalities are needed for bounding cross terms. New sufficient conditions are derived in terms of matrix inequality, which isn’t linear. Furthermore, to calculate controller gain without imposing restrictions on certain matrices to obtain certain LMI conditions, a cone complementary approach proposed by L. El Chaoui in [8] can be made use of to solve nonconvex feasibility problem with LMI conditions. A numerical example is illustrated to demonstrate the effectiveness of this obtained method.

Notation: Throughout the paper the superscript “T” stands for matrix transposition, denotes the n dimensional Euclidean space with vector norm nR • , is the set

of all real matrices, and the notation , for n n

n mR ×

n m× 0P > P R ×∈ means that is symmetric and positive definite. The space of function in that are square integrable

PqR

over is denoted by [0, )∞ 2 [0, )qL ∞ with the norm 2L

• , the space of continuous

functions with the supremum norm : [ , 0] nhφ − → R • is denoted by . Denote

[ , 0]nC h−( ) ( )x x tt θ θ= + ( [ , 0h ]θ ∈ − ).

Received by the editors July 20, 2006 59

60 M. ZHENG, L.CHAI AND S. FEI

2. Problem statement Generally, Lur’e system can be described as follows:

(1)

1 2 1( ) ( ) ( ) ( ) ( ( )) ( )

( )

( ) ( ) ( )

( ) ( ), [ , 0]

T

x t Ax t A x t B u t bf t B w t

t c x

z t Cx t Du t

x t t t

τ σ

σ

φ τ

= + − + + +

=

= +

= ∀ ∈ −

⎧⎪⎪⎨⎪⎪⎩

&

where ( ) nx t R∈2nR

is the system state vector, is the control input, is the exogenous disturbance signal, is the state combination

1( ) nu t R∈3( ) nR∈b

( )w t ∈ z t(objective function signal) to be attenuated, and vectors , . nc R∈ A ,

1A ,

1B ,

2B ,

, and are known matrixes with appropriate dimensions. C D

[ 0 , ]

( ) ( ) (0) 0, 0 ( ), 0f F f f fσ∞

∈ = • = ≤ ≠σ σ σ . Besides, for a prescribed 0γ > , we define the performance index

(2) 2

0( ) ( ( ) ( ) ( ) ( ))T TJ w z t z t w t w tγ

∞

= −∫ dt

We seek a control law (3) 1 2( ) ( )u K x t K x t τ= + − that will asymptotically absolutely stabilize the system(1), and the cost function (2) can achieve ( ) 0J w < for all 2

2 [0, )nw L∈ ∞ .

3. Main results

Define a Lyapunov functional candidate for the time-delay Lur’e system (1) as (4) 1 2 3 4( ) ( ) ( ) ( ) ( )t t t tV x V x V x V x V x= + + + t

where , 1 ( ) ( ) ( )T

tV x x t Px t=0

2 ( ) ( ) ( )t T

t tV x x s Zx s dsd

τ θθ

− += ∫ ∫ & & ,

3 ( ) ( ) ( )t T

t tV x x Qx d

τα α α

−= ∫ , ,

( )

4 0 0( ) ( ) 2 ( ) ( ( ))

t t

tV x f s ds s f s dsσ

β σ σ= +∫ ∫and , , , and the scalar 0P > 0Q > 0S > 0β > . Then, by the Newton-Leibniz

formula, we have ( ) ( ) ( )t

tx t x t x d

ττ α α

−− = − ∫ & . Differentiating the four terms of (4)

with respect to gives us t

1 2 1 1 2 2 1

1 2 2 1

1 2 2

( ) 2 ( ) [( ) ( ) ( ) ( ) ( ( )) ( )]

2 ( ) ( ) 2 ( ) ( ) ( ) 2 ( ) [ ( ( )) ( )]

2 ( ) ( ) 2 ( ) [ ( )] ( ) 2 ( )

T

t

tT T T

t

tT T T

t

V x x t P A B K x t A B K x t bf t B w t

x t PAx t x t P A B K x d x t P bf t B w t

x t PAx t x t Y P A B K x d x t

τ

τ

τ σ

α α σ

α α τ

−

−

= + + + − + +

= − + + +

= + − + + −

∫

∫

&

&

&

A H ∞CONTROL METHOD FOR LUR’E CONTROL SYSTEMS 61

1

2 1 1 2 2

1

( ) [2 ( ) 2 ( ) ] ( ) 2 ( ) [ ( ( )) ( )]

12 ( ) [ ( ) ] ( ) 2 ( ) [ ( ) ] ( )

2 ( ) ( ) 2[ ( ) ( ) ] ( )

2 ( ) [ ( ( )) (

t tT T T

t t

t T T T

t

T T T

T

W x d x t Y x t W x d x t P bf t B w t

x t P A B K Y x t x t P A B K Y W x t

x t Wx t x t Y x t W x d

x t P bf t B w t

τ τ

τ

α α τ α α σ

ττ

τ τ τ τ τ α α

σ

− −

−

− + − + +

= + + + + − + −

− − − + − +

+

∫ ∫

∫

& &

&

)]

−

(5)

where 2 1 1 2 2A A B K A B K= + + + ,

0

2

2 1 1 2 2 1

2 1 1 2 2 1

( ) [ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( )]

[( ) ( ) ( ) ( ) ( ( )) ( )]

[( ) ( ) ( ) ( ) ( ( )) ( )] ( ) ( )

1

tT T T T

t t

t T

t

T

V x x t Zx t x t Zx t d x t Zx t x Zx d

A B K x t A B K x t bf t B w t Z

A B K x t A B K x t bf t B w t x Zx d

τ τ

τ

θ θ θ α α α

τ σ

τ σ α

τ

− −

−

= − + + = −

= + + + − + +

+ + + − + + −

=

∫ ∫

∫

& & & & & & & & &

& &

2 1 1 2 2 1

2 1 1 2 2 1

[( ) ( ) ( ) ( ) ( ( )) ( )]

[( ) ( ) ( ) ( ) ( ( )) ( )] ( ) ( )

t T

t

T

A B K x t A B K x t bf t B w t Z

α α

A B K x t A B K x t bf t B w t x Zx d

ττ τ σ

τ σ τ α

−+ + + − + +

+ + + − + + −

∫& & α α

(6)

(7) 3

1( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( )]

tT T T T

t tV x x t Qx t x t Qx t x t Qx t x t Qx t d

ττ τ τ

τ −= − − − = − − −∫& τ α

and

4

2 1 1 2 2 1

2( ) 2[ ( ) ( ( )) ( ) ( ( ))] [ ( ) ( ( )) ( ) ( ( ))]

2[( ) ( ) ( ) ( ) ( ( )) ( )] ( ( ))

t T T

t t

t T

t

V x t f t t f t x t cf t x t cf t d

A B K I x t A B K x t bf t B w t cf t d

τ

τ

σ σ βσ σ σ β σ ατ

β τ σ στ

−

−

= + = +

= + + + + − + +

∫

∫

& & &

α

(8)

Let ( ) 0tφ = , substitute (5-8) into (4), and from (2) we obtain

2 2

0 0

2

0

1 2 1 20

2

( ) [ ( ) ( ) ( ) ( )] [ ( ) ( ) ( ) ( ) ( )]

[ ( ) ( ) ( ) ( ) ( )]

[ ( ) ( ) ( )] [ ( ) ( ) ( )]

1( ) ( ) ( , ) ( ,

( )T T T T

t

T T

t

T

T

t t

J w z t z t w t w t dt z t z t w t w t V x dt

z t z t w t w t V x dt

Cx t DK x t DK x t Cx t DK x t DK x t

w t w t V x w dt t s

V xγ γ

γ

τ τ

γ ζτ

∞ ∞

∞

∞

∞= − = − + −

≤ − + =

+ + − + + −

− + =

∫ ∫

∫

∫

&

&

&0

) ( ) ( , )t T

tt s dsdt

ττ ζ

∞

−Γ∫ ∫


(9)

where ( , ) ( ) ( ) ( ) ( ( )) ( )T T T Tt s x t x t x s f t w tζ τ σ= − T⎡ ⎤⎣ ⎦& . It’s obvious that ( ) 0τΓ < suggests ( ) 0J w < can be achieved. By Schur complementary lemma, ( ) 0τΓ < is equivalent to the following inequality:

(10)

1 2 7

2 1

7 6

( ) 0T

T

M

M

τ

Ξ Ξ Ξ

ΞΓ = <

Ξ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

M O

1 2 1 2 1( ) ( )T TP A B K A B K P Y Y QΞ = + + + + + + , , 2 1 2 2( ) TP A B K Y WΞ = + − +

3

T TY Wτ τΞ = − −⎡ ⎤⎣ ⎦T

4 2 1 1 2( ) (T T T Tb P c A B K I c A B KβΞ = + + + +, 2 ) 0⎡ ⎤⎣ ⎦ ,

5 1 10 0T T TB P B cΞ = ⎡ ⎤⎣ ⎦ , [ ]6 2 1 2 2( ) 0T

1A B K B K b Bτ τ τΞ = + τ ,

[ ]7 1 2( ) 0 0 0 0T C DK DKΞ = + , 1

TM Q W W= − − − , 2M Zτ= − ,

3 2 TM b c= , 2

24 nM Iγ= − , 1

5M Zτ −= − , 6 nM Iτ= − .

Defining 2

1 n n ndiag X X X I I IΔ = , multiply (10) by and Δ , TΔ

on the left and on the right, respectively, and we obtain

(11)

1 2 7

2 1

7 6

( ) 0T

T

M

M

τ

Ξ Ξ Ξ

ΞΓ = <

Ξ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

%M O

1 2 1 2 1( ) ( )T TAX B U AX B U Y Y QΞ = + + + + + + , 2 1 2 2( ) TA X B U Y WΞ = + − + ,

3

T T TY Wτ τΞ = − −⎡ ⎤⎣ ⎦ , 4 2 1 1 2( ) (T T T Tb c AX B U X c A X B UβΞ = + + + + 2 ) 0⎡ ⎤⎣ ⎦ ,

5 1 10 0T T TB B cΞ = ⎡ ⎤⎣ ⎦ , [ ]6 2 1 2 2( ) 0T

1AX B U B U b Bτ τ τΞ = + τ ,

[ ]7 1 2( ) 0 0 0 0T CX DU DUΞ = + , 1

TM Q W W= − − − , 2M Zτ= − ,

i iM M= , 3, , 6i = L , and Y XYX= , Q XQX= , W XWX= , Z XZX= , , i iU K X= 1, 2i = . Note that if there exist a solution to (11) then there exist a solution

to , which means the system (1) is asymptotically absolute stable. Then we ( ) 0tV x <can obtain the following theorem.


Theorem 1: Considering the retarded Lur’e system (1), for a prescribed 0γ > , the state-feedback controller (3) achieves the system (1) is asymptotically absolute stable and for all nonzero , if there exist ( ) 0J w < 2[0, )qw L∈ ∞ n n× matrices

,0X > 0Q > , 0Z > , Y , W , 1n n× matrices U , i iK X= 1, 2i = , and scalar satisfying (11). Under the aforementioned conditions, the state-feedback gains 0β >

are given by (12) 1

i iK U X −= , 1, 2i = Remark 1: Note that the resulting conditions for the state-feedback controller synthesis problem in Theorem 1 are not LMI condition due to the term

1

5 5

1M M Z XZτ τ−= = − = − X− in (11). An easy way to obtain LMI condition is to simply set Z Xε= , where ε is a positive scalar which can be obtained in calculating (11) until the LMI (11) is feasible. This way will introduce great over-design leading to more conservative results. Fortunately, by following Moon’s idea, and with the help of the results of [8], we can cast this nonconvex problem into a nonlinear minimization problem involving LMI conditions.

First, we define new variable such that L 1XZ X L− ≥ and replace (11) with

(13a)

1 2 7

2 1

7 6

( ) 0T

T

M

M

τ

Ξ Ξ Ξ

ΞΓ = <

Ξ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

%M O

(13b) 1XZ X L− ≥ where 5M Lτ= − . Equation (13) is equivalent to

(14a)

1 2 7

2 1

7 6

( ) 0T

T

M

M

τ

Ξ Ξ Ξ

ΞΓ = <

Ξ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

%M O

(14b) 1 1

1 10

L X

X Z

− −

− −≥

⎡ ⎤⎢ ⎥⎣ ⎦

Then, by introducing new variables L , X , and Z% , the original condition (11) can be represented as

(15a)

1 2 7

2 1

7 6

( ) 0T

T

M

M

τ

Ξ Ξ Ξ

ΞΓ = <

Ξ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

%M O

(15b) 0L X

X Z≥

⎡ ⎤⎢ ⎥⎣ ⎦%


where 1L L−= , 1X X −= , and 1Z Z −=% . Now, using a cone complementary problem [8], we suggest the following

nonlinear minimization problem involving LMI conditions instead of the original nonconvex feasibility problem.

Minimize (tr XX LL ZZ n+ + =% ) 3 subject to the following (16a)=(15a), (16b)=(15b)

(16a) 0L I

I L≥

⎡ ⎤⎢ ⎥⎣ ⎦

, 0Z I

I Z≥

⎡ ⎤⎢ ⎥⎣ ⎦%

, 0X I

I X≥

⎡ ⎤⎢ ⎥⎣ ⎦

If the solution of this minimization problem is , that is, 3n(tr XX LL ZZ n+ + =% ) 3 , we can say from Theorem 1 that the system (1) with the

state-feedback control law (3) is asymptotically absolute stable with a noise attenuation level γ . Note that (11) as a stopping criterion in the following algorithm since it is numerically very difficult in practice to obtain the optimal solution such

(tr XX LL ZZ+ + % ) is exactly equal to . 3nAlgorithm 1: Step 1: Choose a sufficiently large initial γ such that there exists a feasible solution to (16). Set 0γ γ= . Step 2: Find a feasible set 1 2( , , , , , , , , , , )X Y W U U Q Z L L Z X% satisfying (16). Set

0k = . Step 3: Solve the following LMI problem: Minimize ( )k k k k k kTr XX X X LL L L ZZ Z Z+ + + + +% % subject to (16). Step 4: If (11) is satisfied with the solution of Step 3, then set 0γ γ= and return to Step 2 after decreasing γ to some extent. If condition (11) is not satisfied within a specified number of iterations, then exit. Otherwise, set 1k k= + , and go to Step 3.

Similar to the proof of Theorem 1, we can obtain the following theorem.

Theorem 2 Let 2

[ 0, ] ( ) 0 ( ) , 0kf F f f kσ σ σ σ∈ = • < ≤ ≠ . Considering the retarded Lur’e system (1), for a prescribed 0γ > , the state-feedback controller (3) achieves the system (1) is asymptotically absolute stable and ( ) 0J w < for all nonzero

2[0, )qw L∈ ∞ , if there exist n n× matrices , 0X > 0Q > , 0Z > , Y , W , 1n n×matrices , i iU K X= 1, 2i = , and scalar 0β > satisfying ( ) 0τΓ <

). Under the

aforementioned conditions, the state-feedback gain are still given by 1

i iK U X −= , . Here 1, 2i = ( )τΓ

) is taken from ( )τΓ% in (11) by replacing 3M with 3 /M kβ− .

Remark 2 In the proof of theorem 2, we can obtain

(17) 2 ( ( ))

( ) ( ( ))f t

t f tk

σσ σ− ≤ −

from 2

[ 0, ] ( ) 0 ( ) , 0kf F f f kσ σ σ σ∈ = • < ≤ ≠ . Besides, needs to be 4 ( )tV xselected as

( )

4 0 0( ) 2 ( ) ( ) ( ( ))

t t

tV x f s ds s f s dsσ

β σ σ= +∫ ∫ , then we can obtain as following with (17)


(18) 4

2

( ) 2 ( ) ( ( )) ( ) ( ( ))

2 ( ) ( ( )) 2 ( ) ( ( )) ( ) ( ( ))

2[ ( ) ( ( )) ( ) ( ( ))] ( ( ))

tV x t f t t f t

t f t t f t t f t

t f t t f t f tk

σ σ βσ σ

σ σ βσ σ βσ σ

βσ σ βσ σ σ

= +

= + −

+ −

& &

&

&

≤

The rest proof of theorem 2 is as same as that of theorem 1. 4. Numerical examples Example 1: Consider the same Lur’e control system in [2] as following.

[ ][ ]

2 0 0.2 0.5 1 1 0.2( ) ( ) ( 0.1) ( ) ( ) ( ( ))

1 2 0.5 0.2 2 1 0.3

( ) 0.6 0.8 ( )

( ) 0 1 ( ) 0.1 ( )

x t x t x t u t w t f

t x t

z t x t u t

σ

σ

− − − −= + − + + +

− − − −

=

= +

⎧ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎪ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪⎪⎨⎪⎪⎪⎩

&

[0, ]( )

t

where f Fσ∞

∈ . By Theorem 1 and Algorithm 1, and using the software

package LMI Lab, it is obtained that the system can achieve the absolute stability

and for all nonzero ( ) 0J w < 2[0, )qw L∈ ∞ , with the prescribed H ∞

performance index 1.65γ = , and the controller matrices are [ ]1 -0.0575 -0.5738K = ,

[ ]2 -0.1272 0K = .2093 .

5. Conclusion In this paper, for a class of Lur’e control systems with time-delay, a design method of

H ∞ memory state feedback controller has been presented. Feasible design procedures are provided based on the LMI-based convex optimization approach by applying a cone complementary approach proposed by L. El Chaoui. The sufficient conditions are presented which guarantee that the Lur’e systems have H ∞ performance, moreover, the results obtained are less conservative than the reported results due to the bounded real theorem that was derived for Lur’e time-delay systems due to the Shengyuan Xu’s efficient method.

Acknowledgments This work forms part of the project supported by the National Natural Science Foundation of China (Grant No. 60574006) and the Foundation of Doctor (Grant No. 20030286013). References [1] B. Yang, M. Y. Chen, “Delay-dependent criterion for absolute stability of Lur’e type control systems with

time delay”, Control Theory and Application, 18(6)(2001), 929-931


[2] W. H. CHEN, Z. H. GUAN, X. M. LU, X. F. YANG, “Delay-dependent absolute stability of uncertain Lur’e

systems with time delays”, Acta Automatica Sinica, 30(2)(2004), 235-238

[3] Y. WANG, W. L. HU., “Stability analysis of nonlinear networked control systems”, Journal of Southeast

University (Natural Science Edition), Supp(II)(2005), 142-145

[4] E.Fridman, “New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems”,

Systems and Control Letters, 43(2001), 309~319

[5] S. Y. Xu, J. Lam, “Improved delay-dependent stability criteria for time-delay systems”, IEEE Transactions

on Automatic Control, 50(3)(2005), 384-387

[6] J. Yang, X. D. Wang, “Stability of a class of networked control systems”, Proceedings of the 5th World

Congress on Intelligent Control and Automation, HangZhou, (2004), 1401-1405

[7] Lu R Q. Su H. Y. & Chu J., “A descriptor system approach to H ∞ control for a class of uncertain Lur’e

time-delay systems”, Systems, Man and Cybernetics, 2003. IEEE International Conference on Publication

Date: 5-8 Oct. 2003, 1(2003), 199- 204

[8] L. El Chaoui, F. Oustry, M. A. Rami., “A cone complementarity linearization algorithm for static

output-feedback and related problems”, IEEE Trans. Automat. Contr., 42(1997), 1171-1176

[9] Guo, L., “ H ∞ output feedback control for delay systems with nonlinear and parametric uncertainties”,

IEE Pmc. Confml Theory Appl., (2002), 226-236

Min Zheng, Department of Automation, Southeast University, Nanjing, 210096,

P.R. China. Email: [email protected]. Min Zheng is a Ph. D candidate in

the research institute of automation of the department of automation of Southeast

University, and his research interests include control on retarded systems, and

H ∞ control, etc. he has published several papers in these areas.

Lin Chai, Department of Automation, Southeast University, Nanjing, 210096,

P .R. China. Email: [email protected]. Lin Chai is a Ph. D candidate in the

research institute of automation of the department of automation of Southeast

University, and her research interests include control on retarded systems,

adaptive control, and H ∞ control, etc. She has published several papers in these

areas.

S. Fei, Department of Automation, Southeast University, Nanjing, 210096, China



Received by the editors and in revised form December 21, 2006 67


ges 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, PaINFORMATION AND SYSTEMS SCIENCES Volume 2, Pages 67-72

ADVANCES IN ©2007 Institute for Scientific Computing and Information

DEVELOPMENT OF DISTRIBUTED ARCHITECTURE WITH MULTI-CLIENT SINGE SERVER BASED ON pxi FOR COLD-ROLLED

ALUMINUM FOIL

TAOSHENG REN AND WEI JIN

Abstract. In the process of cold rolling, there are many parameters to be controlled, such as AGC (Automatic Flatness Control), AFC (Automatic Gauge Control), Rolling Force etc. With virtual instrumentation and rugged PC-based platform PXI, the good control effect and high performance have been confirmed in the practical measurement and automation system applications. Upon the simple and effective TCP/IP messaging protocol, a multi-client single server distributed architecture, achieving real-time operation and remote data sharing, is developed and described in this paper．In addition, based on the DataSocket, minimizing overhead on the processing of the PXI and simplifying the programming work, the architecture developed is fully confirmed for its powerful network functions in Labview. Key Words. TCP/IP DataSocket Cold rolling PXI

1. Introduction At present, in the design of cold-rolled aluminum foil control systems,

SCADA/HMI (Supervisory Control and Data Acquisition/Human Machine Interface) software is commonly used to complete process control and data acquisition of the industrial. however, with the increase of equipment nodes, the real-time performance of the system often becomes lower, and it also provides some limitations in means of communication between the equipments, its network communication functions can not be satisfactorily completed. With the productive software Labview and RealTime Module, the scalable modular platform PXI, involving embedded controller and RT operation system, has better completed the Data AcQuire (DAQ) tasks. In the network communication, PXI supports most standard interface such as VXI, GPIB, Serial, Network Interface etc. 2. Process Technology

This system is mainly composed of AGC, AFC, MSC (Mill Stand Control) etc, as is shown in Fig. 1.

68 T. REN AND W. JIN

Fig.1 Process of the control system

The simple process is that the strip moving from Unwinder to Winder is rolled by the Regulation Actuators, which is the core of the cold rolling system. However, in the actual processes, the control system is very complex and there are many parameters needed to control, a distributed network is needed. With the commercial platforms PXI and productive graphical programming software Labview, the distributed architecture with Multi-client Single Server integrated systems can be easily accomplished. 3. Network Architecture

In the network communications, a multi-clients single server distributed architecture is used to develop a strong real time, modular easy expansion, supervisory control centralization, and to achieve real-time operation and remote data sharing. The overview of Multi-client Single Server architecture is shown in fig.2.

Fig.2 Overview of Multi-client Single Server architecture

Every station is simply linked by a switch to form a strong Local Area Network, operators and engineers can send commands and parameters to Server (name target), which is responsible for command parser, priority task, data sender etc.

DISTRIBUTED ARCHIYECHURE WITH MULTI-CLIENTS SINGLE SERVER 69

The distributed architecture used in this paper has the following features: The Client (name host) application runs on a remote computer, typically using a

non-deterministic operating system, such as Windows. It sends commands and parameters to the target and receives data and status information sent by the target.

The target application executes deterministic tasks based on the commands received from the host. It can dynamically accept and service any number of incoming connections, also keep track of client requests and service each client in an individual way.

The goal of this application note is to describe a server architecture, shown in fig.3, which can run indefinitely, continuously monitoring for new connections and servicing them accordingly.

ServerClient (n)

Command/ParameterSender

Fig.3 Conceptual diagram of the server architecture

Note: M P T=Medium Priority Task. A reliable TCP/IP-based message protocol is used to enhance the performance,

usability, maintainability and scalability of the system. When a data variable is transmitted by the sender, it is packetized with additional information, so it can be received and decoded correctly on the receiving side. the packet format is shown in Fig.4

Data Size (32 bites) Meta Data ID (16 bites) Data

Fig.4 Packet Format

Every packet includes 48 (32+16)bits of overhead related to the data size and Meta Data ID, according to Meta Data ID to certify the correctness of data transmission. 4. The Programming in Labview

Data Receiver

Connection Monitor

Connection Manager

HostM P T

Command Parser

High Priority

Target

Task


The host application (The HMI is shown in fig.5) is composed of command/parameter sender loop and data receiver loop, commands are originated by the host application's user interface - each front panel control is linked to one or more commands that perform specific operations on the target.

The block diagram of target application, shown in fig.6, has the following parts, TCP Connection Manager, Command Parser, DAQ and Data Sender.

The Connection Manager is a key part of a multi-node architecture. It's role is to store all incoming connections and provide an API (Application Programming Interface) to access those connections in different portions of the server code. It also allows storing context information for each of the connections. When receiving commands from the host, the command parser is responsible for routing the information to the appropriate loop, according to different parameter to write in different RT FIFO. The DAQ part includes A/D、PID Control、D/A etc, by reading the values of the parameter from the RT FIFO to control the automatic system.

Fig.5 HMI of the host application

The next feature is that, to alleviate the overhead of dealing with the PXI, DataSocket communication is adopted to distribute different operation permission. The engineer station centers with senior management authority to control or carry out all functions of the operator station. In the operator station, the limited operation corresponding to the front panel target will be hidden. Using the DataSocket connection,

DISTRIBUTED ARCHIYECHURE WITH MULTI-CLIENTS SINGLE SERVER 71

the engineer can master the operator very well. As the position client-machine has its own interface, with the DataSocket

Connection to connect front panel, target can realize remote data sharing. The operation permission is configured in the DataSocket Server Manager., while we must use the same DataSocket Server in the networks to achieve certain data item sharing. 5. Conclusion

With the PXI-6229 data acquisition cards, real-time operating system and an embedded controller PXI-8186, a multi-client single server is structured in labview, achieving real-time control and remote data sharing. It has the following characteristic such as short develop cycle, modular easy expansion, high performance etc. For the more, the design concepts, for a robust, expandable and easy to maintain, approach for building multi-server multi-client distributed applications.

Fig.6 Block diagram of the target application

Acknowledgements The author wishes to thank Mr. Zhenchao GAO and Ms. Dan YANG and Pan XIN

for helpful discussions throughout the work.


References [1] National Instruments. Labview RT Manual, 2004 [2] National Instruments。Labview User Manual, 2003 [3] Robert H.Bishop, Learning with LabVIEW 7 Express, 2005 [4] http://digital.ni.com/worldwide/china.nsf/main?readform

Taosheng Ren received the B. Eng degree in process control engineering from Northeastern University at Qinhuangdao (NEUQ), China in 2006, His research interests include Virtual Instrument, process modeling and simulation, and process control.

Department of Automation Engineering, Northeastern University at Qinhuangdao, Qinhuangdao, Hebei Province, P. R. China, 066004, email: [email protected]



ges 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, PaVolume 2, Pages 73-80 INFORMATION AND SYSTEMS SCIENCES ADVANCES IN


MEASUREMENT OF DIAMETERS OF ELECTRICAL WIRES USING LASER WITH VIRTUAL INSTRUMENT

WEI JIN, XIAOQIANG WANG and FEN LI

Abstract. Based on Fraunhofer diffraction theory, the measurement technique is described for the

diameters of thin electrical wires using a laser as a lamp-house、a CCD (electric charge coupling

devise) as sensor and the experiment system controlled with virtual instrument. The discussion is the

technique to develop the auto-measure system for the diameters of a wire with virtual instrument

technique. This results in system change and in the photo-detector output for sample time duration

what is determined by the virtual instrument. So the calculating rate is the emphasis. The product of

the time duration gives the diameter of the wire. This technique can also be used to measure the size

of similar objects, it is a non-contact measurement method and can be used in hostile environments. Keywords. Diameter measurement, light diffraction, laser beam electric wires

1. Introduction

The non-contact measurement method for diameter of electrical wires is based on Fraunhofer diffraction theory and a laser beam as a lamp-house. But the experiment system in our laboratory is on manual operation and the data acquired from the meter must log and deal with calculating by hands. The Object for a experiment time duration should be kept in static state, so any change will influence the veracity of result. It is necessary to develop a auto-measurement system to measure a diameters of thin electrical wires within a finite time duration.

Virtual instrument is a kind of blocking programming software with high performance, by which an interface of customer defined measurement system can be established completely. Its software and hardware platform can satisfy various needs of developmental test and measure application, and this is the developing trend exactly of test and measure systems in recent years. Facility and efficiency is the advantage of virtual instrument, the application of virtual instrument in the measurement of laser intensity can give us an accurate value with a neglectable error. The standard graphics interchange format (GIF) program software-LabVIEW, modularly I/O hardware and the technique of developing software and hardware can be used in integrated test system.

74 W. JIN, X. WANG AND F. LI

The development and application of virtual instrument in our laser accuracy measurement system for the single-slit width and the diameter of electrical wire measurement experiment can help us to save time on obtaining a cluster of data and achieve more exact result. In this treatise, we will take the experiment of laser accuracy measurement system as an example to indicate the development process of virtual instrument.

2. Principle of Measurement

When a laser beam is blocked by some opaque object, there will be a phenomenon of diffraction at the edge of object. The opaque object may be thin slot、small circle bore or slender line. The phenomenon of laser diffraction is a kind of representation of wave of light and the character of laser beam passing by a small object. Diffraction makes laser intensity to re-distribute in the space. If the size of opaque object and the wavelengh of the laser beam are close by, the phenomenon of laser diffraction will be observed easily. The application of photoelectric element converting the laser diffraction intensity to the instrument or galvanometer is one of the key methods to measure the laser diffraction in accuracy. The traditional method to measure the diffraction is to install the photoelectric sensor at the top of a support on line-shifting device that can be hand-operated by user to carry out position moving. By the photoelectric galvanometer checking to obtain data, we get the stripe of laser diffraction intensity distribution. By the computation of artificial propagation, obtain measurement value of single slot、small circle bore or slender line etc… . In this experiment we take thin slot laser diffraction intensity distribution and electrical wire diffraction intensity distribution for the examples. The principle diagram of measure equipment is shown in figure 1. The instrument adopted is a laser intensity distribution test instrumentation of WGZ- type, the light source is a helium-neon laser of JGQ—250 type, the wave-length is 632.8nm; Measure instrument is digital galvanometer of WJF type.

MEASUREMENT OF DIAMETERS OF ELECTRICAL WIRES 75

Computing and Information INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, Pages 1-22

(a)

(b)

Fig. 1 The skeleton drawing of test instrumentatio and photograph of the experiment equipment (b)

er of v

（1）

n (a)

Based on principle of Fraunhofer diffraction, it is obvious that the collimated lasertical incidence which past thin slot displays a regular pattern of laser diffraction

intensity distribution as follow:

2

2

0sinθ

θII =

In the formula above there are

Bx=θ

DdB

λπ

=

Where is the width of single slot, d λ is the wave length of light source, D is the distan tween the disposition of sin e slot and photoelectric cell, ce be gl x is th stance

Lase

e di

r mp-house la

Photoelectrical sensor

Galvanometer Instrument

Line shift device

Slot or Electrical wire

Beam of light


between t e center of diffraction stripe and the position of the first dark stripe measure point. The photograph of laser diffraction intensity distribution is shown n figure 2.

h i

Fig. 2 Photograph of laser diffraction intensity distribution

In figure 2, the center, where the laser intensity is maximum, 0=θ , 0II = for

formula (1), called

πθ K=the central primary maximum；When ( .....2,1 ±±=K )，

dDK /λθ =

the symmetry

, that is the dark stripe, that is to say, the stripe is black Th ipes distribute on the left or right of the center is equality along a b

distribute from the center. The width between the center and the first dark stripe, xΔ can be determined by the distance of black strips which 1±=

.lack

e strripexis while st s take

K , that is

dDλ2=Δ ; the position of some black stripe make an inverse proportion with the width

lot d , when d is large, the of single s x is small, the diffraction stripes of all class

below.

(2)

gathered toward the center. So, the calculation formula of single slot width is shown

xDKd λ

=

e positi Therefore, if we measure th on x of black strip of class K, thvalue can be obtained via widt n of single slot. For as much as tha

e measure h calculatio t the wave

s

The virtual instrument technique adopted in this design is the modular hardware , combing with efficient vivid software of NI (National Instrument)

to co

ded into the anal

auto-control、signal auto-acquisition、signal preprocess、quantify calculation and result

length of light is in the range of class hun eds nanometer, so we can measure the physical dimension of minute object. When d is large enough, the phenomena of diffraction will be inconspicuous, in that case we can considered that the light is running in a straight line, a light line or light point can be een only on the central position. 3. Hardware Design

dr

with high performancemplete the automation test work of miscellaneous kinds expediently. In order to make the whole measure process move automatically, the hardware of

measure system adopts the design method of modular instrument, that is diviog input block of data acquisition and the output block of auto-operation control

instrument or devise. Connecting the computer via message interface, completing the



output of testing process. The process system also can connect with other computers on the net via web-bus to exchange information, so as done in figure 3.

In design of data acquisition, i.e. analog input block, we adopt components assembly such as small drift amplifier、 filter and A/D transducer of high precision …etc. The block of output should fulfill the operation steps mentioned

above. Take the

disp

tically. The next step is completing the data by com

Fig. 3 Block composition diagram of the testing system

Software Design The virtual instrument program developed by using Labview is briefly called VI,

which include three m m procedure and the cter. The procedure front panel set the input data or be called

Cont

software with front panel and block diagram program.

lacement control as an example, displacement control block adopt a chip microcomputer to receive the control signal of computer via serial port, then transfer the signal into power signal to control the magnetic stepping motor, drive the sensor of photoelectric element moving in a line.

The displacement control is on moving by 0.1 mm within each step, then collecting the signal of photoelectric sensor, acquiring the data of intensity distribute, and depositing into computer memory automa

puter software, including the calculation of absolute value and average value, the location of central, drawing the curve of intensity distribute and outputting the result of calculation etc.

4.

ain parts: procedure front panel、frame diagradiagram mark/conne

rols and Output of Observation or be called Indicators, that is to simulate the real operation front panel of control instrument. Here we take the example of using the input as the straight-line displacement of light intensity distribution, the front panel is composed of knob、button and switch, which is used to control the operation of displacement direction and auto-steps. The output of light intensity distribution is displayed in chart or diagram. The diagram mark /connecter is a connection when the VI is invoked by another VI, and the connecter mark the input/output port of nodal point data. We need to allocate the port of connecter when programming, in order to be one-to-one correspondence to the input/output on front panel. Figure 4 is the example of

Computer

Instrument Signal

Interface Board

Condition

CRT

Shift

Control Shift Device


In order to utilize the powerful data analyses and process function of Labview, formula editor is adopted to calculate the formula (1) and formula (2) and so on. The FOR circulation is adopted to control the number of calculation times. The SIN module is adopted to calculate SIN function. The standard error module is adopted to calculate the e

of laser diffr

rror value between the sample and the curve of theory. The primary executable file can be invoked in forms of command line, exchange data in form of data file.

As the example of software developing technical application of virtual instrument in real system, we complete the application software in accurate laser measurement experiment. One vision example of testing process is shown in figure 4 (a), which display the measure result of light intensity distribution and the photograph

action of electrical wire in figure 5, correspond to the block diagram program in figure 4 (b).

(a)



(b)

Fig. 4 The example of software (a) front panel (b) block diagram program

Fig. 5 The photograph of laser diffraction of electrical wire

5. Conclusion Based on virtual instrument technology, the equipment for measurement of

diameters of electrical wires has been developed and has realized auto-operation in data acquire and processing. It has the following advantages such as high develop stage in software and hardware, modular expansion, high performance etc. It ensures the measurement more fine in veracity and high efficiency. References [1] National Instruments, Labview RT Manual.2004 [2] National Instruments, Labview User Manual, Texas: National Instruments, 2003 [3] Robert H.Bishop Learning with LabVIEW 7 Express.2005 [4] National Instruments, Using LabVIEW to Create Multithreaded VIs [M] Texas:

National Instruments, 2000. [5] http://digital.ni.com/worldwide


EI JIN received the B. Eng degree in process control, M.Eng degree and PhD

epartment of Automation Engineering, Northeastern University at Qinhuangdao, nc

W

degree from Northeastern University (NEU), China in 1982、1988 and 1996. He is

currently a professor in Northeastern University at Qinhuangdao, China. His

research interests include Virtual Instrument and Intelligent Control &

Measurement.

D

Qinhuangdao, Hebei Provi e, P.R. China, 066004 , email: [email protected]



THE ACCURACY COMPARISON OF CALCULATED METHOD ABOUT LIGHT DISTRIBUTION IN FRAUNHOFER DIFFRACTION

XIAOQIANG WANG, WEI JIN AND FEN LI

Abstract The principle of slit Fraunhofer diffraction is introduced. By the principle of diffraction, the filament diameter could be measured using the optical instrument. This kind of method has some advantages, such as facility, visualize and high degree of accuracy. The method is analyzed in detail. Based on these conclusions, how to realize the purpose of measurement is discussed. Key Words, Fruaunhofer diffraction, Filament diameter, measure, optical instrument

1. Introduction

We could use relatively exact mechanical tools such as vernier callipers and micrometer screw gauge et al to measure the fineness such as filament and fibre diameter. Reading microscope, toolmaker’s microscope and Abbe comparator could be used as well. Because of the diffraction of light, the filament diameter could be measured by the optical instruments based on the principle of diffraction [1]. This kind of method has some advantages such as facility, visualize and high degree of accuracy, and displays its unique function in high-accuracy survey. The purpose of remote-control and non-touch about filament diameter could be realized if we analysis process the corresponding diffraction pattern using computer. This kind of method accords with the modern technology’s requirements for the information transfer, and has the important real significance and the economic value. 2. Principle

Diffraction can be defined as the behavior of a light wave when it encounters an obstacle or a minipore in its propagation. In general, diffraction causes a light to bend around obstacles and make patterns of strong and weak waves radiation out beyond the obstacle. Diffraction is the representation of wave character. The phenomenon could be explained by the Huygens-Fresnel principle. The diffraction of light mainly divides into two kinds: Fresnel diffraction and Fraunhofer diffraction. Using the laser source, the pattern of Fraunhofer diffraction would be obtained in view of the far-field condition. As shown in Fig. 1.

