EditorialNetworked Systems with Incomplete Information

Zidong Wang,1 Bo Shen,2 Hongli Dong,3,4 Xiao He,5 and Jun Hu6

1Department of Computer Science, Brunel University London, Uxbridge, Middlesex UB8 3PH, UK2School of Information Science and Technology, Donghua University, Shanghai 200051, China3College of Electrical and Information Engineering, Northeast Petroleum University, Daqing 163318, China4Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, China5Department of Automation, Tsinghua University, Beijing 100084, China6Department of Applied Mathematics, Harbin University of Science and Technology, Harbin 150080, China

Correspondence should be addressed to Zidong Wang; zidong.wang@brunel.ac.uk

Received 25 December 2014; Accepted 25 December 2014

Copyright © 2015 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this special issue, we have solicited submissions fromelectrical engineers, control engineers, computer scientists,and mathematicians. After a rigorous peer review process,18 papers have been selected that provide overviews, solu-tions, or early promises, to manage, analyse, and interpretdynamical behaviours of networked systems. These papershave covered both the theoretical and practical aspectsof networked system with incomplete information in thebroad areas of dynamical systems, mathematics, statistics,operational research, and engineering.

In this special issue, there is a survey paper on therecent advances of control and filtering problems for Takagi-Sugeno (T-S) fuzzy systems with network-induced phenom-ena. Specifically, in the paper entitled “Analysis, Filtering,and Control for Takagi-Sugeno Fuzzy Models in NetworkedSystems” by S. Zhang et al., the focus is to provide a timelyreview on some recent advances on the T-S fuzzy controland filtering problems with various network-induced phe-nomena. Because of the advantages in dealing with variousnonlinear systems, the fuzzy logic theory has great successin industry applications. Among various kinds of modelsfor fuzzy systems, the T-S fuzzy model is quite populardue to its convenient, simple dynamic structure and thecapability of approximating any smooth nonlinear functionto any specified accuracy within any compact set.This surveydiscusses a variety of T-S fuzzy control and filtering issueswith network-induced phenomena in great detail firstly.Four network-induced phenomena (communication delays,

packet dropouts, signal quantization, and randomly occur-ring uncertainties (ROUs)) are introduced. Both theories andtechniques for dealing with the controller or filter designare systematically reviewed. Then, some latest results on T-S fuzzy control/filtering problems (bilinear T-S fuzzy model,event-based fuzzy control, fuzzy filtering with multiplenetwork-induced phenomena, 𝑙

2–𝑙∞

fuzzy filtering, 𝐻−/𝐻∞

fault detection, 𝐻∞

filtering with unknown membershipfunctions, and nonfragile 𝐻

∞fuzzy filtering) for networked

systems are surveyed and some challenging issues for futureresearch are raised. Finally, some conclusions are drawn andseveral possible related research directions are pointed out.

In the past decades, the stability analysis of the networkedsystems has attracted much research attention. In the workentitled “Uniform Stability Analysis of Fractional-OrderBAM Neural Networks with Delays in the Leakage Terms”by X. Yang et al., the uniform stability analysis is studiedfor a class of fractional-order BAM neural networks withdelays in the leakage terms. By introducing a novel norm,several delay-dependent sufficient conditions are obtained toensure the uniform stability of the proposed system by usinginequality technique and analysis method. Moreover, suffi-cient conditions are established to guarantee the existence,uniqueness, and uniform stability of the equilibrium point.Three simulation examples are given to demonstrate the effec-tiveness of the obtained results. It should be pointed out thatit is possible to extend the main results of this paper to othercomplex systems and establish novel stability conditions

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015, Article ID 852103, 4 pageshttp://dx.doi.org/10.1155/2015/852103

