IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 8

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 8, AUGUST 2010 1781

Networked Control Systems With CommunicationConstraints: Tradeoffs Between Transmission

Intervals, Delays and PerformanceW. P. Maurice H. Heemels, Member, IEEE, Andrew R. Teel, Fellow, IEEE, Nathan van de Wouw, Member, IEEE,

and Dragan Nesic, Fellow, IEEE

Abstract—There are many communication imperfections in net-worked control systems (NCS) such as varying transmission delays,varying sampling/transmission intervals, packet loss, communica-tion constraints and quantization effects. Most of the available lit-erature on NCS focuses on only some of these aspects, while ig-noring the others. In this paper we present a general frameworkthat incorporates communication constraints, varying transmis-sion intervals and varying delays. Based on a newly developed NCSmodel including all these network phenomena, we will provide anexplicit construction of a continuum of Lyapunov functions. Basedon this continuum of Lyapunov functions we will derive boundson the maximally allowable transmission interval (MATI) and themaximally allowable delay (MAD) that guarantee stability of theNCS in the presence of communication constraints. The developedtheory includes recently improved results for delay-free NCS as aspecial case. After considering stability, we also study semi-globalpractical stability (under weaker conditions) and performance ofthe NCS in terms of gains from disturbance inputs to con-trolled outputs. The developed results lead to tradeoff curves be-tween MATI, MAD and performance gains that depend on the usedprotocol. These tradeoff curves provide quantitative informationthat supports the network designer when selecting appropriate net-works and protocols guaranteeing stability and a desirable level ofperformance, while being robust to specified variations in delaysand transmission intervals. The complete design procedure will beillustrated using a benchmark example.

Index Terms— gains, communication constraints, delays,Lyapunov functions, networked control systems (NCS), protocols,stability, time scheduling.

Manuscript received August 15, 2008; Manuscript revised February 19, 2009;accepted January 18, 2010. First published February 08, 2010; current versionpublished July 30, 2010. This work was supported by the National ScienceFoundation under grants ECS-0622253 and CNS-0720842, by the Air ForceOffice of Scientific Research under grant F9550-06-1-0134, by the AustralianResearch Council under the Australian Professorial and Discovery Projectsschemes, and by the European Community through the FP7-ICT-2007-2thematic programme under the WIDE-224168 project. Recommended byAssociate Editor K. H. Johansson.

W. P. M. H. Heemels and N. van de Wouw are with the Deptartment of Me-chanical Engineering, Eindhoven University of Technology, Eindhoven 5600MB, The Netherlands (e-mail: [email protected]; [email protected]).

A. R. Teel is with the Electrical and Computer Engineering Department,the University of California, Santa Barbara, CA 93106–9560 USA (e-mail:[email protected]).

D. Nesic is with the Department of Electrical and Electronic Engineering, TheUniversity of Melbourne, Victoria 3010, Australia (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2010.2042352

I. INTRODUCTION

N ETWORKED control systems (NCS) have received con-siderable attention in recent years. The interest for NCS is

motivated by many benefits they offer such as the ease of main-tenance and installation, the large flexibility and the low cost.However, still many issues need to be resolved before all the ad-vantages of wired and wireless networked control systems canbe harvested. Next to improvements in the communication in-frastructure itself, there is a need for control algorithms that candeal with communication imperfections and constraints. Thislatter aspect is recognized widely in the control community, asevidenced by the many publications appearing recently, see e.g.,the overview papers [23], [47], [54], [56].

Roughly speaking, the network-induced imperfections andconstraints can be categorized in five types:

(i) Quantization errors in the signals transmitted over the net-work due to the finite word length of the packets;

(ii) Packet dropouts caused by the unreliability of thenetwork;

(iii) Variable sampling/transmission intervals;(iv) Variable communication delays;(v) Communication constraints caused by the sharing of the

network by multiple nodes and the fact that only one nodeis allowed to transmit its packet per transmission.

It is well known that the presence of these network phenomenacan degrade the performance of the control loop significantlyand can even lead to instability, see e.g., [10] for an illustrativeexample. Therefore, it is of importance to understand how thesephenomena influence the closed-loop stability and performanceproperties, preferably in a quantitative manner. Unfortunately,much of the available literature on NCS considers only some ofabove mentioned types of network phenomena, while ignoringthe other types. There are, for instance, systematic approachesthat analyse stability of NCSs subject to only one of these net-work-induced imperfections. Indeed, the effects of quantizationare studied in [3], [12], [20], [22], [28], [36], [45], of packetdropouts in [41], [42], of time-varying transmission intervalsand delays in [14], [32], and [10], [16], [24], [27], [35], [55], re-spectively, and of communication constraints in [2], [11], [26],[40].

Since in any practical communication network all aforemen-tioned network-induced imperfections are present, there is aneed for analysis and synthesis methods including all these im-perfections. This is especially of importance, because the design

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1782 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 8, AUGUST 2010

TABLE IREFERENCES THAT STUDY MULTIPLE NETWORK-

INDUCED IMPERFECTIONS SIMULTANEOUSLY

of a NCS often requires tradeoffs between the different types.For instance, reducing quantization errors (and thus transmit-ting larger or more packets) typically results in larger transmis-sion delays. To support the designers in making these tradeoffs,tools are needed that provide quantitative information on theconsequences of each of the possible choices. However, lessresults are available that study combinations of these imper-fections. References that simultaneously consider two types ofnetwork-induced limitations are [5], [9], [13], [18], [25], [29],.[31], [39], [48], [51], [52] which are categorized in Table I.Moreover, [37] consider imperfections of type (i), (iii), (v), [8],[33], [34] study simultaneously type (ii), (iii), (iv), [38] focusseson type (ii), (iii), (v), and [15] incorporates type (i), (ii), and (iv).In addition some of the approaches mentioned in Table I thatstudy varying transmission intervals and/or varying communi-cation delays can be extended to include type (ii) phenomenaas well by modeling dropouts as prolongations of the maximaltransmission interval or delay (cf. also Remark II.4 below).