Lasing light filame

Diffraction Fig. 1 The principle of the measurement with Fraunhofer diffraction

For the single slit Fraunhofer diffraction, the light distribution is [2]:

Received by the editors September 12, 2006 81

82 X. Q. WANG, W. JIN AND F. LI

220 /sin ββII = (1)

λϕπβ /sina= (2)

Where is the width of slit, a λ is the laser wave length, ϕ is the diffraction angle (the angle between the diffracted ray and the optical axis). A theorem called Babinet’s Principle states that the diffraction pattern for an aperture is the same as the pattern for an opaque object of the same shape illuminated in the same manner [2-3]. That is, except for the intensity of the central spot, the pattern produced by a diffracting opening of arbitrary shape is the same as a conjugate of the opening would produce, and would be given by

)()()( PUpUPU ba =+ (3) Where , is the complex amplitude of the diffractional field caused by complementary screen respectively. is the complex amplitude of the free wave field. Because the diameter of the filament and the width of the slit are complementary, we could figure out the diffraction pattern of the filament according to the diffraction pattern of the slit. The distance between the center of the central principal maximum and the first principal maximum is given by

)(PU a )(PU b)(PU o

d

aLd /43.1 λ= (4)

Where is the diameter of the filament, a λ is the laser wave length, is the distance between the filament and diffraction screen.

L

3. Measurement Use high-accuracy (<1µm) solid-state imaging detector CCD to take the picture of Fraunhofer diffraction pattern of the filament in real-time, and store it into computer via interface. After the analysis processing, the size corresponding factor of the picture and actual pattern would be obtained, the distance in the formula (4) could be calculated. The diameter of the filament a would be obtained. Therefore, the purpose of non-touch making measurements of very small objects would be

d

realized. In the analysis processing, the key problem is how to get the central position of the fringe on the CCD picture. This problem could be settled by resolving the fringe and adopting position integral method to calculate the central position. The analysis processing of the CCD diffraction photograph is implemented by software of image processing. The virtual instrument program developed by using Labview is briefly called VI, the testing system and the program construction is respectively shown as Fig. 2, Fig. 3. The coherent work is proceeding. The diffraction fringe is more rarefactional as the filament diameter is smaller, whereas the diffraction fringe is more condensational as the diameter is bigger. In consideration of the computer’s resolution capability for the photograph, the method of linear filtering should be adopted to attain a purpose of background correction. There is the corresponding error for any measurement [4]. This kind of method isn’t exceptional. The error aΔ correlate withλ , and d L

THE ACCURACY COMPARISON OF CALCULATED METHODS 83

dd

LLd

LdLa Δ+Δ+Δ=Δ 2

λλ (5)

CC

D

Imag

e of

diff

ract

ion

Inte

rfac

e

Com

pute

r

Inte

rfac

e

Computation

Fila

men

t dia

met

er a

Calibration

Fig.2 The diagram of the testing system

Fig. 3 The construction of the software program Because the error λΔ of the laser wave length is so small (<10-5λ ) that could be ignored, so

dd

LLd

a Δ+Δ=Δ 2

λλ (Also could be represented by the form of the random error)

Where the error dΔ is relative to the resolution of CCD could be obtained by the error

Con

trol M

enu

Initialization(Calibration)

Obtain the photograph

Adjust size of photograph

Resolving the fringe

Proc

essi

ng

mod

ule

Obtaining the data of

photog

Save

raph

Computation

Images save

Data access

84 X. Q. WANG, W. JIN AND F. LI

analysis according with the corresponding method which is used to calculate the central position of each fringe. Therefore, the error aΔ of the filament diameter could be figured out. According to the formula (4), the measurement accuracy would be higher as the diameter of filament is smaller. Besides the method mentioned above, we adopt other method to measure the filament diameter. At the first, a series of standard library of images based on the formula (4) could be created in computer. The library would be detailed, for instance, the step length of diameter ( ) could be less than 0.01µm. By the computer analysis processing, make a comparison between actual diffraction pattern and the pictures in the image library seriatim, until the picture which error is minimum was found. The diameter corresponding to the picture could be regarded as the filament diameter. The merits of this kind of method are non-external disturbance and high survey accuracy. The key problem that needs to be solved is how to reduce the complicated computation during the comparison process in valid.

a

a

4. Conclusions The filament diameter such as fibre could be measured by the method of Fraunhofer diffraction. In any process of measurement, for increasing the measurement accuracy, these factors such as measurement distance , step length a should be integrated considered. In general, to obtain the optimum resolution capability of computer, the method of linear filtering should be adopted to attain a purpose of background correction. By the method of Fraunhofer diffraction, the purpose of remote-control and non-touch about filament diameter could be realized.

L

References [1] Salvatore G. (2005), “Fraunhofer diffraction by a thin wire and Babinet’s principle” , American J. of Phys.,

Vol. 73, pp. 83-84.

[2] Born M., Wolf E.(1999), Principles of Optics: Electromagnetic Theory of Propagation, Interference and

Diffracion of Light (7th Edition), Cambridge University Press

[3] Powers S. R.(1980), “Some factors affecting the reliability with which fibre size and particle size

distributions may be obtained from light scattering data”, J. Phys. D: Appl. Phys., Vol. 13, pp. 2223-2231.

[4] Shaoqing W., Benzhuo L. and Shihui H. (2005), “Measurement of Filament Diameter Based on

Fraunhofer Diffraction Principle”, J. Jinan University (Sci. & Tech1), Vol. 19, pp. 178-180.

Xiaoqiang Wang, Materials Science & Engineering Department, Northeastern

University at Qinhuangdao. Email: [email protected]. Dr Wang received his

PhD from Lanzhou University in 2004. His research interests are in the areas of

measurement and analysis of thin films/material and surface science.

Northeastern University at Qinhuangdao, Hebei Province, China (066004) [email protected], jinwei @mail.neuq.edu.cn, [email protected]




RESOURCE DISCOVERY MECHANISM BASED ON PEER-TO-PEER PROTOCOLS IN GRID

XUAN WANG ADN LING-FU KONG

Abstract. Resource discovery is a key issue in Grid environments. Centralized approaches to resource discovery have limited scalability. In large scale grid environment, a way to provide Grid scalability and avoid bottleneck is to adopt Peer-to-Peer (P2P) models to implement Grid services. P2P protocols based resource discovery mechanism was introduced. Information nodes with registered resources are organized into a Virtual Organization (VO). Improved P2P resource discovery model was proposed which reduced the forward times of messages among multi-VOs. Considering the effect of clock synchronous and network congest, we defined a dynamic overtime mechanism to adapt network change. The simulation result shows that our mechanism attains good efficiency of resource discovery. Key Words. Grid, Resource discovery, P2P, Virtual organization.

1. Introduction The Grid computing model offers an effective way to build high-performance

computing system. Users can efficiently access and integrate computers, data, and applications, which were distributed geographically. Resource discovery, a crucial part of a grid, refers to the process of finding satisfactory resources for user requests, including resource description, resource organization, resource lookup, and resource selection. Both dynamism and heterogeneity of a grid make resource discovery difficult. There are several projects that have addressed the resource discovery problem in the Grid such as the Monitoring and Discovery System (MDS) of Globus Toolkit[1] and the Matchmaker of Conder-G[2]. They adopt the centralized or hierarchical approaches to support resource discovery. The growing scale of shared resource communities dictates that the resource discovery services should scale well. Centralized approaches to resource discovery have limited scalability.

Peer-to-Peer (P2P) computing is another active research area that shares several computing interests with the Grid. P2P computing, although hampered by a lack of legitimate applications, has acquired a strong footing in certain avenues such as search and storage scalability, decentralization, fault tolerance, anonymity etc. The two environments seem likely to converge in terms of their concerns, as Grids scale and P2P systems address more sophisticated application requirements[3].

In recent years, significant interest has focused on a development effort to align Grid technologies with Web Services. The Grid community introduces the Open Grid Services Architecture (OGSA) which defines Grid Services as an extension of Web Services. The OGSA provides some common operations and supports multiple underlying resource models representing resources as service instances. Since the OGSA model offers an open cooperation model that allows Grid entities to be composed in a decentralized way,

Received by the editors September 18, 2006 This research was supported by Province Science Foundation under grant F2006000281.

85

86 X. WANG AND L. F. KONG

it makes P2P models integrate into Grid environments. This paper constructs a resource discovery system based on P2P protocol that extents

the model of the GT3 information service. Request routing mechanism among multi-VOs is given. Considering the factor of network bandwidth, a dynamic TTL value is defined to adapt network change. Our mechanism performs better scalability than centralized architecture. And the resource discovery algorithm attains better network traffic and response time than flooding algorithm.

2. Related works

The MDS is the Information Services component framework of the Globus Toolkit 3.0(GT3.0). It provides information about Grid resources for use in resource discovery, selection, and optimization. It is based around WSRF (Web Services Resource Framework) standards providing an index service to maintain the set of registered Grid resources in virtual organizations. Index services, or users, can use an enquiry protocol to query directory servers to discover entities and to obtain more detailed descriptions of resources from their information sources. Our research is complementary to MDS, proposing and evaluating mechanisms that can be used to organize these directories in flat, dynamic networks.

Condor-G uses a centralized mechanism for maintaining the resource information; it uses a collector that listens for advertisements of resource availability. Matchmaker in Condor uses a central server to match the attributes in the users specification and those in the service providers’ declaration. The centralized architecture is efficient for the LAN for which Condor was initially designed, but it has a single point of failure and scales poorly.

Iamnitchi et al.[4] discuss decentralized resource discovery for grids. They use a Gnutella[5] like approach and search for resources by guided flooding in an unstructured overlay. Requests are forwarded to one neighbor only based on experiences obtained from previous requests, thus trying to reduce network traffic and the number of requests per peer compared to simple query flooding as used by Gnutella. The approach suffers form higher numbers of required hops to resolve a query compared to our approach and provides no lookup guarantees.

A grid resource discovery model based on route-forwarding is proposed and analyzed in [6]. The model may suffer from scalability problem because all the resource routers are equal peers and resource routing information need propagate across the whole network.

3. Design of architecture 3.1. P2P architecture.

From the perspective of the Globus Toolkit 3.0 information service, the grid is a collection of resource shared by different virtual organizations (VO). Each one indexed by different Index Service holds information about all the underlying resources. However, for scalability reasons, a multi-level hierarchy of Index Services is not appropriate as a general infrastructure for resource discovery in large scale Grid. Since the centralized or hierarchical approaches are inadequate to support discovery of resources that span across many independent Vos[7]. The framework describes here adopt the P2P model to support resource discovery across different VOs quickly.

P2P BASED GRID RESOURCE DISCOVERY MECHANISM 87

FIGURE1. The P2P framework of resource discovery The general architecture of the framework is presented in Figure1. Some independent

domains are represented. Each domain, named Grid Peer (GPeer), is consisted of Peer Mediator (PM), User Agent (UA) and VO. PM is responsible for delivering to or receiving from the messages from other PMs. UA manages user requests and provides request response. Each VO has one PM and uses Index Service of GT3 to manage its underlying resources. Each PM is connected with a set of PMs to exchanges query/response messages in a P2P mode. A connection between two neighbouring PM is a logical state that enables them to directly exchange messages. Direct communication is allowed only between neighbours. Therefore, query messages of one PM only will be forwarded to its neighbours. PM processes query messages by invoking the Index Service of the corresponding VOs. Query responses are sent back directly or along the same path that carried the incoming query message to UA.

User Agent submits local/global queries to PM. A local query searches required resource information in Index Service of local VO. When the desired resources do not exist in locals, UA starts a global query to discover resources located in possibly other VOs. The global query is submitted to local PM and it will be forward to neighbouring PMs in typical P2P networks.

3.2. Resource description and matching.

Searching based pure text underlines P2P systems will not be enough to discover resources in the Grid efficiently. Resource discovery mechanisms need information about the desired resources, which are in the form of metadata. Using metadata can improve the accuracy and efficiency of resource discovery. We adopt established metadata standardization such as Dublin Core to describe resource entities. For example, metadata has to be fresh and time sensitive, especially when a resource is an extra amount of processor time for Grid computation. In this case, it may require checking system resources, such as scheduled jobs and system load which means that the metadata has to be updated at regular intervals. We represent the resources by XML and searches request by a query language XQuery. Figure 2 gives a example of resource description. It illustrates the current system status, such average load and keyboard idle time.


<domain name=”ise.ysu.edu.cn”> <machine name=”client01.ise.ysu.deu.cn”>

<status> <average load> 0.04</average load>. <keyboard idle>02:25:14</keyboard idle> <ValidPeriod>4 minutes<ValidPeriod> <UpdatePeriod>1 minutes</UpdatePeriod> </status>

</machine> </domain>

FIGURE 2. A example of resource description

4. Resource discovery mechanism 4.1. Routing algorithm.

When PM receives a request from UA, it will check its local resources index firstly. If there has satisfactory resources, PM will reserve the resource and return a link handle to UA. Intra-GPeer updating is maintained by Index Services of GT3. Resources refresh their attribute values periodically and PM disseminates the information among the GPeers. Expired resources will automatically be eliminated after a timeout. This soft-state approach allows a fact that resources may join and leave the system without explicit de-registration.

When UA submits a global query, the request will be forwarded to neighbors of current GPeers. The resource discovery in our framework is a P2P router searching process in fact. The number of Grid Service operations that a peer can efficiently manage in a given time interval depends strongly on that overhead. For this reason, standard P2P protocols based on a pervasive exchange of messages, such as Gnutella, are inappropriate on large OGSA Grids where a high number of communications take place among hosts. To overcome this limitation, we modify the Gnutella protocol. We define the Route Mark Table and set dynamic value of Time-To-Live (TTL) to make Grid Services effectives as a way for exchanging messages among Gpeers in a P2P fashion. The resource discovery mechanism is as follows:

1. Each GPeer maintains an XML document that represents the set of resources that are available for other GPeers. This XML document is uniquely identifier. In each Gpeer, its PM maintains a routing table that contains the list of neighbors that are topologically closest. For instance, Gpeer A’s neighbors are stored in a routing table R(A). The router table has one GPeer’s URL at least. Once a GPeer has connected successfully to the network, it communicates with other GPeers by sending and receiving messages. Each node accepts messages destined for other nodes and routes them to their destinations.

2. When a user initiates a request, he submits it to local UA. Based on the user requests, UA generates and submit query message to the local PM. The PM checks its store of XML documents and, if a matching resource is found, it sends a request directly to the node that has this resource. If not, the PM sends a message with a larger TTL and

P2P BASED GRID RESOURCE DISCOVERY MECHANISM 89

forwards the message to another PM, which is chosen randomly from its routing table. This encourages the message to explore all the paths, and also prevents some bad nodes from permanently affecting the search performance. The message contains the following information: source ID, destination ID, the resource requested by the user, message identifiers, and the time-to-live of the message. In addition, for avoiding query round, we add a route mark field in the message. It is used to record those PMs that have been accessed by the same query. The PM forwards the message until either the resource is found, or TTL is zero. The number of PMs that message visited as it tries to find one resource is called the number of hops.

3. Once a resource is found, a reply message is sent to originating PM. The reply message is passed back to the original node either directly, or via each PM that forwarded the request. The reply message may contain the resource itself, or the address of the replying node, possibly with an authentication certificate to grant access to the resource.

4. Once the node that initiated the request receives a reply, a connection is opened to allow the user to access the resource.

4.2. Time-To-Live setting.

To limit congestion and loops in the network, queries contain a TTL, which is decreased at each forward, and queries will not be forwarded when TTL reaches zero. Considering the effect of clock synchronous and network congest, we define a dynamic TTL to adapt network change. The initial TTL value of current PM is T0. When the request is forwarded k times to PMk, its TTL value is Tk. If system adopts the direct responds mode to return query result, the TTL of PMk calculates in formula 1. (1) Tk=Tk-1-PTTk-1,k-ΔTk where PTTk-1,k represents communication latency between PMk-1.and PMk. PTT0,1=0. ΔTk represents the time that deal with forward request from PMk-1 to PMk. If Tk-PTT0,k≤0, PM will discard the request.

5. Conclusion

Resource discovery is a key issue in Grid environments. This paper proposes a resource discovery framework that adopts a P2P approach. It for finding the resources makes a trade-off between the size of the network and efficiency. The inter-GPeers routing algorithm is based on the Gnutella protocol with improved TTL. The modification reduces the response time and network traffic than flooding algorithm. We created a simulation network with 100 GPeers. These GPeers were embedded randomly in 100×100 space. Each GPeer accepted messages destined from other GPeers and routed them to their destinations. Resources show normal distribution in network. The experimental result shows that the average number of discovered resources and the average number of hops per request of our discovery algorithm is better than the flooding algorithm. However, further investigation to assess the performance of our algorithm in a large uncontrolled network is not discussed in this paper, which will be our future work.


References [1] Globus.http://www.globus.org, 2004 [2] J.Frey, T.Tannenbaum, M.Livny, I.Foster and S.Tuecke. Condor-G: a computation management agent for multi-institutional Grids, Proceedings of the Tenth International Symposium on High Performance Distributed Computing, (2001) 55-63 [3] D. Talia, P. Trunfio. Toward a synergy between P2P and Grids. IEEE Internet Computing, (2003) 94-96. [4] A. Iamnitchi, I. T. Foster. On fully decentralized resource discovery in Grid environments. Proceedings of the Second International Workshop on Grid Computing, (2001) 51-62. [5] M. Ripeanu. Peer-to-Peer architecture case study: Gnutella network. Chicago: University of Chicago, http://www.cs.uchicago.edu/matei/PAPERS/gnutella-rc.pdf, 2001 [6] W. Li, Z.Xu, F.Dong and J.Zhang. Grid resource discovery based on a routing-transferring model. Proceedings of the 3rd International Workshop on Grid Computing, (2002) 145-156 [7] Y. Gong, F. Dong, W, Li and Z. Xu. VEGA infrastructure for resource discovery in Grids. Journal of Computer Scientc& Technology, Vol.18, No.4, (2003) 413-422. Department of Computer Science and Engineering, Yanshan University, Qinhuangdao, P. R. China, 066004 Email: [email protected]



ges 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, PaINFORMATION AND SYSTEMS SCIENCES Volume 2, Pages 91-97

ADVANCES IN Computing and Information ©2007 Institute for Scientific

SEMI-BLIND MULTIUSER DETECTION BASED ON IMPROVED PASTD SUBSPACE TRACKING FOR MC-CDMA UPLINK

YAN MENG, JINKUAN WANG, JUN ZHU AND XIN SONG

Abstract In this paper, an adaptive semi-blind multiuser detection (MUD) algorithm based on improved projecting approximation subspace tracking with deflation (PASTd) subspace tracking is proposed for multicarrier code division multiple access (MC-CDMA) uplink where the base station receiver has the knowledge of the spreading sequences of all the users within the cell, but not that of the users from other cells. It is known that the PASTd algorithm has the drawback of slow convergence rate. Based on this, we develop an improved PASTd algorithm and apply it to the adaptive linear hybrid semi-blind multiuser detection. The improved PASTd algorithm guarantees the orthonormality between the estimated eigenvectors such that a fast convergence rate can be achieved. Simulation results show the proposed semi-blind MUD has a fast convergence rate and provides the similar output signal-to-interference-plus-noise-ratio (SINR) and bit error rate (BER) as the Singular Value Decomposition (SVD) semi-blind MUD. Key Words Semi-blind multiuser detection, Subspace tracking, PASTd algorithm, MC-CDMA

1. Introduction

MC-CDMA has received considerable attention for future high-speed wireless systems [1]. The uplink transmission of MC-CDMA suffers from distortion of code-orthogonality among users by the instantaneous frequency response of channels, which causes an increase in the effective multiple access interference (MAI). Thus, multiuser detection is required for reducing its effects.

Recently, various blind multiuser detection techniques have been developed, which can suppress MAI by only exploiting the spreading codes of the desired user [2]. However, in uplinks, the spreading sequences of all the users within the cell are known at the base station receiver, which should assist in further suppressing the MAI. This idea was proposed in [3]-[5], and after that various semi-blind multiuser detectors [6]-[10] have been proposed for CDMA uplink, which can cancel interferers from both known and unknown users, while utilizing the information about known users.

A linear hybrid semi-blind multi-user detector has been proposed in [4]. This detector adopted SVD to obtain the signal subspace, which requires high computational complexity. Many efficient adaptive subspace tracking algorithms with low complexity such as projection approximation subspace tracking (PAST) in [11], PASTd in [11] and orthonormal projection approximation subspace tracking (OPAST) in [12] have been proposed. We found that the convergence rate of the PASTd algorithm is fairly slow. In this paper, we develop an improved PASTd subspace tracking algorithm and use it, along

92 Y. MENG, J. WANG, J. ZHU AND X. SONG

with the closed-form expression for the hybrid semi-blind MUD, to propose an adaptive semi-blind MUD algorithm for MC-CDMA uplink. The improved PASTd subspace tracking algorithm has a faster convergence rate than the PASTd algorithm [11]. The proposed semi-blind MUD offers better performance than PASTd semi-blind MUD and OPAST semi-blind MUD in terms of the convergence rate, SINR and BER. The simulation results are provided to demonstrate the performance of the proposed algorithm.

2. Signal model

We consider a synchronous MC-CDMA communication system in the uplink [8], where K users with known codes are in the cell, and K%

1,users with unknown codes are in

the other cells. The modulating symbols 1kb ∈ − are spread and mapped to different subcarriers by the Inverse Fast Fourier Transform (IFFT), where is the

length of spreading sequence. This signal is transmitted through a multipath fading channel, which is assumed to have paths, hence the channel impulse response (CIR) can be expressed as

N N

L

(1) 1

,0

( ) ( )L

k k ll

h t h t mTδ−

=

= ⋅ −∑ c

where ,k lh is the complex channel gain experienced by the signal of the user in the path, which obeys Rayleigh fading, and cT is the chip-duration. At the receiver,

point FFT is invoked for demodulating samples of the received signal. Hence the received signal can be expressed in vectorial form as

thkthlN N

(2) 1 1

K K

k k k k k kk k

A b A b σ= =

= + +∑ ∑r g g%

%% % n

where kA and k denote the received amplitude and the transmitted symbols of the user within the cell, respectively. k k k is the effective signature waveform of

the user. kC is a diagonal matrix which is composed of the spreading sequence of the user within the cell and k denotes the frequency domain channel transfer function which can be expressed as the point FFT of

bthk =g C H

N

thth

kk H

,0 , 1[ , ] .Hk k k Lh h −=h L ,k kAg %% and

k are the corresponding information of the user in the other cells. denotes a white Gaussian noise vector with zero mean and covariance matrix . b% thk n

NFor convenience and without loss of generality, we assume that the signature

waveforms of all users are linearly independent. Denote 1

I

[ , , ]K=G g gL , 1[ , , ],K=G g g %

% % %L 21diag( , , )2

KA A=A L and 21diag( , , )2

KA A=A %% % L %

2

. The autocorrelation matrix of the received signal r is then given by

(3) 2 2

1 1

2

K K

H H Hk k k k k k N

k k

H HN

E A A σ

σ= =

= + +

= + +

∑ ∑R rr g g g g I

GAG GAG I

%

% % %

% % %

By performing an eigen-decomposition of the matrix , we get R

(4) [ ]H

s ss n H

n n

⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

Λ UR U U

Λ U

where 1diag( , , )s K Kλ λ +=Λ %L contains the K K+ %

] largest eigenvalues of in

descending order and R

1[ , ,s K K+=U u u %L contains the corresponding orthnormal

SEMI-BLIND MULTIUSER DETECTION 93

eigenvectors; ,n N K K% , ]n NK uL conta e N K K− − % orthonormal eigenvectors that correspond to the eigenvalue 2

2σ − −=Λ I and ins th1[ ,K+ +=U u %

σ . The ran space of ge sU is called the signal subspace and its orthogonal complement, the noise subspa isce,

ultiple

spanned by nU .

3.

ulat as the following m

(5)

Semi-blind multiuser detection

Consider an uplink transmission, where the base station receiver has the knowledge of the spreading sequences of users within the cell, while that of users from other cells are unknown. Without loss of generality, we assume user 1 is the desired user. The semi-blind linear hybrid detector for user 1 can be form

2( ) sE A b

ed

.H H

constrained optimization problem

1 1 1 1min .t− m r =m G f

where effG is composed of the ective spreading sequences of all known users and 1[1, 0 , , 0 ] .T K ×= , ∈ ℜf fL By the method of Lagrange multipliers, th

[4]

(6) −

he desired user(2) is in the r followed

(7)

is propose subspace

4. Impr

ithmeig

e solution

1

’s data in

parameters

of (5) in terms of the signal subsp

m of a correlato

king algorithm

ace parameters can beH H=m U Λ U G G U

by a hard limiter

sgn( )Hm r

d to obtain the si

written as 1 1− −Λ

t

gnal

for

c

1 s s s s s s

A linear hybrid semi-blind multiuser detector for demodulating

( )HU G f

b =1 1

It can be seen from (7) that the proposed semi-blind multiuser detection can be expressed by signal subspace parameters. So the subspace methods make the importance of solving a detector turn to be the subspace tracking algorithm. We get the detector when we obtain the signal subspace parameters. In this paper, an improved PASTd subspace traadaptively.

oved PASTd subspace tracking algorithm

There is a lack of orthonormality between the eigenvectors in the PASTd algorithm [11], which induces the slow convergence rate of the PASTd algorithm. To alleviate this drawback, an improved PASTd subspace tracking algorithm is proposed. The basic idea of the improved PASTd algor is to orthonormalize all of the estimated eigenvectors at each iteration. After the thk envector has been extracted, we orthonormalize the eigenvector and the 1kthk − eigenvectors that have been estimated before the thk eigenvector. The same orthonormalization processing is used after we extracted the next eigenvector. Applying this orthonormalization procedure repeatedly, all of the

ctors ha en orthonormalized at each iteration. W e (:,1: 1)i k= −w U is the 1

estimated eigenve ve k

be e assum− eigenvectors at the thi iteration and (:, )i kU is

the thk eigenvect at the thi iteration. An orthonormal tion step or ieen extracted at the i

(:, ) (:, ) Hi ik k= −U U

|| (:, ) ||i k

zth

a

(:w U

can beiteration

k

made as follows after the thk eigenvector has b

, )iw dd

(8) = U

ng algorithm is sum

(9) (10) (:, ) (:, ) /i ik k dd=U U The im m follows prove spad PASTd sub ce tracki arized as


For 1, ,i M= 2,L(:,1i ) i=x r

K

For 1, 2, ,k K= + %L(:, ) (:, )Hy kU x

1i i i−=( , ) ( , )k k k kβ= +Λ Λ

k

k k

21 | |i i y−

(:, ) (:, ) [ (:, )k k= +U U xi

1 1(:, ) ] / ( , )Hi i i i i ik k y y− −− Λ

, (:, ) (:, )i i ik k k yiU

(: 1)i + = −x U If 2k ≥ x

(:,1: 1)i k= −w U (:, )k k= −U(:, )iU (:, )H

i iww U End

k

dd|| (:, ) ||idd k= U

) (:,k k(:,i ) /i=U U End End

where β is the forgetting factor, M is the number of iterations and K K+ % is the number of users. The improved PASTd algorithm guarantees the orthonormality of the weight

5. Simul

matrix spanned by the signal subspace at each iteration.

as th

e desired re

ation results

We consider a synchronous MC-CDMA uplink system, where five users with known codes are in the cell, while two users with unknown codes are in the other cells. A multipath channel with three paths is considered .The channel coefficients are randomly generated according to a complex Gaussian distribution. The channel coefficients of users in the cell are available at the receiver. The user 1 is specified e desired user. All of the interference users have the same power. There are six 10dB MAI’s in the channel, all relative to th user’s signal. The signatu sequence of desired user is Wash codes of length 32N = . The forgetting factor β is set to be 0.995. The performance of the proposed semi-blind MUD based on the improved PASTd (newPASTd) is compared with that of semi-blind MUD based on SVD, OPAST in [12], and PASTd in [11]. The performance measure is the output SINR given in [13] and BER. The data plotted is the averages over 100 Monte Carlo simulations and 3000

iterations in

each simulation.

Figure 1. Output SINR versus number of iterations


Fig.1 illustrates the SINR performance of the detectors versus number of iterations

where the SNR of user 1 is 20 dB. It shows that SVD MUD, newPASTd MUD and PASTd MUD can track subspace efficiently; with a fast convergence rate, OPAST MUD reaches the steady state with low values of the SINR; the convergence rate of PASTd MUD is very slow so it is unsuitable for real-time system. It is obvious that newPASTd MUD has a high convergence rate and high stable SINR, and almost has identical pe ormance with SVD MUD.

steady SINR and has strong ability to eliminate MAI.

rf

Figure 2. Output SINR versus SNR

The SINR performance versus the SNR of user 1 is plotted in Fig.2. In Fig.2, it can be seen the SINR of OPAST MUD and PASTd MUD are much lower than that of SVD MUD and newPASTd MUD; the newPASTd MUD presents the same performance as SVD MUD in each SNR. It shows that the newPASTd MUD has high

Fig.3 displays the BER performance of four detectors where the SNR of user 1 is 20dB.

Figure 3. Output BER versus number of iterations


It shows the BER performance of PASTd MUD and OPAST MUD are very poor; the pe

ate and high SNR, the BER of newPASTd MUD is little higher than that of SVD MUD.

6.

is close to the SVD semi-blind MUD in terms of the convergence rate, SINR and BER.

References

ng channels”, IEEE Transactions on Vehicular Technology, Vol.48, No.5, pp.1584-1595,

ion: A subspace approach”, IEEE Transactions

e Record of

CDMA”, IEEE Journal on

[5] Host-Madsen A., and Kyung-Sean C., “MMSE/PIC multiuser detection for DS/CDMA systems with

rformance of newPASTd MUD is close to that of the SVD MUD. Fig.4 presents the BER performance versus the SNR of user 1. 3000 symbols are used

to calculate the steady-state BER with various inputs SNR. It can be seen that the BER performance of OPAST MUD and PASTd MUD are poor; newPASTd MUD performs like SVD MUD in low SNR, for moder

Figure 4. Output BER versus SNR

Conclusion

In this paper, an adaptive hybrid semi-blind multiuser detector based on the improved PASTd subspace tracking is proposed for MC-CDMA uplink. We have developed an improved PASTd subspace tracking algorithm and applied it to the hybrid semi-blind multiuser detection. The improved PASTd algorithm has a fast convergence rate. The simulation results have shown that the proposed semi-blind detector outperforms the semi-blind MUD based on OPAST and PASTd, and

[1] Hara S., and Prasad R., “Design and performance of multicarrier CDMA system in frequency selective

Rayleigh fadi

Sept.1999.

[2] Wang X., and Poor H.V., “Blind multiuser detect

Information Theory, Vol.44, No.2, pp. 677-690, 1998.

[3] Host-Madsen A., “Semi-blind multi-user detectors for CDMA: subspace methods”, Conferenc

the Asilomar Conference on Signals, Systems & Computers, Vol.2, pp.1858-1862, Nov. 1998.

[4] Wang X., and Host-Madsen A., “Group-blind multiuser detection for uplink

Selected Areas in Communications, Vol.17, No.11, pp.1971-1984, Nov.1999.


inter- and intra-cell interference”, IEEE Transactions on Communications, Vol.47, No. 2, pp. 291-299,

Feb.1999.

[6] Xu Z., and Cheng R.S., “Multistage semi-blind adaptive multiuser detector for reverse link in CDMA

systems”, IEEE International Conference on Communications, Vol.1, pp.16-20, June 2001.

[7] Kafle P.L., and Sesay A.B., “Iterative semi-blind multiuser detection for coded MC-CDMA uplink

system”, IEEE Transactions on Communications, Vol.51, No.7, pp. 1034 -1039, July 2003.

[8] Hua W., and Hanzo L., “Semi-blind and group-blind multiuser detection for the MC-CDMA uplink”,

IEEE Vehicular Technology Conference, Vol.59, No.3, pp.1727-1731, 2004.

[9] Zhang G., Bi G., and Zhang L, “Group-blind intersymbol multiuser detection for downlink CDMA with

multipath”, IEEE Transactions on Wireless Communications, Vol.4, No.2, pp.434-443, March 2005.

[10] Kafle P.L., and Sesay A.B., “Iterative semiblind multiuser receiver for a space-time block-coded

MC-CDMA uplink system”, IEEE Transactions on Vehicular Technology, Vol.53, No.3, pp.601-610, May

2004.

[11] Yang B., “Projection approximation subspace tracking”, IEEE Transactions on Signal Process, Vol.43,

No.1, pp.95-107, 1995.

[12] Abed-Meraim K., Chkeif A., and Hua Y., “Fast orthonormal PAST algorithm”, IEEE Signal Processing

Letters, Vol.7, No.3, pp.60-62, 2000.

[13] Caamano A. J., Segovia-Vargas D., and Ramos J., “Blind adaptive Krylov subspace multiuser detection”,

IEEE Vehicular Technology Conference, Vol.4, pp.2338-2341, 2001.

Department of Automation Engineering, Northeastern University at Qinhuangdao, Qinhuangdao, Hebei Province, P.R. China, 066004



AN ADAPTIVE MODEL FOR LIVE MEDIA SERVICE GRID

YUHUI ZHAO, YUYAN AN, DEGUO YANG, HUI WANG, YUAN GAO

Abstract For supporting adaptive live media applications， a live media service Grid(LMSG) is developed, which adopts an extensible middleware architecture in a grid computing environment. This paper mainly describes its adaptive model in component level for robust, concurrent, distributed and available streaming services. An adaptive model is able to detect object anomalies of the component, reconfigure inter-component and intra-components, repair it, and then test the adapted object. The model is structured to the service layer and the adaptive layer separately, and the service layer provides functional services to other components, whereas the adaptive layer encapsulates the available adaptive mechanism for monitoring objects in the service layer and repairing the sick objects detected.

Keywords, Live Media Service Grid, adaptive service, overlay network, Service Component

1. Introduction

Although there are many web cast and distributed collaboration productions [1, 2]. Because of the lack of adaptive service model, several problems remain unsolved. First, they are lack of a QoS-awareness and scalability live media feasible delivery protocol, which is fitted to the synchronous group communication. Second, there is not an efficient multicast model for ISP to benefit from the service providing. Third, applications that manage a production environment are difficult to build because the detailed knowledge of the environment is required. Forth, applications are often tightly coupled with the environment and cannot adapt well to changes. Fifth, an application written for one environment cannot be easily reused in another environment.

Unlike [2,3,4,5,6,7], We are developing the live media service Grid (LMSG)[8] to address the above challenges. LMSG aims to bring together large communities of users across networks to enable them to work together closely and successfully. The LMSG service was provided by the service broker(SvB) nodes that are strategically deployed by ISPs and run the LMSG middleware. LMSG middleware is the components library for supporting live media applications in the streaming grid service environment.

This paper mainly discusses the adaptive model and its adjustment mechanism. An adaptive model is used to match the components, reconfigure inter-component and intra-components, repair it, and then test the adapted object. The model is structured to the service layer and the adaptive layer separately. The former provides functional services to other components, whereas the latter encapsulates the dependable adaptive mechanism for monitoring objects in the service layer and repairing the sick objects detected.

The adjustment mechanism is based on two fundamental notions. First, components should directly connect to those components from which they are likely to get Received by the editors January 10, 2006 98

A HIERARCHICAL MODEL AND EMPIRICAL INVESTIGATION 99

satisfactory streams. Second, components may use past history to determine the components from which they are likely to download satisfactory streams. Components connect to those components that have high scores, and disconnect from components with low scores.

2. Adaptive Model

LMSG is a concurrent and distributed streaming system, which can be described by means of distributed components referred to as subsystems, their connectors, and configuration. Components are designed as distributed component types, each of which defines functionality that is relatively independent of those provided by other components. The component is self-contained, that is, it can be compiled, instantiated and linked separately into a distributed system. The connector acts on behalf of components in terms of communication between components and encapsulates the details of inter-component communication. Components and connector deployed in the different SvBs.

2.1 Adaptive Component Architecture

The LMSG components are composed of the service layer and the adaptive layer. The service layer provides full functionality to service requests from other components and notifies the status of messages passing between objects in that layer to the adaptive layer. The service layer of a component communicates with service layers of other components through connectors to provide functions to and request them from other components. The service layer is composed of tasks (active objects), passive objects accessed by tasks, and connectors between tasks. A connector not only encapsulates a mechanism synchronizing message communication between tasks, but also notifies messages passing it to the adaptive layer. Passive objects accessed by tasks (e.g., entity objects) also notify messages arrived from tasks to invoke their operations to the adaptive layer. With the message notification from the service layer, the adaptive layer of the component monitors objects in the service layer to detect any anomalous behavior of objects. Once the adaptive layer detects an anomalous object in the service layer and presumes that the object needs to be treated, the component changes its mode from the normal phase to the adaptive phase. In the adaptive phase, the service layer may not provide services any more or provide partial services by means of remaining healthy objects, whereas the adaptive layer involves repairing the sick object in the service layer. The adaptive layer reconfigures objects in the service layer of the component and, if needed, notifies the object sickness to other components requiring services from the sick component to minimize the impact on the components from the sick component. It then starts repairing the sick object.

The adaptive layer of each component depicted in Fig. 1 is composed of the Reconfiguration Plan Generator, Repair Plan Generator, Self-Adaptive Controller, Monitor, Reconfiguration Executor, and Repair Executor, which are responsible for detection, reconfiguration and repair of objects in the service layer. The Monitor contains state charts for each task thread in the service layer, which model dynamic behavior of the task thread. With messages notified by both connectors between tasks and passive objects accessed by tasks in the service layer, the Monitor supervises the behavior of tasks, connectors and passive objects accessed by tasks using state charts for the task

100 Y. ZHAO, Y. AN, D. YANG, H. WANG AND Y. D. GAO

threads.

Fig. 1. The SvB Component architecture in LMSG

The Reconfiguration Plan Generator maintains configuration information of objects in the service layer, generating a reconfiguration plan in response to changes to the status of objects in the service layer within the component, and it also involves a list of other components, whose objects may need to be reconfigured to minimize the impact from the paralyzed object in the neighboring component and to provide services continuously without stopping any more than necessary. To achieve this, it mains information on the interconnection with other components in the system, and generates a reconfiguration plan if there are changes in the configuration of objects in other components. The Repair Plan Generator maintains knowledge of repair plans specific to each object such as a task, connector, and passive object accessed by tasks in the service layer of a component, and generates repair plans for repairing sick objects.

The Reconfiguration Executor substantially carries out the reconfiguration plan generated by the Component Reconfiguration Plan Generator to reorganize the objects in the component in response to sick objects in the component or different components. The Repair Executor performs the repair plan generated by the Repair Plan Generator to treat abnormal objects and test them after repairing in order to check whether the objects just repaired work normally. The Self-Adaptive Controller of a component coordinates the Reconfiguration Plan Generator, Component Repair Plan Generator, Monitor, Component Reconfiguration Executor, and Repair Executor to conduct self-adaptive processes for the sick objects, cooperating with the Component Self-Adaptive Controllers of other components for reconfiguration against the object anomalies.