2 Abstract and Applied Analysis

with less conservatism by using more up-to-date techniques.The 𝜇-stability issue is discussed in “Global 𝜇-Stability ofImpulsive Complex-Valued Neural Networks with LeakageDelay and Mixed Delays” by X. Chen et al. for complex-valued neural networks (CVNNs) with leakage delay, discretedelay, and distributed delay under impulsive perturbations.The 𝜇-stability is the concept for the purpose of unifyingthe exponential stability, power-rate stability, and log-stabilityof neural networks. CVNN is an extension of real-valuedneural network which has been applied in physical systemsdealing with electromagnetic, light, ultrasonic, and quantumwaves. Based on the homeomorphism mapping principle ofcomplex domain, a sufficient condition for the existence anduniqueness of the equilibrium point of the addressed CVNNsis proposed in terms of linear matrix inequality (LMI). Byconstructing appropriate Lyapunov-Krasovskii functionalsand employing the free weighting matrix method, severaldelay-dependent criteria for checking the global 𝜇-stabilityof the CVNNs are established in LMIs. As direct applicationsof these results, several criteria on the exponential stability,power-stability, and log-stability are obtained. In the paperentitled “A Switched Approach to Robust Stabilization ofMultiple Coupled Networked Control Systems” by M. Yu etal., multiple coupled networked controlled systems (NCSs)with norm-bounded parameter uncertainties and multipletransmissions are considered. All the nodes in the proposedsystems act over a limited bandwidth communication chan-nel. The state information of every subsystem is split intodifferent packets and there is only one packet of the subsystemthat can be transmitted at a time. Based on the toking busprotocol, the nodes are arranged logically into a ring andtransmit the corresponding packets in a prefixed circularorder. Then, the proposed multiple NCSs can be modelledas periodic switched systems. Furthermore, the robust stabi-lization issue is dealt with by applying the switched systemtheory. State feedback controllers are constructed in terms ofLMIs. A numerical example is given to show that the coupledNCSs considered can be effectively stabilized via the designedcontroller.

Control and fault estimation problems for stochasticsystems have been of interest of many researchers during thepast decades. In the paper entitled “Robust 𝐻

∞Control for

a Class of Discrete Time-Delay Stochastic Systems with Ran-domlyOccurringNonlinearities” by Y.Wang et al., the robust𝐻∞

problem is studied for a class of discrete time-delaystochastic systems with randomly occurring nonlinearities(RONs). It is assumed that all the system matrices containthe parameter uncertainties. The stochastic disturbances areboth state- and control-dependent, and the RONs satisfy thesector boundedness conditions. The purpose of the problemproposed is to design a state feedback controller such that,for all admissible uncertainties, nonlinearities, and time-delays, the closed-loop system is robustly asymptoticallystable in the mean square, and a prescribed 𝐻

∞disturbance

rejection attenuation level is also guaranteed. By using theLyapunov stability theory and stochastic analysis tools, aLMI approach is developed to derive sufficient conditionsensuring the existence of the desired controllers, where theconditions are dependent on the lower and upper bounds

of the time-varying delays. The explicit parameterizationof the desired controller gains is also given. The problemof 𝐻∞

control for network-based 2D systems with missingmeasurements is investigated in “𝐻

∞Control for Network-

Based 2D Systems with Missing Measurements” by X. Buet al. A state feedback controller is designed such that theclosed-loop 2D stochastic system is mean-square asymptoticstability and has 𝐻

∞disturbance attenuation performance.

A sufficient condition is derived in terms of LMIs technique,and formulas can be given for the control law design. Theresult is also extended to more general cases where thesystem matrices contain uncertain parameters. Numericalexamples are also provided to show the effectiveness ofproposed approach. In the work entitled “Krein Space-Based𝐻∞

Fault Estimation for Discrete Time-Delay Systems” byX. Song and X. Yan, the finite-time 𝐻

∞fault estimation

issue is investigated for linear time-delay systems where thedelay appears in both state output and measurement output.Firstly, the design of finite horizon 𝐻

∞fault estimation is

converted into a minimum problem of certain quadraticform. Then, a sufficient and necessary condition for theexistence of the desired 𝐻

∞fault estimator is derived by

employing the Krein-space theory. A solution of the desired𝐻∞

fault estimator is obtained by recursively computinga partial difference Riccati equation which has the samedimension as the original systems. Therefore, solving a highdimension Riccati equation is avoided compared with theconventional augmented method. A numerical example isgiven to demonstrate the effectiveness of the approach.