Another paper that studies three different types of networkimperfections is written by Chaillet and Bicchi [6]. This paperstudies NCS involving both variable delays, variable trans-mission intervals and communication constraints, and uses amethod for delay compensation. The delay compensation isbased on sending a larger control packet to the plant containingnot just one control value at one particular time instant, butcontaining a control signal valid for a given future time horizon.For this particular control scheme, [6] provides bounds on thetolerable delays and transmission intervals such that stabilityof the NCS is guaranteed. Also in the present paper we willstudy NCS corrupted by varying delays, varying transmissionintervals and communication constraints, while packet dropoutscan be included as well (in the way explained in Remark II.4below). In other words, this paper considers network-inducedimperfections of type (iii), (iv) and (v). After developing anovel NCS model incorporating all these types of networkphenomena, we will present allowable bounds on delays andtransmission intervals guaranteeing both stability and perfor-mance of the NCS. However, in contrast with [6], we considerthe more basic emulation approach in the spirit of [5], [11],[38], [39], [51], [52], which encompasses no specific delaycompensation schemes. The work in [6] is of interest, as it aimsat allowing larger delays by including specific delay compen-sation schemes, at the cost of sending larger control-packetsand requiring time-stamping of messages. The features ofcompensation and time-stamping of messages are not neededin our framework. Another distinction with [6] is related to theadmissible protocols that schedule which node is allowed totransmit its packet at a transmission time. Our work appliesfor all protocols satisfying the UGES property (see belowfor an exact definition) and not only for so-called invariably

UGES protocols (cf. [6]), which exclude the commonly usedRound-Robin (RR) protocol.

One of the main contributions of this paper is that we ex-plicitly construct a continuum of Lyapunov functions based onthe standard delay-free conditions as adopted in [5], [11], [38],[39], [51], [52]. This continuum of Lyapunov functions leadsto tradeoff curves between the maximally allowable transmis-sion interval (MATI) and the maximally allowable delay (MAD)guaranteeing stability of the NCS. These tradeoff curves will de-pend on the specific communication protocol used, so that theyeven allow for the comparison of different protocols. In addi-tion to stability, which is only a basic property that has to besatisfied by the control loop, there are often additional require-ments with respect to the performance of the NCS. This paperalso studies the performance in terms of gains between spe-cific exogenous inputs (e.g., disturbances) and controlled out-puts of the system. We will show how performance of theNCS depends on the MATI, the MAD and the protocol used,leading to tradeoff curves as well. This design methodology andthe method to compute the tradeoff curves will be demonstratedon the case study of the batch reactor that has developed over theyears as a benchmark system for NCS, see e.g., [5], [38], [39].Next to stability and performance, also semiglobal practicalstability results will be presented that can be obtained underweaker conditions.

The paper is organized as follows. The current section willend with introducing some notational conventions and con-cepts. Next, in Section II we present a general NCS modelingframework that extends the NCS models in [5], [11], [38], [39],[51], [52] to include both communication constraints as well asvarying transmission delays and transmission intervals. In Sec-tion III we will transform this new NCS model into the hybridsystem framework as introduced in [17] as this will facilitatefurther analysis. Also the stability and performance concepts asused in this paper are defined in this section. In Section IV wewill derive the Lyapunov-based conditions that determine boththe maximally allowable transmission interval (MATI) and themaximally allowable delay (MAD) guaranteeing global asymp-totic stability and performance. We also present the resultson semiglobal practical stability in this section. In Section Vwe show how the Lyapunov functions can be constructed onthe basis of the widely adopted non-delay conditions in [5],[11], [38], [39], [51], [52] and show that the non-delay case is aparticular case of general framework. To demonstrate how thedeveloped methods can be used for explicitly computing MATIand MAD guaranteeing stability or certain performance, weapply the framework to the benchmark problem of the batchreactor [5], [38], [39]. Finally, we state the conclusions and ourideas for future work.

The following notational conventions will be used in thispaper. will denote all nonnegative integers, denotes allpositive integers, denotes the field of all real numbers and

denotes all nonnegative reals. By and we denotethe Euclidean norm and the usual inner product of real vectors,respectively. For a collection of real vectors with

, we denote the column vector ob-tained by stacking the vectors , on top of eachother by . For a symmetric matrix ,

HEEMELS et al.: NETWORKED CONTROL SYSTEMS WITH COMMUNICATION CONSTRAINTS 1783

denotes the largest eigenvalue of . By and we denote thelogical ’or’ and ’and,’ respectively. A functionis said to be of class if it is continuous, zero at zero and strictlyincreasing. It is said to be of class if it is of class and it isunbounded. A function is said to be ofclass if it is continuous, is of class for eachand is nonincreasing and satisfiesfor each . A function issaid to be of class if it is continuous, and for each ,

and belong to class . We write forthe standard exponential function.

We recall now some definitions given in [17] that will be usedfor developing a hybrid model of a NCS later. For the motivationand more details on these definitions, one can consult [17].

Definition I.1: A compact hybrid time domain is a setwith and

. A hybrid time domain is a set suchthat is a compact hybrid time domainfor each .