2.2 Self-Adaptive Components’ Connections

The components communicate with each other via connectors between them in LMSG. In the normal phase, the service layer request services from the service layers of other


components, or provide services to them through connectors. When detecting an object in the service layer to be sick, the adaptive layer reconfigures objects using a reconfiguration plan generated by its Reconfiguration Plan Generator, and notifies its neighboring components of the object sickness through connectors. The notification messages are delivered to the adaptive layers of neighboring components, which reconfigure objects in the service layers of the components using their reconfiguration plans generated by their Reconfiguration Plan Generators to minimize the impact from the neighbor’s sickness.

Fig. 2 depicts self-adaptive architecture for resilient concurrent and distributed systems. When the adaptive layer of the Component1 (in Fig. 2) detects a sick object in the service layer, it generates a reconfiguration plan responding to the sick object. Based upon the reconfiguration plan, the adaptive layer reconfigures objects in the service layer and then may notify the adaptive layers of the Component2 and the Component3 via Connector1, Connector3, Connector2, and Connector6 (in Fig. 2), which reconfigure their objects in the service layers using their reconfiguration plans. After repairing the sick object, software architecture is also dynamically reconfigured as the sick object is returned to normal.

Fig. 2. Self-Adaptive Component-Based Software Architecture for LMSG

3 Available Adaptive Mechanisms

Usually the adaptation of components is carried out in a sequence: (1) detection of object anomaly, (2) reconfiguration before repairing the anomalous object, (3) repairing of the sick object, (4) testing of the repaired object, and (5) finally reconfiguration after repairing of the sick object. We present the local dependability scores and the connection dependability between the components, which address how to decide what kind of component is dependable, and what kind of component is anomalous.

3.1 Local Dependability Scores

LMSG’s adaptive model is based on two notions:


1. A component is distributed and on different brokers, which should directly connect to those SvBs from which it is more dependable.

2. A component may use its SvB’s past history to estimate the likelihood of a future successful QoS session.

In the Adaptive mechanism, a component encodes its past history with a set of local dependability values [1]. Component i stores a local dependability value for each component it has interacted with. If sat(i,j)is the number of satisfactory transactions component i has had with component j, and unsat(i,j) is the number of unsatisfactory transactions component i has had with component j, then we define the local dependability value as

Dij = sat(i,j) – unsat(i,j). Component i may deem a transaction unsatisfactory if, for example, the streaming is

inauthentic or tampered with, or if the streaming is either slow or interrupted. The local dependability vector Vd associated with component i contains all Dij values where j varies over all components in the network. In our implementation, each component maintains a hash table containing the local satisfactory values of all its acquaintances. An acquaintance is then defined as an entry in the hash table. If component i has never interacted with component j then there will be no entry in component i’s table for component j, and thus Dij = 0.

3.2 Connection Availability

Given a LMSG system, it is presented as an undirected graph G=(C, E), where C is the set of the components, and E is the set of connections. We define the Availability worthiness of the system to be:

1 1( , )

v v

iji j

A connection i j D= =

= ×∑∑

Where v is the number of components in C, and connection(i,j) =1, if (i,j)∈E, otherwise connection(i,j) = 0. If a system’s components can connected to more components, their dependability will have a high A value.

The Adaptive mechanism is a component-level greedy algorithm for maximizing A and it proceeds as follows. Component i serves in LMSG by attempting γ random connections. After connecting an authentic streaming from component j N(i), component i with ni＜τ, connections sends the connection request R(i,j) to component j. Here a limit τ is the number of components to which they connect in order to conserve bandwidth, if ni =τthen component i sends a connection request if one of the following holds:

∉

1. Component j ∉ N(i) achieves a higher local Availability value than one of component i’s neighbors (message R(i,j) is sent);

2. Component j∈N(i) is assigned a lower local Availability value than some acquaintance k ∉N(i) of component i (message R(i,k) is sent).

The first scenario describes an authentic connection from component j, while the second occurs after an inauthentic connection from component j. In both cases, if component i’s connection request is granted then it will disconnect from its neighbor with the lowest local Availability value. A special case occurs if a neighbor of component i is assigned a negative local dependability score and component i is not able to connect to an acquaintance. In this situation component i immediately disconnects


from that neighbor and attempts a connection to a random component. Component j will only accept component i’s connection request if component j’s local

availability value Aij of component i is non-negative and one of the following conditions is true:

1. Component j has no more thanτconnections. 2. Component j’s local availability value Aij of component i is greater than at least one

existent local availability value of the neighbors. In the second case component j will replace its lowest availability valued neighbor

with component i. Component j would have previously made a connection request to component i because of component i’s desirable Aij value. It is assumed that component i had denied the connection request R(j,i) but then later sent the connection request R(i,j).

4. Conclusion

We developed a adaptive model for live media service Grid(LMSG) as an extensible middleware architecture for supporting live media applications. It is a component level model for robust, concurrent, distributed and dependable streaming services, and able to detect object anomalies of the component, reconfigure inter-component and intra-components, repair it, and then test the adapted object. At last, we discribed the Available Adaptive Mechanisms for monitoring objects in the service layer and repairing the sick objects detected. References

[1] Machnicki, E., and Rowe, L.A. : Virtual director: Automating a webcast. In Proceedings of the SPIE

Multimedia Computing and Networking 2002, Vol. 4673, San Jose, CA, January 2002

[2] Perry, M., and Agarwal, D. : Remote control for videoconferencing. In Proceedings of the 11th

International Conference of the Information Resources Management Association, Anchorage, AK, May

2000

[3] Richard Hsiao, Sheng-De Wang. ”Jelly: A Dynamic Hierarchical P2P Overlay Network with Load Balance

and Locality,” icdcsw, vol. 04, no. 4, pp. 534-540, 24th 2004.

[4] B. Zhang, S. Jamin, and L. Zhang, “Host multicast: A framework for delivering multicast to end users,”

presented at the INFOCOM’02, New York, June 2002.

[5]S. Jain, R. Mahajan, D. Wetherall, and G. Borriello. Scalable self-organizing overlays. Technical Report

UW-CSE 02-06-04, University of Washington, 2002.

[6]B.B hattacharjee,and C. Kommareddy. Scalable application layer multicast. Proc. Of ACM SIGCOMM,

2002. [1]5.

[7] D.A. Tran, K.A Hua. “ZIGZAG: an efficient peer-to-peer scheme for media streaming”, INFOCOM 2003.

Twenty-Second Annual Joint Conference of the IEEE Computer and Communications Societies. IEEE,

Vol.2, Iss., 30 March-3 April 2003 Pages: 1283- 1292 vol.2

[8]Yuhui Zhao, Yuyan An, Cuirong Wang, and Yuan Gao. A QoS-Satisfied Interdomain Overlay Multicast

Algorithm for Live Media Service Grid. GCC 2005, LNCS 3795, pp. 13–24, 2005


Yuhui Zhao, a Doctor associated in Northeastern University at Qinhuangdao,

066000, Qinhuangdao, China. Email: [email protected]. His research

interests are in the areas of Service Oriented Architecture, Overlay Network and

Streaming Media.

Yuyan An, a lecturer in Qinhuangdao Foreign Language Professional College,

Qinhuangdao, 066311, China. Her research interests are in the areas of Network

and Streaming Media.

Received by the editors June 2, 2007 and in revised form December 21, 2006 105


s 1-22 INFORMATION AND SYSTEMS SCIENCE Volume 1, Number 1, PageVolume 2, Pages 105-113 INFORMATION AND SYSTEMS SCIENCES ADVANCES IN


A FAST AND ROBUST ALGORITHM FOR PARAMETER ESTIMATION OF DISTRIBUTED SOURCE

YINGHUA HAN, JINKUAN WANG, XIN SONG AND YANFENG ZHANG

Abstract. In mobile communications, local scattering in the vicinity of the mobile results in angular spreading as seen from a base station antenna array. In this paper, we consider the problem of estimating the parameters [direction of arrival (DOA) and angular spread] of a spatially distributed source. A two-step procedure enabling decoupling the estimation of DOA from that of the angular spread is proposed. The DOA estimate can be obtained from the relationship between subarrays without any searching. The so-obtained DOA estimate does not depend on any assumption on the spatial distribution of the source and is hence robust to mismodeling. Additionally, instead of a 2-D search, the proposed algorithm requires only one step 1-D search for estimating the angular spread, which has much reduced complexity and computation. Numerical examples illustrate the proposed method is not only effective, but also enjoys better performance compared with DSPE algorithm. Key Words. Distributed source, the central DOA, angular spread, subarrays

1. Introduction

Most conventional direction-finding techniques based on the assumption that the source energy is concentrated at discrete angles that are referred to as the source DOAs. However, in several applications such as sonar, radar, and wireless communications, such a point source assumption can be irrelevant because signal scattering phenomena may result in angular spreading of the source [1]-[3]. The signals do not arrival at a discrete angle but instead at a continuum of angles due to multipath from local scattering at the source transmitters. In such cases, a distributed source model is more realistic than the point source one.

As opposed to a point source, a distributed source can be parameterized by the central DOA and angular spread. The angular spread of the source is modeled by a random weighting function, which, in turn is described by an underlying, parametric angular weighting function centered at the central DOA. Gaussian, uniform, and Laplacian shape angular weight function have all been previously proposed in the literature.

In wireless communication systems with antenna arrays at base stations, in particular, depending on the environment of the mobile, the base-mobile distance and the base station height, angular spreads up to 10 ° can be commonly observed in practice[4]-[5].Depending on the relationship between the channel coherency time and the observation period, signal components arriving from different directions exhibit varying degrees of correlation, ranging from totally incorrelated (incoherent) to fully correlated (coherent) cases

106 Y. HAN, J. WANG, X. SONG AND Y. ZHANG

Several distributed source location techniques have been proposed in the recent literature. Based on generalization of the signal and noise subspace concepts to distributed sources, an algorithm called the distributed source parameter estimator (DSPE) has been proposed [6], which can be applied to both coherently and incoherently distributed source. In [7], a maximum likelihood algorithm has been proposed for localization of Gaussian distributed sources. The computational complexity of this method grows exponentially with the number of sources. Similar to DSPE, an algorithm called DISPARE has been presented for localization of incoherently distributed source [8]. In [9], an algorithm has been presented for localization of a single uniformly distributed source. In [10], a subspace fitting method has been proposed for estimating the parameters of distributed sources. The covariance matching estimation technique is proposed in [11], which develops a new algorithm for incoherently distributed sources.

In the present paper, we develop a new algorithm for DOA and angular spread estimation for coherently distributed source, which allows decoupling the estimation of the DOA from that of angular spread. The DOA estimate can be obtained on the basis of the eigen-structure of the two subarrays without searching. Due to the approximate closed form of the steering vector, the generalized signal and noise subspace concepts is applied to estimate angular spread. In addition, instead of a 2-D search, the proposed algorithm requires only one step 1-D search for estimating the angular spread, which reduces complexity and computation.

2. Problem formulation

Assume that the signals of q narrowband stationary sources impinge on an array of M sensors. The complex envelope of the array output can be written as

(1) ( ) ( ) ( )1

q

ii

t t=

= +∑ tX S n

where is the array snapshot vector, ( )tX ( )i tS is the vector that described the contribution of the ith signal source to the array output, and ( )tn is the vector of sensor noise.

In point source modeling, the baseband signal of the ith source is modelled as

(2) ( ) ( ) ( )i it s t iθ=S a

where ( )is t is the complex envelope of the ith source, iθ is its DOA, and ( )iθa is the corresponding steering vector. In distributed source modeling, the source energy is considered to be spread over some angular volume. Hence, ( )i tS is written as

(3) ( ) ( ) ( ), ,i it tϑ

dϑ ς ϑ ψ ϑ∈Θ

= ∫S a

where ( ) ( )2 1 sin2 sin[1 , ]j Mje e π ϑπ ϑϑ − − Δ− Δ= , L，a), , t

is the steering vector of the point source, is a complex random angular-temporal signal intensity which can be expressed

as ( iς ϑ ψ

(4) ( ) ( ) ( ), ,i it s tς ϑ ψ ϑ ψ= l ；

under the coherently distributed source assumptions, ψ is the location parameter. Examples of the parameter vector are the mean and standard deviation of a source with Gaussian angular distribution.

A FAST AND ROBUST ALGORITHM FOR PARAMETER ESTIMATION 107

The steering vector of distributed source is defined as

(5) ( ) ( ) ( ) dψ ϑ ϑ ψ ϑ= ∫b a l ；

As a common example of the coherently distributed source, assume that the deterministic angular weighting function ( )ϑ ψ l ； has the Gaussian shape

(6) ( ) ( )2

2

1; , exp22θ

θθ

ϑ θϑ θ σ

σπσ

⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠

l

Hereθ is the central DOA, θσ is angular spread. In practical situations, the true covariance matrix of ( )tX is unavailable but can be

estimated. Therefore, the sample covariance matrix with N snapshots is defined as

(7) ( ) ( )1

1 NH

XXt

t tN =

= ∑R X X)

3. Parameter estimation of coherently distributed source

In this section, we make use of a uniform linear array, which enable us to decouple the estimation of the central DOA from that of angular spread.

A. Decouple the central DOA from angular spread

The closed form of the steering vector can be written as

(8)

( ) ( )( )( )( )

( )( )

2

2

2

2

, exp 2 1 sin

exp 2 1 cos

1 exp22

exp 2 1 sin

m

m

j m

j m

d

j m h

θ

π

π

θθ

θ σ π θ

π θ θ

θ θσπσ

π θ

−

≈ − Δ −⎡ ⎤⎣ ⎦

− Δ −

⎛ ⎞−⎜ ⎟

⎝ ⎠= − Δ −

∫

b

%

%%

or

(9) ( ) ( ), θθ σ θ≈b a h

in vector notation, where is the Schur-Hadamard or elementwise product and the mth element of h is

(10) ( )( )2

2

2

2

exp 2 1 cos

1 exp22

m j m

d

π

π

θθ

π θ θ

θ θσπσ

−= − Δ −

⎛ ⎞−⎜ ⎟

⎝ ⎠

∫h %

%%

h is a 1M × real-valued vector depending on the angular weighting function ( )ϑ ψ l ； . Clearly, is a real-valued function sinceh ( );θ θ ψ+%l is an even function ofθ% because of the symmetry assumption.

Using the integral formula [12]


(11) ( ) ( )

( )

2 2

2

2

exp exp

expexp

4

q x jp x dx

jppqq

λ

λπ

∞

−∞ − +⎡ ⎤⎣ ⎦

⎛ ⎞= −⎜ ⎟

⎝ ⎠

∫

Equation (10) can be written as

(12) ( )( )22 2 2 2exp 2 1 cosm m θπ θσ≈ − Δ −h

So (8) has the following form

(13) ( ) ( )( )

( )( )22 2 2 2

, exp 2 1 sin

exp 2 1 cosm

j m

m

θ

θ

θ σ π θ

π θσ

≈ − Δ −⎡ ⎤⎣ ⎦

− Δ −

b

From (9) and (13), it is clear that ( )θa only inclues the central DOA and angular spread is included in . The central DOA and angular spread are decoupled. h

B. Estimate the central DOA

Assume that we have a criterion function, which has the central DOA and angular spread as unknown parameters. When we estimate the two parameters based on the criterion function, we have to generally solve a 2-D optimization problem. Furthermore, the computational complexity grows rapidly. On the other hand, if we have some preliminary information on either the central DOA or angular spread, it is anticipated that we can estimate the other by 1-D searching of the criterion function with much reduced complexity and computation.

The basic idea behind the proposed algorithm is to exploit the so-called array displacement invariance structure, i.e., two identical subarrays are separated by a common displacement disΔ .

For simplicity, the discussion herein will be restricted to uniform linear array. As shown in Figure.1, the first K and last K elements can be used to form two identical subarrays with displacement ( ) , 1 1dis D M K K MΔ = − ≤ ≤ − . Thus, the steering vectors for these two subarrays can be expressed as

(14) 1 1=b J b

and (15) 2 2=b J b

where 1J and 2J are selection matrices

(16)

1

2

K M K

K

M K K

K

−

−

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

J I 0

J 0 I

Clearly, 1J b picks the first K rows of , whereb 2J b chooses the last K rows of . bFrom (9)，we have


(17) ( )( )1 2 1 1 2 2H H T=b b a h a h

It is clear that the central DOA and angular spread included in the phase angle and the magnitude, respectively. Using the phase angle, we can obtain the central DOA estimate.

In the sample covariance matrix case, the elements in (14) and (15) are replaced by approximate ones. It follows that sℜ = ℜU b or , where s =U bT sU is signal subspace

Subarray1 K=M-1

D= and is a full-rank matrix. Then we have the signal subspaces of the two subarrays as follows

T

(18) 1 1

2 2

U = b TU = b T

For simplicity, the case with one distributed source is considered. So the central DOA can be estimated using the phase angle of the following,

(19) 1 2arg Hθ ⎡ ⎤= ⎣ ⎦U U)

C. Estimate angular spread

We attempt to generalize the signal and noise subspace concepts. Consequently, one intuitively appealing and reasonable cost function is given as follows with θ

)obtained in

(19)，

(20) 1arg max H Hn nθ

θ σσ =

b U U b)

where nU is noise subspace. Here has the approximate closed form of the steering vector as in

b(13). So it avoids integral when searching compared with DSPE [6], which is

much reduced complexity and computation. Instead of 2-D searching jointly, the so-called θσ) can be obtained by 1-D searching with the estimate θ

)obtained by (19).

We summarize the proposed algorithm, 1) Compute the sample covariance matrix using (7). 2) Trough singular value decomposition of XX

)R , obtain the sample signal

subspace sU and noise subspaceU , and subsequently, 1U and 2U . n3) Estimate the central DOA from (19). 4) Estimate angular spread from (20) using θ θ=

)obtained at step3.

4. Simulation results

In this section, we investigate the performance of the proposed algorithm through some simulation experiments. Assume a uniform linear array with 8 sensors, separated by a half wavelength.

To get a better insight into the effect of angular spread for judging the demand for an

Subarray2dis

FIGURE 1. Two subarrays from an eight-element uniform linear array


estimation scheme, we first compute the effect of angular spread. We numerically evaluate the ratio ( ) ( ) 1r m b m b= , where ( )b m and 1 are the mth

and the first element of the steering vector. Figure.2 shows the ratio for a wavefront impinging from the central DOA for different angular spread of Gaussian distribution. It can be seen that the effect of angular spread reduces the mutual correlations between the antenna elements. If we use the point source model, which corresponds to discrete DOAs , it will results in estimate errors.

b

15θ = o

In the sequel, we provide numerical results to compare the performances of our proposed method with those of DSPE in [6].

In the first example, we examine the estimation performance of the proposed algorithm in comparison with the DSPE for a Gaussian-shaped distributed source with the central DOA 15°and angular spread 5°. A Monte Carlo simulation of 40 independent runs with 500 snapshots for each trial has been performed. The root-mean-square-error (RMSE) values of the central DOA and angular spread estimated by the proposed algorithm and DSPE are illustrated at different SNR in Figures. 3 and 4. As it can be seen from these figures, for estimating the central DOA, the proposed method based on the relationship between subarrays has better estimation performance as compared with DSPE algorithm. When estimating angular spread, the approach proposed herein performs as well as DSPE but with much reduced complexity and computation.

FIGURE 2. Ratio of a steering vector for different angular spread

Finally, the influence of angular spread is examined in Figures. 5 and 6. The parameter

angular spread θσ is varied between 3-10°, whereas the central DOA is fixed to and SNR is 10dB. For estimating the central DOA, the proposed algorithm

outperforms DSPE. It is clear that the angular spread estimate obtained from the one step 1-D searching is as accurate as 2-D searching jointly.

15θ = o


5. Conclusions

In this paper, we considered the estimation of parameters of a coherently distributed source with a view to provide a statistically and computationally efficient algorithm. Towards this end, we utilize the relationship between the two subarrays to estimate the central DOA first. Due to the approximate closed form of the steering vector, the generalized signal and noise subspace concepts is applied to estimate angular spread. We show that the original 2-D minimization problem can be replaced by one step 1-D searching problems, which lows computational cost. Additionally, the estimate of the central DOA can be obtained without requiring any assumption on the angular weighting function.

FIGURE 3. RMSE versus SNR for the cetral DOA estimate

FIGURE 4. RMSE versus SNR for the angular spread estimate


FIGURE 5. RMSE versus angular spread for the central DOA estimate

References [1] Raich R, Goldberg J and Messer H. “Bearing Estimation for a Distributed Source: Modeling, Inherent

Accuracy Limitations and Algorithms”, IEEE Trans. Signal Processing, 48(2) (2000), 429-441.

[2] Besson O, and Stoica P. “Decoupled estimation of DOA and angular spread for a spatially distributed

source”, IEEE Trans. Signal Processing, 48(7)(2000), 1872-1882.

FIGURE 6. RMSE versus angular spread for the angular spread estimate

[3] Bengtsson B, and Ottersten. “Low-complexity estimators for distributed sources”, IEEE Trans. Signal

Processing, 48(8)(2000), 2185-2194.

[4] Pedersen K I, Mogensen PE and Fleury B H. “A stochastic model of the temporal and azimuthal dispersion

seen at the base station in outdoor propagation environments”, IEEE Trans. Veh. Technology, 49(3)(2000),

437-447.

[5] Pedersen K I, Mogensen PE and Fleury B H. “Spatial channel characteristics in outdoor environments and

their impact on BS antenna system performance”, Proc.Veh. Technol. Conf., vol.2, pp.719-723, May 1998.


[6] Valaee S, Champagne B and Kabal P. “Parametric Localization of Distributed Sources”, IEEE Trans. Signal

Processing, 43(9)(1995), pp.2144-2153.

[7] Trump T, and Ottersten B. “Estimation of nominal direction of arrival and angular spread using an array of

sensors” , Signal Processing, 50(4)(1996), pp.57-69.

[8] Meng Y, Stoica P and Wong KM, “Estimation of the directions of arrival of spatially dispersed signals in

array processing”, IEE Proc. Radar. Sonar Navig., 143(1)(1996) , pp.1-9.

[9] Shahbazpanahi S , Valaee S and Nayeb M I. “Distributed source parameters estimation”, Proc.Commun.

Fifth Iranian Conf. Elect. Eng., 6-304-6-311, May 1997.

[10] Bengtsson M and Ottersten B. “A generalization of weighted subspace fitting to full-rank models”,

IEEE Trans on Signal Processing, 49(5)(2001), pp.1002-1012.

[11] Shahbazpanahi S, Valaee S and Gershman A B. “A covariance fitting approach to parametric localization

of multiple incoherently distributed sources”, IEEE Trans on Signal Processing, 52(3)(2004), pp.592-600.

[12] Gradshteyn I S and Ryzhik I M. Table of Integrals, Series, and Products. Academic Press, Orlando, FL,

1980. Department of Automation Engineering, Northeastern University at Qinhuangdao, Qinhuangdao, Hebei Province, P.R. China, 066004

ADVANCES IN c© 2008 Institute for ScientificINFORMATON AND SYSTEMS SCIENCES Computing and InformationVolume 2, Pages 114–121

SPACE-TIME SMOOTHING TECHNOLOGY FOR JOINT ANGLEAND FREQUENCY ESTIMATION

FULAI LIU1,2, JINKUAN WANG2, XIYUAN ZHOU1 AND CHUN LI3,1

Abstract. High-resolution parameter estimation techniques, such as MUSIC

and ESPRIT etc, have recently been applied to electronic warfare. In this

paper, we consider the problem of estimating the frequencies and directions of

multiple narrow-band sources, namely, the joint angle and frequency estimation

(JAFE) problem. We discuss the ESPRIT algorithm based on the space-time

smoothing technology of a uniform linear array, for the JAFE. Simulation re-

sults are presented verifying the efficiency of the method.

Key Words. ESPRIT, temporal smoothing, spatial smoothing, space-time

smoothing, eigenvalue decomposition, joint angle and frequency estimation.

1. Introduction

In electronic warfare, it is a major task to estimate the frequencies and anglesof incidence of a number of narrow-band signals impinging on a passive sensorarray. It is well-known that the ESPRIT algorithm is a high-resolution signalparameter estimation method, which has been widely used in many joint parameterestimation problems, such as joint azimuth and elevation angle estimation (2-DDOA estimation)[1], joint DOA and delay estimation [2], joint frequency and 2-Dangle estimation [3], and joint angle-frequency estimation (JAFE) [4], etc. In thispaper, we use space-time smoothing technology to estimate the angle and frequencyof incident narrow-band signal sources, and discuss its estimation performance.According to references [5,6], we give a new parameters matching approach whichreplaces the joint diagonal method.

Notations:The operator (·)H denotes complex conjugate transpose. The operator (·)T

stands for transpose operation. The superscript (·)−1 presents matrix inverse. Thesuperscript (·)† denotes matrix pseudo-inverse (Moore-Penrose). c denotes the prop-agation velocity of light. ⊗ is the Kronecker matrix product. 0 is a column vectorof zeros.

2. Data model

Consider a uniform linear array (ULA) of M elements equispaced by d, as shownin figure 1. The signal received at the kth antenna is

xk(t) =d∑

i=1

ak(θi)ej2πfits(t) + wk(t), k = 1, · · · ,M

Received by the editors June 1, 2006 and, in revised form, December 22, 2006.2000 Mathematics Subject Classification. 35R35, 49J40, 60G40.

114

SPACE-TIME SMOOTHING TECHNOLOGY FOR JAFE 115

LM

LM

LM

Figure 1. The uniform linear array and spatial smoothing.

where ak(θi) is the antenna response of the kth antenna to a signal from directionθi, wk(t) is noise output of the kth antenna. Assume P is the sample rate, thenthe data sample of the array is

x(n

P) =

d∑

i=1

a(θi)expj 2π

Pfinsi(

n

P) + w(

n

P)

where a(θi) is the array response vector of the ith source, and w( nP ) is the noise

vector collecting the samples of the noise terms at the output of each antennaelement. In matrix form, this can be written as

(1) x(n

P) = AΦns(

n

P) + w(

n

P)

where Φ = diagφidi=1, φi = ej(2π/P )fi , A is an M × d matrix collecting the d

steering vectors, and the vector s(t) is a stack of the d signals. Assume that wehave collected N samples of the array output x(t) at the rate P into the M × Ndata matrix X, i.e.,

(2) X = A[s(0) Φs(1P

) · · · ΦN−1s(N − 1

P)] + W ∈ CM,N .

3. Space-Time Smoothing Technology for JAFE

3.1. Temporal Smoothing. In this section, we consider a data stacking technol-ogy (referred to as temporal smoothing) that adds structure to the data model forthe implementation of the JAFE algorithm.

An m-factor temporally smoothed data matrix constructed by stacking m tem-porally shifted versions of the original data matrix. This results in the followingmM ×N −m + 1 matrix:

(3) Xm =

A[s(0) Φs( 1P ) · · · ΦN−ms(N−m

P )]AΦ[s( 1

P ) Φs( 2P ) · · · ]

...AΦm−1[s(m−1

P ) Φs(mP ) · · · ]

+ Wm

where Wm represents the noise term constructed from W in a similar as Xm isobtained from X. Assume that the signals are narrow band, i.e.,

s(t) ≈ s(t +1P

) ≈ · · · ≈ s(t +m− 1

P)

116 F. LIU, J. WANG, X. ZHOU AND C. LI

In this case, all the block rows in the right-hand term of (3) are approximatelyequal, which means that Xm has the factorization:

(4) Xm ≈

AAΦ

...AΦm−1

[s(0) Φs( 1

P ) · · ·] + Wm.= AmFm + Wm ∈ CmM,N−m+1

where Am, throughout the sequel, which is referred to as the extended array steeringmatrix, is given by

(5) Am =

AAΦ

...AΦm−1

∈ C

mM,d

and

(6) Fs =[s(0) Φs( 1

P ) · · · ΦN−ms(N−mP )

] ∈ Cd,N−m+1

is a matrix collecting N −m + 1 samples of the d sources.According to reference [4], we have the following theorem.Theorem 3.1: Consider an M elements antenna array impinged by d < M

narrowband far-field signals. Assume that all the signals have distinct (different)center frequencies. Suppose that the signals are divided into r groups, such thatthe signals from each group have the same DOA. Let pi, for i = 1, · · · , r, representthe number of sources in the ith group. Then, the m-factor temporally smootheddata matrix Xm of (4) is full rank d if and only if m ≥ maxipi.

Remark: The data matrix in (2) is rank deficient when two or more signalshave the same DOA. But theorem 3.1 shows the temporal smoothing can restorethe rank of the data matrix. In addition, we have obtained a model with much thesame structure as in the classical ESPRIT algorithm for DOA estimation but withA replaced by Am. Therefore, the frequency estimation can be obtained accordingto ESPRIT algorithm.

3.2. Spatial Smoothing. Employing spatial smoothing technology in the spatialdomain, coherent signals can be separated. Assume that the ULA of M sensorsis subdivided into L subarrays as shown in Fig.1. Thus, the number of elementsper subarray is ML = M − L + 1. For l = 1, · · · , L, let the ML ×M matrix Jl bea selection matrix that selects part of the M ×N data matrix X that correspondto the lth subarray. Then, a spatially smoothed ML × LN data matrix XL isconstructed as

(7) XL =[J1X J2X · · · JLX

] ∈ CML,LM

Using the structure of X in (2), we can re-express (7) as

XL =[J1A J2A · · · JLA

]Fs

. . .Fs

+ WL

where WL is a noise term that has also been shuffled in a similar way as XL. LetA1 contain the rows of A that correspond to the first subarray. Then, from theshift-invariance property, we have the following relation for k = 1, · · · , L

JkA = J1AΘk−1 = A1Θk−1


where Θ = diagϑidi=1, ϑi = exp−j2πd c

fisin(θi), c is the propagation speed.

Theorem 3.2: Consider an M element antenna array impinged by d narrow-band far-field signals. Assume that all the signals have distinct (different) DOAs.Suppose that the signals are divided into r groups, such that the signals from eachgroup have the same center frequencies. Let, for i = 1, · · · , r, qi represent thenumber of sources in the ith group. Then, the L-factor spatially smoothed datamatrix XL, with ML > d, (4) is full rank d if and only if L ≥ maxiqi.

3.3. Space-Time Smoothing Technology for JAFE.

3.3.1. Space-Time Smoothing Data Model. Start with the temporal smooth-ing data Xm given in (3). Let ML be the number of antenna elements in the sub-arrays of the spatially smoothed data. Let the selection matrix Jl ∈ RmML,mM

(l = 1, · · · , L) select part of the data matrix Xm that corresponds to the lth subar-ray. Then, an (m, L) factor space-time smoothing data matrix Xm,L is constructedas

(8) Xm,L =[J1Xm · · · JLXm

] ∈ CmML,L(N−m+1)

Using the structure of Xm from (3), this can be factored as

(9) Xm,L =[J1Am · · · JLAm

]Fs

. . .Fs

+ Wm,L

where Wm,L is a noise term that has also been shuffled in a similar way as xm,L. LetA′

m = J1Am ∈ CmML,d. Then, from the shift invariance structure of Am ∈ CmM,d,it follows that

JkAm = J1AmΘk−1 = A′mΘk−1 k = 1, · · · , L.

Thus, Xm,L can be written in a compact form as

(10) Xm,L = A′m

[Fs ΘFs · · · ΘL−1Fs

]+ Wm,L = A′

mFL + Wm,L.

Finally, performing forward-backward averaging on the above data, we get themML × 2L(N −m + 1) data matrix

(11) Xm,L,fb = [Xm,L conj(IIXm,L)]

where II is an exchange matrix that reverses the ordering of the rows of Xm,L.

3.3.2. Estimation Algorithm. The signal eigenvectors Us can be computed viathe ”largest” vectors of the matrix R = Xm,L,fbXH

m,L,fb.Since RUs = RXm,L,fb,there must exist a unique nonsingular matrix T such that Us = A′

m,L,fbT. Webegin the estimation of the parameters by defining two types of selection matrices:a pair to select submatrices for estimating Φ and a pair for estimating Θ. Theselection matrices are given by

Jx(θ) = Im ⊗ [IML−1 01

]Jy(θ) = Im ⊗ [

01 IML−1

]

Jx(φ) =[Im 01

]⊗ IMLJy(φ) =

[01 Im

]⊗ IML

Let

(12) Ux,φ = Jx(φ)Us Uy,φ = Jy(φ)Us

(13) Ux,θ = Jx(θ)Us Uy,θ = Jy(θ)Us

These data matrices have the following structures

(14) Ux,φ = A1T Uy,φ = A1ΦT


(15) Ux,θ = A2T Uy,θ = A2ΘT

where A1 and A2 are both submatrices of A′m,L,fb.

From (14) and (15), we have the following equations:

EΦ = U†x,φUy,φ = TΦT−1(16)

Eθ = U†x,θUy,θ = TΘT−1(17)

It is seen that the data matrices EΦ and Eθ are jointly diagonalizable by the samematrix T. After T has been found, we have estimates of (φi, ϑi) for each of thed sources. This provides us with frequency and angle estimates:

fi = arg(φi)P

2πθi = asin(arg(ϑi)

fi

2πc).

3.3.3. Estimated Value Pairing. In order to estimate the frequency and angleof the ith incident source, we need to determine the correct pairs of (φi, θi). By(16) and (17), we have the following equations:

(18) Ψ1 = EΦ + E−1Φ = T(Φ + Φ−1)T−1

(19) Ψ2 = Eθ + E−1θ = T(Θ + Θ−1)T−1

Form (18) and (19), it is seen that the eigenvalues of Ψ1 and Ψ2 are cos functionsand become real numbers. Hence, we define Ψ as follows:

(20) Ψ = Ψ1 + jΨ2 = T[(Φ + Φ−1) + j(Θ + Θ−1)]T−1

The above equation shows that we can estimate the frequency fk and angle θk ofthe the kth incident signal by the real and imaginary parts of the kth eigenvalue ofΨ, respectively.

3.3.4. Summary of The Proposed Method. In this subsection, we summa-rize the joint frequency and angle estimation based on the space-time smoothingtechnology.

Summary of the JAFE:1) Form the sample matrix X = [x(1), · · · ,x(N)] by collecting N samples of the

ULA output, where x(k) represents the kth snapshot of the ULA output.

2) Get the temporal smoothing data matrix Xm =

x(1),x(2), · · · ,x(N −m + 1)x(2),x(3), · · · ,x(N −m + 2)

...x(m),x(m + 1), · · · ,x(N)

.

3) Get the space-time smoothing data matrix Xm,L =[J1Xm,J2Xm, · · · ,JLXm

]by the temporal smoothing data matrix Xm, where Jk is the selection matrix.

4) Estimate the signal subspace Us by computing the eigenvalue decompositionof Xm,LXH

m,L.5) Estimate the matrices Ux,φ, Uy,φ, Ux,θ, Uy,θ. making us of them, we calculate

the matrices Eφ = U†x,φUy,φ, Eθ = U†

x,θUy,θ.6) Compute Ψ = (Eφ + E−1

φ ) + j(Eθ + E−1θ ).

7) Calculate λk(k = 1, · · · , d), as the eigenvalues of Ψ.8) Estimate fk and θk from fk = P

2π × cos−1(Reλk/2) and θk = sin−1− fk

2πc ×cos−1(Imλk/2).


−10 −5 0 5 10 15 200

1

2

3

4x 10

6

SNR

RM

SE

/Hz

Frequency estimation

−10 −5 0 5 10 15 20

0.5

1

1.5

RM

SE

/deg

DOA estimation

Figure 2. RMSE curves for uncorrelated signals.

−10 −5 0 5 10 15 200

2

4

6

8

10

12

14x 10

7

SNR

RM

SE

/Hz

Frequency estimation

−10 −5 0 5 10 15 200

20

40

60

80

SNR

RM

SE

/deg

DOA estimation

Figure 3. RMSE curves for four coherent sources.

4. Simulation results

In this section, we construct several simulations to evaluate the space-timesmoothing technology for joint angle and frequency estimation. Assume that theuniform linear array consists of M = 9 elements spaced by d = 1m. Suppose


that there are four signals impinging on the ULA. We use root-mean-square-error(RMSE), which is defined as

(21)

√√√√E4∑

k=1

(fk − fk)2 or

√√√√E4∑

k=1

(θk − θk)2

as the performance measure. All results provided are based on 100 independentruns. In the simulations, the temporal and spatial smoothing factors are chosen tobe m = 3 and L = 3, respectively.

In the first test, we consider four uncorrelated signals. Let the DOAs and carrierfrequencies be [−20, 0, 20, 45] and [1, 1.1, 1.2, 1.3] × 108Hz, respectively. Fig.2plots the RMSEs for the four uncorrelated signals of the space-time smoothingmethod as a function of signal-to-noise ration (SNR). We can see that the RMSEcurves become flat when SNR≥0dB.

In the second test, we consider four same frequency coherent sources. Let theDOAs and carrier frequencies be [−20, 0, 20, 45] and [1.2, 1.2, 1.2, 1.2]× 108Hz,respectively. Fig.3 gives the performance for the coherent sources. It is shown thatthe RMSE curves become flat when SNR≥2dB.

5. Conclusion

In this paper, we have discussed the space-time smoothing technology for jointangle and frequency estimation. Using the space-time smoothing technology, wewere able to jointly estimate the angles and frequencies for uncorrelated or coherentsignal sources.

An efficient estimated parameters pairing method has been addressed as well.Therefore, we can avoid the complex joint diagonal processing, which reduces thecomputation complexity.

References

[1] A.J. vand der Veen, P. Ober, and E. Deprettere, Azimuth and elevation computation in highresolution DOA estiamtion, IEEE Transactions on SIgnal Processing, vol.40, pp.1828-1832,1992.

[2] A. van der Veen, M. Vanderveen, and A. Paulraj, Joint angle and delay estiamtion using shift-invariance techniques, IEEE Transactions on Signal Processing, vol.46, pp.405-418, 1998.

[3] M.D. Zoltowski, C.P. Mathews, Real-time frequency and 2-D angle estimation with sub-nyquist spatio-temporal sampling, IEEE Transactions on Signal Processing, vol.42, pp.2781-2794, 1994.

[4] A.N. Lemma, A.J. van der Veen. Analysis of joint angle-frequency estimation using ESPRIT,IEEE Transactions on Signal Processing, vol.51, pp.1264-1283, 2003.