Circulant type matrices have significant applicationsin network systems. In the work entitled “Equalities andInequalities for Norms of Block Imaginary Circulant Oper-ator Matrices” by X. Jiang and K. Hong, the block imaginarycirculant operator matrices are studied. Firstly, by combiningthe special properties of block imaginary circulant operatormatrix with unitarily invariant norm, several norm equalitiesare obtained. It should be pointed out that the norm in con-sideration is the weakly unitarily invariant norm. The usualoperator norm and Schatten 𝑝-norm are included.Then, sev-eral pinching type inequalities are presented by the triangleinequality and the invariance property of unitarily invariantnorms. Furthermore, some special cases and examples areconsidered. Circulant and left circulant matrices with Fermatand Mersenne numbers are considered in “Exact InverseMatrices of Fermat and Mersenne Circulant Matrix” by Y.Zheng and S. Shon. Moreover, the exact determinants andthe inverse matrices of Fermat and Mersenne left circulantmatrix are given.The nonsingularity of these special matricesis discussed. In the paper entitled “Norms and Spread ofthe Fibonacci and Lucas RSFMLR Circulant Matrices” byW. Xu and Z. Jiang the norms and spread of Fibonacci rowskew first-minus-last right (RSFMLR) circulant matrices areinvestigated as well as the Lucas RSFMLR circulant matrices.Firstly, these two kinds of special matrices are defined. Then,the lower and upper bounds for the spectral norms of thesematrices are proposed as well as the upper bounds for thespread of these matrices. Afterwards, some corollaries relatedto norms of Hadamard and Kronecker products of thesematrices are obtained, respectively. The determinants and

Abstract and Applied Analysis 3

inverses of Tribonacci circulant type matrices are discussedin “Explicit Form of the Inverse Matrices of TribonacciCirculant TypeMatrices” by L. Liu and Z. Jian.The definitionof Tribonacci circulant typematrices is given firstly.Then, theinvertibility of Tribonacci circulant type matrices is studied.Based on constructing the transformation matrices, both thedeterminant and the inversematrix are derived. Furthermore,by utilizing the relation between left circulant, 𝑔-circulantmatrices, and circulant matrix, the invertibility of Tribonaccileft circulant and Tribonacci 𝑔-circulant matrices is alsostudied. Finally, the determinants and inverse matrices ofthese matrices are presented, respectively. A future researchdirection is pointed out at last. In the paper entitled “Analysisof the Structured Perturbation for the BCSCB Linear System”by X. Tang and Z. Jian, the analysis problem associated withthe BCSCB matrix is considered. The BCSCB matrix is anextension of the circulant matrix and skew circulant matrix.Firstly, the form of the BCSCB matrix is obtained based onthe style spectral decomposition of the basic circulant matrixand the basic skew circulant matrix. Then, the structuredperturbation analysis for BCSCB linear system is proposed,which includes the condition number and relative error ofthe BSCSB linear system. A new approach is presented toderive the minimal value of the perturbation bound, whichis only related to the perturbation of the coefficient matrixand the vector. Simultaneously, the algorithm for the optimalbackward perturbation bound is developed.

As is well known, the analysis of issues on networksystems has important significance. In the paper entitled “Onthe Incidence Energy of Some Toroidal Lattices” by J.-B. Liu,et al., the closed-form formulae expressing the incidenceenergy of the 3.12.12 lattice and triangular kagomé latticeare derived as well as 𝑆(𝑚, 𝑛) lattice. The calculations ofthe energy of graphs become a popular topic of research.However, it is not an easy task to deal with the problemof the asymptotic incidence energy of various lattices withthe free boundary. By utilizing the applications of analysisapproach with the help of software calculation, the explicitasymptotic values of the incidence energy in these lattices arederived simultaneously. This developed method can be usedwidely to handle the asymptotic behaviour of other latticesand can obtain some useful results simultaneously. For thepurpose of studying the distribution of evolving networks, akind of evolving network is proposed in “Asymptotic DegreeDistribution of a Kind of Asymmetric Evolving Network” byZ. Li et al., where the model is a combination of preferentialattachment model and uniform model. The distributionof the number of vertices with given degree is studiedas well as the asymptotic degree distribution. It is shownthat the proportional degree sequence obeys power law,exponential distribution, and other forms according to therelation of the degree and parameter 𝑚. In the work entitled“Partial Synchronizability Characterized by Principal Quasi-Submatrices Corresponding to Clusters” by G. Zhang et al.,a partial synchronization problem is studied in an oscillatornetwork. In order to investigate the partial synchronization,the concept on a principal quasi-submatrix corresponding tothe topology of a cluster is proposed. A novel criterion onpartial synchronization is developed based on the analysis