Definition I.2: A hybrid trajectory is a pair (dom , ) con-sisting of a hybrid time domain dom and a function definedon dom that is absolutely continuous in on

for each .Definition I.3: For the hybrid system given by the state

space , the input space and the data , whereis continuous, is locally

bounded, and and are subsets of , a hybrid trajectory(dom , ) with : dom is a solution to for a locallyintegrable input function if

1) For all and for almost all dom, we have and .

2) For all dom such that dom , we haveand .

Hence, the hybrid systems that we consider are of the form

We sometimes omit the time arguments and write

(1)

where we denoted as . We also note that typi-cally and, in this case, if we have thateither a jump or flow is possible, the latter only if flowing keepsthe state in . Hence, the hybrid model (1) may have non-uniquesolutions.

In addition, for , , we introduce the normof a function defined on a hybrid time domain dom

with possibly and/or , by

(2)

provided the right-hand side exists and is finite. In case isfinite, we say that . Note that this definition is essentially

identical to the usual norm in case a function is defined on asubset of .

II. NCS MODEL AND PROBLEM STATEMENT

In this section, we introduce the model that will be used to de-scribe NCS including both communication constraints as wellas varying transmission intervals and transmission delays. Thismodel will form an extension of the NCS models used beforein [38], [39] that were motivated by the work in [52]. All theseprevious models did not include transmission delays. We con-sider the continuous-time plant

(3)

that is sampled. Here, denotes the state of the plant,denotes the most recent control values available at the

plant, is a disturbance input and is the outputof the plant. The controller1 is given by

(4)

where the variable is the state of the controller,is the most recent output measurement of the plant that

is available at the controller and denotes the controlinput. The functions , are assumed to be continuous andand are assumed to be continuously differentiable. At times

, , (parts of) the input at the controller and/or theoutput at the plant are sampled and sent over the network. Thetranmission/sampling times satisfyand there exists a such that the transmission intervals

satisfy for all ,where denotes the maximally allowable transmission in-terval (MATI). At each transmission time , , the pro-tocol determines which of the nodes is grantedaccess to the network. Each node corresponds to a collection ofsensors or actuators. The sensors/actuators corresponding to thenode that is granted access collect their values of the entries in

or that will be sent over the communication net-work. They will arrive after a transmission delay of time unitsat the controller or actuator. This results in updates of the cor-responding entries in or at times , . The situ-ation described above is illustrated for and in Fig. 1 for thesituation of two nodes that obtain access to the network in analternating fashion.

It is assumed that there are bounds on the maximal delay inthe sense that , , whereis the maximally allowable delay (MAD). To be more precise,we adopt the following standing assumption.

Standing Assumption II.1: The transmission times satisfy, and the delays satisfy

, , whereis arbitrary.

The latter condition implies that each transmitted packet ar-rives before the next sample is taken. This assumption indicatesthat we are considering the so-called small delay case as op-posed to the large delay case, where delays can be larger thanthe transmission interval. The inequalities and

1Extensions of the theory presented below to the case of time-dependent sys-tems (3) and time-dependent controllers (4) are straightforward.


Fig. 1. Illustration of a typical evolution of � and �� in case of two nodes thatget access to the network in an alternating manner.

can be taken non-strict with the understandingthat in case the update instant coincides with the nexttransmission instant , the update is performed before thenext sample is taken. The updates at satisfy

(5)

where denotes the vector with and. Hence, with . If the

NCS has nodes, then the error vector can be partitionedas . The functions and are now up-date functions that are related to the protocol that determineson the basis of and the networked error which node isgranted access to the network. Typically when the -th node getsaccess to the network at some transmission time we havethat the corresponding part in the error vector has a jump at

. In most situations, the jump will actually be to zero,since we assume that the quantization effects are negligible. Forinstance, when is sampled at time , then we have that

. However, we allow for more freedom inthe protocols by allowing general functions . See [38], [39] formore details. We will refer to as the protocol.

In between the updates of the values of and , the network isassumed to operate in a zero order hold (ZOH) fashion, meaningthat the values of and remain constant in between the up-dating times and for all :

(6)

To compute the resets of at the update times ,we proceed as follows:

In the third equality we used that due tothe zero order hold character of the network. We also implicitly

employed our Standing Assumption II.1 as we used that therealways occurs an update before the next sample is taken

.A similar derivation holds for , leading to the following

model for the NCS:

(7a)

(7b)

where with , , areappropriately defined functions depending on , , andand . See [38] for the explicit expressions of and

, which also reveal how we use the differentiability conditionson and imposed earlier.

Standing Assumption II.2: and are continuous and islocally bounded.

Observe that the system

(8)

is the closed-loop system (3) and (4) without the network (i.e.,and in (3) and (4)).

The problem that we consider in this paper is formulated asfollows.

Problem II.3: Suppose that the controller (4) was designedfor the plant (3) rendering the closed-loop (3) and (4) without thenetwork, i.e., and , (or equivalently, (8))stable in some sense. Determine the value of and sothat the NCS given by (7) is stable as well when the transmissionintervals and delays satisfy Standing Assumption II.1

Remark II.4: Of course, there are certain extensions that canbe made to the above setup. The inclusion of packet dropoutsis relatively easy, if one models them as prolongations of thetransmission interval. Indeed, if we assume that there is a bound

on the maximum number of successive dropouts, the sta-bility bounds derived below are still valid for the MATI givenby , where is the obtained valuefor the dropout-free case. Another extension of the frameworkin this paper could be the inclusion of quantization effects. Thisstep is more involved. It might be based on recent work in [37]that unifies the areas of networked and quantized control sys-tems without communication delays. It can be envisioned thatthe results presented here can be combined with the frameworkin [37] leading to an overall methodology capable of handlingall types of networked phenomena that were mentioned in theintroduction. The specific conditions and types of quantizers forwhich this methodology is effective are subject of future re-search and some preliminary results are reported in [21]. An-other possible extension of interest is the consideration of thelarge-delay case (in which the delays can be larger than thetransmission interval). This would require a more involved NCSmodel that does not have the periodicity between transmissionand update events as implied by Standing Assumption II.1. Thisis a hard problem, which will be considered in future research.