[5] M.D. Zoltowski, M.H. Haardt, C.P. Mathews. Closed-form 2-D angle estimation with rectan-gualr array in element space or beamspace via unitary ESPRI, IEEE Transactions on SignalProcessing, vol.44, pp316-328, 1996.

[6] Toru Kuroda, Nobuyosishi Kikuma, Naoki Inagaki. DOA estimation and pairing method in2D-ESPRIT using triangular antenna array. Electronics and Communications in Japan, Part1, vol.86, No.6, pp.59-68, 2003.


Fulai Liu was born in Hebei, China, in 1975. He received his B.S. degreefrom Hebei Normal University, Shijiazhuang, China, in 1999, the M.S. degreeand Ph.D. degree from Northeastern University, Shenyang, China, in 2002 and2005, respectively. He is currently working as a postdoctor in CommunicationTelemetry and Telecontrol Research Institute, Shijiazhuang,China.

His research interests include adaptive signal processing and parameter es-timation in multipath environment, wireless communication.

Jinkuan Wang was born in 1957. He received the M. Eng. Degree fromNortheastern University, Shenyang, China, in 1985 and the Ph.D. degree fromthe University of Electro-Communications, Japan, in 1993. As a special mem-ber, he joined the Institute of Space and Astronautical Science, Japan, in 1990.And he worked as a engineer in Research Department of COSEL company,Japan, in 1994.

He is currently a professor in the School of Information Science and Engi-neering at Northeastern University, China, since 1998. His main interests arein the area of intelligent control and adaptive array.

Xiyuan Zhou was born in Hebei, China, in 1945. He has been a profes-sor at Communication Telemetry and Telecontrol Research Institute, Shiji-azhuang,China.

His research interesting includes Wireless Communications, Electronic War-fare, Network Attack and Defense.

Chun Li was born in Hebei, China, in 1971. She received B.E. degree fromHebei Industry University, China in 1993 and M.E degree from CommunicationTelemetry and Telecontrol Research Institute, Shijiazhuang,China in 1996.

She is a senior engineer in 54th Research Institute of CETC. Her researchinteresting includes direction finding, array signal processing and location tech-nology.

1Communication Telemetry and Telecontrol Research Institute, Shijiazhuang,China.

2Northeastern University, 110004 Shenyang, China.

3Key Lab of Radar Signal Processing, Xi’dian University, Xi’an, China.E-mail : [email protected], [email protected]


DISSIPATIVE CONTROL FOR A CLASS OF NONLINEARSYSTEMS

LI YANG1, QINGLING ZHANG2, ZHANZHI QIU3, AND XIAOGUANG YANG2,4

Abstract. This paper considers the dissipative control of the first-order sys-

tems which is determined by the nonlinear ordinary differential equation, and

then provides the sufficient conditions of the dissipative control for the closed-

loop systems of this kind of the systems via the state feedback controller and

the dynamic output feedback controller. Finally, we give the foundation of the

state feedback dissipative controller, and then the dynamic output feedback

dissipative controller. At the end we testify the feasibility of the theorems by

the numerical example.

Key Words. nonlinear system, supply rate, dissipative control.

1. Introduction

Most physical and chemical processes that dominate the field of chemical en-gineering are inherently nonlinear, and they are typically modeled by systems ofnonlinear ordinary (ODEs) and/or partial differential equations (PDEs) which ob-jective is to capture the dynamic behavior of the actual processes as accurately aspossible[1]. The achievement of systems which are defined by ODEs are abundant,as it is introduced in the references, see, e.g. [1,7]. However the basic problemof the research is to guaranty the performance and stability, which is based onthe Lyapunov theory at present. The key to the Lyapunov stability theory is toconstruct the perfect Lyapunov functions. Furthermore the process of constructingthe Lyapunov function is exactly the process of making the systems passive[2]. Thepassiveness is a special case of the dissipation of the systems, for this reason, it isimportant to discuss the dissipative control of the systems in details. The presentresearch work has gained a lot of achievements in this aspect [2−7]. In the paper[2] it is discussed that the robust decentralized dissipative control for the couplednonlinear system with uncertainties, and it has provided a useful way to introducethe dissipative control for these types of nonlinear systems. The paper [8] has alsoprovided the more comprehensive theoretical foundation for the nonlinear systemscontrol theory. On this basis, we discuss the dissipative control for the first-ordernonlinear system, then we introduce the dissipation of the closed-loop system viathe state feedback controller and dynamic output feedback controller. In this pa-per we give the basics of the dissipative state feedback controller and dissipativedynamic output feedback controller. Finally, an easy numerical example is given totest and verify the feasibility of the theorems. The paper is structured as follows.The next section, Section 2, we introduce notations and review the most impor-tant definitions. And then, Section 3, we prove two theorems to get the dissipative

Received by the editors September 1, 2006 and, in revised form, December 22, 2006.2000 Mathematics Subject Classification. 35R35, 49J40, 60G40.This research was supported by National Nature Science Foundation of China (60574011).

122

DISSIPATIVE CONTROL FOR A CLASS OF NONLINEAR SYSTEMS 123

state feedback controller and dissipative output feedback controller respectively. InSection 4 we testify the feasibility of the theorems by the numerical example. Thefinal Section 5 contains conclusions and recommendatians.

2. Mathematics Preliminaries

Because some nonlinear or linear dynamic processes models are related to thesystems defined by a kind of ODEs, we consider the following first-order system ofnonlinear ODEs:

(1)

x = f(x) + g1(x)u + g2(x)ω ,z = h1(x) + d1(x)u ,y = h2(x) + d2(x)ω .

where x ∈ Rn is the state, y ∈ Rm is the measured output, u ∈ Rr is the controlledinput, z ∈ Rq is the controlled output, ω ∈ Rq is the exogenous input, and f(x),g1(x), g2(x), h1(x), h2(x), d1(x), d2(x) are known matrices functions with appro-priate dimensions.

Definition 1[9] The system (1) is said to be the zero-state detectable, if for anyt ≥ 0 and u(t) = 0, y(t) = 0, it is implicated that limt→∞ x(t) = 0.

Definition 2[10] The system (1) with a C1 storage function V (x) is said to bedissipative with respect to the supply rate r(ω, z), if there exists a nonnegativedefinite function V (x) with V (0) = 0 such that for all x and all ω

(2) V (x) ≤ r(ω, z) ∀t ≥ 0

holds. If the dissipation inequality (2) holds with strict inequality, then we say thesystem is strictly dissipative.

In this paper, we focus on the quadratic supply rate:

(3) r(ω, z) =12ωTJω + ωTSz +

12zTRz ,

where J , R and S are matrices with appropriate dimensions, and J = JT > 0,R > 0.

For convenience, we make the following assumptions:Assumption1 f(x), h1(x) is the zero-state detectable.Assumption2 hT

1 Rd1 = 0.In this paper, we consider the dissipative control problem of the system (1) as

follows: find the state feedback controller u = α(x) and dynamic output feedbackcontroller respectively, in order that the closed loop of system (1) is dissipative viathe controller.

3. Main results

3.1. Dissipative control via state feedback. We discuss the state feedbackdissipative control problem for the system just as the following:

(4)

x = f(x) + g1(x)u + g2(x)ω ,z = h1(x) + d1(x)u .

Theorem 1Given matrices J , R and S with J = JT > 0, R > 0, system (4) isconsidered to be subject to the assumptions. If there exists nonnegative definitefunction V (x) ∈ C and V (0) = 0 such that the following matrix inequality holds

(5) LV f + LV g1α + LV g2ω +12(Sh1)TJ−1(Sh1) ≤ 0 ,

124 L. YANG, Q. ZHANG, Z. QIU, AND X. YANG

where LV f = Vx · f , then the system (4) is dissipative with respect to the supplyrate (3), and the dissipative state feedback controller is as follows:

(6) u = α(x) = −ϕ−1(ωTSd1)T , ϕ =12dT1 Rd1 .

Proof: Firstly we want to prove the following inequality holds:

V − 12ωTJω − ωTSz − 1

2zTRz

= Vxx− (12ωTJω + ωTSz +

12zTRz

= Vx(f + g1u + g2ω)− r(ω, z)

= LV f + LV g1u + LV g2ω − r(ω, z) .

That is

(7) LV f + LV g1u + LV g2ω − 12ωTJω − ωTSz − 1

2zTRz ≤ 0 .

Take ϕ = 12dT

1 Rd1, u = α(x) = −ϕ−1(ωTSd1)T, then

[uTϕ12 + (ωTSd1)ϕ−

12 ][uTϕ

12 + (ωTSd1)ϕ−

12 ]T

= uTϕu + uT(ωTSd1)T + (ωTSd1)u + (ωTSd1)ϕ−1(ωTSd1)T

= uTϕu + ωTSd1u ,

[J12 ω + J−

12 (Sh1)]T[J

12 ω + J−

12 (Sh1)]

= ωTJω + ωTSh1 + (Sh1)Tω + (Sh1)TJ−1(Sh1)

= ωTJω + 2ωTSh1 + (Sh1)TJ−1(Sh1) .

Thus

LV f + LV g1u + LV g2ω − 12ωTJω − ωTSz − 1

2zTRz

= LV f + LV g1u + LV g2ω − 12ωTJω − ωTS(h1 + d1u)− 1

2(h1 + d1u)TR(h1 + d1u)

= LV f +LV g1u+LV g2ω− 12ωTJω−ωTSh1−ωTSd1u− 1

2hT

1 Rh1− 12(d1u)TR(d1u)

≤ LV f+LV g1u+LV g2ω−12[J

12 ω+J−

12 (Sh1)]T[J

12 ω+J−

12 (Sh1)]

+12(Sh1)TJ−1(Sh1)− [uTϕ

12 + (ωTSd1)ϕ−

12 ][uTϕ

12 + (ωTSd1)ϕ−

12 ]T

From the above, if (5) holds, then V (x) ≤ r(ω, z). Thus system (4) is dissipative.


3.2. Dissipative control via dynamic output feedback. For system (1), con-sidering the dynamic output feedback controller just as the following:

(8)

ζ = f(x) + g(ζ)y ζ(0) = 0 ,u = α(ζ) α(0) = 0 .

We can obtain the closed-loop system via the controller (8) :

(9)

x = f(x) + g1(x)α(x) + g2(x)ω ,

ζ = f(ζ) + g(ζ)[h2(x) + d2ω] ,z = h1(x) + d1(x)α(x) .

Then the integral form of (9) is

(10)

xe = fe + g1eαe + g2eωe ,ze = h1e + d1eαe .

where

xe = [xT ζT]T, fe = [fT fT0 ]T, f0 = f + gh2, g1e = diagg1, 0,

g2e = diagg2, g0, g0 = gd2, αe = [αT 0]T, ωe = [ωT ωT]T,

ze = [zT 0]T, d1e = diagd1, 0, h1e = [hT1 0]T.

The quadratic energy supply rate function associated with system (10) is definedby

(11) re(ωe, ze) =12ωT

e Jeωe + ωTe Seze +

12zTe Reze ,

where Je = diagJ, J0, Se = diagS, S0, Re = diagR, R0, Je, Se, Re areblocks opposite angles matrices, and the sense of J , S, R is just as the above, andJ0, S0, R0 are matrices with appropriate dimensions.

IfU(xe)− re(ωe, ze)

(12) = U(xe)− 12ωT

e Jeωe − ωTe Seze − 1

2zTe Reze .

That is

(13) Uxe(fe + g1eαe + g2eωe)− r(ωe, ze) ≤ 0 .

Let Uxe = ∂Φ∂xe

, then LΦfe = ∂Φ∂xe

fe.Thus (13) is equal to the following:

LΦfe + LΦg1eαe + LΦg2eωe − r(ωe, ze) ≤ 0 .

i.e.LΦfe + LΦg1eαe + LΦg2eωe − 1

2ωT

e Jeωe − ωTe Se(h1e + d1eαe)

−12(h1e + d1eαe)TRe(h1e + d1eαe) ≤ 0 .

Thus similar to the theorem 1,

LΦfe + LΦg1eαe + LΦg2ωe − 12ωT

e Jeωe − ωTe Seh1e

−ωTe Sed1eαe − 1

2(d1eαe)TRe(d1eαe) ≤ 0

only if

LΦfe + LΦg1eαe + LΦg2eωe +12(Seh1e)TJ−1

e (Seh1e) ≤ 0 .

126 L. YANG, Q. ZHANG, Z. QIU, AND X. YANG

Let ∂U(x,ζ)∂x = ∂Θ

∂x , ∂U(x,ζ)∂ζ = ∂Ξ

∂ζ , then the above inequality can be divided into twoparts:

LΘf1 + LΘg1α + LΘg2ω + 12 (Sh1)TJ−1(Sh1) ≤ 0 ,

LΞ(f + gh2) + LΞg0ω ≤ 0 .

In one word, we can obtain the following theorem:Theorem 2 With the assumptions, if there exists U(x, ζ) ∈ C, U(0, 0) = 0, such

that the following inequalities holds:

LΘf1 + LΘg1α + LΘg2ω + 12 (Sh1)TJ−1(Sh1) ≤ 0 ,

LΞ(f + gh2) + LΞg0ω ≤ 0 .

where LΘf = ∂Θ∂x f , LΞ(f + gh2) = ∂Ξ

∂ζ (f + gh2), and ∂U(x,ζ)∂x = ∂Θ

∂x , ∂U(x,ζ)∂ζ =

∂Ξ∂ζ , then the closed-loop system (10) is dissipative with supply rate (11), and thedynamic output feedback controller (8) is dissipative output feedback controller ofsystem (1).

4. Numerical example

The parameters of system (4) are as follows:

f(x) =[−x1

−x2

], g1(x) =

[x

321 0

0 x322

], g2(x) =

[x

121 0

0 x122

], d1(x) =

[x1 00 x2

],

h1(x) =[00

], ω(x) =

[x

121

x122

].

And the parameters of supply rate (3) are J = R = S = I, then the supply rate(3) is

r(ω, z) =12ωTω + ωTz +

12zTz .

Let V (x) = xTx, then V (x) satisfies the conditions of the theorem 1. By checking:the above parameters satisfy the inequality (5), therefore, because of the theorem1, we can obtain the dissipative state feedback controller of system (4)

u(x) = −2[x−2

1 00 x−2

2

] [x1 00 x2

] [x

121

x122

]= −2

[x− 1

21

x− 1

22

].

And the closed-loop system, which is obtained by system (4) via this state feedbackcontroller, is dissipative with the above supply rate.

5. Conclusion

In practice, real systems can’t be often described by linear models, they arenaturally nonlinear models. And the dynamic processes can be described moreeffectively by a kind of first-order systems. Therefore it is necessary to study thecontrol theory of this kind of nonlinear systems. In this paper, we introduce thebasics of the dissipation of the closed-loop system via state feedback controller anddynamic output feedback controller, and obtain the dissipative feedback controllers.At the end, we prove the feasibility of the theorem by a numerical example. Theother control properties are the problems to be studied in the future.

Acknowledgments

This research was supported by National Nature Science Foundation of China(60574011).


References

[1] Nikolaos Kazantzis, Theresa Good. Invariant manifolds and the calculation of the long-termasymptotic response of nonlinear processes using singular PDEs. Computers and ChemicalEngineering, 26 (2002): 999-1012.

[2] Zhu Yonghong, Jiang Changsheng, Fei Shumin. Robust decentralized dissipative controlfor nonlinear interconnected system with uncertainties. Journal of Southeast University,32(2002), no.6, 899-904.

[3] Mei Shengwei, Shen Tielong. Passivation Control of Nonlinear Systems with Disturbances.Control Theory and Applications, 16(1999), no.6, 797–801.

[4] Shoulie Xie, Lihua Xie. Robust dissipative control for linear systems with dissipativeuncertainty and nonlinear perturbation. Systems & Control Letters, 29(1997), no.5, 255-268.

[5] Guan Xinping, Hua Changchun, Duan Guangren. Robust dissipation control of uncertaintime-delay system. System Engineering and Electronics, 24(2002), no.1, 48-51.

[6] Liu Yongqing, Xie Xiangsheng. Robust stability of uncertain singular systems with timedelay. Journal of South China University of Technology, 24(1996), 44-50.

[7] Nikolaos Kazantzis, Costas Kravaris. Nonlinear observer design using Lyapunov’s auxiliarytheory. Systems & Control Letters, 34(1998), 241-247.

[8] Mu Chundi, Mei Shengwei, Shen Tielong. New developments in robust nonlinear controltheory. Control Theory and Applications, 18(2001), no.1, 1-6.

[9] Arjan vander Schaft (Sun Yuan-zhang, Liu Qian-jin, Yang Xin-lin Traslation from theEnglish Language), L2-Gain and Passivity Techniques in Nonlinear Control. TsinghuaUniversity Press & Springer Press, 2002.

[10] Fu Yusun, Tian Zuohua, Shi Songjiao. Robust dissipative control for nonlinear systems.Control Theory and Applications, 17(2000), no.2, 915-918.

1Department of Mathematics, Liaoning University, Shenyang, Liaoning, 110036, P. R. China

2Institute of Systems Science, Northeastern University, Shenyang, Liaoning, 110004, P. R.China

3Software Technology Institute, Dalian Jiaotong University, Dalian, Liaoning, 116028, P. R.China

4Department of Mathematics, Dalian Maritime University, Dalian, Liaoning, 116026, P. R.China


RESEARCH ON COAL PULVERIZING SYSTEM OF THEPOWER PLANT

QING-LI WANG1, YUAN-WEI YUAN1, AND ZHAN-ZHI QIU2

Abstract. The powder manufacturing system has an interacting and close

coupling control plant. It is a MIMO apparatus with the pure delay perfor-

mance, great inertia performance and non-linear character. It is difficulty to

get the exact mathematic model for the system with the great change of the

coal quality. Based on these characteristics, the method in the literature [1] is

adopted to present a T-S model in the simulator of our simulation plant. The

fuzzy control method with T-S model is used in the powder manufacturing sys-

tem of the power plant. The problem in traditional fuzzy control system, such

as multiple control rules, the implementation difficulty etc, are solved well. A

new control scheme is presented for multivariable or higher dimensional non-

linear fuzzy control system in theory and practice. The results of experiment

indicate that if the heat engine plant which has the middle warehousing pow-

der manufacturing system with this control strategy in practice, the automatic

level of the powder manufacturing system of the power plant and economic cri-

terion will be improved. The simulation shows that the control of the system is

available. The scheme improves the performance of the powder manufacturing

system of the power plant greatly and has great value in theory and practice.

Key Words. T-S fuzzy model, powder manufacturing system, multivariable

nonlinear fuzzy control system, higher dimensional nonlinear fuzzy control sys-

tem

1. Introduction

In this paper, the control strategy of the middle warehousing powder manu-facturing system in the heat-engine set is considered. The middle warehousingpowder manufacturing system is extensive used by many medium and small-sizedheat-engine set. It is the large-sized major equipment in the heat-engine plant. Butthe automatic control system has no place in practice. The powder manufacturingsystem that controlled by hand in Long-time may result many problems in ballmill, such as coal-full, coal-off, temperature excursion, coal-loss etc. On the otherhand, the control system can’t run with the maximum capacity for long time. So,keeping the safe running of the ball mill and reducing the power consumption of thespecific coal mill, increasing the input rate of the automatic control system are veryimportant projects of the electric power plant. Therefore the study on the controlstrategy of the middle warehousing powder manufacturing system has importantpractical purpose and theoretical meaning.

Received by the editors January 1, 2004 and, in revised form, March 22, 2004.2000 Mathematics Subject Classification. 35R35, 49J40, 60G40.This research was supported by the National Natural Science Foundation of China (60274009).

128

RESEARCH ON COAL PULVERIZING SYSTEM OF THE POWER PLANT 129

The middle warehousing powder manufacturing system has an interacting andclose coupling control plant which is a MIMO apparatus with the pure delay perfor-mance, great inertia performance and non-linear character. The dynamic character-istic of the system is very complex. It is difficulty to get the exact mathematic modelof the system. Since the mathematic model change greatly following the changeof the coal quality, the traditional mutually independent PID control scheme can’terase the coupling among the loop. In the smith forecast control and Dahlin algo-rithm, the exact mathematic model of the controlled plant are also needed. It isdifficult to achieve the required control performance.

Fuzzy neural network control scheme is a new control method which first ap-pearance is in the last after century. It is a non-linear control scheme in essence.The exact mathematic model isn’t required. Based on the performance of the fuzzyneural network control scheme, T-S modeling fuzzy neural network is adopted inthe powder manufacturing system in electric plant. The control effect is well. Thescheme is available in practice and worth being extended.

2. Generalizing of the ball mill middle warehousing powder manufactur-ing system

Figure 1 describes a ball mill middle warehousing coal pulverizing system. Theworking processes are as follows: the hot wind made by boiler air fore warmer iscontrolled by hot wind gate (R); The cold wind gate is closed when the systemnormal working. It is used to cold the coal mill when the system is stopped. Therecirculate wind that comes from the mill fan and enters in the ball mill is controlledby the recirculate gate (Z). The raw coal given by the coal feeder is send to the mainbody of the ball mill with hot wind together. The mixture of wind and separatorthrough the pipeline 1. Then the coarse pulverized coal is send back to the mainbody through the pipeline 2. The qualified pulverized coal float in the wind streamenters in the pulverized-coal collector. That is the ready-made pulverized coal. Itis stored into the powdered coal storage with the pulverized coal that comes fromthe neighbor warehouse together. Then, it is blown in the hearth by the hot wind

Figure 1. Ball mill middle warehousing coal pulverizing system

that comes from the pulverized fuel feeder and burning. The gas given by thepulverized-coal collector contains some powder. It is split when it is comes from

130 Q. WANG, Y. JING AND Z. QIU

the pipeline 4 and the mill fan. Some enters in the hearth through the pipeline 6and burning. The other is send to the main body of the coal mill by the recirculatewind gate. The purpose is reducing the temperature of the mixture of wind andpowder that comes from the main body. And the negative pressure in the coal millentrance is maintained in the prescribed limit. According to the mentioned aboveball mill middle warehousing powder manufacturing system of working process,control system needs to meet with three requirements. 1) guarantee coal quantityin the coal mill approaches to the optimally quantity. Theoretical analysis andexperiences show coal mill’s energy consumption, which is irrelative with powderquantity. Therefore, guarantee of coal quantity in the coal mill approaches to theoptimally quantity in order to make up powder coal as much as possible is key ineconomical factor. Storing coal quantity in the coal mill is controlled by coal feedquantity. 2) Increase desiccant’s temperature to increase dry power as much aspossible , but not exceeding scheduled value. Usually, transient-state deviation isnot much than 8oC, and steady-state deviation is not much than 2oC. Therefore,temperature of coal mill’s vent is controlled by opening value of hot wind. 3) Controlnegative pressure in the coal mill entrance which is a little less than surroundingpressure by opening value of recirculating gate in order to achieve coal mill theoptimal ventilation quantity for carrying coal powder to coal powder barn andpreventing leaking. However, because the system is difficult to be controlled, neitherthree single circuits can achieve to desired control. As follows: 1) Coal quantity inthe coal mill is difficulty to be measured instantly. Many people have endeavored tomeasure method [2, 3], but they are far form desired objection. Up to now, almostsystems’ coal quantity stored is represented by differential pressure of coal mill’sentrance-exit. But the differential pressure of coal mill’s entry-exit is influenced byrecirculating air. So the optimal stored coal quantity is not ensured. 2) The systemis close coupling great inertia delay and pure delay three inputs and three outputsmultivariable system. Adjustment mode of three single circuits can not obviouslysatisfy the needs of production practice and the manually operation is difficult toimplement. 3) The system is severely nonlinear. If coal quantity exceeds certainlevel, the phenomenon that the coal blocks up or coal milling leaks will occur, thuswill make against the safety in production and healthy environment.

3. T-S fuzzy model of milling system [4]

The T-S fuzzy model of powder manufacturing system is used in paper [1]. Thephysical parameters are defined as follows:

u1—coal feed quantity,u2—ventilation quantity,u3—entry temperature,y1(kg/s)—coal storage quantity,y2(oC)—exit temperature,y3(kg/s)—coal powder exit quantity,y4(Pa)—exit differential pressure.1. T-s fuzzy model of coal mill machine about coal storage quantity

(1) R1 : If

y1(k − 1)u1(k − 1)y3(k − 1)

is A1

118912.20511.589

then y1(k) = −52.4805 + 0.9529y1(k − 1) + 18.7712u1(k − 1)− 9.0125y3(k − 1)


(2) R2 : If

y1(k − 1)u1(k − 1)y3(k − 1)

is A1

1199.712.68411.736

then y1(k) = −78.854 + 0.9449y1(k − 1) + 18.521u1(k − 1)− 5.6673y3(k − 1)

2. T-S fuzzy model of coal mill machine about coal exit quantity

(3) R1 : If

y3(k − 1)y1(k − 1)u2(k − 1)

is A1

11.5891189

−19.239

then y3(k) = 1.1302 + 0.1424y3(k − 1) + 0.0029y1(k − 1) + 0.2793u2(k − 1)

R2 : If

y3(k − 1)y1(k − 1)u2(k − 1)

is A2

11.7361199.716.492

then y3(k) = 0.9343 + 0.1846y3(k − 1) + 0.0026y1(k − 1) + 0.2836u2(k − 1)

3. T-S fuzzy model of coal mill machine about exit temperature

(4) R1 : If

y4(k − 1)u1(k − 1)u3(k − 1)

is A1

71.08610.766223.71

then y2(k) = 3.4676 + 0.8915y2(k − 1)− 0.2324u1(k − 1) + 0.0308u3(k − 1)

R2 : If

y4(k − 1)u1(k − 1)u3(k − 1)

is A2

3794.516.4921199.7

then y2(k) = 3.7362 + 0.9158y2(k − 1)− 0.2441u1(k − 1) + 0.0226u3(k − 1)

4. T-S fuzzy model of coal mill machine about exit differential pressure

(5) R1 : If

y4(k − 1)u2(k − 1)y1(k − 1)

is A1

3743.819.2391189

then y4(k) = −5974.8 + 0.01y4(k − 1) + 359.52u2(k − 1) + 2.4278y1(k − 1)

R2 : If

y4(k − 1)u2(k − 1)y1(k − 1)

is A2

3794.516.4921199.7

then y4(k) = −6338.9 + 0.01y4(k − 1) + 359.26u2(k − 1) + 2.7362y1(k − 1)


Figure 2. Control model of fuzzy neural network

4. System parameters optimization

The node in the network is different from the general nerve cell. In essence itimplements nonlinear mapping from the input to the output. Followed BP network,design and regulating the parameter used the method of error inverted transmissionso as to approximate the output of controlled device to the expected output. Fuzzyneural network needs the learning parameter, importantly joint weight ωbk of thefinal layer and the central value ωc and the breadth ωd of membership grade func-tion. Given yd(k) is the systematic expected output (given controlled quantity),y(k) is the systematic k time output. Defining the error function is

(6) E =12

m∑

i=1

ydi − yi2

where, m is learning sample data, ydi is the given output. yi is the output ofthe controlled device. Using the method of error transmission, learning process isimplemented. Given learning rule of network joint weight is

∆ω ∝ (−∂E

∂ω)

ω(k + 1) = ω(k) + η(−∂E

∂ω) + α[ω(k)− ω(k − 1)]]

where, η is learning speed, η > 0, α are smoothing factor, 0 < α < 1, which canenhance the rate of convergence and smooth change of the weight value. It can beseen that if partial derivative ∂E

∂ω can be worked out, adjustment can be got too.Calculation proceeds from the output layer to the front layer.

The adjustment of (H) layer weight value ωb, the output of the (H) layer is

(7) u =r∑

k=1

Ogk · ωbk


Due to (6) and (7)

∂E

∂ω=

∂E

∂yi

∂yi

∂u

∂u

∂ωbk=

m∑

i=1

(ydi − yi)∂yi

∂u·Og

k

where, ∂yi

∂u can be approximated if the controlled devices are unknown.

∂yi

∂u≈ yi[u(k + 1)]− yi[u(k)]

u(k + 1)− u(k)=

yi[u(k) + ∆u(k)]− yi[u(k)∆u(k)

Define that δhk is the inverted error signal of (H) layer

δhk = − ∂E

∂Ihk

= − ∂E

∂Oh

∂Oh

∂Ihk

=∂E

∂yi

∂yi

∂Ohf ′(Ih

k ) = −f ′(Ihk ) ·

∑(ydi − yi) · ∂yi

∂u

Let ∂E∂ω = ∂E

ωbk, we have

− ∂E

ωbk=

(−∂E

Ihk

)∂Ih

k

∂ωbk= δh

kOgk

weight value adjustment formula of (H) layer is

(8) ωbk(k + 1) = ωbk(k) + ηδhk ·Oh

k + α[ω(k)− ω(k − 1)]

(G) layer has no adjustment of weight value, which only need to calculate theinverted error signal δg

k,

δgk = − ∂E

∂Igk

= − ∂E

∂Ogk

· ∂Ogk

∂Igk

= − ∂E

∂Ihk

· ωbk · ∂Ogk

∂Igk

,

or

δgk = f ′Ig

k · δhkωbk.

Given δNk is the inverted transmission error signal of others intermediate layer,

using the same method upside, the computational formula of the inverted trans-mission error coefficient in (B) (C) (E) (G) layer:

(9) δNk = f ′(IN

k ) ·∑

i

δN+1i · ωN,N+1

ki

The computational formula of the inverted transmission error coefficient in (D)(F) layer:

(10) δNk = f ′(IN

k ) ·∑

i

δN+1i · ωN,N+1

ki [∏

ωN,N+1ji ·ON

j ]

where, in the (D) layer f ′(INk ) = −2Id

k · Odk, in the (F) layer number zero joint is

f ′(Ifo ) = (Of

o )2, others linear joint is f ′(INk ) = 1. From the above derived result

it can be seen that the adjustment formula of joint weight value of the network isachieved.

(11) ωN−1,Nik (k + 1) = ωN−1,N

ik (k) + ηδNk ·ON−1

i + α∆ωN−1,Nik

where, ωN−1,N denotes the joint weight value from the (N-1) layer to the (N) layer.


5. design controller of milling system on fuzzy neural network

Fuzzy neural network controller of milling system consists of three subsystems,which respectively are the outlet temperature control subsystem of milling coal ma-chine, the outlet subatmospheric pressure control subsystem of milling coal machineand load control subsystem. Control method is related to control mode, controllerconstruction and control rules.5.1 Fuzzy control on T-S [2, 3, 4]

T-S fuzzy control mode is a systematic method which produce fuzzy rules fromthe given input and output data set. Typical fuzzy rules in the T-S fuzzy modeconsist of following rules with the IF-THEN formal.

Ri : If x1 is A1 and L and xr is Air

then yi = ai0 + ai1x1 + L + airxr, for i = 1, 2, L, K

where, K is rule amount, xj is input, yi is the output of the number i, Aij is thefuzzy set for counterpart of the inputxj , aij is the linear parameter of the backrule. For the input zk of r dimension sample point, the general output of T-S fuzzymode:

(12) yk =∑k

i=1 [µ(zk)yi(zk)]∑ki=1 µ(zk)

=∑

[wi(zk)yi(zk)]

where,yi(zk) = ai0 + ai1zki + L + air, µ(zk) is the active degree for zk to rule Ri,which can be denoted by

(13) µi(zk) = Ai1(zk1)×Ai2(zk2)× L×Air(zkr), i = 1, 2, L,K

(14) wi(zk) =µi(zk)∑Ki=1 µi(zk)

i = 1, 2, L,K

Global dynamic characteristics of nonlinear system attributes to the integralityof the rule base. And local characteristics can be corresponded without rules. Theformer mode can also describe multi-input and multi-output system (MIMO), whichusually transform multi-input and signal output to model building respectively.From the algorithm it can be known that for the given inputs, membership gradeand the corresponding output can be calculated only according to membershipgrade function, then the output of the controller can be worked out using theweighted average algorithm. Because membership functions and output functionand weighted calculations are all simple function relationships, algorithm is visual,brevity and fast which is the most preferred method on fuzzy modelings of samples.5.2 Input-output fuzzification

According to the given T-S fuzzy control model, input-output quantities arefuzzificated which respectively are u1 (coal feed quantity), u2 (ventilation quan-tity), u3 (entry temperature), y1(kg/s) (coal storage quantity), y2(oC) (exit tem-perature), y3(kg/s) (coal milling exit quantity), y4(Pa) (exit differential pressure).Given NB, NM, NS, O, PS, PM, PB are described as fuzzy subset of language valueof input-output variables. Where NB, NM, NS, O, PS, PM and PB are respectivelynegative large, negative middle, negative small, zero, positive small, positive mid-dle, positive large. Given domains of u1, u2, u3, y1, y2, y3, y4 are respectively x1,x2, x3, x4, x5, x6, x7 which are divided seven ranges according to the quantity,


respectively -3, -2, -1, 0, +1, +2, +3. then,

x1 = −3,−2,−1, 0, 1, 2, 3, x2 = −3,−2,−1, 0, 1, 2, 3,x3 = −3,−2,−1, 0, 1, 2, 3, x4 = −3,−2,−1, 0, 1, 2, 3,x5 = −3,−2,−1, 0, 1, 2, 3, x6 = −3,−2,−1, 0, 1, 2, 3],x7 = −3,−2,−1, 0, 1, 2, 3.

membership grade function curve of language variable u1, u2, u3, y1, y2, y3, y4

presents as figure 3

0 1 2 3 - 1 - 2 - 3

N B P B P M P S N S N M

x 1 x 3 x 2

0 . 5 0 . 5

O 1

Figure 3. Membership function of language variable

5.3 Systematic non-fuzzificationInput-output fuzzy quantities of fuzzy control are u1, u2, u3, y1, y2, y3, y4.

According to approximate range of input quantity and specified amount of fuzzysubset, input quantities are fuzzificated using fuzzy subset positive big (PB), pos-itive middle (PM), positive small (PS), zero (ZO), negative small (NS), negativemiddle (NM), negative big (NB). Fuzzy language subset membership function ofinput quantity adopts triangle model. In theory the back piece of T-S fuzzy modelis polynomial expression. In order to calculate briefly and emulation-adjustment,first-order T-S fuzzy model was chosen which can implement control for the systemby simulation. A lot of methods can be used to solve fuzzy questions. In this papercentroid method are adopted.

6. Simulation research

1. Simulation result about differential pressure fixed value of disturbance

Figure 4. Closed loop respond curve about the fixed minuspressure of ball mill disturbing the entry minus pressure of ballmill

2. Simulation result about exit temperature of disturbance


Figure 5. Closed loop respond curve about the fixed exit tem-perature of ball mill disturbing the exit temperature of ball mill

3. Simulation result about fixed burthen of disturbance

Figure 6. Closed loop respond curve about the fixed burthen ofball mill disturbing the burthen of ball mill

4. Simulation result about fixed entry minus pressure and fixed exit temperatureand fixed burden of disturbance.

Figure 7. Closed loop respond curve about the fixed entry minuspressure and fixed exit temperature and fixed burden of ball milldisturbance


7. Conclusion

For nonlinear model of the heat-engine plant milling system, T-S fuzzy controlmodel is established. By neural network optimizing parameter, fuzzy control isachieved. The results of simulations and testing of the electric power plant’s emu-lator show the validity of the control performance and energy-saving. Particularly,contradiction between the exit temperature and coal powder explosion is resolvedand the optimal point is found which compromised among coal storage quantity,exit temperature and exit minus pressure.

References

[1] Takagi T, Sugeno M. Fuzzy identification of systems and its application to modeling andcontrol, IEEE Trans Sys Man Cybem, 1985, 15(1): 116-132.

[2] Chiu S. L. Fuzzy model identification based on cluster estimation, Journal of Intelligent andFuzzy system, 1994, (2): 264-278.

[3] Kukolj D Levi E. Identification of complex system based on neural and takagi-sugeno fuzzymodel, IEEE Trans Sys Man Cybem (B), 2004, 34(1): 272-282.

[4] CHEN Shao-bing, ZHANG Tie-jun, XU Zhi-gao LENG Wei. A Dynamic Fuzzy ModellingMethod for Ball Pulverizing Systems Based on Mechanism Analysis, Chinese Journal of PowerEngineering 2005, 25(2): 243-248.

1College of Information Science and engineering, Northeastern University, Shengyang, 110004,China

2Software Technology Institute, Dalian Jiaotong University, Dalian 116028, ChinaE-mail : [email protected], [email protected], [email protected]

URL: http://www.neu.edu.cn


DECENTRALIZED ROBUST CONTROL FOR THE UNCERTAINNONLINEAR SYSTEMS WITH INPUT SATURATION

SONG TONG1, QINGLING ZHANG1, YUECHAO MA1,2 AND CHAO LIU1

Abstract. By utilizing of the Lyapunov stability theory and the matrix theory,

decentralized robust problem for a class of uncertain nonlinear systems with in-

put saturation and a class of uncertain nonlinear similar systems is investigated.

Furthermore the decentralized robust controllers are designed.

Key Words. input saturation, decentralized control, robust stabilization.

1. Introduction

The stabilization problem for linear system is investigated these years in [1]-[3].Input saturation is a common property for control systems. Global stabilizationproblem and semi-global stabilization problem for linear system with input satura-tion are investigated in [4]-[5], the systems studied in [4]-[5] are neither nonlinearnor composite systems. Linear composite systems with input saturation are inves-tigated in [6]-[7], the systems studied in [6]-[7] do not have nonlinear terms. Inthe past few years few researchers pay attention to the decentralized stabilizationproblem for uncertain nonlinear composite large scale systems with input satura-tion, hence such system is worthy investigating theoretically and practically. Inthis paper a class of decentralized stabilization problem for uncertain nonlinearcomposite large scale systems with input saturation will be investigated. By meansof the Lyapunov stability theory and the matrix theory, designs of decentralizedrobust controllers for this class of systems are presented. Similar composite systemsare extensively applied in fields of power system, system of multi-arms robots andsystem of mutual two-handstand cycloids etc.. According to the property of sim-ilar systems, some concise conditions of decentralized robust control for uncertainnonlinear composite large scale systems with input saturation are presented in thispaper. Compared with the previous related papers, nonlinearity and mutuality ofthe investigated systems is considered in this paper. Consequently, results obtainedin this paper generalize those in the previous related papers.

2. Definition and Problem Formulation

Some symbols used in this paper are given as follows.R,Rn,Rn×mrepresent the set of real number, the set of real n dimensional vectors

and the set of n×m matrixes, respectively; ‖ · ‖ represents the norm of spectrum,V ω

n represents the set of analytic field of n dimensional vectors defined in set E.A > 0(A < 0) denotes A is a positive(negative) definite matrix.