of principal quasi-submatrices corresponding to the clusters.The proposed criterion is not distinctly dependent on theintercluster couplings or the topology matrix of the wholenetwork. If a network is composed of a large number ofnodes, the enormous amount of calculation can be reduced byreplacing the couplingmatrixwith several quasi-submatrices.Therefore, this criterion provides a novel index of partialsynchronizability. It is shown that different types of partialsynchronization occur in a star-global network when thecoupling strength is increased. The proposed approach forpartial synchronization might be applicable to the complexnetworks with networked induced phenomena. A backboneextraction heuristic with incomplete information (BEHwII)is investigated in “Extracting Backbones from WeightedComplex Networks with Incomplete Information” by L. Qianet al. The presence of the backbone is the signature orthe abstraction of the nature of complex systems and canprovide huge help for understanding them inmore simplifiedforms. For the purpose of extracting backbones from large-scale weighted networks, a novel filter-based approach ispresentedwhich only needs incomplete information and theninvokes the iteratively local search scheme for improving theefficiency. First, a strict filtering rule is designed to determineedges to be preserved or discarded.Then, a local searchmodelis proposed to examine part of edges in an iterative way.Experimental results on four real-life networks demonstratethe advantage of BEHwII over the classic disparity filtermethod by either effectiveness or efficiency validity.

Recently, the application of networked systems hasattracted a great deal of research interest. The data com-munication networks play an important role during thedevelopment of smart grid. Since the data communicationnetwork in smart gird is affected by plenty of decisive factors,different decision-making problems are presented accordingto the variable factors. In the paper entitled “Real-TimePricing Decision Making for Retailer-Wholesaler in SmartGrid Based on Game Theory” by Y. Dai and Y. Gao, a novelgame-theoretical decision-making scheme is investigated forelectricity retailers and wholesaler in the smart grid withdemand side management (DSM). The interaction betweentwo retailers and their wholesaler is modelled by a two-stagedynamic gamewhere the competition between two retailers isconsidered. According to the different action order betweenretailers and their wholesaler, two different game modelsare developed. The subgame perfect Nash equilibrium (SPE)for this game is determined through backward induction. Itis shown that the wholesaler wants to decentralize certainmanagement powers to the retailers through analysing theequilibrium revenues of the retailers for different situations.Imposing legal restrictions on the wholesaler’s discretionarypolicy suggests that the time-inconsistency problem is mit-igated. The packing problem of unit equilateral trianglesis investigated in “A New Quasi-Human Algorithm forSolving the Packing Problem of Unit Equilateral Triangles”by R. Wang et al. The packing problem of unit equilateraltriangles offers broad prospects in different fields includingthe network resource optimization. This problem is non-deterministic polynomial (NP) hard and has the feature ofcontinuity. A novel quasi-human algorithm for solving this

4 Abstract and Applied Analysis

problem is proposed according to the characteristic of theunit equilateral triangles and in the base of analysis of thegeneral triangles packing problems. Time complexity analysisand the calculation results indicate that the proposedmethodis a polynomial time algorithm,which provides the possibilityto solve the packing problem of arbitrary triangles.

Acknowledgments

This special issue is a timely reflection of the research pro-gress in the area of networked systems with incomplete infor-mation. We would like to acknowledge all authors for theirefforts in submitting high-quality papers. We are also verygrateful to the reviewers for their thorough and on-timereviews of the papers.

Zidong WangBo Shen

Hongli DongXiao HeJun Hu

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