III. REFORMULATION IN A HYBRID SYSTEM FRAMEWORK

To facilitate the stability analysis, we transform the aboveNCS model into the hybrid system framework as developed in[17]. This hybrid systems framework was also employed in [5],where a similar model was obtained without the incorporation ofdelays. To do so, we introduce the auxiliary variables ,

, and to reformulate the model interms of flow equations and reset equations. The variable isan auxiliary variable containing the memory in (7b) storing thevalue for the update of at the update instant

, is a counter keeping track of the transmission, is atimer to constrain both the transmission interval as well as thetransmission delay2 and is a Boolean keeping track whether thenext event is a transmission event or an update event. To be pre-cise, when the next event will be related to transmissionand when the next event will be an update. The Booleanwill be used to guarantee in the model below that the transmis-sion and update events are alternating in the sense that before anext sample is taking the previous update is implemented in theNCS.

The hybrid system is now given by the flow equations

(9)

and the reset equations are obtained by combining the “trans-mission reset relations,” active at the transmission instants

, and the “update reset relations”, active at the updateinstants , given by

when

(10)

with the mapping given by the transmission resets (when)

(11)

and the update resets (when )

(12)

Two comments on this model are in order. First of all, the roleof is to exclude (instantaneous Zeno) solutions tosatisfying , and

for or with , when. Also when similar solutions exist that

are only resetting (sometimes called ‘livelock’ in hybrid systemstheory [50]). However, can be taken arbitrarily small tostill allow for small transmission intervals.

Secondly, the choice for when is irrelevant from amodeling point of view. However, it was selected here as

2We could also have introduced two timers, one corresponding to the trans-mission interval and one to the transmission delay. However, it turns out thatthe NCS can be described using only one timer, which has the advantage of re-sulting in a more compact hybrid model.

, because it will simplify the analysis later. By takingthe hybrid system above is in the

form (1).Definition III.1: For the hybrid system with ,

the set given by issaid to be uniformly globally asymptotically stable (UGAS) ifthere exists a function such that, for each

, and any initial condition , ,, , ,

with3

, all corresponding solutions satisfy

(13)

for all in the solution’s domain. The set is uniformlyglobally exponentially stable (UGES) if can be taken of theform for some and

.Remark III.2: The factor multiplying in the right-hand

side of (13) is motivated by the fact that for small values ofsolutions of exist that have many resets without

progressing too fast. Actually, the limit case has ’livelock’solutions that are only resetting (with remaining 0 and

) as discussed above. The scales the right-hand side of (13)for this effect.

We also introduce the concept of uniform semiglobal prac-tical asymptotical stability.

Definition III.3: For the hybrid system with ,the set is said to be uniformly semiglobally practically asymp-totically stable (USPAS) with respect to and , if thereexists such that for any pair of positive numbers

there exist and such thatfor each , each initial condition ,

, , , ,with

, ,, and each corresponding solution we have

(14)

for all in the solution’s domain.In the presence of disturbance inputs in we might

be interested in reducing its influence on a particular controlledoutput variable

(15)

in terms of the induced gain, as formally defined below. Thehybrid model expanded with the output (15) is denotedby .

Definition III.4: Consider with and let begiven. The hybrid system is said to be stable with gain

, if there is a -function such that for any ,any input and any initial condition ,

3Note that the next condition is just saying that �� in theterminology of (1).


, , , ,with

, each corresponding solutionto satisfies

(16)

IV. STABILITY AND PERFORMANCE ANALYSIS

In this section we focus on the analysis of UGAS and UGES,USPAS, and stability.

A. Stability Analysis

In order to guarantee UGAS or UGES, we assume the exis-tence of a Lyapunov function for the reset equa-tions (11) and (12) satisfying

(17a)

(17b)

for all and all and the bounds

(18)

for all , and for some functionsand and .

In Section V we will show how a function satisfying (17)and (18) can be derived from the generally accepted conditionson the protocol as used for the delay-free case in [5], [38].To solve Problem II.3, we extend (17) and (18) to the followingcondition.

Condition IV.1: There exist a functionwith locally Lipschitz for all

and , a locally Lipschitz function ,-functions , , and , continuous functions

, positive definite functions and and constants, , for , and such that:

• for all and all (17) holds and (18) holdsfor all ;

• for all , , , and almostall it holds that

(19)• for all , , and almost all

(20)

and(21)

The inequalities (19) and(20) are similar in nature to the delay-free situation as studied in [5] and are directly related to the

gain conditions from to as adopted in [38]. Althoughthese conditions may seem difficult to obtain at first sight, this isnot the case. We will demonstrate this in Section VI, where the

complete computational set-up for determining the parametersin the above conditions is provided. See also Remark V.2 for amore detailed discussion.