If a function σ : Rm → Rm satisfies following conditions :(1) σ(u) is decentralized, s.t. σ(u) = [σ1(u1), σ2(u2), · · ·, σm(um)]T .(2) σj is semi-global Lipschitz. j = 1, 2, · · ·, m.

This research was supported by National Natural Science Foundation of China (No. 60574011)and Natural Science Foundation of Liaoning Province China (No. 20052022).

138

DECENTRALIZED ROBUST CONTROL FOR THE UNCERTAIN NONLINEAR SYSTEMS 139

(3) sσj(s) > 0,∀s > 0.(4) min lim

s→0+

σj(s)s , lim

s→0−σj(s)

s > 0.

(5) lim|s|→∞

inf(|σj(s)|) > 0.

then σ is a saturation function.Lemma 1[7]. if σ is a saturation function, then the following holds.

(1) ‖12s− σ(s)‖ ≤ 1

2‖s‖, ∀s ∈ Rm

Consider the following composite large scale system:

(2) xi = Aixi + ∆fi(xi) + Bi(σi(ui) + ∆gi(xi)) +N∑

j=1,j 6=i

Hij(xij), i = 1, 2, · · ·, N.

where Ai ∈ Rni×ni , Bi ∈ Rni×mi , σi : Rmi → Rmi is a saturation function,∑Nj=1,j 6=i Hij(xij) ∈ V ω

ni(Ω) is a mutual term (Ωi is a neighborhood of xi = 0,

Ω=Ω1×Ω2×· · ·×ΩN is a neighborhood of x = 0), ∆fi(xi) is an incompatible anduncertain term, ∆gi(xi) is a compatible and uncertain term. Without loss of gen-erality, here Hij(0) = 0 (i, j = 1, 2, · · ·, N, i 6= j).

A special form of composite system is also studied as follows:

(3) xi = Axi + ∆fi(xi) + B(σi(ui) + ∆gi(xi)) +N∑

j=1,j 6=i

Hij(xij), i = 1, 2, · · ·, N.

Equation (3) is an uncertain similar system with input saturation, which is ap-plied extensively.

Problem. Under what condition can system (2) find a feedback control ui =Kixi, i = 1, 2, · · ·, N satisfying followings:

Problem (1) For the ith closed-loop subsystem

(4) xi = Aixi + ∆fi(xi) + Bi(σi(ui) + ∆gi(xi))

is asymptote stable with respect to xi = 0 in Ωi (Ωi is a neighborhood of xi = 0)Problem (2) For a complete closed-loop subsystem, xi = Axi + ∆fi(xi) +

B(σi(ui) + ∆gi(xi)) +∑N

j=1,j 6=i Hij(xij), (i = 1, 2, · · ·, N) is asymptote stable withrespect to x = 0 in Ω (Ω=Ω1×Ω2×· · ·×ΩN is a neighborhood of x = 0)

Lemma 2. Supposing (Ai, Bi), (i = 1, 2, · · ·, N) is stable, Qi ∈ Rni×ni , Qi > 0,Ri ∈ Rmi×mi , Ri > 0, then the Ricaati Equation

(5) ATi Pi + PiAi − PiBiR

−1i BT

i Pi + Qi = 0

has a unique positive definite solution Pi.Lemma 3[10] If Hij(xij) is a field of n dimensional vectors in Rp and H(0) = 0,

there will be a n× p analytical function matrix Rij(xj) satisfying:

(6) Hij(xj) = Rij(xj)xj , j = 1, 2, · · ·, N, j 6= i

3. Main Results

Firstly some basic assumptions are introduced.Assumption 1

(7) ‖∆fi(xi)‖ ≤ αi‖xi‖, ‖∆gi(xi)‖ ≤ βi‖xi‖.

140 T. ZHANG, Y. MA, AND C. LIU

Assumption 2(Ai, Bi), (i = 1, 2, · · ·, N) is stable,

let Qi = Qi + PiBiBTi Pi + PiPi + (α2

i + β2i )I,Qi > 0.

Theorem 1We suppose that system (2) is nonliner and assumption (1) and (2) hold simul-

taneously. If matrix WT (x) + W (x)(W (x) = Wij(xj)N×N ) is positive definite inΩ,

(8) Wij(xj) =

1− ‖(Q−

12

i )T PiBi‖‖R−1i BT

i PiQ− 1

2i ‖, i = j

−2λM ((Q−12

i )T PiRij(xj)Q− 1

2i ), i 6= j

where Qi and Ri are chosen based on necessity, Pi satisfies (5), Rij(xj) satisfies(6), there will be a feedback control,

(9) ui = −R−1i BT

i Pixi, i = 1, 2, · · ·, N.

the above feedback control ui is a solution of problem (1) and (2).Proof. For the ith subsystem (4), we set a positive definite function Vi(xi) =

xTi Pixi, derivate Vi(xi) by the track of system (4), consider (1) and (5),

Vi = xiT Pixi + xT

i Pixi

= xTi (PiAi + AT

i Pi)xi − 2xTi PiBiσi(R−1

i BTi Pixi) + 2xT

i Pi∆fi(xi)

+2xTi PiBi∆gi(xi)

= xTi (−Qi − PiBiB

Ti Pi − PiPi + (α2

i + β2i )I + PiBiR

−1i BT

i Pi)xi

−2xTi PiBiσi(R−1

i BTi Pixi) + 2xT

i Pi∆fi(xi) + 2xTi PiBi∆gi(xi)

= −xTi Qixi + 2xT

i PiBi(12R−1

i BTi Pixi − σi(R−1

i BTi Pixi))

+(2xTi Pi∆fi(xi)− xT

i PiPixi − α2i x

Ti xi)

+(2xTi PBi∆gi(xi)− xT

i PiBTi xi − β2

i xTi xi).

here

(10) ‖12R−1

i BTi Pixi − σ(R−1

i BTi Pixi)‖ ≤ 1

2‖R−1

i BTi Pixi‖

2xTi Pi∆fi(xi)− xT

i PiPixi − α2i x

Ti xi

= 2xTi Pi∆fi(xi)− ‖xT

i Pi‖2 − α2i ‖xi‖2

≤ 2‖xTi Pi‖‖∆fi(xi)‖ − 2αi‖xT

i Pi‖‖xi‖ ≤ 0.(11)

2xTi PiBi∆gi(xi)− xT

i PiBiBTi Pixi − β2

i xTi xi

= 2xTi PiBi∆gi(xi)− ‖xT

i PiBi‖2 − β2i ‖xi‖2

≤ 2‖xTi PiBi‖‖∆gi(xi)‖ − 2βi‖xT

i PiBi‖‖xi‖ ≤ 0.(12)

Consider (10), (11) and (12)


Vi ≤ −‖xTi Qixi‖+ ‖2xT

i PiBi‖‖R−1i BT

i Pixi‖= −xT

i (Q12i )T Q

12i xi + ‖xT

i (Q12i )T (Q−

12


i PiQ− 1

2i Q

12i xi‖

≤ −‖Q12i xi‖2 + ‖(Q−

12


i PiQ− 1

2i ‖‖Q

12i xi‖2

= −(1− ‖(Q−12


i PiQ− 1

2i ‖)‖Q

12i xi‖2(13)

Because WT (x) + W (x) is positive definite in Ω, Wii > 0, i = 1, 2, · · ·, N . Con-sequently, Vi is negative definite in Ω, problem (1) is solved.

For system (2), set a positive definite function,

V (x1, x2, · · ·, xN ) =N∑

i=1

xTi Pixi

Consider (1), (5) and (6), derivate V (x1, x2, · · ·, xN ) with respect to t by thetrack of system (2),

Vi =N∑

i=1

(xiT Pixi + xT

i Pixi)

=N∑

i=1

xTi (−Qi + PiBiR

−1i BT

i Pi − PiPi − PiBiBTi Pi

−(α2i + β2

i )I)xi −N∑

i=1

2xTi PiBiσi(ui) +

N∑

i=1

2xTi Pi∆fi(xi) +

N∑

i=1

2xTi Pi

+N∑

i=1

2xTi PiBi∆gi(xi) +

N∑

i=1

2xTi Pi

N∑

j=1,j 6=i

Rij(xj)xj

= −N∑

i=1

xTi Qixi +

N∑

i=1

2xTi PiBi[

12R−1

i BTi Pixi − σi(R−1

i BTi xi)]

+N∑

i=1

(2xTi Pi∆fi(xi)− xT

i (PiPi + α2i I)xi) +

N∑

i=1

[2xTi PiBi∆gi(xi)

−xTi (PiBiB

Ti Pi + β2

i I)xi] +N∑

i=1

2xTi Pi

N∑

j=1,j 6=i

Rij(xj)xj

Consider (10), (11) and (12)

Vi ≤ −N∑

i=1

[‖Q12i xi‖2(1−

N∑

i=1

‖(Q−12

i )T PiBi‖‖B−1i BT

i PiQ− 1

2i ‖)

+N∑

j=1,j 6=i

λM ((Q−12

i )T PiRij(xj)Q− 1

2j )‖Q

12i xi‖‖Q

12j xj‖]

= −Y T W (x)Y

Vi ≤ −12Y T (WT (x) + W (x))Y

142 T. ZHANG, Y. MA, AND C. LIU

where Y = (‖Q121 x1‖, ‖Q

122 x2‖, · · ·‖, Q

12NxN‖)T , because WT (x)+W (x) is positive

definite in Ω, V is negative definite in Ω. Consequently, problem (2) is solved.Subsequently, the nonlinear composite large scale system with input saturation

will be investigated. Due to the structural property of system (3), following simplesufficient condition for decentralized robust stabilization is obtained.

Deduction 1. For system (3), we suppose that assumption (1) and (2) hold. Ifmatrix WT (x) + W (x) (W (x) = (Wij(xj))N×N ) is positive definite in Ω, where

(14) Wij(xj) =

1− ‖(Q− 1

21 )T P1B1‖‖R−1

1 BT1 P1Q

− 12

1 ‖, i = j

−2λM ((Q−12

1 )T P1Rij(xj)Q− 1

21 ), i 6= j

Q1, R1 and P1 satisfy (5), Rij(xj) satisfies (6), then a feedback control

(15) ui = −R−11 BT

1 P1xi, i = 1, 2, · · ·, N.

is a solution of problem (1) and (2).Design Steps of decentralized stabilization controller:(1) set positive definite matrixes Qi, Ri, obtain a solution Pi of the Riccati

equation (5).(2) find Rij(xj) satisfying (6), estimate Ω under the case that WT (x) + W (x)

is positive definite in it.(3) design stabilization controller (9).

4. Numerical Example

Now uncertain nonlinear mutual large scale system with input saturation termwill be considered.

(16)

(x11

x12

)=

( −4 00 −5

)(x11

x12

)+ ∆f1(x1) +

(11

)(σ1(u1) + ∆g1(x1)) + 1

4

(x11x21

0

)

(˙x21

˙x22

)=

( −4 00 −5

)(x21

x22

)+ ∆f2(x2) +

(11

)(σ2(u2) + ∆g2(x2)) + 1

4

(x21x12

0

)

where x1 = (x11, x12)T , x2 = (x21, x22)T ; ‖∆fi(xi)‖ ≤ 12‖xi‖, ‖∆gi(xi)‖ ≤√

32 ‖xi‖, i = 1, 2.

By computation,

R12 =14

(x21 00 0

), R21 =

14

(x12 00 0

)

βi =√

32

, αi =12, i = 1, 2

set

Q =(

8 00 10

), R−1 = 3I

with solving the Riccati Equation (5), we can obtain

P =(

1 00 1

)


then by computation

W (x) =(

0.3250 −0.0625‖x21‖−0.0625‖x12‖ 0.3250

)

setΩ = (x11, x12, x21, x22) | |xij | < 1, i, j = 1, 2

then WT (x) + W (x) is positive definite in Ω, so the decentralized robust stabi-lization controller which makes system as well as its isolated subsystem stabilizeis

ui = −(3, 3)xi, i = 1, 2

5. Conclusion

In this paper, a class of uncertain nonlinear system with input saturation and aclass of uncertain nonlinear similar system with input saturation are investigated.By utilizing of transform of the Riccati equation and the matrix theory, the decen-tralized robust stabilization controller is designed. Finally a numerical example ispresented to demonstrate the effectiveness of the proposed method in this paper.Furthermore, results obtained in this paper generalize those in the previous relatedpapers.

Acknowledgments

The author thanks the anonymous authors whose work largely constitutes thissample file. This research was supported by National Natrual Science Foundationof China and Natural Science Foundation of Liaoning Province China.

References

[1] Anderson, B.D.O and Y.Liu. Controller reduction : concepts and approaches [J]. IEEETrans.Automat. Contr, 1989, AC- 34(8): 802-812.

[2] Guanghong Yang, Siying Zhang. Stabilizing controllers for uncertain symmetric compositesystems [J]. Automatica, 1995, 31(2): 337-340.

[3] Guanghong Yang, Wang J L. Reliable controller design for linear systems [J]. Automatica,2001, 37(5): 717-725.

[4] Sussmann H J, Sontag E D and Yang Y D. A General Result on the stabilization of linearsystems using Bounded Controls[J].IEEE Trans. On Automatic Control, 1994, 39(12): 2411-2425.

[5] Saberi A, Lin Z and Teel A R. Control of Linear systems with saturating Actuator [J]. IEEETrans. On Automatic Control, 1996, 41(3): 368-378.

[6] Ding Zhai, Qingling Zhang, Yuepeng Chen and Guoyi Liu. Decentralized robust stabilizationfor uncertain composite system with input saturation [J]. Journal of Northeastern University(Natural Science), 2004, 25(4): 309-312.

[7] Ding Zhai, Qingling Zhang. Decentralized control for composite systems with input saturation[J]. Control Theory and Application, 2003, 20(2): 280-282.

[8] Siying Zhang. Symmetry and similar structure of complex systems [J]. Control Theory andApplication, 1994, 11(2): 231-237.

[9] Guanghong Yang, Siying Zhang. Stability for large-scale composite systems with similarity[J]. Automatica, 1995, 31(2): 337-340.

[10] Banks S P, Aljurani S K. Lie algebra and Stability of nonlinear Systems[J]. Int.J.Control.1994, 60(3): 315-329.

1Institute of systems science, Northeastern University, Shenyang Liaoning 110004, China

2College of science, Yanshan University, Qinhuangdao, Hebei 066004, China.E-mail : [email protected]


BIFURCATION ANALYSIS OF A FOOD CHAIN MODELWITH B-D FUNCTIONAL RESPONSE IN VARYING

ENVIRONMENT

JING LIU, DONGYA ZHANG AND SHUQIN YANG

Abstract. This paper studies a food chain model with B-D functional response

in the environment varying with time . The existence of semi-trivial solutions

is obtained and the stability of the solutions is proved by Floquet theory. The

existence of periodic solutions is determined by the simple eigenvalue bifur-

cation theory. The numerical simulation for the model is given by applying

Matlab software and the results of numerical simulation are consistent with

the theoretic analysis.

Key Words. Chemostat; Varying environment; B-D functional response; Flo-

quet theory; Bifurcation.

1. Introduction

The chemostat is a laboratory apparatus used for the continuous culture of mi-croorganisms [1]. It can be used to study interactions between different populationsof microorganisms, and has the advantage that the parameters are readily measur-able. It is applied broadly and easy to be controlled in practice and can be testedtheoretically. These advantages attracted the notation from many researchers. Thechemostat model has a lot of forms because of the multiformities of ecosystems. Sin-gle food chain model and competition model are common forms [2-11]. It can bedescribed with ordinary differential equations, reaction-diffuse equations and timelag differential equations in mathematics. The questions being considered aboutthe chemostat model involve: the existence of steady solutions, global and localstability, the existence and stability of periodic solutions and the persistence of sys-tems. The mathematical knowledge used to study a chemostat involve the theoriesof differential equation(equations), the bifurcation, the index of fixed point and thetopological degree. The approaches are qualitative analysis and numerical simu-lation methods. In view of different species having different traits, the functionalresponses in the model have different forms. There are two common types: one typeis Hassell-Rahinowitz just related to the prey and another is Beddington-DeAngelistype related to both of the prey and the predator. Literature [4] studied the foodchain model with H-V functional response, proved the existence of the periodic pos-itive solutions by the simple eigenvalues bifurcation theory, and proved the stabilityof bifurcation solutions by Crandall-Rabinowitz theory. Literature [5] considered amodel describing predator-prey interactions in a chemostat that incorporates gen-eral response functions and indepedent removal rates by Hopf bifurcation theory.The food chain model with the substratum varying with time is studied in [8] inwhich the periodic oscillations of substrate, preys and predators are proved and

2000 Mathematics Subject Classification. 34K13, 37G10, 92D25.This research was supported by the National Science Foundation of China (NO 10471013).

144

BIFURCATION ANALYSIS OF A FOOD CHAIN MODEL 145

simulated numerically by using standard techniques of bifurcation theory. Twokinds of bifurcation with different bifurcation parameters are obtained. Literatures[10,11] study the reaction-diffusion equations in the chemostat unstirred, and getthe conditions for the coexistence solutions by topological degree theory in thesteady states. Literature [12] gives the deductive process of the B-D functionalresponse which is obtained as the dependence of preys and predators on each otherin a finite space is considered. In this paper, we study the food chain consisting ofsubstrate, preys and predators with B-D functional response in a chemostat stirredhomogeneously under the environment varying with time. We use coefficient bas a bifurcation parameter to prove. the stability of the semi-trivial solutions byFloquent theory [14] in section 2 and prove the existence of periodic solutions tothe single species by the simple eigenvalue bifurcation theory [13]. In section 3,we obtain the conditions for the existence of periodic solutions to the food chainmodel. Finally, we give numerical simulation to the model by Matlab software, andget the respective figures.

2. Model and the analysis of coexistence conditions

A mathematical model of the homogeneously stirred chemostat consisting ofsubstrate, prey and predator is

(1)

dSdt

= (S0 − S)θ −m1x1f1(S, x1),dx1dt

= x1(m1f1(S, x1)− θ)−m2x2f2(x1, x2),dx2dt

= x2(m2f2(x1, x2)− θ),S(0) = S0 ≥ 0, xi(0) = xi0 ≥ 0, i = 1, 2.

where S, xi(i = 1, 2) represent the densities of the substrate, the preys and thepredators at time t respectively. The parameter θ represents the amount of mediumflowing into or out of the chemostat. m1,m2 represent respectively the maximumrate of the food uptake of preys and predators. fi(i = 1, 2) are the B-D functionalresponses (particularly, f1 represents the rate at which a individual prey consumessubstrate; f2 represents the rate at which the prey are killed by per predator.),these are, f1(S, x1) = S

1 + α1S + β1x1, f2(x1, x2) = x1

1 + α2x1 + β2x2where α1, α2

represent the time of consuming unit substrate by a prey and killing one prey by apredator respectively, β1, β2 represent the reciprocals of the maximum number ofpreys and the maximum number of predators respectively. S0 is the concentrationof the substrate in the feed and is generally considered as a constant under thesteady environment. But since the concentration of substrate will change with thealternating of seasons or days and nights, we suppose that S0 is the function of timet: S0(t) = a+ be(t), where e(t) is a periodic function with period 2π and |e(t)| ≤ 1;a and b are constants and b 6= 0( a > |b| such that the concentration of substrate isnonnegative). In this paper, we let a = 1, then the system (1) becomes

(2)

dSdt

= (1 + be(t)− S)θ − x1m1S1 + α1S + β1x1

,

dx1dt

= x1(m1S

1 + α1S + β1x1− θ)− x2m2x1

1 + α2x1 + β2x2,

dx2dt

= x2(m2x1

1 + α2x1 + β2x2− θ),

S(0) = S0 ≥ 0, xi(0) = xi0 ≥ 0, (i = 1, 2).

146 J. LIU, D. ZHANG AND S. YANG

By the biologic meaning of the model, the state space of the system (2) is thenonnegative cone

R3+ = (S, x1, x2) ∈ R3 : S ≥ 0, x1 ≥ 0, x2 ≥ 0.

The coexistence means that the concentrations of substrate, preys and predatorsare positive. For the simple food chain model, the coexistence predicates the successof the cultivation of predators.

Firstly, we consider the chemostat with substrate and only one prey species. Themodel is as follows

dSdt

= (1 + be(t)− S)θ − x1m1S1 + α1S + β1x1

,

dx1dt

= x1( m1S1 + α1S + β1x1

− θ),

S(0) = S0 ≥ 0, x1(0) = x10 ≥ 0.

(3)

We define λ∗ = 12π

∫ 2π

0S∗

1+α1S∗ dt, where S∗ is the unique solution with period 2πof the equation

dy

dt+ θy = θ(1 + be(t)).

Lemma 2.1. If m1θ < λ∗−1, then 2π-periodic solution (S∗, 0) of system (3) is

asymptotically stable; if m1θ > λ∗−1, then (S∗, 0) is unstable and the system (3) has

a positive and unstable 2π-periodic solution (S(t), x1(t)) satisfying 0 < S(t) < S∗(t)and x1(t) > 0.Proof. Suppose that Bi(i = 1, 2) are the Banach spaces consisting of the2π-periodic functions which are continuous and differentiable satisfying ‖x‖Bi

=‖x‖∞ + ‖x′‖∞, and Bi(i = 1, 2) are the Banach spaces consisting of 2π-periodicfunctions satisfying ‖y‖Bi = ‖y‖∞. Let B = B1 × B2, B = B1 ×B2, I : B → B bethe natural embedding from B to B.

(i) proof of the stability of (S∗, 0)We consider the variational equations at (S∗, 0) of system (3)

(y′1y′2

)=

−θ − m1S

∗1 + α1S

∗

0 m1S∗

1 + α1S∗ − θ

(y1

y2

),

the fundamental solution matrix of the above equations is

A(t) =

exp(−θt) exp(

∫ t

0( m1S

∗1 + α1S

∗ − θ)dt)

0 − exp(∫ t

0( m1S

∗1 + α1S

∗ − θ)dt)

.

Since the period of A(t) is 2π, A(t + 2π) = CA(t). Let t = 0, we obtain

C = A(2π)A(0) =

exp(−2πθ) exp(

∫ 2π

0( m1S

∗1 + α1S

∗ − θ)dt)

0 − exp(∫ 2π

0( m1S

∗1 + α1S

∗ − θ)dt)

.

By the Floquet theory, we know (S∗, 0) is unstable when characteristic exponent∫ 2π

0( m1S

∗1 + α1S

∗ − θ)dt > 0, namely, m1θ

> λ∗−1; (S∗, 0) is stable when characteristic

exponent∫ 2π

0( m1S

∗1 + α1S

∗ − θ)dt < 0, namely, m1θ

< λ∗−1.

(ii) proof of that (λ∗−1, S∗, 0) is a bifurcation point


Let λ = m1θ

, define T : R+ × B → B as

T (λ, S, x1) =(

S′

x′1

)−

(1 + be(t)− S)θ − x1m1S

1 + α1S + β1x1

x1(m1S

1 + α1S + β1x1− θ)

,

Obviously T (λ, S∗, 0) ≡ 0 for any λ ∈ R. Since

T(S,x1)(λ∗−1, S∗, 0)y0 =

(y′1y′2

)−

−θ − θS∗

(1 + α1S∗)λ∗

0 θS∗(1 + α1S

∗)λ∗ − θ

(y1

y2

),

we obtain N(T(S,x1)(λ∗−1, S∗, 0)) = span(y1, y2)T , where N denotes the zero

space of T(S,x1)(λ∗−1, S∗, 0), y0 = (y1, y2)T ∈ B and (y1, y2) is the solution of

equations(4)

(y′1y′2

)=

−θ − θS∗

(1 + α1S∗)λ∗

0 θS∗(1 + α1S

∗)λ∗ − θ

(y1

y2

).(4)

Solving the equations, we obtain

y1

y2

=

−e−θt

∫ t

0θS∗

(1 + α1S∗)λ∗ exp(

∫ τ

0θS∗

(1 + α1S∗)λ∗ dτ)dt

exp(∫ t

0θS∗

(1 + α1S∗)λ∗ − θ)dt

,

obviously y1 < 0 and the dimension of BN = ker(T(S,x1))(λ∗−1, S∗, 0)) is 1. The

adjoint equation of (4) is(

ω′1ω′2

)= −

( −θ 0− θS∗

(1 + α1S∗)λ∗

θS∗(1 + α1S

∗)λ∗ − θ

)(ω1

ω2

).

Let (ω1

ω2

)=

(0

exp(− ∫ t

0( θS∗(1 + α1S

∗)λ∗ − θ)dt)

),

then N(T ∗(S,x1)(λ∗−1, S∗, 0)) = span(ω1, ω2)T , where T ∗(S,x1)

(λ∗−1, S∗, 0) is theadjoint operator of T(S,x1)(λ

∗−1, S∗, 0). The range of the operator T(S,x1)(λ∗−1, S∗, 0)

is given by

R(T(S,x1)(λ∗−1, S∗, 0)) = (y1, y2) ∈ B :

∫ 2π

0

y2ω2dt = 0

which is of codimension one.Now, we will prove that

T(λ,S,x1)(λ∗−1, S∗, 0)(y1, y2)T /∈ R(T(S,x1)(λ

∗−1, S∗, 0))

as (y1, y2)T ∈ XN\0. Therefore, we should testify that∫ 2π

0

T(λ,S,x1)(λ∗−1, S∗, 0)

(y1

y2

) (ω1

ω2

)dt 6= 0.

Through calculations, we know∫ 2π

0

T(λ,S,x1)(λ∗−1, S∗, 0)

(y1

y2

)(ω1

ω2

)dt =

∫ 2π

0

θS∗

1 + α1S∗ dt = 2πθλ∗ > 0.


By the simple eigenvalues bifurcation theory in [13], we obtain that (λ∗−1, S∗, 0)is a bifurcation point. Hence, there exists δ > 0 and a function (λ(ε), ϕ1(ε), ϕ2(ε)) :(−δ, δ) → R+ × Z of C1 type such that λ(0) = λ∗−1, ϕ1(0) = 0, ϕ2(0) = 0,(ϕ1(ε), ϕ2(ε)) ∈ Z, where Z satisfies B = BN ⊕ Z and λ = λ(ε), S = S∗ + ε(y1 +ϕ1(ε)), x1 = ε(y2 + ϕ2(ε)) are the solutions of T (λ, S, x1) = 0.

Because of y1 < 0, y2 > 0, the above expressions are the positive periodic solu-tions of system (3) when ε is larger than 0 and sufficiently small.Lemma 2.2. (i) If m1

θ< λ∗−1, then limt→∞ x1(t) = 0 and limt→∞ x2(t) = 0, so

(S∗, 0, 0) is asymptotically stable.(ii) If m1

θ> λ∗−1 or m2

θ> 1, then limt→∞ x2(t) = 0.

(iii) Since (S, x1) bifurcated from (λ∗−1, S∗, 0) is the 2π-periodic solution of sys-tem (3), we know that (S, x1, 0) is the 2π-periodic solution of system (2). Letµ∗ = 1

2π

∫ 2π

0x1

1 + α2x1dt, then (S, x1, 0) is asymptotically stable if m2

θ< µ∗−1 and

(S, x1, 0) is unstable if m2θ

> µ∗−1.

Proof. The conclusion (i) is right directly from Lemma 2.1.From the third equation of the system (2), we know that if m2

θ> 1, thendx2

dt< 0,

so conclusion (ii) is right.Now, we will prove conclusion (iii). The variational equations at (S, x1, 0) of

system (2) are

y′1y′2y′3

= B

y1

y2

y3

,

B =

−θ − m1x1(1+β1x1)

(1+α1S+β1x1)− m1S

(1+α1S+β1x1)+ m1Sx1β1

(1+α1S+β1x1)20

m1x1(1+β1x1)(1+α1S+β1x1)2

m1S(1+α1S+β1x1)

− m1Sx1β1(1+α1S+β1x1)2

− θ − m2x1(1+α2x1)

0 0 m2x1(1+α2x1)

− θ

.

The Floquet theory tells us that the characteristic exponent∫ 2π

0( m2x11 + α2x1

−θ)dt

of characteristic multiplier ρ = exp(∫ 2π

0( m2x11 + α2x1

− θ)dt) decides the stability of

(S, x1, 0). We know that (S, x1, 0) is unstable when∫ 2π

0( m2x11 + α2x1

− θ)dt > 0,

namely, m2θ

> µ∗−1 and (S, x1, 0) is stable when∫ 2π

0( m2x11 + α2x1

− θ)dt < 0, namely,m2θ

< µ∗−1.By the results of Lemma 2.1 and Lemma 2.2, we know that all microorganism

will coexist only under the conditions m1θ

> λ∗−1, m2θ

> µ∗−1.

3. The existence of coexistence periodic solution

By the definition of µ∗, we know that µ∗ relates to α2, x1. Hence, µ∗ relates tob. The change of parameter b will lead to the change of µ∗, so result in the changeof stability of (S, x1, 0). Let α1, α2, β1, β2,m1, θ fixed and m2 = m2(b), where b isthe bifurcation parameter. We want to get a bifurcation solution from (S, x1, 0).The bifurcation solution corresponds to the positive 2π-periodic solution of system(2).

When assuming b = b0, we have

12π

∫ 2π

0

m2x1

1 + α2x1dt− θ = 0,


d

db(

12π

∫ 2π

0

m2x1

1 + α2x1dt− θ)|b=b0 > 0,

so we know that (S, x1, 0) will lose its stability when b is larger than b0.Theorem. When m1

θ> λ∗−1, m2

θ> µ∗−1, system (2) has positive 2π-periodic

solutions.Proof. Suppose that Bi(i = 1, 2, 3) are the Banach spaces consisting of 2π-periodic functions which are continuous and differentiable satisfying ‖x‖Bi

= ‖x‖∞+‖x′‖∞, and Bi(i = 1, 2, 3) are the Banach spaces consisting of 2π-periodic functionsatisfying ‖y‖Bi = ‖y‖∞. Let B = B1× B2× B3, B = B1×B2×B3, I : B → B bethe natural embedding from B to B.

Define T : R+ × B → B as

T (b, S, x1, x2) =

S′

x′1x′2

−

(1 + be(t)− S)θ − x1m1S1 + α1S + β1x1

x1( m1S1 + α1S + β1x1

− θ)− x2m2x11 + α2x1 + β2x2

x2( m2x11 + α2x1 + β2x2

− θ)

,

The linearized matrix at (S, x1, 0)(where m2 = m2(b0) ) is

A =

−θ − m1x1(1+β1x1)

(1+α1S+β1x1)− m1S

(1+α1S+β1x1)+ m1Sx1β1

(1+α1S+β1x1)20

m1x1(1+β1x1)(1+α1S+β1x1)2

m1S(1+α1S+β1x1)

− m1Sx1β1(1+α1S+β1x1)2

− θ −m2(b0)x1(1+α2x1)

0 0 m2(b0)x1(1+α2x1)

− θ

,

so T(S,x1,x2)(b0, (S, x1, 0))~U = ~U ′ − A~U , where ~U = (u1, u2, u3)T . When ~U ′ = A~U ,we obtain

~U0 =

u1(t)u2(t)

exp(∫ t

0(m2(b0)x11 + α2x1

− θ)dt)

,

it is no need to find u1(t) and u2(t) exactly as they are not required in the followinganalysis.

We also know that

N(T(S,x1,x2)(b0, (S, x1, 0))) = span~U0,and

dim(ker(T(S,x1,x2)(b0, (S, x1, 0)))) = 1.

Let T ∗(S,x1,x2)(b0, (S, x1, 0)) be the adjoint operator of T(S,x1,x2)(b0, (S, x1, 0)) ,

thenN(T ∗(S,x1,x2)

(b0, (S, x1, 0))) = span ~W0,where ~W0 = (w1, w2, w3)T 6= 0 and is the solution of adjoint equation of ~U ′ = A~U .Let

~W0 =

00

exp(− ∫ t

0(m2(b0)x11 + α2x1

− θ)dt)

,

so the range of the operator T(S,x1,x2)(b0, (S, x1, 0)) is given by

R(T(S,x1,x2)(b0, (S, x1, 0))) = (u1, u2, u3) ∈ B :∫ 2π

0

u3w3dt = 0,

which is of co-dimension one.


Now, we will prove that T(b,S,x1,x2)(b0, (S, x1, 0))~U0 /∈ R(T(S,x1,x2)(b0, (S, x1, 0)))when ~U0 ∈ B\0. For this, we should testify that

∫ 2π

0T(b,S,x1,x2)(b0, (S, x1, 0))~U0

~W0dt 6= 0. By calculations and the above assumptions, we obtain∫ 2π

0

T(b,S,x1,x2)(b0, (S, x1, 0))~U0~W0dt =

d

db(

12π

∫ 2π

0

m2x1

1 + α2x1dt− θ)|b=b0 > 0.

By the simple eigenvalues bifurcation theory in [13], we obtain that there existsδ > 0 and a function (b(ε), φ(ε), ϕ(ε), η(ε)) : (−δ, δ) → R+ × Z of C1 type. Suchthat b(0) = b0, φ(0) = 0, ϕ(0) = 0, η(0) = 0, (φ(ε), ϕ(ε), η(ε)) ∈ Z where Z satisfiesB = N(T(S,x1,x2)(b0, (S, x1, 0)))⊕ Z, and b(ε), S + ε(u1 + φ(ε)), x1 + ε(u2 + ϕ(ε),ε(u3 + η(ε))) are the solutions of T (b, S, x1, x2) = 0. The above expressions are thepositive periodic solutions of system (2) when ε is larger than 0 and sufficientlysmall.

4. Numerical simulation

In this section, we use Matlab softwares to check the above conclusions. If m1θ

<

λ∗−1, then preys and predators do not survive. If m1θ

> λ∗−1 and m2θ

< µ∗−1,then only preys survive. If m1

θ> λ∗−1 and m2

θ> µ∗−1, then preys and predators

all survive. We take the initial value ratio of substrate, preys and predators as1 : 1 : 1 and set b = 1/2, e(t) = sint, θ = 2, α1 = 1, β1 = 0.25, α2 = 2, β2 = 0.5 toapproximate the values. From calculation, we obtain λ∗−1 = 2.0373, µ∗−1 = 4.0520.The figures and introductions as follows

0 20 40 60 80 1000

0.5

1

1.5

T

micr

ioor

gani

sm

su1u2

Figure 1. m1 = 4,m2 = 4.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

T

micr

ioor

gani

sm

su1u2

Figure 2. m1 = 5,m2 = 4.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

T

micr

ioor

gani

sm

su1u2

Figure 3. m1 = 7,m2 = 6.

0 20 40 60 80 1000

0.5

1

1.5

T

micr

ioor

gani

sm

su1u2

Figure 4. m1 = 7,m2 = 12.


(i) In Fig. 1, let m1 = 4, m2 = 4, then m1 < 2λ∗−1, m2 < 2µ∗−1. We can seethat preys and predators don’t survive which is consistent with our theories.

(ii) In Fig. 2, let m1 = 5, m2 = 4, then m1 > 2λ∗−1,m2 < 2µ∗−1. We can seethat only preys survive which is consistent with our theories.

(iii) In Fig. 3, let m1 = 7, m2 = 6, then m1 > 2λ∗−1,m2 < 2µ∗−1. We can seethat only preys survive which is consistent with our theories.

(iv) In Fig. 4, let m1 = 7, m2 = 12, then m1 > 2λ∗−1,m2 > 2µ∗−1. We can seethat preys and predators all survive which is consistent with our theories.

From Fig. 3 and Fig. 4, we obtain that predators will survive when the growthratio of predators is higher than the growth ratio of preys [15].

Acknowledgments

The author thanks the anonymous authors whose work largely constitutes thissample file. This research was supported by the National Science Foundation ofChina(No.10471013).

References

[1] Chen L X and Chen J. Nonlinear dynamical system[M], Beijing: Scientific and TechnicalPublisher, 1993, 165-183.

[2] Smith H I. Competitive coexistence in an oscillating Chemostat[J], SIAM. J. Appl. Math,1981, 40(3): 498-521.

[3] Liu J and Zheng S N. Qualitative analysis of a kind of model with competition in microorganismcontinuous culture[J], J. Biomath., 2002, 17(4): 399-405.

[4] Liu J and Zheng S N. Bifurcation analysis to a food chain model in chemostat with timevarying environment[J], J. Biomath., 2006 21(1): 89-96.

[5] El-sheikh M M A and Mahrof S A A. Stability and bifurcation of a simple food chain in achemostat with removal rates[J], Chaos, Solitons and Fractals, 2005, 23: 1475-1489.

[6] Zhu L M and Huang X C. Bifurcation for a functional yield chemostat when on competitorproduces a toxin[J], J.Math.Anal.Appl., 2007, 329: 891-903.

[7] Wang F Y and Zhang S W. Bifurcation and complexity of Monod type predator-prey systemin a pulsed chemostat[J], Chaos, Solitons and Fractals, 2006, 27: 447-458.

[8] Pang G P and Wang F Y. Analysis of a Moond-Haldene type food chain chemostat with periodi-cally varying substrate[J], Chaos, Solitons and Fractals, 2007, doi:10.1016/ j.chaos.2007.01.018.In Press.

[9] Shinji Nakaoka and Yasuhiro Takeuchi. Competition in chemostat-type equations with twohabitats[J], Mathematical Biosciences, 2006, 201: 157-171.

[10] Zheng S N and Liu J. Coexistence solutions for a reaction-diffusion system of un-stirred chemo-stat model[J], Applied mathematics and computation, 2003, 145: 579-590.

[11] Liu J and Zheng S N. A reaction-diffusion system arising from food-chain man unstirred chemo-stat[J], J. Biomath., 2002, 17(3): 263-272.

[12] Chris C, Donald L, DeAngelis, Jerald S and Olson B. Effects of Spatial Grouping on theFunctional Response of Predators[J], Theoretical Population Biology, 1999, 56: 65-75.

[13] Crandall M G and Rabinowitz P H. Bifurcation from simple eigenvalues[J], J. Funct Anal.,1971, 8: 321-340.

[14] Li Y. The method and application of nonlinear dynamical system[M], Beijing: China Commu-nications Press, 1998, 138-140.

[15] Kooi B M, Boer M P and Kooijman S A L M. Resistance of a food chain to invasion by a toppredator, Mathematical Biosciences[J], 1999, 157: 217-236.

Department of Mathematics, Dalian Maritime University, Dalian, Liaoning Province, 116026,China.