Consider now the differential equations

(22a)

(22b)

where and , are the real constantsas given in Condition IV.1. Observe that the solutions to thesedifferential equations are strictly decreasing as long as

, .Theorem IV.2: Consider the system that satisfies Con-

dition IV.1. Suppose satisfy

(23a)

(23b)

for solutions and of (22) corresponding to certain choseninitial conditions , , with

, and as in Condition IV.1. Thenfor the system with the set is UGAS in thesense of Definiton III.1. If in addition, there exist strictly positivereal numbers , , , and such that ,

, , ,and , , then this set is UGES.

Proof: The solutions to the differential (22) are scaledas and for new functions , . Thedifferential (22) and the conditions (23) transform into

(24)

and(25a)

(25b)

We consider now the function

(26)

and show that this constitutes a suitable Lyapunov function forthe system , which can be used to conclude UGAS andUGES under the stated conditions.

Below, with some abuse of notation, we consider the quantitywith

as in (9) even though is not differentiable with respect toand . This is justified since the components in corre-sponding to and are zero. We will first show that

whenever the system with resets.When and a jump occurs, we have that

and obtain, using (11), that


Similarly, when we have that and obtain,using (12)

We also have, for all and almost all , that

The above proves that forms a Lyapunov function for thesystem with and UGAS and UGES follows nowusing standard Lyapunov arguments as are provided in [5] forthe delay-free case. This completes the proof.

From the above theorem quantitative numbers for andcan be obtained by constructing the solutions to (22) for

certain initial conditions. By computing the value of the in-tersection of and the constant line providesaccording to (23a), while the intersection of and givesa value for due to (23b). Different values of the initialconditions and lead, of course, to different solu-tions and of the differential (22) and thus also to differentLyapunov functions in (26). Hence, a continuum of Lyapunovfunctions is obtained by varying the initial conditions and

. Moreover, each different choice of and pro-vides different and . As a result, tradeoff curves be-tween and can be obtained that indicate when sta-bility of the NCS is still guaranteed. This will be illustrated inSection VI, where the complete analysis framework will be ap-plied to a benchmark example.

Remark IV.3: The existence of strictly positive andsuch that (23) holds follows from in (17)

as this implies the existence of , with. Moreover, when using

in case , we recover the explicit formulafor the MATI obtained in [5] (which improved earlier results in[38], [39]) in the sense that we have

(27)where . Hence, for the delay-free case

we recover the results in [5] as a special case.

B. Uniform Semiglobal Practical Stability

Under a version of Condition IV.1, which is weaker at var-ious points, we can obtain semi-global practical asymptoticalstability results with respect to and for the zero-inputsystem of .

Condition IV.4: There are a functionwith locally Lipschitz for all

and , -functions , , , and suchthat

• for all and all (17) holds and (18) holdsfor all ;

• for all , , all and almost allit holds that

(28)

• The origin of the network-free and zero-input systemis globally asymptotically stable.

Theorem IV.5: Consider the system with thatsatisfies Condition IV.4. The set is USPAS with respect to

and in the sense of Definition III.3.Proof: We will prove the USPAS property with re-

spect to and considering . Sinceis globally asymptotically stable and is con-

tinuous, we can apply a converse Lyapunov theorem [7] thatyields the existence of a continuously differentiable function

and a such that

(29)

We consider now the Lyapunov function

(30)

using and as in the proof of Theorem IV.2 and takingthe constant such that with as in (17). Nowexploiting (17), yields that in case of and

In case of and , we have

Finally, considering the evolution of along the flow ofwith gives using (18) and (28) for

, all and almost all


where

Note that is a continuous function due to the contin-uous differentiability of and the continuity of and

. Moreover, for all . Usingnow Lemma 2.1 in [46] guarantees, for each pair ofstrictly positive numbers , the existenceof such that for almost all in

itholds thatIn a similar way as in the proof of Theorem IV.2, the USPASproperty can now be derived by straightforward reasoning.

C. Stability Analysis

For the stability analysis, we replace Condition IV.1 bythe following.

Condition IV.6: There exist a functionwith locally Lipschitz for all

and , a locally Lipschitz function ,-functions , , and , continuous functions

, and constants , , for ,and such that:

• for all and all (17) holds and(18) holdsfor all ;

• for all , , , ,and almost all it holds that

(31)• for all , , , and almost

all

(32)

for some and , and

(33)

The main difference between Condition IV.6 and ConditionIV.1 is the presence of the disturbance input and the de-pendence of on both and instead of on only. More-over, comparing (20) and (32), we observe the additional term

in the right-hand side of (32), which isneeded to obtain a bound on the gain between and .

Theorem IV.7: Consider the system that satisfies Con-dition IV.6. Suppose satisfy (23) for solu-tions and of (22) corresponding to certain initial condi-tions , , with ,

and as in Condition IV.6. Then the systemis stable with gain .

Proof: In a similar manner as in the proof of Theorem IV.2,we obtain that the function in (26) satisfies

(34)

whenever there is a reset and

(35)

during the flow of the hybrid system . Since we canwithout loss of generality assume that by scalingto . Let be a solution to with correspondingoutput for initial condition and input . Denotethe hybrid time domain of by domwith and possibly and/or . Let be thecorresponding output also considered on dom . Reformulating(34) by using the hybrid time domain dom gives that for each

we have

(36)

and integrating (35) yields for each andwith that

(37)

Computing now the norm of gives

As a consequence, we have that .Due to the bounds (33) and(18) on and , respectively, wecan bound as for a suitable

. This proves that the system is stable with gain, where the -function in (16) can be taken as

.Remark IV.8: Essentially, in the above proof we constructed

a so-called storage function [53] given by as in (26) for thesystem with supply rate during the flowphases and supply equal to 0 during the resets. In the analysisof passivity and stability these concepts are exploited forvarious classes of systems in, for instance, [4], [19], [49], [53].