E-mail : [email protected]


DECISION OF PRICING FOR MULTI-PORT IN THE SAME HINTERLANDBASED ON GAME THEORY

SHUQIN YANG1, ZAN YANG2 AND WEI LIU1

Abstract. The dimension of hinterland range of a port is an important factor for determiningthe transportation price. The factors influencing the transportation price involve costs, ratesand tariffs. The cost and tariff are relatively stable, but the rates are various and changeable.The competition for attracting freight resource among the ports facing the same hinterlandis drastic seriously. Reducing rates is a common strategy of a port to stay competition. Butthe reducing of rates blindly must break the ordered market and destroy the benefits of allaspects. The competition will end at Nash equilibrium. If all the ports take cooperationaltitude to negotiate the rates by the range of their hinterlands, it will be a double-win sit-uation at Nash negotiation solution. Generally speaking, the rate charged by the port withenough freight resource is adjusted lower a little and that of the port with less resource ispromoted a little through the negotiation. This paper pays attention only on the influenceof the hinterland range on the port price and uses an example to indicate the necessity ofthe cooperation of ports. We consider the hinterland as a circular disc and the ports facingthe hinterland as the points on the circumference, and use game theory to discuss the resultsunder two situations: cooperation and non-cooperation. By comparing two results, we seethe advantage of cooperation.

Key Words. Hinterland; Payoff function; Nash equilibrium; Nash negotiation solution.

1. Introduction

With the rapid development of economy, the competition among the ports becomesmore and more drastic. To stay competition every port has to take measures such as reduc-ing costs, lowering rates and improving service and so on. These measures must reduce notonly the incomes of local company but also other company’s profits. On the other hand,with the economic globalization, the sufficiently large scale of a port becomes more andmore important. The trend of economic behavior is the partial cooperation replacing ofentirely competition. A current issue is the growth of multi-ports alliances which aims toavoid malicious competition[1,2].

In the port strategies, pricing strategy is commonly used in the short-period competi-tions. Particularly, when the economic situation of hinterland is not so good and multi-portsface the same resources, the pricing level of a port influences directly the profits.

The port pricing strategy in competition is always analyzed qualitatively from the angleof policy and pricing management system[3-5]. Professor Shi Xin studied the price strat-egy quantitatively by expressing the price as the function of market possession rate[6-9].Professor Zhu Daoli introduced a pricing method according to the hinterland range anddiscussed the rationality of some behavior of the ports[10].

Generally, the overall carrier pricing function revolves around costing, rates, and tar-iffs. The costs and tariffs do not change a lot; the rate is variable which allow carriers and

Received by the editors July 6, 2007.2000 Mathematics Subject Classification. 91A40.

152

DECISION OF PRICING FOR MULTI-PORT 153

shippers the flexibility to tailor rate structures to meet market needs[11]. A port can appro-priately reduce the various types of rates to attract freight suppliers. But the reduction ofrates blindly brings malicious competitions which destroy the profits of every aspect. Thebest way to avoid this situation is to cooperate with other ports to determine proper ratesby the range of goods resources.

In our paper we express the hinterland range as a circular disc and the ports are locatedon the circle which makes the problem more difficult but more approaching general sit-uation. We use them to discuss the different results in two conditions: cooperation andnon-cooperation. In the situation of non-cooperation, we have to solve a nonlinear pro-gramming. The example proves that the price is higher and the profit is more in cooperationthan in non-cooperation.

2. Construct of models

Suppose that there are three ports represented by A, B,C facing the same hinterland, thetransport rates charged by three ports are P1, P2, P3 respectively. Freight suppliers chooseone port by the price and the land transportation fee. Introduce a circular disc with theradius R to represent the hinterland, locate three ports on the circle such that the length ofthe arc between two ports is proportional to the real distance. Set up a plane coordinatesystem with the center as the origin O(Figure 1). Let port A on the x-axis, port B on thefirst quadrant, and the port C on the fourth quadrant. This assumption is rational becausethe ports distribute in port group and three ports in the same economic region are not faraway geographically. Let the angle between OA and OB be α, between OA and OC be β,then the polar coordinates of point A is (R, 0), point B is (R, α), point C is (R,−β). Assumethat the distribution of suppliers is homogeneous in the disc. The choice of the ports isrelated to the distance from the commodity site to the port. The freight in the left half diskis transferred to a port through the center, hence the distances to three ports are equal andthe possibilities by three ports are equal, therefore we just discuss the influence of the righthalf disk hinterland on the prices of three ports. If the transportation fee each unit and eachdistance by land is F, how do the ports determine the prices such that the profits attain themaximum.

F 1. the circular disc as the hinterland

If the hinterland of port A is two sectors: 1.0 ≤ θ ≤ ω1, 2.−ω2 ≤ θ ≤ 0, then thehinterland of port B is 3. ω1 ≤ θ ≤ π

2 and the port C is 4. − π2 ≤ θ ≤ π2 . The lower the price

of port A is, the greater the angles ω1,ω2 are.

154 S. YANG, Z. YANG AND W. LIU

The square of the distance from a point M(r, θ)(0 ≤ r ≤ R,− π2 ≤ θ ≤ π2 ) which is the

position of the freight to the point D(R, ϕ) on the circle is

d2 = R2 + r2 − 2Rrcos|θ − ϕ|(Figure2)

F 2. the circular disc as the hinterland

The average distance from the point in the sector 0 ≤ r ≤ R, θ1 ≤ θ ≤ θ2 to the point Don the circle is approximated by the mathematical expectation of the distance

d =

√R2 + r2 − 2Rrcos|θ − ϕ|

, where the distribution of (r,R) is uniform distribution on the square 0 ≤ r ≤ R, θ1 ≤ θ ≤θ2. Hence

Ed2 =1

R(θ2 − θ1)

∫ θ2

θ1

dθ∫ R

0(R2 + r2 − 2Rrcos|θ − ϕ|)dr

=R2

3(θ2 − θ1)

∫ θ2

θ1

(8sin6 |θ − ϕ|2

+ 3sin2(θ − ϕ) + cos3|θ − ϕ|)dθ(1)

We approximate the distance by Ed which error is Variance of d.By (1) the distances between the sectors 1, 2 and ports A, B,C are calculated approxi-

mately as follows:

d1A =

√43

R2 − R2

ω1sinω1

d2A =

√43

R2 − R2

ω2sinω2

d1B =

√43

R2 − R2

ω1sinα − R2

ω1sin(ω1 − α)

d2B =

√43

R2 +R2

β − ω1sinω2 − R2

β − ω1sinβ

d1C =

√43

R2 − R2

ω1sinβ − R2

ω1sin(ω1 + β)


and

d2C =

√43

R2 +R2

ω2sinα − R2

ω2sin(ω2 + α)(2)

Where ω1,ω2 are the central angles of sectors 1, 2 which are changing with the change ofprices P1, P2, P3, hence the hinterland of three ports are different.

The fees paid by suppliers for transferring unit commodity from sectors 1, 2 to threeports are respectively:

fi1 = P1 + di1 · F i = 1, 2,

fi2 = P2 + di2 · F i = 1, 2,

fi3 = P3 + di3 · F i = 1, 2

where fik, (k = 1, 2, 3) , is the transportation fee from sector i to port k.The fee transporting one ton-kilogram cargo from sector 1 to port A is less than that

from sector 1 to port B , that is

P1 + d1AF ≤ P2 + d1BF

Solving the equation for ω, we get approximately

ω1 ≤ α

2− 2(P1 − P2)√

3FRsinα2

(3)

And from P1 + d2AF ≤ P3 + d2C F we get approximately

ω2 ≥ β

2− 4(P1 − P3)√

3FRsinα2

(4)

Then the profits of three ports are proportional to the area of the hinterland of the portand respectively are

u1(P1, P2, P3) =R2

2P1(α + β − 2(P1 − P2)√

3FRsinα2

− 4(P1 − P3)√3FRsin β

2

)(5)

u2(P1, P2, P3) =R2

2P2(

π

2− α +

2(P1 − P2)√3FRsinα

2

)(6)

and

u3(P1, P2, P3) =R2

2P3(

π

2− β +

4(P1 − P3)√3FRsinα

2

)(7)

where P1, P2, P3 are the rates of transporting cargo charged by ports.Set R√

3Fsin α2

= a, 2R√3Fsin β

2= b, then the profits are simplified as

u1(P1, P2, P3) = P1((α + β)R2

2+ aP2 + bP3 − (a + b)P1),(8)

u2(P1, P2, P3) = P2((π − 2α)R2

4+ aP1 − aP2),(9)

and

u3(P1, P2, P3) = P3((π − 2β)R2

2+ bP1 − bP3).(10)


3. Analysis of the model by game theory

3.1. Nash equilibrium in a non-cooperative game. There is a set of n players calledplayer 1, player2,· · ·, player n . For each player k, there is a set S k of actions called the setof strategy. For xk ∈ S k, k = 1, 2, · · ·, n, X = (x1, x2, · · ·, xn) which belongs to the Cartesianproduct S = S 1 × S 2 × · · · × S n] is called the combined strategy.

If for each player k, there exists a mapping f from S to R, then we call the mapping thepayoff function of player k .

Under non-cooperative competition, if player knows that the other’s strategies are x1, · ··, xk−1, xk+1, · · ·, xn , the player’s objective is to find a strategy xk ∈ S k such that

(x1, x2, · · ·, xn) ∈ S

and

f (x1, · · ·, xk−1, xk, xk+1, · · ·, xn) = maxyk

fk(x1, · · ·, xk−1, yk, xk+1, · · ·, xn)(11)

here yk ∈ S K , (x1, · · ·, xk−1, yk, xk+1, · · ·, xn) ∈ SWe call the set

Rk = x = (x1, x2, · · ·, xn) ∈ S , and(x1, x2, · · ·, xn)makes(11)holdrational response set[12].

Every player expects his strategy in the rational response set. We call the point in theset

(x1, x2, · · ·, xn) ∈ R1

⋂R2

⋂· · ·

⋂Rn

Nash equilibrium of non-cooperative game[13]. At Nash equilibrium, any player cannot change his strategy rashly. Otherwise, his benefit will decrease. Let’s consider thenon-cooperation of ports.

If the ports non-cooperate, the aim for every aspect is to purchase the maximum profit.The competition finished at Nash equilibrium which is calculated by

∂u1∂P1

= 0∂u2∂P2

= 0∂u3∂P3

= 0

The unique solutions are

P∗1 =(π − α − β)R2

6(a + b)

P∗2 =(5aπ + 3bπ − 8aα − 6bα − 2aβ)R2

24a(a + b)(12)

P∗3 =(5bπ + 3aπ − 6αβ − 8bβ − 2bα)R2

24b(a + b)

and the profits of three ports are

ui(P∗1, P∗2, P

∗3) i = 1, 2, 3(13)

computed by (8),(9),(10).Sometimes, the results are not best for all the three aspects. If they cooperate to negoti-

ate the price, the result is out of expectation though it isn’t stable sometimes. We use Nashnegotiation solution to prove this.


3.2. Nash negotiation solution in cooperative game. Suppose that each player wouldlike to cooperate such that the payoff increases and knows the strategies and profits ofothers. This is the situation of cooperation.

We adopt the denotations in non-cooperative game.For any

(x1, x2, · · ·, xk, · · ·, xn) ∈ S

we define Pk(X) = xk , that is, Pk is a mapping from S to S k which is the projection ofS on S k.

Setting S ∗k = Pk(S ) , we define a function as follows:

f ∗k : S ∗k −→ R

such thatf ∗k (xk) = inf

(x1,···,xn)∈P−1k (xk)

fk(x1, · · ·, xn)

holds for any xk ∈ S ∗k . f ∗k (xk)represents the worst result of the player k when he choosesthe strategy xk . If there exists xk ∈ S ∗k such that

f ∗k (xk) = maxyk∈S ∗k

f ∗k (yk)

this indicates that player k chooses strategy by the minmax principle. The strategy isthe best one in the worst results which is the most conservative behavior and is one ofdecision-making ways when the player K has no sufficient information about others. Let

αk = supyk∈S ∗k

f ∗k (yk)

then the vector α = (α1, α2, · · ·, αn) ∈ Rn is called the bottom strategy[ 12] where the playerk obtains the worst result if he doesn’t give up the game.

The setF = ( f1(x), · · ·, fn(x)) : x ∈ S

is called the profit set of n players. If f = ( f1, · · ·, fn) , then the set

D = x ∈ S : f (x) ≥ αis called the rational set. Each player expects the profit is better than that in the worst result.The final solution of the game must be in the rational set D.

The strategy x ∈ S is called Pareto optimal if there doesn’t exist a strategy y such thatf (x) > f (y). f (x) = a ∈ F is called a Pareto optimality.

LetF∗ = ( f1(x), · · ·, fn(x)) : x ∈ D, andisParetooptimal.

Finding a cooperative strategy is converted to determine which Pareto optimality in F∗

is the result of the game.In 1950, John Nash proposed the following method for solving the problem:a = (a1, ···, an) ∈ F∗ is a Nash negotiation solution of a cooperative game if the function:d : F∗ −→ R+defined by

d(b) = (b1 − α1)(b2 − α2) · · · (bn − αn), b ∈ F∗

attains the maximum at a ∈ F∗ .Determining F∗ directly is not easy specially when n is bigger. Thus we use the follow-

ing identical technique:Find the maximum point x∗ of the function

d(x) = ( f1(x) − α1) · · · ( fn(x) − αn),


prove x∗ ∈ D , and compute the Nash negotiation solution

a = ( f1(x∗), · · ·, f2(x∗))

The following is theory of existence of Nash negotiation solution [12].Theorem Suppose that F is a compact convex set on Rn, if there exists a ∈ F such thatai > αi, i = 1, · · ·, n, then the function

d(b) = (b1 − α1)(b2 − α2) · · · (bn − αn)

attains maximum value at one and only one point in the set

F∗ = b ∈ F : b ≥ αWe use the conclusions above to discuss the cooperative game of the ports.If the ports cooperate and negotiate the rate charged by each port, then the least expec-

tation profits of three ports are

α1 = maxP1

minP2,P3

u1 = max P1((α + β)R2

2− (a + b)P1) =

(α + β)2R4

16(a + b)

α2 = maxP2

minP1,P3

u2 = max P2((π − 2α)R2

4− aP2) =

(π − 2α)2R4

64a

α3 = maxP3

minP1,P2

u3 = max P3((π − 2β)R2

4− bP3) =

(π − 2β)2R4

64bFind the values of P1, P2, P3 such that the function

d = (u1 − α1)(u2 − α2)(u3 − α3) = (P1((α + β)R2

2+ aP2 + bP3) − (a + b)P1) − (α + β)2R4

16(a + b)

·(P2((π − 2α)R2

4+ aP1 − aP2) − (π − 2α)2R4

64a)(P3(

(π − 2β)R2

4+ bP1 − bP2) − (π − 2β)2R4

64b)

attains maximum. (P1, P2, P3) is Nash negotiation solution of the problem. Generally, thissolution is bigger than Nash equilibrium.

We take an example. Suppose that R = 300, α = π10 , β = π

12 , F = 2 then the feasibleregion is

D = (P1, P2, P3)|0 ≤ Pi ≤ 100 i = 1, 2, 3,and the payoff functions are

u1(P1, P2, P3) = 32P1(1.5708 + 0.1443P2 + 0.5577P3 − 0.702P1)

u2(P1, P2, P3) = 32P2(0.5236 − 0.1443P2 + 0.1443P1)u3(P1, P2, P3) = 32P3(1.0472 + 0.5577P1 − 0.5577P3)

By formula (12) ,we calculate Nash equilibrium which is(20.4487, 61.2589, 33.6446),the profits of three ports by (13) are (1.56 ∗ 106, 0.96 ∗ 106, 1.16 ∗ 106).

Solving the non linear programming for Nash negotiation solution (P1, P2, P3)

max(P1, P2, P3) = (u1 − α1)(u2 − α2)(u3 − α3)= max(50.2656P1 + 4.6176P1P2 + 17.8464P1P3 − 22.464P2

1 − 89199.75)

× (16.7296P2 − 4.6176P22 + 4.6176P1P2 − 1564851.45)

× (33.4848P3 + 17.8464P1P3 − 17.8464P23 − 652855)

s.t. 0 ≤ Pi ≤ 100 i = 1, 2, 3by comparing the values step by step , we get (P1, P2, P3) = (29.6304, 65.8763, 37.0053),the profits are(1.65 ∗ 106, 2.4 ∗ 106, 1.82 ∗ 106).

Compared with Nash equilibrium, the prices of three ports are higher and the profits areincreasing. This is the result of negotiating to avoid hurting each other.


4. Conclusion

The final resolution of malicious competition is cooperation with each other. When thecompetition is drastic, negotiation is a way to avoid hurting each other.

When considering the range of freight origin we suppose that the commodity distributeshomogeneously without distinguishing the difference between cargos. In fact, the pricedecided by ports is for some certain kind of cargo and several freight origins. We cancompare the cost of every supplier to different port to negotiate the price which is simplewhen the number of suppliers is finite.

References

[1] Belenky A S. Cooperative games of choosing partners and forming coalitions in the marketplace[J], Mathe-matical and computer modeling, 2002, 36: 1279-1291.

[2] John A A. Research joint ventures: A cooperative game for competitors[J], European journal of operationalresearch, 2002, 136: 591-602.

[3] Feng X J and Yan Y X. Strategy choice of port logistics enterprises in price war[J], Journal of Traffic andTransportation Engineering, 2005, 5(2): 113-116.

[4] He Z Q et al. Profit distribution tactics of inter-modals transportation[J], Journal of Traffic and TransportationEngineering, 2005, 5(3): 122-126.

[5] Shi X. An analysis of the market structure and competition strategy in port competition(J), Navigation ofChina, 1998, 2: 89-93.

[6] Shi X. A game model of port competition (J), Systems engineering - theory and practice, 1998, 9: 27-33.[7] ShiXin. An entry/deterrence of duopoly in port competition[J], Journal of Traffic and Transportation Engi-

neering, 2001, 1(2): 120-123.[8] Shi X. Analysis on determinants of the port collution[J], Journal of Shanghai Jiaotong University, 2001,

35(6): 943-946.[9] Zhu D L and J A B. The competition strategies between regional container terminals[J], Journal of fudan

university(natural science), 2006, 45(5): 555-566.[10] Edward J B et al. Management of Transportation, Sixth Edition[P], Tsing Hua University Press, 2006, 12:

276-297.[11] Ye Q X. Guidance material on mathematical modeling contest for university students (first book) [P], Hunan

Education Press, 1998, 4: 191-207.

1 Department of Mathematics, Dalian Maritime University, Liaoning Dalian, 116026 P. R. China

2College of Transportation and Logistics, Dalian Maritime University, Liaoning Dalian, 116026 P.R.ChinaE-mail: [email protected]: [email protected]


RELAXED STABILITY CONDITIONS FOR NONLINEAR FUZZYTIME DELAY SYSTEMS

XIAOGUANG YANG1,2, QINGLING ZHANG1, LI LI3, XIAODONG LIU3

AND SHUQIN YANG2

Abstract. It is known that the stability condition of a nonlinear time-delay

system depends on the existence of the common matrices P and S which satisfies

all Lyapunov inequalities. In general, the common matrices P and S can be

found by means of linear matrix inequalities (LMIs) method. However, if the

number of rules of a fuzzy system is large, the common matrices P and S may

not exist or may not be found even using LMI. Therefore, in this paper, the

state space is divided into several subregions and the local common matrices Pj

and Sj for each subregion-j is found. Then the number of Lyapunov inequalities

to be satisfied by the corresponding common matrices Pj and Sj become much

fewer such that the stability condition of a nonlinear time-delay system is more

relaxed.

Key Words. Time-delay system, fuzzy control, relaxed conditions, linear

matrix inequality (LMI).

1. Introduction

Due to its conceptual simplicity and easy implementation, fuzzy control has beenapplied in miscellaneous industrial fields. It can readily represent nonlinear systemor uncertain systems by an appropriate transformation [11, 12].Tanaka and Sugeno[13] presented the T-S fuzzy model, in which the consequent parts are representedby linear state equations, and studied the related stability issues. Thereafter, thestability condition of T-S fuzzy model became a topic of extensive research, e.g.[14]-[19]. Another approach using Takagi-Sugeno fuzzy model has also been inten-sively used [15]-[18].These models utilize rules where the conclusion part is a localmodel has been derived by means of several methods: the Lyapunov approach [9],switching system theory [10], linear system with modelling uncertainties [21], etc.Of course, stability conditions derived are only sufficient ones, as these works donot take into account the antecedent part of the rules. Some works try to take intoaccount the antecedent part, allowing to obtained less conservative results [22, 23].

In this paper, we extend the above procedure to be nonlinear system with timedelay. This paper deals with relaxed stability conditions for nonlinear retardedsystems. It is well known that most stability analyses of nonlinear retarded systemsare based on Lyapunov stability theorem. For a T-S fuzzy system with time-delay,if there are common matrices P and S satisfying all Lyapunov inequalities, thestability condition of a nonlinear time-delay system is guaranteed [2]. The common

This work is supported in part by the National Natural Science Foundation of China underGrant 60534010 60575039.

160

STABILITY CONDITIONS FOR NONLINEAR FUZZY TIME DELAY SYSTEMS 161

matrices P and S are always solved by linear matrix inequality (LMI) method. Eventhough MATLAB LMI toolbox [20] solves the LMI problem efficiently, the feasiblecommon matrices P and S may not be found if the large number of rules of thefuzzy system with time-delay.

We choose a set of local common matrices Pj and Sj to replace the global commonmatrices P and S to relax the stability conditions of nonlinear fuzzy systems withtime-delay. If there are local common matrices Pj and Sj satisfying the associatedLyapunov inequalities of some local region, then the stability of the subsystemis guaranteed, i.e., Pj and Sj can ensure the local stability for some subregion-j.Under some extra conditions, the local stability can imply global stability of theT-S fuzzy system with time-delay.

In this paper, we study the stability conditions of nonlinear fuzzy systems withtime-delay again but consider these problems from the viewpoint of fired rules.Besides, a piecewise smooth quadratic (PSQ) Lyapunov function is adopted toderive more relaxed stability conditions. In general, when a input get into therule base of the fuzzy system, only partial rules are fired. Those unfired rulesare not necessary to be considered for the local stability. Those partial Lyapunovinequalities to be satisfied by each local common matrix Pj and Sj are fewer thanthe total Lyapunov inequalities which are satisfied by global common matrices Pand S. Therefore, under some additional conditions, a set of local common matricesPjs and Sjs can guarantee the global stability and relax the stability conditionsof nonlinear fuzzy systems with time-delay. The reduced number of the Lyapunovinequalities depends on both the number of the premise variables of each fuzzy ruleand the number of fuzzy sets for each premise variable.

The organization of this paper is as follows: Section 2 reviews the existing sta-bility conditions of nonlinear fuzzy systems with time-delay. In Section 3, themain results are proposed to relax those conditions stated in Section 2. Finally aconclusion is given in Section 4.

2. Basic stability conditions for fuzzy time-delay model

The T-S fuzzy time-delay model is composed of r plant rules that can be repre-sented as follows.

Plant Rule i:IF ξ1 is M1i and . . . and ξp is Mpi , THEN

x(t + 1) = A1ix(t) + A2ix(t− τ) + Biu(t)(1)y(t) = C1ix(t) + C2ix(t− τ)(2)x(t) = ψ(t), t ∈ [−τ0, 0] , i = 1, 2, · · · , r(3)

where x (t) ∈ Rn is the state vector, y (t) ∈ Rq is the output, u (t) ∈ Rm is theinput, A1i, A2i, Bi, C1i, C2i are some constant matrices of compatible dimensions,r is the number of IF- THEN rules, and ξ (t) = [ξ1 (t) , · · · , ξp(t)] are the premisevariables. It is assumed that the premise variables do not depend on the inputvariables u (t) explicitly. τ (t) ≤ τ0 is the bounded time-varying delay in the stateand it is assumed that

τ (t) ≤ β < 1

162 X. YANG, Q. ZHANG, L. LI, X. LIU AND S. YANG

i.e., the derivative of the time-varying delay function is continuous and bounded,which is a natural supplementary condition. ψ(t) ∈ Cn,τ0 is a vector-valued initialcontinuous function.

Given a pair of (x (t) , u (t)),the final outputs of the fuzzy systems are inferredas follows:

(4) x(t + 1) =r∑

i=1

λi(ξ (t))(A1ix(t) + A2ix(t− τ) + Biu(t))

(5) y(t) =r∑

i=1

λi(ξ (t))(C1ix(t) + C2ix(t− τ))

where λi(ξ(t)) = βi(ξ(t))r∑

j=1βj(ξ(t))

, βj(ξ(t)) =p∏

k=1

Mkj(ξj(t)),Mkj(ξj(t)) is the grade of

membership of ξj(t) in Mkj . It is assumed that

r∑

i=1

λi(ξ(t)) = 1, λk(ξ(t)) ≥ 0, k = 1, · · · , r,∀t

First we will derive the stability conditions for the unforced discrete system (4)with time delay

(6) x(t + 1) =r∑

i=1

λi(ξ (t))(A1ix(t) + A2ix(t− τ))

Some sufficient conditions for ensuring delay-independent stability of time delaysystem (6) can also be derived using the Lyapunov approach.

Lemma 1. [1] Given two matrices A ∈ Rm×n, B ∈ Rm×n and two positivedefinite matrices P ∈ Rm×m, Q ∈ Rn×n such that

AT PA−Q < 0 and BT PB −Q < 0thenAT PB + BT PA− 2Q < 0

Theorem 1. [2] The equilibrium of the discrete-time fuzzy system with timedelay described by (6) is asymptotically stable in the large if there exist two commonmatrices P > 0, S > 0 such that

(7)[

AT1iPA1i − P + S AT

1iPA2i

AT2iPA1i AT

2iPA2i − S

]< 0, i = 1, · · · , r

Theorem 1 shows that we have to find two common matrices P and S to satisfyr inequalities (7) to guarantee the stability of the T-S discrete-time fuzzy systemwith time delay. LMI method is always used to find the common matrices P andS.

3. Main results

Practically, when a state vector x(t) comes into the system (1), only partial rulesare fired. That is, some weights λis in the summation of (6) are zeros. Those unfiredrules would not influence the stability of the system. Therefore, in any instants,the common matrices P and S only need to satisfy β Lyapunov inequalities with


β rules fired by x(t). It should be noted that β is less than or equal to r, i.e.β ≤ r.According to the property of fired rules, Theorem 1 may be relaxed.

At the beginning, some assumptions must be brought out.Assumption (a) [4] each premise variable has g fuzzy sets (M1i, · · · ,Mgi) which

are complete and concise [3] in the corresponding universal set.Assumption (b) [4] each input datum of xj is a singleton.Now, according to the analysis above, (6) in group-j can be rewritten as the

following form:

(8) x(t + 1) =β∑

i=1

λi(ξ (t))(Aj1ix(t) + Aj

2ix(t− τ))

where Aj1i, A

j2i denote any Aj

1i, Aj2i in group-j. Based on the analysis [4-8], the

following theorem shows that the local stability condition of the system also impliesthe global stability of the system if Assumption (c) exists.

Assumption (c) ([6] and [7]) α is finite, where α is the number of the timeinstants at which the state trajectory of system (6) meets the boundary of any twosubregions.

Theorem 2. For a T-S discrete-time fuzzy system with time delay describedby (6) under Assumption (a), (b), (c) is asymptotically stable if there exist twocommon matrices Pjs > 0, Sjs > 0 such that(9)[

(Aj1i)

T PjAj1i − Pj + Sj (Aj

1i)T PjA

j1i

(Aj2i)

T PjAj1i (Aj

2i)T PjA

j2i − Sj

]< 0, i = 1, · · · , β, j = 1, · · · , g

where Aj1i, A

j2i denote any Aj

1i, Aj2i in group-j, Pj and Sj denote common positive

matrices for the group-j.Proof. For the system (6), under Assumption (a), (b), select the Lyapunov

function to be piecewise smooth quadratic as follows:

(10) V (x(t)) =g∑

j=1

δjVj (x(t))

where

(11) Vj (x(t)) = xT (t)Pjx(t) +t−1∑

σ=t−τ

xT (σ)Sjx(σ)

where Pj > 0 and Sj > 0 is to be selected. It is evident that there exist σ1 and σ2

such that

σ1 ‖x (t)‖2 ≤ V (x (t)) ≤ σ2 ‖x (t)‖2

For example,taking σ1 = λmin (Pj) and σ2 = λmax (Pj) + τλmax (Sj), then wehave

∆V (x(t)) = V (x(t + 1))− V (x(t)) =β∑

i=1

λ2i x

T[(AT

ij)PjAij − Pj

]x +

β∑i=1

β∑j=1

λiλj xT

[(AT

ij)PjAjj + (ATjj)PjAij − 2Pj

]x


where

x(t) =[(

x(t)x(t− τ)

)], Aij =

[Aj

1i Aj2i

], Pj =

[Pj − Sj 0

0 Sj

]

With the aid of Lemma 1, (9) implies ∆V (x(t)) < 0. In other words, the fuzzysystem with time delay described by (6) is asymptotically stable in each subregion.It should be noted, so for, x(t) and x(t+1) are at the same subregion in (8). If x(t)and x(t + 1) are at different subregions, from Theorem 3.1 of [7], the fuzzy system(6) is still asymptotically stable if Assumption (c) holds.

Theorem 3. For a T-S discrete-time fuzzy system with time delay describedby (6), suppose Assumption (a), (b) holds, then it is asymptotically stable, if thereare a set of positive definite matrices Pjs and Sjs(j = 1, · · · , g) , such that(12)[

(Aj1i)

T PjAj1i − Pj + Sj (Aj

1i)T PjA

j1i

(Aj2i)

T PjAj1i (Aj

2i)T PjA

j2i − Sj

]< 0, i = 1, · · · , β, j = 1, · · · , g

Proof. Choose a Lyapunov function as (10). In Theorem 2, (9) implies thesystem to be asymptotically stable in each subregion. If the state of the systemgoes across two different subregions from time instant t to t + 1, we have

∆V (x(t)) = V (x(t + 1))− V (x(t))

=β∑

i=1

λ2i x

T[(AT

ij)PjAij − Pj

]x

+β∑

i=1

β∑

j=1

λiλj xT

[(AT


]x(13)

Since x(t) and x(t + 1) are at the adjacent subregions Gj and Gj respectively.The corresponding matrices are Pj and Pj , so we can rewrite (14) as

∆V (x(t)) =β∑

i=1

λ2i x

T[(AT

ij)PjAij − Pj

]x

+β∑

i=1

β∑

j=1

λiλj xT

[(AT


]x(14)

With the aid of lemma 1, (9) implies ∆V (x(t)) < 0. Thus T-S fuzzy systemwith time delay described by (6) is asymptotically stable.

Remark 1. It is more possible to find feasible Pjs and Sjs for satisfying (12)than to find the common matrices P and S to satisfy (9). Since if the commonmatrices P and S for (9), Pj = P (j = 1, · · · , g) are feasible solutions for (12). Ifthe common matrices P and S do not exist, however, those Pjs and Sjs in (12)might still exist. It implies that Theorem 3 is more relaxed than Lemma 1.

Remark 2. If (12) is not considered,then x(t) and x(t + 1) can be at anydifferent subregions respectively. (12) is replaced by

(15) (ATij)PlAij − Pj < 0, l, j = 1, 2, · · · , g

where l, j are for any subregions. It is obvious that finding those Pj in (12) is mucheasier than finding all Pl in (15).


4. Simulation example

In this section, a numerical example is given to illustrate the stability ex-amination and stablization for T-S fuzzy discrete time-delay systems.

Example. Consider a T-S fuzzy discrete time-delay systems as follows:Rule-i:If x1 (k) is Mi1, x2 (k) is Mi2, · · · , xm (k) is Mim,then

(16) x (k + 1) = A1ix (k) + A2ix (k − h)

A11 =[

0.76 0.360.09 0.76

],A12 =

[0.75 0.220.09 0.75

],A13 =

[0.75 0.200.09 0.75

],A14 =

[0.75 0.090.09 0.75

],

A15 =[

0.75 0.110.10 0.75

],A16 =

[0.75 0.100.10 0.75

],A17 =

[0.75 0.090.23 0.75

],A18 =

[0.76 0.090.25 0.78

],

A21 =[

0.76 0.250.09 0.76

],A22 =

[0.75 0.180.09 0.75

],A23 =

[0.75 0.110.11 0.75

],A24 =

[0.75 0.100.11 0.75

],

A25 =[

0.75 0.110.09 0.75

],A26 =

[0.75 0.090.23 0.77

],A27 =

[0.76 0.090.27 0.78

],A28 =

[0.76 0.090.36 0.79

].

According to Theorem 3, using MATLAB LMI toolbox, we can find eight localcommon matrices.

P1 = 1.0e−012∗[

0.1253 −0.3292−0.3292 0.2193

], S1 = 1.0e−012∗

[0.0627 −0.1670−0.1670 0.1257

],

P2 = 1.0e−011∗[

0.0467 −0.5917−0.5917 0.0691

], S2 = 1.0e−011∗

[0.0146 −0.2956−0.2956 0.0520

],

P3 = 1.0e−012∗[

0.0371 −0.6453−0.6453 0.0340

], S3 = 1.0e−012∗

[ −0.0108 −0.3275−0.3275 −0.0155

],

P4 = 1.0e−011∗[

0.0501 −0.1062−0.1062 0.0076

], S4 = 1.0e−012∗

[0.1922 −0.5362−0.5362 0.0377

].

such that (12) is satisfied. By Theorem 3, the equilibrium of the T-S fuzzy discretetime-delay system is globally asymtotically stable. Fig.1 is the simulation result.

5. Conclusions

In this paper, a new idea to relaxed stability conditions is addressed for a T-S fuzzy time-delay system. According to the idea of fired rule group, the rulebase is divided into several rule groups. Under this division, a set of Pjs andSjs satisfying the local Lyapunov inequalities respectively are adopted instead ofcommon matrices P and S satisfying all Lyapunov inequalities. Theorem 2 and 3are the main criteria for the stability conditions for T-S fuzzy time-delay system.While the number of rules is large, the proposed results have their superiorityobviously.


1 2 3 4 5 6 7 8 9 10−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time sec

stat

e

x1x2

Figure 1. The responses of the state of the time-delay fuzzy system

References

[1] M. Johansson, A. Rantzer, K.E. Arzen, Piecewise quadratic stability of fuzzy systems, TEEETrans. Fuzzy Syst. Vol.7,713-722,1996.

[2] Y. Y. Cao , P. M. Frank, “Analysis and synthesis of nonlinear time-delay systems via fuzzycontrol approach,” IEEE Trans. Fuzzy Syst., Vol.8, No.2,200-211, 2000.

[3] L. X. Wang, A Course in Fuzzy Systems and Control, Tsinghua University Press, 2003.[4] Wen-June Wang, Chung-Hsun Sun, Relaxed stability and stabilization conditions for a T-S

fuzzy discrete system, Fuzzy Sets and Systems Vol.156,208-225,2005.[5] S. G. Cao, N. W. Rees, Stability analysis and design for a class of continuous-time fuzzy

control system, Internat. J. Control Vol.64,1069-1087,1996.[6] S. G. Cao, N. W. Rees, Lyapunov-like stability theorems for discrete-time fuzzy control

systems, Internat. J. Sys. Sci.Vol.28,297-308,1997.[7] S. G. Cao, N. W. Rees, Further results about quadratic stability of continuous -time fuzzy

control system, Internat. J. Sys. Sci. Vol.28,397-404,1997.[8] S. G. Cao, N. W. Rees, G. Feng, Quadratic stability analysis and design of continuous - time

fuzzy control systems, Internat. J. Sys. Sci. Vol.27,193-203,1996.[9] M. A. L. Thathachar, P. Viswanath, On the stability of fuzzy systems, IEEE Trans. Fuzzy

Systems Vol.5,No.1 ,145-151,1997.[10] K. Kiriakidis, A. Grivas, A. Tzes, Quadratic stability analysis of the Takagi-Sugeno fuzzy

models, Fuzzy Sets and Systems Vol.98,1-14,1996.[11] K. Tanaka, M. Sano, A robust stabilization problem of fuzzy control systems and its applica-

tion to backing up control of a truck-trailer, IEEE Trans. Fuzzy Syst. Vol.2,No.2,119-134,1994.[12] K. Tanaka, M. Sugeno, Stability analysis and design of fuzzy control system, Fuzzy Sets and

Systems Vol.45,135-156,1992.[13] J.M. Zhang, R. H. Li, P. A. Zhang, Stability analysis and synthesis design of fuzzy control

system, Fuzzy Sets and Systems Vol.120,65-72,2001.[14] H. O. Wang, K. Tanaka, M. F. Griffin, An approach to fuzzy control nonlinear systems:

stability analysis and design issues, IEEE Trans. Fuzzy Systems Vol.4,No.1, 14-23,1996.[15] K. Tanaka, T. Ikeda, H. O. Wang, Fuzzy regulators and fuzzy observers: relaxed stability

conditions and LMI-base designs, IEEE Trans. Fuzzy Systems Vol.6,No.2, 250-265,1998.[16] X.J. Ma, Z. Q. Sun, Y. Y. He, Analysis and design fuzzy controller and fuzzy observers, IEEE

Trans. Fuzzy Systems Vol.6,No.1,41-51,1998.[17] J .Joh, Y.H. Chen, R. Langari, On the stability issues of linear Takagi-Sugeno fuzzy models,

IEEE Trans. Fuzzy Systems Vol.6,No.3,402-410,1998.[18] X. Yang, X. Liu, Q. Zhang and P. Liu, Stability analysis for discrete T-S fuzzy systems,

International Journal for Information & Systems Sciences,Vol.1,No.3-4, 339-346,2005.


[19] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modellingand control,” IEEE Trans. Fuzzy Syst., Man, Cybern. Vol. SMC-815, 116-132,1985.

[20] P. Gahinet, A. Nemirovski, A. Laub, M. Chilali, LMI control toolbox, The Math WorksInc.,1994.

[21] J. P. Marin, Necessary and sufficient quadratic stability conditions for a class of fuzzy sys-tem, Actes Rencontres Francophone surla Logique Floue et ses Applications, LFA’95, Paris,Cepadues-Editions, 240-247(in French),1995.

[22] Y. Blanco, W. Perruquetti, P. Borne, Non quadratic stability of nonlinear systems in theTakagi-Sugeno form, in: Proc. ECC, Porto, Portugal, 3917-3922,2001.