V. CONSTRUCTING LYAPUNOV AND STORAGE FUNCTIONS

In this section we will construct Lyapunov and storage func-tions and as in Condition IV.1, Condition IV.6 and Con-dition IV.4 from the commonly adopted assumptions in [5],[38], [39], [51], [52] for the delay-free case. We will start with


constructing Lyapunov and storage functions as in ConditionIV.1 and Condition IV.6, respectively. For the delay-free case,one considers in [5], [38] protocols satisfying the followingcondition:

Condition V.1: The protocol given by is UGES (uniformlyglobally exponentially stable), meaning that there exists a func-tion that is locally Lipschitz in its secondargument such that

(38a)

(38b)

for constants and .Additionally we assume here that

(39)

for some constant4 and that for almost all andall

(40)

for some constant . For all protocols discussed in [5],[38], [39], [51], [52] such constants exist. In Lemma V.4 below,we specify appropriate values for these constants in case of theoften used Round Robin (RR) and the Try-Once-Discard (TOD)protocols (see [38], [52] for their definitions). We also assumethe growth condition on the NCS model (7)

(41)

where , if we are looking for a Lyapunov func-tion establishing UGAS, and

(42)

where , if we are looking for a storagefunction establishing stability. In both cases is aconstant. Building upon slightly modified conditions as usedfor the delay-free case in [5] given by the existence of a locallyLipschitz continuous function satisfying thebounds

(43)

for some -functions and , and, in case of constructinga Lyapunov function, the condition

(44)for almost all and all with , and, incase of constructing a storage function, the condition

(45)

4In principle this constant can be taken non-negative. However, as all proto-cols available in the literature satisfy � � �, we take � � � to reduce somenotational burden later.

for almost all and all and all ,we can derive functions and satisfying Condition IV.1and Condition IV.6, respectively. The constants in (44) satisfy

, where is sufficiently small.Remark V.2: Condition V.1 and inequality (44) are essen-

tially the same as in [5] with . The constantis selected small to sacrifice only a little of the

gain from to . In [38, Thm. 4] the inequality (44) wasactually formulated in terms of an gain, while we use aLyapunov-based formulation here. As gains are establishedoften using Lyapunov functions, this seems to be a natural refor-mulation (see also [5, Rem. 2]). The only additional conditionwe add here is (39), which holds for all protocols considered in[5], [11], [38], [39], [51], [52] as is demonstrated in Lemma V.4below for the RR and TOD protocols.

Theorem V.3: Consider the systems and , re-spectively, such that

• Condition V.1, (39) with and (40) with constanthold;

• Equation (41) is satisfied for some function(and (42) for some function

, respectively) and ;• there exists a locally Lipschitz continuous function

satisfying the bounds (43) for some-functions , , and (44) with and

(and (45) with ,and , respectively).

Then, the functions given by

(46)

(47)

satisfy Condition IV.1 (and Condition IV.6, respectively) with, , , ,

, and ,(and , respec-

tively), , with as in Condition V.1

(48)

and some positive constants , (and ,respectively).

Proof: We only prove the theorem for establishing Con-dition IV.1, as the case for Condition IV.6 is analogous. Thecondition (17) with of the form (46) is equivalent to


Using (38b) and (39), this follows trivially. The condition (17b)is identical to

which is true as . Based on (46) we obtain fourcases for (19):

• Case 1: and

(49)

• Case 2: and

(50)

• Case 3: and

(51)

• Case 4: and

(52)

Hence, (19) holds for withand , as in (48). Here we used that

as .Finally, to obtain (20) observe that for the case , we

have due to (44)

(53)

Similarly, for we obtain

(54)

Take and multiply the inequalities (53) and(54) by , which give for

and for

Note that the bounds on as in (18) with linear bounding func-tions and can be easily obtained from the fact thatsatisfies (38a) with linear functions. This completes the proof.

To apply the above theorem for a given protocol we need toestablish the values , , , and . The followinglemma determines these constants for the well-known RR andTOD protocols. See [38], [52] for the exact definitions of theseprotocols.

Lemma V.4: Let denote the number of nodes in the network.For the RR protocol there is a that islocally Lipschitz in its second argument satisfying (38), (39)and (40) with , , ,

and . For the TOD protocolthere is a that is locally Lipschitzin its second argument satisfying (38), (39) and (40) with

, ,and .


Proof: The constants , , are derived for both pro-tocols in [38] and corresponding Lyapunov functions . Forthe RR protocol can be taken with

, , and is thenumber of nodes in the network, see Example 3 in [38]. Thisimplies that

Using and the identity

, we obtain that

.

Since for the TOD protocol,follows from the above by taking for

all .In the next theorem, we show how to obtain Condition IV.4,

used for establishing USPAS in Theorem IV.5, from the com-monly adopted conditions in the delay-free case. As the proof issimilar in nature as the proof of Theorem V.3, it is omitted.

Theorem V.5: Consider the system . Assume that Con-dition V.1 and (39) with hold. Moreover, assume thatthere exists a -function such that

(55)

for almost all and all . Assume that the originof the network-free and zero-input system is

globally asymptotically stable. Then, the function given by(46) satisfies Condition IV.4 with , ,

for some positive constants and andas in Condition V.1.The conditions in Theorem V.5 are precisely those used in

the delay-free case for obtaining USPAS as adopted in [5, Thm.2]. To obtain the constants , , and for the RR orTOD protocol again Lemma V.4 above can be employed.