[23] S. C. Tong, T. Wang, Y. P. Wang, J. T. Tang, Design and stability analysis of a fuzzy controlsystem, Science Press, 2004.

1Institute of Systems Science, Northeastern University, Shengyang 110004 P.R. China

2 Department of Mathematics Dalian Maritime University, Dalian 116026 P.R.China

3Research Center of Information and Control, Dalian University of Technology, Dalian 116024P.R. China



FUZZY RULE EXTRACTION FROM FUZZY DECISION TREES∗

XINGHUA FENG, XUELIAN XU AND XIAODONG LIU

Abstract. In this paper, we develop the AFS fuzzy decision trees proposed by

Liu and Pedrycz (Applied Soft Computing, 2007, 7: 325-342). A new approach

is presented to extract the rule-base from fuzzy decision trees and pruning the

rule-base. We evaluate the performance of the proposed algorithms using three

well-known benchmark data sets—the Iris data, the Wine classification data,

and the Wisconsin Breast cancer data to verify their effectiveness.

Key Words. Fuzzy decision trees, Fuzzy rules, AFS fuzzy logic, Knowledge

representation and Classification.

1. Introduction

Decision trees [14, 15] are one of the most popular methods for solving prob-lems in pattern recognition, system modeling, and data mining. Up to now, manyapproaches have been proposed for decision trees, such as ID3, C4.5 [14, 15], etc.,focusing on discretization of data attributes, tree construction, pruning, and re-finement. A decision tree is a flow-chart-like tree structure attribute, each branchrepresents an outcome of the test, and each leaf node is labeled by a class or classdistribution [21]. In fuzzy decision trees, every node except for their root comesequipped with a fuzzy set that can be represented as a conjunction of several fuzzylinguistic terms, where the conjunctions are implemented by some standard opera-tors encountered in fuzzy logic. It becomes apparent that to a significant extent thefuzzy decision trees are pre-determined by the membership functions of the fuzzyterms and the fuzzy logic operators. In [5] the author has developed the fuzzydecision trees in the framework of AFS (Axiomatic Fuzzy Set) logic. The fuzzysets (membership functions) are obtained by the AFS structure and the trainingexamples, and the AFS fuzzy logic is used to deal with conjunctions of fuzzy propo-sitions.

In this paper, we focus on the inference for decision assignment and pruning therule-base extract from the fuzzy decision tree. We study and discuss the approachfor rule extraction and pruning the rule-base. Besides, we test and analyze thealgorithm using three well-known real data sets in UCI Repository for MachineLearning Data-Bases [10].

The rest of this paper is organized as follows: Section 2 gives an overview ofAFS theory. In Section 3, we discuss the new algorithm for generation of fuzzyrules from AFS-decision tree. Section 4 is concerned with numeric experiments andsome analysis. The conclusions of the study are included in Section 5.

Received by the editors June 1, 2006 and, in revised form, December 22, 2006.2000 Mathematics Subject Classification. 35R35, 49J40, 60G40.* This research is supported by the Natural Science Foundation of China under Grant 60575039

60534010 and the National Key Basic Research and Development Program of China under Grant2002CB312200 .

168

FUZZY RULE EXTRACTION FROM FUZZY DECISION TREES 169

2. Overview of the AFS theory

In this section, we recall the notations and present several pertinent results ofAFS theory concerning this paper.

Definition 1. ([6]) Let M be a non-empty set. Then the set EM∗ is defined byEM∗ = ∑i∈I(

∏m∈Ai

m)|Ai ∈ 2M , i ∈ I, I is any non-empty indexing set .A binary relation R on EM∗ is defined as follows: for any

∑i∈I(

∏m∈Ai

m),∑j∈J(

∏m∈Bj

m) ∈ EM∗,[∑

i∈I(∏

m∈Aim)

]R

[∑j∈J (

∏m∈Bj

m)]⇐⇒ (i) ∀Ai (i ∈ I), ∃Bh (h ∈ J) such

that Ai ⊇ Bh; (ii) ∀Bj (j ∈ J), ∃ Ak (k ∈ I), such that Bj ⊇ Ak.

Theorem 1. ([7]) Let M be a non-empty set. Then (EM,∨,∧) forms a completelydistributive lattice under the binary compositions ∨ and ∧ defined as follows. Forany

∑i∈I(

∏m∈Ai

m),∑

j∈J(∏

m∈Bjm) ∈ EM,

∑

i∈I

(∏

m∈Ai

m) ∨∑

j∈J

(∏

m∈Bj

m) =∑

k∈ItJ

(∏

m∈Ck

m),(1)

∑

i∈I

(∏

m∈Ai

m) ∧∑

j∈J

(∏

m∈Bj

m) =∑

i∈I,j∈J

(∏

m∈Ai∪Bj

m),(2)

where for any k ∈ I t J (the disjoint union of I and J , i.e., an element in I andan element in J are always regarded as different elements in I t J), Ck = Ak ifk ∈ I, and Ck = Bk if k ∈ J .

(EM,∨,∧) is called the EI (expanding one set M) algebra over M—one kindof AFS algebra. For ψ =

∑i∈I(

∏m∈Ai

m), ϑ =∑

j∈J(∏

m∈Bjm) ∈ EM , ψ ≤ ϑ

⇐⇒ ψ ∨ ϑ = ϑ ⇔ ∀Ai (i ∈ I), ∃Bh (h ∈ J) such that Ai ⊇ Bh.

Definition 2. ([8]) Let ζ be an attribute on the universe of discourse X. Rζ

is called a binary relation (i.e., Rζ ⊂ X × X) of ζ if Rζ satisfies: x, y ∈ X,(x, y) ∈ Rζ ⇔ x has attribute ζ at some degree and the degree of x having ζ islarger than or equal to that of y, or x has attribute ζ at some degree and y does notat all.

Definition 3. ([8]) Let X be a set and R be a binary relation on X. R is calleda sub-preference relation on X if for x, y, z ∈ X, x 6= y, R satisfies the followingconditions:1. If (x, y) ∈ R, then (x, x) ∈ R;2. If (x, x) ∈ R and (y, y) /∈ R, then (x, y) ∈ R;3. If (x, y), (y, z) ∈ R, then (x, z) ∈ R;4. If (x, x) ∈ R and (y, y) ∈ R, then either (x, y) ∈ R or (y, x) ∈ R.A concept ζ is called a simple concept on X if Rζ is a sub-preference relation.Otherwise ζ is called a complex concept on X. Where Rζ is the binary relation ofζ defined by Definition 3.

Definition 4. ([7]) Let X, M be sets and 2M be the power set of M . Let τ :X × X → 2M . (M, τ, X) is called an AFS structure if τ satisfies the followingconditions:AX1: ∀(x1, x2) ∈ X ×X, τ(x1, x2) ⊆ τ(x1, x1);AX2: ∀(x1, x2), (x2, x3) ∈ X ×X, τ(x1, x2) ∩ τ(x2, x3) ⊆ τ(x1, x3).X is called universe of discourse, M is called an attribute set and τ is called astructure.

170 X. H. FENG, X. L. XU AND X. D. LIU

Let X be a set of objects and M be a set of attributes on X. If τ : X×X → 2M

is defined as follows: for any (x, y) ∈ X ×X

(3) τ(x, y) = m|m ∈ M, (x, y) ∈ Rm ∈ 2M ,

where Rm (defined by Definition 2) is the binary relation of simple attribute m ∈ M .It can be easy to check τ defined by (3) is structure.

Definition 5. ([9]) Let (M, τ,X) be an AFS structure and for any A ⊆ X, |A| bethe number of elements in A. For any fuzzy concept η =

∑i∈I(

∏m∈Ai

m) ∈ EM ,the membership function of η is defined as follows. For any x ∈ X,

µη(x) = supi∈I

|Aτi (x)||X|(4)

where Aτ (x) = y ∈ X |τ(x, y) ⊇ A.The membership function defined by (4) only depends on the distribution of the

original data and has been successfully used to cluster analysis [8].

3. Generation of fuzzy rules from AFS decision tree

In this section, we discuss the AFS decision tree, the algorithm for decisionassignment and pruning the rule-base extract form the fuzzy decision tree.

3.1. Basic notions. Let us recall some notions and definitions pertaining to thestudy presented in [5]. Table 1 includes a collection of the training examples.

Table 1 Collection of training examples

Training Inc Emp CreditExamples U1 U2 Y

u1j u2

j yj

e1 0.20 0.15 0.00e2 0.35 0.25 0.00e3 0.90 0.20 0.00e4 0.60 0.50 0.00e5 0.90 0.50 1.00e6 0.10 0.85 1.00e7 0.40 0.90 1.00e8 0.85 0.85 1.00

Here ei is a training example, Ui is the universe of discourse of the fuzzy variableor attribute Vi, Y is the universe of discourse of decision variable Dc. Beforewe define the fuzzy-build and fuzzy-inference procedures, let us introduce someadditional notations.

1) The set of fuzzy variables or attributes is denoted by

V = V1, V2, . . ., Vn.where Vi is a fuzzy variable over the universe of discourse Ui, i = 1, 2, . . ., n.

2) For each variable Vi ∈ V

• data of training example j is uij ∈ Ui.

• Di denotes the set of fuzzy terms for fuzzy variable Vi .• vi

p ∈ Di denotes the fuzzy term for the variable Vi. (e.g., vIncsmall, as necessary

to stress the variable or with anonymous values—otherwise p alone may beused).


3) The set of fuzzy terms for the decision variable is denoted by Dc. Each fuzzyterm vc ∈ Dc is a fuzzy set over universe of discourse Y .

4) The set of training examples is

E = ej | ej = (u1j , u

2j , . . ., u

nj , yj).

5) M is the set of all fuzzy terms

M = Dc ∪ (n⋃

i=1

Di)

(M , τ , E) is the AFS structure and EM is the EI algebra over M . In general,for each pair vc

p, vcq ∈ Dc, vc

p 6= vcq, µvc

p∧vcq(e) < ε, ∀e ∈ E (ε is a very small positive

number), for any e ∈E. This implies that the fuzzy terms in Dc implement a fuzzyclassification on E.

6) For each node N of the fuzzy decision trees• FN denotes the set of fuzzy restrictions on the path leading to N , e.g.,

FN = [ Inc is small ], [Imp is large]• V N is the set of attributes appearing on the path leading to N

V N = Vi | ∃p ([Vi is vip] ∈ FN ).

• βN is a fuzzy set in EI algebra EM, and µβN (ej) is the membership degreeof training sample ej in node N , where

βN = vip | ∃p ([Vi is vi

p] ∈ FN ) ∈ EM.

• N |vip denotes the particular child of node N created by the use of the fuzzy

attribute Vi to split N and following the edge vip ∈ Di.

• SNVi

denotes the set of N ’s children when Vi ∈ (V − V N ) is used for thesplit.

(5) SNVi

= (N |vip) | vi

p ∈ DNi , DN

i = vip ∈ Di | ∃ej ∈ E, µβN∧vi

p(ej) > δ;

• PNvc

denotes the example count for decision vc ∈ Dc in node N , where

PNvc =

|E|∑

j=1

µβN∧vc(ej), PN =∑

vc∈Dc

PNvc ,

PN |vi

p

vc =|E|∑

j=1

µβN∧vip∧vc(ej), PN |vi

p =∑

vc∈Dc

PN |vi

p

vc .(6)

• PN and IN denote the total example count and information measure fornode N , where IN is the standard information content

(7) IN = −∑

vc∈Dc

PNvc

PNlog

PNvc

PN.

• GNVi

= IN − ISNVi denotes the information gain when using the fuzzy at-

tribute Vi to split N , where

(8) ISNVi =

∑

vip∈DN

i

PNvi

p∑vi

p∈DNi

PNvi

p

IN |vip


is the weighted information content.

3.2. Procedure to Build an AFS Decision Tree.

3.2.1. Node splitting criterion. First, the attributes need to be fuzzify intofuzzy terms. Here just need the order relation on the attributes, for example, aattribute Vi could be fuzzify into two fuzzy terms vi

large: “the value of Vi is large”and vi

small: “the value of Vi is small”. The growth process of the tree is pursued bythe maximum information gain. The information gain of the use of fuzzy attributeVi to split the current node N is GN

Vi= IN − ISN

Vi , where IN and ISNVi are defined

by (7) and (8). The fuzzy attribute Vi which exhibits the maximum informationgain at the current node N is applied to split N . The children of the node N is theset DN

i (defined by (5)).

3.2.2. Growth of the tree and Stopping condition. Let N be a node and thethreshold δ > 0. βN

δ is the δ cut set of fuzzy set βN ∈ EM

(9) βNδ = e ∈ E | µβN (e) > δ

where µβN (e) is the membership degree of example e in node N . A given nodecan be expanded if the examples belong to βN

δ do not have a unique classification,if not satisfied, it prevents us from expanding the node. The second stoppingcondition is self-evident: The current node N can be expanded if V N 6= V . Anothertermination option is to monitor the information gain along the nodes of the treeonce being built, in case of that the fuzzy attribute Vi exhibits the maximuminformation gain at the current node N , but the information gain GN

Vi< 0, or the

set of the children at node N is an empty set, DNi = ∅, then we stop expanding this

node. The first two stopping criterions are a sort of precondition: if not satisfied,we stop expanding the node. The third one comes in a form of some postcondition:to make sure if it is satisfied, we have to expand the node first and then determineits value, if not satisfied, we should backtrack and refuse to expand the particularnode.

3.2.3. Rule extraction. Each path starting from the root up to a classifiernode(leaf node) is converted to a rule. The antecedent part of the rule is a fuzzyset–the conditions leading to the leaf, that is βN ∈EM, where N is the leaf nodeby the corresponding path. It is important to label the rules, we use a new methodto determine the class label of rules. Training example ej has its membership ineach rule calculated as some fuzzy set βN ∈EM. Let ξi ∈ EM be a rule and thethreshold δ > 0.

Eξi = e| e ∈ E, µξi(e) ≥ µξj (e),∀ j, j 6= iThe consequent of the rule ξi, i.e., the class assigned by the rule, is the same asthe class label of the main class contained by Eξi . Especially, for rule ξi, if Eξi isempty set, we put the βN

δ instead of Eξi and label this particular rule by the classlabel of the main class contained by βN

δ , N is the leaf node by the correspondingpath. In other words, the rule ξi is poor ability for classification on training data,it is redundant rule for the training data. Actually, rules have empty Eξi are notall redundant for the testing data, so we give class labels to them.


3.3. Pruning the Rule-Base. The proposed rule extraction method, however,does not rule out the possibility of the redundant rules as well as bad rules, whichshould be removed from the rule-base for a better performance and efficiency. Wepropose a very simple but effective scheme for rule-base minimization. We classifythe training data with the rule-base as the follows:

1) Remove each rule from the rule-base, and classify the training data withthe corresponding remaining rule-base.

2) Delete the rule, whose the corresponding remaining rule-base have the max-imal increase of accuracy.

3) Repeat 1) and 2) and stop the pruning if have a decrease of accuracy byremoving any rule of the rule-base.

4) Combine the rules which have the same consequent with the AFS fuzzylogic operator “∨” (refer to (1)). Therefore, with a training data set whichhave c classes, we can make a rule-base with c rules whose antecedents areξ1, ξ2, · · · , ξc respectively.

3.4. Inference for decision assignment. For any e ∈ U1 × U2 × · · · × Un,ξ =

∑i∈I Ai ∈ EM ,

µξU (e) = supi∈I

(inf

g∈UeAi

(µAi(g))

)

(10) µξL(e) = supi∈I

|LeAi|

|X|where Ue

Ai⊆ E, Le

Ai⊆ E, i ∈ I are defined as

UeAi

= ej ∈ E|(ej)m ≥ em, ∀m ∈ AiLe

Ai= ej ∈ E|(ej)m ≤ em, ∀m ∈ Ai

We call ξL the lower bound of fuzzy set ξi and ξU serves as the upper bound offuzzy set ξ. When we get a new sample e ∈ U1 × U2 × · · · × Un, we calculate themembership degrees of e belonging to ξL

k , k = 1, 2, ..., c by (10) and select the classwith the maximum membership degree of e belonging to.

4. Experimental studies

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

memb

ership

degree

s of ea

ch class

the testing data of Breast−W(70)

class 1class 2

Figure 1. The best result of ten experiments on the Wisconsin Breastcancer data set


0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

memb

ership

degree

s of ea

ch class

the testing data of Breast−W(70)

class 1class 2

Figure 2. The worst result of ten experiments on the WisconsinBreast cancer data set

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

the testing data of Iris(15)

memb

ership

degree

s of ea

ch class

class 1class 2class 3

Figure 3. The best result of ten experiments on the Iris data set

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

memb

ership

degree

s of ea

ch class

the testing data of Iris(15)


Figure 4. The worst result of ten experiments on the Iris data set

In this section, three well-known data sets are used in our experiments. Theycome from the Machine Learning repository [10]. For each experiment data, tencomplete ten-fold cross validations were carried out, the cases were partitioned intoten equal-sized subsets with similar class distributions. Each subset in turn wasthen used as test data for the AFS decision tree inference systems generated fromthe remaining nine subsets. In all our simulations, all missing attribute values werereplaced by the averages of the corresponding attributes.

4.1. Experiment 1. Wisconsin Breast cancer diagnosis problem: The data setconsists of 699 patterns which are classified two classes, each example is described by


0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

memb

ership

degree

s of ea

ch class

the testing data of Wine(18)


Figure 5. The best result of ten experiments on the Wine data set

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

memb

ership

degree

s of ea

ch class

the testing data of Wine(16)


Figure 6. The worst result of ten experiments on the Wine data set

nine features. In the data set, the values of the sixth feature of 16 examples are miss-ing. E is the training data set, each of the nine attributes Vi is fuzzify into two fuzzyterms. The set of fuzzy terms for attribute Vi is Di = vi

small, vilarge, and the set of

fuzzy terms for the decision variable (class attribute) is Dc = vcClass1, v

cClass2. Let

M = m1,m2, ..., m20 be the set of simple concepts on U . Where m2i−1 = vismall

with the semantic meaning “the value on Vi is small”, m2i = vilarge with the seman-

tic meaning “the value on Vi is large”(i = 1, 2, · · · , 9) and m19 = vcClass1,m20 =

vcClass2. Now, we can establish the AFS structure (M, τ, E), where τ is defined by

(3).The tree under threshold δ = 0.36 have 171.3 (average) leaf nodes, so we extract

a rule-base with 171.3 rules. The average classification rate is 97.41% on trainingset, then, we prune the rule-base by the above algorithm. The rule-base reducesto 15.3 rules(average), for this case, the percentage of correct classification by therule-base becomes 97.71% (average) on training set. The percentage of correctclassification of each experiment is shown in table 2, and the membership degreesof testing samples belonging to each class are shown in Fig.1 (the highest accuracyrate) and Fig.2 (the lowest accuracy rate).

Table 2 The percentage of misclassification of experimentExperiment i 1 2 3 4 5 6 7 8 9 10

Error (%) 7.14 2.86 1.43 1.43 8.57 4.29 4.29 4.29 4.29 1.45

4.2. Experiment 2. Iris Plants Database: The data set consists of 150 patternswhich are classified three classes, each example is described by four features, each


of the four attributes Vi is fuzzify into three fuzzy terms. The set of fuzzy terms forattribute Vi is Di = vi

small, vimid, v

ilarge, and the set of fuzzy terms for the decision

variable (class attribute) is Dc = vcClass1, v

cClass2, v

cClass3. Let Let m3i−2 = vi

small

with the sematic meaning “the value on Vi is small”, m3i = vilarge with the sematic

meaning “the value on Vi is large”, m3i−1 = vimid with the sematic meaning “the

value is closer to the mediacy on Vi” (i = 1, 2, 3, 4) and m13 = vcClass1,m14 =

vcClass2,m15 = vc

Class3. Thus, the set of all fuzzy terms is M = Dc ∪ (4⋃

i=1

Di) =

m1,m2, · · · ,m12,m13,m14,m15, E is the training data, and the AFS structureis (M, τ, E), where τ is defined by (3).

The tree with the value of threshold δ = 0.75, have 3 (average) leaf nodes, so weextract a rule-base with 3 rules. The average classification rate is 96.67% on trainingset, then, we prune the rule-base. After that, the percentage of correct classificationby the rule-base becomes 96.81% (average) on training set. The membership degreesof the testing samples belonging to each class are shown in Fig.3 (the highestaccuracy rate) and Fig.4 (the lowest accuracy rate).

4.3. Experiment 3. Wine recognition data: The data set consists of 178 pat-terns which are classified three classes, each example is described by thirteen fea-tures. E is the training data, each of the thirteen attributes Vi is fuzzify into threefuzzy terms. The set of fuzzy terms for attribute Vi is Di = vi

small, vimid, v

ilarge,

and the set of fuzzy terms for the decision variable (class attribute) is Dc =vc

Class1, vcClass2, v

cClass3. Let m3i−2 = vi

small with the sematic meaning “the valueon Vi is small”, m3i = vi

large with the sematic meaning “the value on Vi is large”,m3i−1 = vi

mid with the sematic meaning “the value is closer to the mediacy on Vi”(i = 1, 2, · · · , 13) and m40 = vc

Cclass1,m41 = vcClass2,m42 = vc

Class3. Thus, the set

of all fuzzy terms is M = Dc ∪ (13⋃

i=1

Di) = m1, m2, · · · ,m39,m40,m41,m42. Now,

we can establish the AFS structure (M, τ,E), where τ is defined by (3).The tree is done with the value of threshold δ = 0.63, and have 14.8 (average)

leaf nodes, so we extract a rule-base with 14.8 rules. The average classification rateis 96.07% on training set, After pruning the rule-base, the rules reduce to 6.8 andthe percentage of correct classification by the rule-base becomes 97.69% (average)on training set. The membership degrees of the testing samples belonging to eachclass are shown in Fig.5 (the highest accuracy rate) and Fig.6 (the lowest accuracyrate).

5. Conclusion

The performance of the rule-base extracted from AFS-decision tree is shown intable 3. Comparing with the case Not Pruned, the pruning of the rule-base greatlyimproves the performance of the rule-base .

Table 3 Performance of the rule-base on different data setsData sets Error : Training data Error : Testing data Number of rules

Not Pruned Pruned Not Pruned Pruned Not Pruned Pruned% % % %

Breast-W 97.41 97.71 95.71 95.99 171.3 15.3

Iris 96.67 96.81 96.00 96.00 3 3

Wine 96.07 97.69 93.26 95.51 14.8 6.8


References

[1] J. E, Graver, M. E.Watkins, Combinatorics with Emphasis on the Theory of Graphs. NewYork, inc.:Springer-Verlag, 1977.

[2] C. Z. Janikow, Fuzzy decision trees: Issues and Methods, IEEE Transactions on Systems,Man, and Cybernetics-Part B: Cybernetics 28(1) (1998) 1-14.

[3] K. H. Kim, Boolean matrix Theory and Applications, Inc.: Marcel Dekker, 1982.[4] T. Kohonen, Self-Organization and Associative Memory, Berlin, Germany: Springer-Verlag,

1989.[5] X. D. Liu, W. Pedrycz, The Development of Fuzzy Decision Trees in the Framework of

Axiomatic Fuzzy Set Logic, Applied Soft Computing, 2007,7:325-342.[6] X. Liu , The Fuzzy Theory Based on AFS Algebras and AFS Structure, J. of Mathematical

Analysis and Applications, 217(1998):459-478.[7] X. Liu, The Fuzzy Sets and Systems Based on AFS Structure, EI Algebra and EII algebra,

Fuzzy Sets and Systems, 95(1998):179-188.[8] X. Liu, W. Wang, T. Chai, The Fuzzy Clustering Analysis Based on AFS Theory, IEEE

Transactions on Systems, Man and Cybernetics, Part B, 35(5)(2005):1013-1027.[9] X. Liu, T. Chai and W. Wang, Approaches to the Representations and Logic Operations for

Fuzzy Concepts in the Framework of Axiomatic Fuzzy Set Theory I, II, Information Sciences,177(2007):1007-1026,1027–1045.

[10] C. J. Merz and P. M. Murphy. (1996) UCI Repository for Machine Learning Data-Bases.Dept. of Information and Computer Science, University of California, , Irvine, CA. [Online].Available: http://www.ics.uci.edu/ mlearn/MLRepository.html

[11] M. Mizumoto, Fuzzy controls under various fuzzy reasoning methods, Inf. Sci. 45 (1988)129–151.

[12] W. Pedrycz, G. Vukovich, logic-oriented fuzzy clustering, Pattern Recognition Letters, 23(2002) 1515–1527.

[13] W. Pedrycz, Zenon A. Sosnowski, The design of decision trees in the framework of granulardata and their application to software quality models, Fuzzy Sets and Systems. 123 (2001)271–290.

[14] J. R. Quinlan, Decision trees as probabilistic classifiers, in Proc. 4th Int. Workshop MachineLearning, 1987, pp. 31–37.

[15] J. R. Quinlan, C4.5: Programs for Machine Learning. San Mateo, CA: Morgan Kaufmann,1993.

[16] S. R. Safavian and D. Landgrebe, A survey of decision trees classifier methodology, IEEETransactions on Systems, Man, and Cybernetics 21(1991) 660–674.

[17] C. Schaffer, Overfitting avoidance as bias, Mach. Learn., 10 (1993) 153–178.[18] S. Sestino and T. Dillon, Using single-layered neural networks for the extraction of conjunctive

rules and hierarchical classifications, J. Appl. Intell., 1(1991) 157–173.[19] G. J. Wang, Theory of topological molecular lattices, Fuzzy Sets and Systems, 47 (1992)

351–376.[20] X. Z. Wang, D. S. Yeung and E. C.C. Tsang, A comparative study on heuristic algorithms for

generating fuzzy decision trees, IEEE Transactions an Systems, Man, and Cybernetics–PartB: Cybernetics, 31(2) (2001) 215–226.

[21] D. S. Yeung, X. Z.Wang, and E. C. C. Tsang, Learning weighted fuzzy rules from exampleswith mixed attributes by fuzzy decision trees, in Proc. IEEE Int. Conf. on Systems, Man,and Cybernetics, Tokyo, Japan, Oct. 12–15, 1999, pp. 349–354.

[22] L. A. Zadeh, Fuzzy sets, Inf. Contr., 8(1965) 338–353.[23] L. A. Zadeh, Fuzzy logic and approximate reasoning, Synthese, 30 (1975) 407–428.

Dalian Maritime University, Dalian 116026, P. R. ChinaE-mail : feng [email protected]


IMAGE ENCRYPTION BASED ON REVERSIBLECELLULAR AUTOMATA

ZHIHUA FENG, YUNJIE ZHANG, AND YONGCHAO LIU

Abstract. Internet and multimedia technology develop more and more fast,

image encryption become important to ensure information security. This pa-

per presents a new approach for image encryption based on one dimensional

reversible cellular automata (RCA). The encryption method can encrypt im-

ages with higher security because RCA rules and randomness of the data are

used in the process of encryption. When using RCA to encrypt, it converts the

given gray level image into binary image and rearranges the data as a sequence

of 0-1 data. Simulation results for gray level images show that the proposed en-

cryption method satisfies the properties of confusion and diffusion, and almost

perfect guess of encryption key makes decryption impossible, this encryption

method works well as its expectation.

Key Words. Cellular Automata (CA), Reversible Cellular Automata (RCA),

Image, and Encryption.

1. Introduction

With the rapid spread of Internet applications and multimedia applications, im-ages are used more and more widely in our daily life. Because of the easiness ofimage copying or downloading from Internet, security has become an importantissue in communication and storage of images, and encryption is one of the waysto ensure security. Image encryption has applications in Internet communication,multimedia systems, medical imaging, telemedicine, and military communication,etc.. There already exit several image encryption methods, they include SCANbased methods, chaos based methods, three structure based methods, MPEG basedmethods, and other miscellaneous methods[1,2,3,4].

Reversible cellular automata (RCA) is used on image encryption. There aremainly one dimensional RCA and two dimensional RCA. On two dimensional RCA,its computation-universality can be proved by embedding universal logic elements.But two dimensional RCA is larger state and uses too much computation andmemory space. On one dimensional RCA, it’s security can meet image encryptioncommunication and storage, satisfies both confusion and diffusion properties. Theone dimensional RCA shows that the method is flexible and reliable, and has theadvantage of small calculation amount and less computer storage requirements.

In this paper, we proposed a new encryption method based on one dimensionalRCA, data privacy, data integrity and non-repudiation are considered in this en-cryption algorithm. One dimensional RCA can ensure higher security of images inthe process of communication and storage.

One dimensional RCA needs two sequences of data composed of 0 or 1 as theinitial configuration, and after forward n-1 steps of iterations according to some

Received by the editors January 1, 2007 and, in revised form, March 22, 2007.

178

IMAGE ENCRYPTION BASED ON REVERSIBLE CELLULAR AUTOMATA 179

RCA evolution rule the n-th configuration is formed. When using RCA to encrypt,we convert the given gray level image into binary image and rearrange the data as asequence of 0-1 data, then the rearranged data together with a sequence of randomdata form the RCA initial configuration. After forward steps of iterations accord-ing to some RCA evolution rule the final data formed. Then we convert the finaldata into decimal data and show the data as a gray level image what is encrypted.The key must be known to both sender and receiver before the communication ofimage[5,6,7].

This paper is organized into four sections: Section 2 presents a brief introductionto RCA. Section 3 describes the RCA based image encryption method and givesthe simulation results. Section 4 presents conclusions about the proposed imageencryption method.

2. Reversible Cellular Automata (RCA)

2.1. Cellular Automata(CA). A one dimensional CA consists of a line of cells,with each cell carrying a value 0 or 1. The value αi of a cell at each position i isupdated in discrete steps according to an identical deterministic rule depending ona neighborhood of cells around it:

(1) α(t+1)i = φ(α(t)

i−r, α(t)i−r+1, ..., α

(t)i−1, α

(t)i , α

(t)i+1, ... , α

(t)i+r).

where α(t)i is the value of the i -th cell (the state of a cell) in step t ; r is a radius

of the neighborhood, the neighborhood size is 2r+1; and there are 22r+1 possibleconfigurations of the neighborhood; φ is the evolution rule of the CA. This meansthat the total number of rules with the neighborhood of radius r is 222r+1

. So there

Figure 1. CA rule.

are 223= 256 radius one rules. Rules are usually named using standard convention

invented by Wolfram, radius one rule definition is shown on the following formula:

(2) (β1, β2, β3, β4, β5, β6, β7, β8)bin = (8∑

i=1

βi28−i)dec.

An example of radius one rule definition is shown on Figure 1.According to above (2), we can get result:

(3) 0× 27 + 0× 26 + 0× 25 + 1× 24 + 1× 23 + 1× 22 + 1× 2 + 0× 20 = 30.

so the decimal rule number is 30.In this example we can see that the neighborhood size is 3 and there are 8 possible

configurations of the neighborhood. For each configuration of the neighborhood,a state of the central cell in the next step is defined. For example, if the state ofsome cell is 1 and the state of its left and right neighbor is 1, then the next state

180 Z. FENG, Y. ZHANG, AND Y. LIU

of the cell is 0. The next state values defined for each possible configuration of theneighborhood form a binary number 00011110 (decimal 30). When dealing withfinite CA, cyclic boundary conditions are usually applied, which means that CAcan be treated as a ring. When given the initial configuration of CA, by updatingvalues in all cells, it can be transformed into a new configuration according to someCA rule. Changing values of all cells by the defined rule step by step are called CAiterations[8,9].

2.2. Reversible Cellular Automata (RCA). A CA is reversible, if its globalmap is one-to-one, that is if every configuration not only has one successor butalso has one predecessor. When analyzing the original CA, it turns out that onlya small number of rules have property of being reversible. Among all 256 CAs ofradius one only six are reversible. This is why we consider classic of CA with rulesspecially created to be reversible.

Wolfram represented RCA in which the new state of a cell is determined not onlyby the cell itself and its neighbors one step back but by the cell two steps back. Forα

(t+1)i = φ(α(t)

i−r, α(t)i−r+1, ..., α

(t)i−1, α

(t)i , α

(t)i+1, ..., α

(t)i+r), in the original CA value

α(t+1)i of i -th cell in configuration t+1 depends on its state and the states of its

r neighbors in configuration t. In RCA the value of the central cell α(t−1)i in step

t-1 is considered. Although those RCA are named following the original CA byWolfram, the RCA are not the invert rules of the original ones. In fact those RCAare rather new kind ones having the two steps back cell as an additional neighborthat affects the current state of the cell. In the strict sense, those RCA are timereversal invariant of which rule has built-in invertibility. Example of such a ruledefinition is shown on Figure 2, the decimal rule number is 30R.

Figure 2. RCA rule 30R.

We can note that the Rule 30R does exactly the same behavior when the cellwas 0 two steps back, whereas state of new cell is flipped when the cell itself was1 two steps back. Since a RCA rule depends now on two steps back, an initialconfiguration must be composed of two successive configurations. To run a RCAbackward, we only need to copy two steps back cells to the next step. The cellularautomaton will rebuild the initial data as shown on Figure 3.


Figure 3. Run rule 30R in reverse.

3. The proposed RCA based image encryption

3.1. RCA for encryption. Reversible rules meet the following criteria: they arenumerous and they exhibit complex behavior, so they can be used in encryption.When using RCA, first we set up a block of pseudo random data as configurationC0 and the data to be encrypted as configuration C1. Both C0 and C1 form initial

Figure 4. Encryption using RCA.


configuration of CA. Encryption is done by forward iterations of CA by fixed num-ber of steps according to some reversible rule. This process is shown in Figure 4.The configuration Cn is the encrypted final data.

Decryption is simply a backward iteration of CA with initial configuration com-posed of final data and ciphertext.

The same rule is used as in encryption. Decryption process is shown in Figure5. We can acquire plaintext by n-1 steps iteration backward.

Figure 5. Decryption using RCA.

The rule used both in encryption and decryption forms a secret key. The randomconfiguration is used as a part of the initial configuration of CA, whenever thesame message is encrypted using the same rule, different ciphertext and final dataare produced. Using RCA to encrypt the message has a very important propertythat changing value of one randomly chosen bit in the plaintext or in the keyproduces change of nearly half of the ciphertext and final data after certain numberof iterations, RCA in encryption can ensure the messages greater security.

3.2. RCA-based image encryption. Assuming a gray level image to be en-crypted and decrypted, its data may be denoted by a n× n matrix[10,11,12] :

(4)

a11 a12 · · · a1j · · · a1n

a21 a22 · · · a2j · · · a2n

......

......

ai1 ai2 · · · aij · · · ain

......

......

an1 an2 · · · anj · · · ann

.

The value of aij is between decimal 0-255, and can be expressed by 8 bit binary.In order to use one dimensional RCA, we convert (4) into one 1× n2 matrix:

(5)(

a11 a21 · · · an1 a12 a22 · · · an2 · · · a1n a2n · · · ann

).

The RCA-based image encryption is shown in Figure 6.


Figure 6. RCA based image encryption.

From Figure 6 we can see the image encryption method clearly. When given ann×n original gray level image that to be encrypted, first we convert every decimalvalue into binary value, second we rearrange the binary value as a string composedof 0 or 1 and the length of the string is n× n× 8. The string is named C1. At thesame time we generate a sequence of random data composed of 0 or 1. The randomdata can be looked as a string with the same length of C1 and named C0. Both C0

and C1 form the initial configuration of RCA. After forward n-1 steps of iterationsaccording to some reversible rule, the final data Cn−1 and Cn are formed. Thenwe convert the final data Cn composed of 0 or 1 into n × (n× 8) binary data. Atlast we turn the binary data into n × n decimal data and show the figure that isthe encrypted image.

Figure 7 shows the RCA based image decryption process. The decryption issimply a backward iteration of encryption. Let the final data and the ciphertextin encryption as the initial data of the decryption. The same rule is used as inencryption.

The rule used both in encryption and decryption forms a secret key. The advan-tage of the method is that it has higher security. However, it has the disadvantagethat if the sender and receiver forget the secret key, the image will not be restoredforever even a part of the image.

3.3. Simulation results. Various properties of the proposed RCA-based imageencryption method that includes confusion and diffusion properties is tested[13,14].Note that in all the following experiments, all images are of size 256× 256. Figure


Figure 7. RCA based image decryption.

8(a) shows Lena image that is used for testing the performance of the proposedRCA-based image encryption method. We generated a sequence of random 0-1data formed the initial data, then used RCA rule 30 as the key. After 13 stepsevolution iterations and convert between binary data and decimal data, we get theencrypted image of Lena what is show in Figure 8(b).

Figure 8. (a) Lena image. (b) Encrypted Lena image.

Histograms of Lena image (Figure 9(a)) and the encrypted Lena image (Fig-ure 9(b)) show that the encrypted image gets uniformly distributed pixels. Thefact illustrates that the proposed RCA-based image encrypted method satisfies theconfusion property. The encrypted image performs the process of decryption to


Figure 9. (a)Histogram of image. (b)Histogram of Encrypted image.

Figure 10. Encrypted Lena image to test diffusion property.

produce the decrypted image, and the decrypted image is exactly identical to theoriginal Lena image.

In order to explain security of RCA image encryption method, we also generateanother sequence of 0-1 random data and use the same key to encrypt the Lenaimage. Figure 10 shows the two encrypted images have no similarities even thoughtheir original images are same. So the RCA-based method satisfies the diffusionproperty.

4. Conclusions

This paper presented a new method for image encryption based on RCA. Theproposed encryption method is loss less encryption of image. Using this encryptionmethod, number of possible of security keys is large, confusion and diffusion prop-erties are satisfied, and almost perfect guess of encryption key makes decryptionimpossible. RCA image encryption has capability to encrypt large blocks of anyone dimensional digital data. The encryption uses only integer 0-1 arithmetic andit can be easily implemented in hardware and computer.

References

[1] Chuang, T., Lin, J., A New Multiresolution Approach To Still Image Encryption, PatternRecognition Image Anal, 9 (3), 1999.

[2] Scharinger, J., Fast Encryption Of Image Data Using Chaotic Kolmogorov Flows, ElectronicImaging, 17, 1998.

[3] Chuang, T., Lin, J., New Approach To Image Encryption, Electronic Imaging, 4, 1998.


[4] He, M. Z., Cai, L. Z., Liu, Q., et., Multiple Image Encryption And Watermarking By RandomPhase Matching, Optics Communications, 247, 2005.

[5] Lukac, R., Plataniotis, K. N., A Cost-Effective Encryption Scheme For Color Images, Real-Time Imaging, 11, 2005.