VI. CASE STUDY OF THE BATCH REACTOR

In this section we will illustrate how the derived conditionscan be verified and how this leads to quantitative tradeoff curvesbetween , and performance for the case study ofthe batch reactor. This case study has developed over the yearsas a benchmark example in NCS, see e.g., [5], [11], [38], [52].The functions in the NCS (7) for the batch reactor are given bythe linear functions and

. The batch reactor, which isopen-loop unstable, has inputs, outputs,plant states and controller states and nodes (onlythe outputs are assumed to be sent over the network). See [38],[52] for more details on this example. We included here alsoa disturbance input that takes values in and a controlledoutput , taking values in , given by

. The numerical values for , , as provided in[38], [52], are given in (56). In [38], [52] and were absent,as they considered the disturbance-free case. We selected for theremaining matrices , , and in this case study thenumerical values as in (56), shown at the bottom of the page.

This means that the disturbance is such that affects thefirst and third state of the reactor and, affects the second andfourth state. The controlled output is chosen to be equal to themeasured output , see [38], [52].

(56)


Fig. 2. Batch reactor functions � , � � �� with� �� .

A. Stability Analysis

To apply the developed framework for stability analysis, wefirst ignore the controlled output and set . Moreover, wetake and in(41). To verify (44) we take and consider a quadraticLyapunov function to compute the gain from

to (or actually a value close to thegain by selecting small) by minimizing subject to thelinear matrix inequalities (LMIs)

(57a)

(57b)

Minimizing subject to the LMI (57) with usingthe SEDUMI solver [43] with the YALMIP interface [30] pro-vides the minimal value of . This value of ap-plies for both the TOD as well as the RR protocol in (44) since

and due to the proofof Lemma V.4. From Lemma V.4 also the values for the con-stants , , and can be obtained. Then Theorem V.3can be applied to construct suitable Lyapunov functions for theclosed-loop NCS system. This results for the TOD protocol inthe values , , and

. Note that and are the same as found in [5]and [38] for the delay-free case (up to some small numerical dif-ferences). In case we now takeas , we recover exactly the resultsin [5], see Fig. 2. Indeed, checking the conditions (23) gives

and , as also found in [5].Besides this delay-free limit case, the above numerical values

provide much more combinations of that yieldstability of the NCS by varying the initial conditions and

. Actually, each pair of initial conditions provides a dif-ferent Lyapunov function as in (26) and different valuesfor as discussed after the proof of Theorem IV.2.To illustrate this, consider Fig. 3, which displays the solutions

, , to (22) for initial conditions and. The solutions , are determined

using Matlab/Simulink [using zero crossing detections to de-termine the values of and accurately according to

Fig. 3. Batch reactor functions � , � � �� with � �� and� �� .

Fig. 4. Tradeoff curves between MATI and MAD.

(23)]. The condition (23a) indicates that is determinedby the intersection of and the constant line with value

and condition (23b) states that is determinedby the intersection of and (as long as ).For the specific situation depicted in Fig. 3 this would resultin and , meaning thatUGES is guaranteed for transmission intervals up to 0.008794and transmission delays up to 0.005062. Interestingly, theinitial conditions of both functions and can be used tomake design tradeoffs. For instance, by taking larger, theallowable delays become larger (as the solid line indicated by‘o’ shifts upwards), while the maximum transmission intervalbecomes smaller as the dashed line indicated by “ ” will shiftupwards as well causing its intersection with (dotted lineindicated by “ ”) to occur for a lower value of . Hence, oncethe hypotheses of Theorem IV.2 are satisfied, a continuum ofLyapunov functions is available leading to different combi-nations of MATI and MAD. This shows that tradeoff curvesbetween and can indeed be constructed. Followingthis procedure for various increasing values of , whilekeeping equal to , provides the graphin Fig. 4, where the particular point and

corresponding to Fig. 3 is highlighted. Notethat the graph ends where as the developedmodel does not include delays larger than the transmissioninterval.


In case of the RR protocol we obtain the values, , and if

we invoke Theorem V.3 directly. However, improved values forand can be obtained by exploiting the special structure in

the matrix as was also done in [38, Ex. 3]. This is achievedby deriving directly the condition (19) instead of following thegeneral approach based on (49)–(52) using (40) and (41) in theproof of Theorem V.3. Indeed, using thatfor a diagonal matrix (with the values as in theproof of Lemma V.4 on the diagonal), we can derive directlyfor almost all and

where we used that as both matricesare diagonal. Using this sharper result in (49)–(52) instead of(40) and (41), we obtain the improved values ,

, and , similar to [5]and [38], which leads for the delay-free case to(recovering the result in [5], which outperforms the valuesfound in [38], [39], [51]). The tradeoff curve between MATIand MAD is also given in Fig. 4. In this figure also the delay-freecase with and is visualized. Thesetradeoff curves can be used to impose conditions or select asuitable network with certain communication delay and band-width requirements (note that MATI is inversely proportionalto the bandwidth).

Also different protocols can be compared with respect to eachother. In Fig. 4, it is seen that for the task of stabilization ofthe unstable batch reactor the TOD protocol outperforms theRR protocol in the sense that it can allow for larger delaysand larger transmission intervals. The difference between thetradeoff curves for different protocols is caused by differentvalues for the parameters , , and (see LemmaV.4), which in turn induce different values for the parameters

, , and (see Theorem V.3) and thus different solu-tions to the differential equations (22). This results in differentcombinations of and that guarantee stability due toTheorem IV.2.