[6] Ateniese, G., Blundo, C., Visual Cryptography For General Access Structure, Informationand Computation, 129(2), 1996.

[7] Lukac, R., Plataniotis, K. N., Image Representation Based Secret Sharing, Communicationsof the CCISA, 11, 2005.

[8] Lin, C.C., Tsai, W. H., Visual Cryptography For Gray-Level Images By Dithering Techniques,Pattern Recognition Letters, 24 (1-3), 2003.

[9] Lukac, R., Plataniotis, K. N., Cost-Effective Encryption Of Natural Image, Proceedings of22nd Biennial Symposium on Communications, 2004.

[10] Chang, H., Liu, J., A Linear Quadtree Compression Scheme For Image Encryption, SignalProcess: Image Commune, 10 (4), 1997.

[11] Chang, L.,et., Encrypting Of Binary Images With Higher Security, Pattern Recognition Let-ters, 19(5), 1998.

[12] Yang, C. N., Chen, T. S., Aspect Ratio Invariant Visual Secret Sharing Schemes With Mini-mum Pixel Expansion, Pattern Recognition Letters, 26(2), 2005.

[13] Chen R. J., Lai, Y. T., Lai, J. L., Image Encrypted System Using Scan Patterns And 2-DCellular Automata, The IEEE Asia-Pacific Conference on Circuits and Systems, 2004.

[14] Maniccam, S. S., Bourbakis, N. G., Image And Video Encryption Using Scan Patterns,Pattern Recognition, 37, 2004.

Department of Mathematics, Dalian Maritime University, Dalian, 116023, China. School ofMathematics and Physics, Dalian Jiaotong University, Dalian, 116028, China


URL: http://www.dlmu.edu.cn

Department of Mathematics, Dalian Maritime University, Dalian, 116023, ChinaE-mail : [email protected] and [email protected]

URL: http://www.dlmu.edu.cn


H∞ CONTROL FOR PARAMETER UNCERTAIN T-S FUZZY SYSTEMSWITH TIME-DELAY IN STATE AND CONTROL INPUT

LI LI1, WEI LIU2 AND XIAODONG LIU1

Abstract. This paper focuses on the problem of H∞ fuzzy control for a class of

nonlinear uncertain delay systems in both state and control input through Takagi-Sugeno

(T-S) fuzzy model approach. Sufficient conditions for the existence of H∞ state feedback

controller for the systems are presented on the basis of the linear matrix inequalities(LMIs)

approach. Design examples of robust H∞ controller are given to illustrate the effectiveness

of approaches proposed in this paper.

Keywords: H∞ control; State feedback; Time-delay; Uncertain; LMI

Key Words.

1. Introduction

With the development of fuzzy systems, some fuzzy control design methods haveappeared in the field of fuzzy control. Among various kinds of fuzzy control methods, Takagiand Sugeno (T-S) [1] proposed a design and analysis method for overall fuzzy systems, inwhich the qualitative knowledge of a system is represented by a set of local T-S fuzzy models.Local dynamics in different state-space regions is represented by linear models and the overallmodel of the system is represented as the merge of these linear models. Therefore, it has aconvenient dynamic structure such that some well-established linear systems theory resultscan be applied for the theoretical analysis and design of the overall closed-loop controlledsystem. The idea is that for each local linear model, a linear feedback control is designedand the resulting overall controller, which is nonlinear in general, is fuzzy blending of eachindividual linear controller. Up until now, a lot of research results on T-S fuzzy systemshave been reported. Particularly in recent years, linear matrix inequality (LMI) based designapproaches for T-S fuzzy models have been developed [2],[3], [4], [5], [6].

There are two major fields: one is to design a robust controller for a system withtime-delay norm-bounded uncertainty [5]; the other is to design an H∞ controller for asystem with disturbance attenuation within a prescribed level [7]. Since the uncertaintiesand time-delays are frequently a source of instability and encountered in various engineeringsystems such as chemical processes, long transmission lines in pneumatic systems. Thestudy of uncertain or time-delay systems have received considerable attention over thepast years [8]. Many researchers have studied on the problem of robust H∞ time-delaycontrol with parameter uncertainty. But not only their works but also other results wereconservative in pre-determination of some starting values determined whether there is apositive-definite solution, and were not considered uncertain and time-delay in both state andcontrol input. Currently, the delay systems in state and control input have been reported [6].However they did not deal with robust H∞ control with parameter uncertain delay systemsin both state and control input. For a linear uncertain system with time-varying delay inall states and control inputs, it is more complicated to obtain the controllers. Therefore,our result deal with controller design methods of more generalized uncertain time-delaysystems. Furthermore, It is shown that the existence of time-delays as well as parameteruncertainties are frequently the main cause of deterioration of system performance andinstability of systems.

187

188 LI LI1, WEI LIU2 AND XIAODONG LIU1

In this paper, the problem of robust H∞ fuzzy control designs for T-S fuzzy uncertainsystems with time-delay in both state and control is addressed. Attention is focused on thedevelopment of a time-delay approach for designing fuzzy state feedback controllers. Boththe problems of robust stabilization and robust H∞ control are investigated. As for therobust H∞ stabilization problem, the controllers are required to guarantee the asymptoticstability of the resulting closed-loop system for any delay smaller than a given bound. Con-trol design methods in the sense of time-delay stability are developed for solving the robustcontrol problems, and all results are presented in terms of LMI form.

2. Problem formulation and preliminaries

Consider a nonlinear system with time-delay that can be represented by thefollowing fuzzy model:

Plant Rule i :IF θ1(t) is Mi1, · · · ,and θl(t) is Mil,Then

.x(t) = (Ai + ∆Ai)x(t) + (Aid + ∆Aid)x(t− τ1) + (Bid + ∆Bid)u(t− τ2) + Bwiw(t)(1)z(t) = Dix(t) + D1iu(t) + D2iw(t)x(t) = ϕ(t), t ∈ [−τ, 0], i = 1, 2, · · · , r

where x ∈ Rn and u ∈ Rm are the state and control input,respectively; Ai, Aid, Bi, Bid, Bwi

are constant real matrices with appropriate dimensions; r is the number of plant rules; θi(t)and Mij are, respectively, the premise variables and the fuzzy sets. It is assumed that thepremise variables are independent of the input variables u(t). τ1, τ2 > 0 is a real positiveconstant representing the time delay. The matrices ∆Ai,∆Aid,∆Bi and ∆Bid denote theuncertainties in system and they are of the form

[∆Ai, ∆Aid, ∆Bid] = MF (t)[Nia, Ni1, Ni2]

where M,Nia, Ni1, Nib, Ni2 are known constant matrices and F (t) is an unknown matrixfunction with the property FT (t)F (t) ≤ I. For the simplicity, let us introduce the followingnotations:−Ai = Ai + ∆Ai,

−Aid = Aid + ∆Aid,

−Bid = Bid + ∆Bid, hi(θ(t)) = hi, hj(θ(t− τ2)) = hτ2

j .

By fuzzy blending the overall fuzzy model is inferred as follows:

.x(t) =

r∑

i=1

hi(θ(t))[−Aix(t) +

−Aidx(t− τ1) +

−Bidu(t− τ2) + Bwiw(t)]

z(t) =r∑

i=1

hi(θ(t))[Dix(t) + D1iu(t) + D2iw(t)](2)

where θ(t) = [θ1(t), · · · , θl(t)]T , µi : Rl → [0, 1], i = 1, 2, · · · , r is the membership functionof the system with respect to the ith plant rule, and hi(θ(t)) = µi(θ(t))

r∑i=1

µi(θ(t)). In this paper,

we assume that :µi(θ(t)) ≥ 0 for i = 1, 2, · · · , r andr∑

i=1

µi(θ(t)) > 0 for all t. Therefore,

hi(θ(t)) ≥ 0 (i = 1, 2, · · · , r) andr∑

i=1

hi(θ(t)) = 1.

Based on the parallel distributed compensation, the following fuzzy control law isemployed to deal with the problem of stabilization via state feedback.

Control Rule i :IF θ1(t) is Mi1, · · · ,and θl(t) is Mil,then

u(t) = Fix(t), i = 1, 2, · · · , r.

Hence, the overall fuzzy control law is represented by

H∞ CONTROL FOR PARAMETER UNCERTAIN T-S FUZZY SYSTEMS 189

(3) u(t) =r∑

i=1

hi(θ(t))Fix(t) , i = 1, 2, · · · , r.

where Fi(1, 2, · · · , r) are the local control gains. The aim is to determine the local feedbackgains Fi such that the closed-loop system (6) is stable

.x(t) =

r∑

i=1

r∑

j=1

hihτ2j [

−Aix(t) +

−Aidx(t− τ1) +

−BidFjx(t− τ2) + Bwiw(t)]

z(t) =r∑

i=1

r∑

j=1

hihj [(Di + D1iFj)x(t) + D2iw(t)](4)

3. Main results

In the sequel,we will present our main results on robust stabilization and robustH∞ controller. Before stating our main results, we first give one lemma which will be usedin the proofs of our results. For system (4), choose Lyapunov function as follows:

(5) V (x) = xT (t)Px(t) +∫ t

t−τ1

xT (s)R1x(s)ds +∫ t

t−τ2

xT (s)PR2Px(s)ds

where P, R1, R2 are positive symmetric definite matrices with appropriate dimensions.To address time-delay stability analysis, the following lemma is first presented.

Lemma 3.1. [9] (1) For any x, y ∈ Rn×n and for any positive symmetric definite matrixP ∈ Rn×n, the following inequality holds

(6) 2xT y ≤ xT P−1x + yT Py

(2) Let A, D, E and F be real matrices of appropriate dimensions with ‖F‖ ≤ 1, Thenthe following inequations hold

(a) For any scalar ε > 0

(7) DFE + ET FT DT ≤ ε−1DDT + εET E

(b) For any matrix P = PT > 0 and scalar ε > 0 such that εI − EPET > 0

(8) (A + DFE)P (A + DFE)T ≤ APAT + APET (εI − EPET )−1EPAT + εDDT

(c) For any matrix P = PT > 0 and scalar ε > 0 such that P − εDDT > 0

(9) (A + DFE)T P−1(A + DFE) ≤ AT (P − εDDT )−1A + ε−1ET E

3.1. Time-delay robust stabilization. Initially, we consider the system (4) with w ≡0 and associated with the control law (3) then the resulting closed-loop system can beexpressed as follows:

(10).x(t) =

r∑

i=1

r∑

j=1

hihτ2j [

−Aix(t) +

−Aidx(t− τ1) +

−BidFjx(t− τ2)], i = 1, 2, · · · , r.

Thus, the time-delay stabilization result of T-S fuzzy model is summarized in the followingtheorem.

Theorem 3.1. There exists a fuzzy control law (3) such that the closed-loop fuzzy system(4) is quadratically stable if there exist matrices W1 > 0 , W2 > 0 , εi > 0 (i = 1, 2, 3), Mi,1 ≤ i ≤ r, and symmetry matrix X > 0, such that the following LMIs hold:

(11)

Ωii X Ξ13ii Ξ14ii

∗ −W1 0 0∗ ∗ Ξ33ii 0∗ ∗ ∗ Ξ44ii

< 0 i = j = 1, 2, · · · , r.


(12)

Ωij X Ξ13ij Ξ14ij

∗ −1/2W1 0 0∗ ∗ Ξ33ij 0∗ ∗ ∗ Ξ44ij

≤ 0, i < j = 1, 2, · · · , r

whereΞ13ii =

[XNT

ia ATid

], Ξ33ii = diag −ε−1

1 I Γ1 ,Ξ14ii =[

NTi1 Mi

T BTid MT

i NTi2

],

Ξ44ii = diag −ε2I Γ2 −ε3I ,Ξ13ij =[

XNTia XNT

ja ATid AT

jd

],

Ξ33ij = diag −ε−11 I −ε−1

1 I Γ1 Γ1 ,Ξ14ij =

[NT

i1 NTj1 Mj

T BTid Mi

T BTjd MT

i NTi2 MT

i NTi2

],

Ξ44ij = diag −ε2I −ε2I Γ2 Γ2 −ε3I −ε3I ,Γ1 = −(W1 − ε2MMT ), Γ2 = −(W2 − ε3MMT )Ωii = AiX + XAT

i + R2 + ε−11 MMT

Ωij = AiX + XATi + AjX + XAT

i + 2R2 + 2ε−11 MMT

If this is the case, the local feedback gains Fi are given by Fi = MiX−1, i = 1, 2, · · · , r.

Proof. Choose Lyapunov function (5) ,Then the derivative of Lyapunov function (5) alongthe closed-loop system (4) is given by.

V =.x

T (t)Px(t) + xT (t)P.x(t) + xT (t)R1x(t)− xT (t− τ1)R1x(t− τ1) + xT (t)PR2Px(t)

−xT (t− τ2)PR2Px(t− τ2)

= 2r∑

i=1

r∑

j=1

hihτ2j xT (t)P [

−Aix(t) +

−Aidx(t− τ1) +

−BidFjx(t− τ2)] + xT (t)R1x(t)

−xT (t− τ1)R1x(t− τ1) + xT (t)PR2Px(t)− xT (t− τ2)PR2Px(t− τ2)(13)

By Lemmma 1(1), we have

(14) 2xT (t)P−Aidx(t− τ1) ≤ xT (t)P

−AidR

−11

−A

T

idPx(t) + xT (t− τ1)R1x(t− τ1)

(15)

2xT (t)P−BidFjx(t− τ2) ≤ xT (t)P

−BidFjP

−1R−12 P−1FT

j

−B

T

idPx(t)+xT (t− τ2)PR2Px(t− τ2)

Thus, substituting (14) and (15) into (13) yields

.

V =r∑

i=1

r∑

j=1

hihτ2j xT (t)PAi + AT

i P + ∆ATi P + P∆Ai + P (Aid + ∆Aid)R−1

1 (Aid + ∆Aid)T P

+P (Bid + ∆Bid)FjP−1R−1

2 P−1FTj (Bid + ∆Bid)T P + R1 + PR2Px(t)(16)

By Lemmma 1(2) ,we have

(17) 2P∆Ai = 2PMF (t)Nia ≤ ε−11 PMMT P + ε1N

TiaNia

(Aid + ∆Aid)R−11 (Aid + ∆Aid)T = (Aid + MF (t)Ni1)R−1

1 (Aid + MF (t)Ni1)T

≤ ATid(R1 − ε2MMT )−1Aid + ε−1

2 NTi1Ni1(18)

(Bid + ∆Bid)FjP−1R−1

2 P−1FTj (Bid + ∆Bid)T

= (Bid + MF (t)Ni2)FjP−1R−1

2 P−1FTj (Bid + MF (t)Ni2)T

≤ (BidMj)T (R2 − ε3MMT )−1(BidMj) + ε−13 MT

j NTi2Ni2Mj(19)


Substituting (17),(18) and (19) into (16) ,we have

.

V ≤r∑

i=1

r∑

j=1

hihτ2j xT (t)PAi + AT

i P + R1 + PR2P + ε−11 PMMT P + ε1N

TiaNia

+PATid(R1 − ε2MMT )−1AidP + ε−1

2 PNTi1Ni1P + P (BidMj)T (R2 − ε3MMT )−1(BidMj)P

+ε−13 PMT

j NTi2Ni2MjPx(t)

=r∑

i=1

hihτ2i xT (t)∆iix(t) +

r∑

i=1

r∑

j=1

hihτ2j xT (t)∆ijx(t)

where

∆ii = PAi + ATi P + R1 + PR2P + ε−1

1 PMMT P + ε1NTiaNia + PAT

id(R1 − ε2MMT )−1AidP

+ε−12 PNT

i1Ni1P + P (BidMi)T (R2 − ε3MMT )−1(BidMi)P + ε−13 PMT

i NTi2Ni2MiP(20)

∆ij = PAi/2 + ATi /2P + PAj/2 + AT

j /2P + R1 + PR2P + ε−11 PMMT P + ε1/2(NT

iaNia + NTjaNja)

+1/2[PATid(R1 − ε2MMT )−1AidP + ε−1

2 PNTi1Ni1P + PAT

jd(R1 − ε2MMT )−1AjdP

+ε−12 PNT

j1Nj1P ] + 1/2[P (BidMj)T (R2 − ε3MMT )−1(BidMj)P + ε3PMTj NT

i2Ni2MjP

+P (BjdMi)T (R2 − ε3MMT )−1(BjdMi)P + ε−13 PMT

i NTj2Nj2MiP ](21)

Denote X = P−1, Mi = FiX, W1 = R−11 ,W2 = R2,and premultiply and postmultiply to

(20) and (21) by positive-definite matrix P−1, respectively. Then, by Schur complement,P−1∆iiP

−1 < 0 and P−1∆ijP−1 < 0 is equivalently changed to (11) and (12).This com-

pletes the proof. ¤3.2. Robust H∞ Control. In the section,we focus on the design of a time-delay robustH∞ control law for system (4).

Theorem 3.2. Consider the system (4).Given scalars υ > 0 ,this system is robustly stablewith disturbance attenuation υ for any constant time delay τ1, τ2 if there exist matricesW1 > 0, W2 > 0, εi > 0 (i = 1, 2, 3) and Mi, 1 ≤ i ≤ r, and symmetry matrix X > 0, suchthat the following LMIs hold:

(22)

Θii X Ψ13ij Ψ14ij

∗ −W1 0 0∗ ∗ Ψ33ij 0∗ ∗ ∗ Ψ44ij

< 0, i = 1, 2, · · · , r

(23)

Θij X Ψ13ij Ψ14ij

∗ −1/2W1 0 0∗ ∗ Ψ33ij 0∗ ∗ ∗ Ψ44ij

≤ 0, i < j = 1, 2, · · · , r

Ψ13ii =[

XNTia AT

id NTi1

], Ψ33ii = diag −ε−1

1 I Γ1 −ε2I ,Ψ14ii =

[Mi

T BTid MT

i NTi2 XDT

i + MTi DT

1i Λ2ii

],Ψ44ii = diag Γ2 −ε3I −I Λ3i ,

Ψ13ij =[

XNTia XNT

ja ATid AT

jd NTi1 NT

j1

],

Ψ33ij = diag −ε−11 I −ε−1

1 I Γ1 Γ1 −ε2I −ε2I ,Ψ14ij =

[Mj

T BTid Mi

T BTjd MT

j NTi2 MT

i NTj2 Λ1ij Λ1ji Λ2ij Λ2ji

],

Ψ44ij = diag Γ2 Γ2 −ε3I −ε3I −I −I Λ3i Λ3j , Θii = Ωii

Θij = (Ai + Aj)X + X(Ai + Aj)T + 2W2 + 2ε−11 MMT

Λ1ij = XDTi + MT

j DT1i, Λ1ji = XDT

j + MTi DT

1j ,Λ2ii = Biw + XDTi D2i + MT

i DT1iD2i,

Λ2ij = Biw + XDTi D2i + MT

j DT1iD2i, Λ2ji = Bjw + XDT

j D2j + MTi DT

1jD2j ,

Λ3i = −υ2I + D2iDT2i,Λ3j = −υ2I + D2jD

T2j

If this is the case, the local feedback gains Fi are given by Fi = MiX−1, i = 1, 2, · · · , r.


Proof. Consider the system (4) with the control law (3), Similar to the theorem 1, we have

(24).

V ≤r∑

i=1

r∑

j=1

hihτ2j [xT (t)Πx(t) + wT BT

wiPx(t) + xT (t)PBwiw(t)]

where

Π = PAi + ATi P + R1 + PR2P + ε−1

1 PMMT P + ε1NTiaNia + PAT

id(R1 − ε2MMT )−1AidP

+ε−12 PNT

i1Ni1P + P (BidMj)T (R2 − ε3MMT )−1(BidMj)P + ε−13 PMT

j NTi2Ni2MjP

let us introduce

J =∫ ∞

0

[zT (t)z(t)− υ2wT (t)w(t)]dt

Then for any nonzero w(t) ∈ L2[0,∞) and under zero initial condition, we have

J =∫ ∞

0

[zT (t)z(t)− υ2wT (t)w(t) +.

V (x(t))]dt− V (x(τ))

≤∫ ∞

0

[zT (t)z(t)− υ2wT (t)w(t) +.

V (x(t))]dt(25)

and further substituting (24) into (25)

J ≤∫ ∞

0

r∑

i=1

r∑

j=1

hihτ2j [xT (t)(Di −D1iFj)T (Di −D1iFj)x(t)− (υ2I −D2iD

T2i)w

T (t)w(t)

+xT (t)Πx(t) + wT (BTwiP + DT

2iDi + DT2iD1iKj)x(t) + xT (t)(PBwi + DT

i D2i

+FTj DT

1iD2i)w(t)]dt

let ζ(t) = [xT (t) wT (t)]T , then

(26) J ≤∫ ∞

0

ζ(t)T Σijζ(t)dt

where Σij is defined

(27) Σij =[

Π + (Di + D1iFj)T (Di + D1iFj) PBwi + DTi D2i + FT

j DT1iD2i

∗ −υ2I

]

Denote X = P−1, Mi = FiX, W1 = R−11 ,W2 = R2, and premultiply and postmultiply to

(27) by positive-definite matrix diag Q = [P−1, I], respectively. then(28)

QΣijQ =[

XΠX + (XDTi + MT

j DT1i)(DiX + D1iMj) Bwi + XDT

i D2i + MTj DT

1iD2i

∗ −υ2I + D2iDT2i

]

(29)

i.e.QΣiiQ =[

X∆iiX + (XDTi + MT

i DT1i)(DiX + D1iMi) Bwi + XDT

i D2i + MTi DT

1iD2i

∗ −υ2I + D2iDT2i

]

(30) QΣijQ =

Υij Bwi + XDTi D2i + MT

j DT1iD2i Bwj + XDT

j D2j + MTi DT

1jD2j

∗ −υ2I + D2iDT2i 0

∗ ∗ −υ2I + D2jDT2j

where

Υij = X∆ijX + (XDTi + MT

j DT1i)(DiX + D1iMj) + (XDT

j + MTi DT

1j)(DjX + D1jMi)

If Σij < 0, it implies J < 0. After some manipulation using Schur complement, (29)< 0 and(30)< 0 are equivalently changed to (22) and (23). So we have Σij < 0. It implies


(31) J ≤∫ ∞

0

ζ(t)T Σijζ(t)dt < 0

for any nonzero w(t) ∈ L2[0,∞) and all τ > 0. Through (31) ,we have ‖z(t)‖2 ≤ υ ‖w(t)‖2for any any nonzero w(t) ∈ L2[0,∞). Therefore, when Σij < 0 for t ≥ 0, the system (4) isquadratically stable.

This completes the proof. ¤

We would like to emphasize that, to the best of our knowledge, so far there has beenno results appeared in literature for the fuzzy controller design in case of input delay forrobust H∞ control system.

4. Example

In this section, we use three examples to illustrate the effectiveness and merits ofour results. Example 1 is selected from [10] which can be examined by Theorem 2. Example2 is selected from [5] in order to illustrate the more effectiveness of the design method inTheorem 2.

Example 4.1. Consider the following fuzzy system:

.x(t) =

2∑

i=1

hi[(Ai + ∆Ai)x(t) + (Aid + ∆Aid)x(t− τ1) + (Bid + ∆Bid)u(t− τ2) + Bwiw(t)]

z(t) =2∑

i=1

hi[Dix(t) + D1iu(t) + D2iw(t)]

with

A1 =[

1 −0.51 0

], A2 =

[ −1 −0.51 0

], A1d = A2d =

[0 −0.2

0.2 0

], B1d = B2d =

[0.20.1

],

Bw1 = Bw2 =[

0.30.2

], D1 = D2 =

[3 1

], D11 = D12 = 1, D21 = D22 = 0.5 ,

he uncertainties in the system can be modelled as

∆Ai = ∆Aid,= MF (t)N1 with M =[

0.30.1

], N1 =

[0.1 0.1

]

and

∆Bid = MF (t)N2 with M =[

0.30.1

], N2 = 0.1

whereF (t) = sin(t)

we takeh1 = sin2(x1 + 0.5) and h2 = cos2(x1 + 0.5), F (t) = sin(t),

The disturbance input asw(t) = 1.5sin(2t) ∗ e−0.05t

Then solving LMIs (40) and (41) for υ = 1 and σ = 0.5 gives the following feasible solution:

P =[

0.4960 0.21910.2191 0.4144

], F1 =

[ −3.3585 −1.2179], F2 =

[ −2.7864 −1.0265],

ε1 = 0.2184, ε2 = 5.8176, ε3 = 0.2145, ε4 = 4.7502


0 10 20 30 40 50 60−2

−1

0

1

2

3

4

Time(Sec)

stat

e

x1x2

Figure 1. The responses ofthe state

0 10 20 30 40 50 60−1.5

−1

−0.5

0

0.5

1

1.5

Time(Sec)

z(t)

and

w(t

)

w(t)z(t)

Figure 2. The responses ofz(t) and w(t)

The simulation is run for an initial function x(t) =[

2 3]T . The simulation results

are shown in Figs.1-2. Fig.1shows the response of the closed-loop system. In Fig.2, thecontrolled output and disturbance input are shown. It can be seen that with the fuzzy controllaw (3) the closed-loop system is robustly stable with disturbance attenuation υ = 1 .

5. Conclusion

In this paper, we have developed a robust fuzzy H∞ controller design method foruncertain nonlinear delay systems in both state and control input. As described by Takagiand Sugeno model with uncertainties. Based on the notion of quadratic stabilization withdelay rate and an L2 norm bound, sufficient conditions to solve the robust fuzzy H∞ controlproblem have been obtained and solutions have been produced in terms of LMIs. An exam-ples of robust fuzzy H∞ controllers for uncertain nonlinear systems is used to illustrate thedesign procedures.

References

[1] T.Takagi and M. Sugeno, ”Fuzzy identification of systems and itd applications to modeling and control,”IEEE Trans.Syst.Man.Cybern., Vol SMC-15.no.1,pp.116-132, Jan.1985.

[2] K.Tanska and H.O.Wang,”Fuzzy control systems design and analysis.” in S Linear Matrix InequalityApproach. New York: Wiley,2001.

[3] L.xiaodong and Z.Qingling. New approaches to H∞ controller designs based on fuzzy observers for T-Sfuzzy systems via LMI. Automatica, 39:1571-1582, 2003.


[4] L.xiaodong and Z.Qingling. Approaches to quadratic stability conditions and H∞ control designs forT-S fuzzy systems IEEE Trans. Fuzzy System 11(6)(2003) 830-839.

[5] Chen, B., Liu,X.P., Delay-dependent robust H∞ control for T-S fuzzy systems with time-delay. IEEETrans.Fuzzy System 13(4)(2005) 544-556.

[6] Lin, C., Wang,Q.G. and Lee, T.H., Delay-dependent LMI conditions for stability and Stabilization ofT-S fuzzy systems with bounded time-delay. Fuzzy sets and systems, in press.

[7] Doyle, J.c., K. Gliver, P.P. Khargonekar and B. A. Francis. State-space solutions to standard H2 andH∞ control problems IEEE Trans Automat. Control, AC-34(1989),831-847.

[8] L.Yu,J.Chu,H.Su,Robust memoryless H∞ controller design for linear time-delay systems with norm-bounded time-varying unertainty,Automatica 32(12)(1996)1759-1762.

[9] Cao, Y.Y., Sun, Y.X., &Lam,J . (1998). Delay dependetn robust H∞ control for uncertain systemswith time varying delays. IEEE proceedings: Control Terory and Applications, 143(3), 338-344.

[10] K.Tanaka and H.O. Wang, Fuzzy control systems design and analysis:a linear matrix inequality ap-proach. Canada(2001).

1Research Center of Information and Control, Dalian University of Technology, Dalian 116024 P. R.China

2 Department of Mathematics,, Dalian Maritime University, Dalian 116026 P. R. ChinaE-mail : [email protected]

INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCE

ADVANCES IN INFORMATION AND SYSTEMS SCIENCES Volume 2 Pages 196-200


FUZZY CLUSTERING STUDY ON THE DRIFTING SEA-ICE

IN THE NORTH OF LIAONING GULF

NAXIN CHEN AND YUNJIE ZHANG Abstract The impact of sea-ice is the most serious oceanic disaster in the north of Liaodong Gulf in China, where the tide is irregular semi-diurnal and the wind often occurs during the winter. The drifting floes, with a certain speed because of current and wind, threaten the safety of the engineering structures in the sea area. The degree of damage is related to the thickness , the area as well as the amount of drifting sea-ices。According to the average speed of drifting sea-ices and the change in wind direction in a 3-day cycle, we choose a typical research area of 34km×26km. After collecting a density dataset, obtained from a remote sensing image taken in a severer year for sea-ice condition, we applied The Fuzzy C-Means (FCM) clustering algorithm in the classification of the data set. And then the density was converted to the thickness of sea-ice. Finally, the area/amount/proportion of sea-ice, corresponding to various thicknesses, is stated according to different risks. And a method for assessing the risk of sea –ices in a large area is given in this paper.

Key Words, drifting sea-ices, FCM, clustering. 1. Introduction

Among various oceanic disasters, the impact of sea-ice is the most serious one in the north part of Liaodong Gulf. The sea-ice load of engineering structures is the designed governed load.

Besides the compression load, there are other forms of acting force, such as the vibrating load on flexible constructions, and the load from hummock on slope structures. In order to evaluate the impact from ice load, the thickness and the strength of sea-ice should be considered first. In fact, the area should be another important index because it governs kinetic energy[1]. With a limited speed, oscillating load does not occur if small floes don’t have enough energy, and neither does the basic load as a result of compression. Furthermore, if the thickness is no less than 26 cm[2], the failure of press and bending doesn’t occur, and the energy can be transferred because floes usually move together closely. Therefore, If the strength of floes is considered as a constant (design strength), evaluating the damage of sea-ice should concentrate on thickness and area. 2 . The collection of the data set

Under the condition of technology available, it is practical for us to obtain the data of sea-ice in a large sea area from a remote sensing image. According to the statistics, the winter, when the remote sensing image was taken, is the most serious one for sea-ice condition in the recent 20 years. Therefore, it is reasonable to use the data set collected from the large-scale image.

The increase in the speed of drifting floes should be 0.1m/s, as the wind velocity is 6.0m/s [3]. If the wind with a velocity of 9.0m/s lasts 48 hours, a floe can drift through a distance of 26km. According to the statistics obtained from an observation, the average speed of drifting sea-ice is 0.38m/s, so the drifting distance is about 8km in an irregular semi-diurnal sea area. As a result, we chose a field of 34km×26km as the research area. All the floes in this area may damage the engineering structures located at the boundary. Received by the editors June 30, 2006 196

FUZZY CLUSTERING STUDY ON THE DRIFTING SEA-ICE 197

Fig.1 The distribution map of sea ice in research area

In the Fig.1, the darkest small areas should be sea water, where the density of pixels is

32; On the other hand, the brightest ones should be the thickest sea-ice, where the density is 215. Based on the design thickness of 40cm[4] for an ice condition appeared once in 20 twenty years, the thickest sea-ice should be 40cm thick in this sea area. At the same time, According to an investigation conclusion, there are little level ices with a more than 30cm[2] thickness. Therefore, the thickness of sea ice located in the brightest plot could be 30 cm too.

After confirming the scope of sea-ice thickness, we applied The Fuzzy C-Means clustering algorithm in the classification of the density data set. After that, the density of pixels in each clustering center is converted to the thickness of sea ice according to a linearity formula. Finally, the amount of various types of sea-ice with different impact is provided. 3. Fuzzy C-Means Clustering

Given a set of data, Fuzzy C-Means clustering (FCMC) performs clustering by iteratively searching for a set of fuzzy partitions and the associated clustering centers that represent the structure of the data as best as possible. The FCMC algorithm relies on the user to specify the number of clusters present in the set of data to be clustered. Given the

number of clusters c, FCMC partitions the data into c fuzzy

partitions by minimizing the within group sum of squared error objective function as follows (eqn 3.1).

, ,2,1 xxx nX L=

∑∑ −= =

=n

k

c

i

m

m vxUJ ikikVU1 1

2

)(),( ∞≤≤ m1 eqn 3.1

where Jm(U,V) is the sum of squared error for the set of fuzzy clusters represented by

the membership matrix U, and the associated set of cluster centers V. • is some inner

product-induced norm. In the formula, VX ik−2

represents the distance between the

198 N. CHEN AND Y. ZHANG

data X k and the cluster center V i . The squared error is used as a performance index

that measures the weighted sum of distances between cluster centers and elements in the corresponding fuzzy clusters. The number m governs the influence of membership grades in the performance index. The partition becomes fuzzier with increasing m and it

is proven that the FCMC algorithm converges for any ),1( ∞∈m [5]. The necessary

conditions for eqn 3.1 to reach its minimum are Eqn 3.2

k i, c

1j v jxk

vixk1) 1

∀∀∑= −

−−2/(m

Uik

−

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎜⎝⎛

⎟⎠⎞

Eqn 3.2

And Eqn 2.3

( )( )∑

∑

=

== n

k

m

n

kk

m

ik

iki

U

xUv

1

1 Eqn 3.3

In each iteration of the FCMC algorithm, matrix U is computed using eqn 3.2 and the associated cluster centers are computed as eqn 3.3. This is followed by computing the square error in eqn 3.1. The algorithm stops when either the error is below a certain tolerance value or its improvement over the previous iteration is below a certain threshold. 4. The analysis on the result of clustering

According to the principle of FCMC, each class is made up the measurements which are closest to the center. We can use the corresponding thickness of the density in each clustering center to represent the measurements contained in each class. In addition, we also change the number of clusters to make the corresponding thicknesses of clustering centers to be the value with engineering characteristics, such as 10cm, 20cm and 26cm.

From some practical investigations, we found that very thin floes (with a thickness no more than 3cm) can’t damage structures and that it is difficult for them to form a hummock while piling up due to the lack of strength of itself. When colliding with structures, thin floes (with a thickness from 3cm to 10cm) is often cracked, meanwhile, the kinetic energy is consumed by the friction. However, thick floes (with a thickness no less than 26cm) can ceaselessly impact the engineering structures. In conclusion, the floes with different thickness damage structures in different ways. Therefore, the amount of various floes in the area is very important for the evaluation on the risk of sea-ice disaster. The results of clustering and some statistical data are given in Table 1-- table 4

Table 1 The list of clustering results and some statistical data

(as the thickest drifting ice is 40-cm-thick.)

Clusters Clustering centers

Thickness of sea-ice

Number of pixels

Area /km2

Percentage of the whole

Volumes /104m3

FUZZY CLUSTERING STUDY ON THE DRIFTING SEA-ICE 199

/cm area /% 1 49.39 3.80 447733 179.09 20.26 699.80

2 68.75 8.03 463588 185.44 20.98 1490.92

3 87.21 12.07 452289 180.92 20.47 2179.63

4 103.44 15.62 423332 169.33 19.15 2643.84

5 121.70 19.61 233913 93.57 10.58 1843.18

6 151.98 26.23 124215 49.69 5.62 1301.55

7 183.50 33.12 64930 25.98 2.94 857.94

Table 2 The list of large and thick drifting ices (as the thickest drifting ice is 40-cm-thick.)

No. Number of pixels

Average thickness/cm

Area /km2

Volume /104m3

1 22872 30.59 2.29 70.00

2 51512 28.03 5.15 144.39

3 22734 27.67 2.27 62.89

4 18371 27.17 1.84 49.91

Table 3 The list of clustering results and some statistical data

(as the thickest drifting ice is 30-cm-thick.)

Clusters Clustering centers

Thickness /cm

Number of pixels

Area /km2

Percentage ofthe whole area /%

Volumes /104m3

1 48.45 2.70 393958 157.58 17.83 435.44

2 63.10 5.10 285752 114.30 12.93 582.39

3 74.67 7.00 309390 123.76 14.00 868.10

4 88.58 9.27 363081 145.23 16.43 1344.24

5 101.27 11.36 334836 133.93 15.15 1519.09

6 114.38 13.51 246887 98.75 11.17 1335.73

7 133.80 16.69 132818 53.13 6.01 888.35

8 160.60 21.08 100553 40.22 4.55 847.92

9 189.70 25.85 42725 17.09 1.93 440.59

Table 4 The list of large and thick drifting ices (as the thickest drifting ice is 30-cm-thick..)

No. Number

of pixels

Average

thickness/cm

Area

/km2

Volume

/104m3

1 22872 22.94 2.29 52.50

200 N. CHEN AND Y. ZHANG

2 51512 21.02 5.15 108.29

3 22734 20.75 2.27 47.17

4 18371 20.37 1.84 37.43

Table 5 The list of sea ice with the different impact on offshore structures

Max of

thickness/cm risk Thickness

/cm Area /km2

Volume /104m3

Percentage of the whole

area / % light 3.80 179.09 699.80 20.26

severe 8.03～19.21

629.22 8157.57 71.18 40

severest 26.23～33.12

75.67 2159.49 8.56

lighter 2.70～5.10

271.88 1017.83 30.76

severer 7.00～20.18

595.02 6803.43 67.31 30

severest 25.85 17.09 440.59 1.93 5. Conclusion

In conclusion, under such an ice condition, the list of sea-ice with different impact on

structures is given in table 5. Further more, there exist some huge ice covers in the area, the area of the biggest one is 1.84km2~5.15 km2 and the thickness of the ices is 20~30cm.

REFERENCES [1] Li Zhijun, Wang Yongxue. Design criteria of Bohai sea ice-control engineering[J]. Ocean Engineering, 2000, 18(1): 61-64. [2] Li Zhijun, Ding Dewen, Sui Jixue, et al. Theoretical analysis on the rafted ice thickness in Liaodong Gulf[J]. Marine Environmental Science, 1997, 16(4): 21-25. [3] Su Jie, Wu Huiding, Liu Qinzhen, et al. A coupled ice- ocean model for the Bohai Sea I. Study on model and parameter [J]. Acta Oceanologica Sinica, 2005, 27(1):19-26. [4] Ding Dewen, et al. An Introduction Engineering Sea Ice [M]. Beijing: China Ocean Press, 1999: 188. [5] Chong, A.,Gedeon,T.D., Koczy, L.T.， A Hybrid Approach for Solving the Cluster Validity Problem, 14th International Conference on Digital Signal Processing, Volume 2, 1-3 July 2002. Department of Mathematics, Dalian Maritime University Dalian 116026, P. R. China [email protected]

ISSN 1916-176XISBN 978-1-55195-221-5

Printed in Canada 2007

©ISCI 2007

Information, Optimizations and Systems Controls in Engineering

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