B. Gain Analysis

To apply the developed framework for gain analysis, theeffect of the additional disturbance input on the controlledvariable is studied.

Fig. 5. Pareto optimal curves for �� for the TOD protocol.

We take andin (41). To verify (45) we

compute simultaneously an upper bound on the gain fromto and on the gain

from to by finding Pareto minimal values for and subjectto the matrix inequalities in the matrix and multiplieras given in (58), shown at the bottom of the page. These matrixinequalities demonstrate that there will be a tradeoff betweenperformance in terms of the gain from to reflected in

on the one hand and the size of and on the otheras reflected in . Recall that directly influences MATI andMAD through and in (48). The tradeoff betweenand can still be made by varying and whilstguaranteeing a certain gain for the NCS.

To make these observations quantitative, we fix at variousvalues and search for the smallest value of such that thereexist and satisfying (58). Note that (58) is an LMIwhen is fixed and hence, can be solved efficiently. As a lowerbound on we take the gain from to of the systemwithout the network . This value is found by solvingthe standard -gain/ LMI for linear systems (see e.g., [1])using, again, the SEDUMI solver and the YALMIP interface.This yields . Using this as a lower bound on the con-sidered values of the desirable performance level in terms of the

gain , we search minimal values for (corresponding to theselected value of ) under the feasibility of the LMI (57). Theseminimal values yield the “Pareto optimal curves” for asshown in Fig. 5. This figure demonstrates that the minimal valueof approaches 15.9165 for large values of , which was thegain from to as computedin Section VI-A for the stability analysis. This is expected as thevalue 15.9165 corresponds to the smallest value of found in

(58)


Fig. 6. Tradeoff curves between MATI and MAD for various levels of the �gain of the NCS with the TOD protocol.

Section VI-A if only stability is required (without any additionalperformance conditions, so the gain approaches in-

finity). The other extreme, when approaches infinity, recoversthe situation where the values approach the asymptotic valueof being the optimal network-free gain fromto .

As we can see, a smaller gain requires a larger valueof , which will in turn result in smaller values for and

. We will demonstrate this for the TOD protocol. The re-sults for the RR protocol can be obtained using the same ap-proach. The computed values of are now used in (45) andthe values for the constants , , and can be obtainedfrom Lemma V.4, as in the stability analysis above. Then The-orem V.3 can be applied to construct a continuum of suitablestorage functions for the closed-loop NCS system choosing acertain combination of in a similar manner as for the sta-bility analysis. This leads to combinations ofsuch that the NCS has an gain from to smaller than fortransmission intervals smaller than and communicationdelays smaller than . The tradeoff curves for various levelsof the performance are provided in Fig. 6. For the valueof we (almost) recover the tradeoff plot as in Fig. 4for the TOD protocol, because, loosely speaking, the gainrequirement is very mild as is a relatively large valuethat practically approaches the condition that the NCS should beUGES only. In the other extreme, if very high requirements aregiven with respect to robustness to disturbances (in the senseof a very low gain from to approaching the network-free

gain ), the values for MATI and MAD that guar-antee this gain are approaching 0. This is clearly shown inFig. 6 as the tradeoff curve for the value corre-sponds to values of and close to zero. The limit case

would actually correspond to and .Hence, the control and network engineers have to make cleardesign tradeoffs between MATI, MAD, robustness in terms of

performance and the choice of the protocol. The providedframework supports the engineers to make these design choicesin a quantified manner.

VII. CONCLUSION

In this paper, we presented a framework for studying thestability of a NCS, which involves communication constraints(only one node accessing the network per transmission), varyingtransmission intervals and varying transmission delays. Basedon a newly developed model, a Lyapunov-based characteriza-tion of stability was provided and explicit bounds on the max-imally allowable transmission interval (MATI) and the max-imally allowable delay (MAD) were obtained that guaranteestability of the NCS. We explicitly showed how a continuumof Lyapunov functions can be constructed from the commonlyadopted conditions for the delay-free case. The application ofthe results on a benchmark example showed how tradeoff curvesbetween MATI and MAD can be computed providing designersof NCS with proper tools to support their design choices. In-terestingly, recently developed improvements in [5] leading tosharper bounds for the MATI (the non-delay case) are a spe-cial case of this more general framework. Additionally, we ana-lyzed the performance of NCS and provided the theoreticalframework that shows how MATI, MAD and can be tradedquantitatively against each other. Under weaker conditions, weprovided also semiglobal practical stability results.

Future work will involve the consideration of the large delaycase (delays larger than the transmission interval) and the devel-opment of an analysis framework that also includes quantizationeffects. Extending the current framework so as to include quan-tization effects would provide the means to study NCS incor-porating all the types of network phenomena mentioned in theintroduction (as packet dropouts can be modeled using a prolon-gation of the MATI as discussed in Remark II.4). We foresee thatsuch an extension would be extremely valuable to network andcontrol system designers, provided that the extended frameworkleads to quantitative tradeoff curves between the various net-work parameters (MATI, MAD, quantization error, bandwidth,etc.) and the performance of the overall control loop. Some pre-liminary results in this direction can be found in [21].

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W. P. Maurice H. Heemels (M’06) received theM.Sc. degree in mathematics (with honors) and thePh.D. degree (with honors) from the EindhovenUniversity of Technology (TU/e), Eindhoven, TheNetherlands, in 1995 and 1999, respectively.

From 2000 to 2004 he was with the Electrical Engi-neering Department, TU/e, as an Assistant Professorand from 2004 to 2006 with the Embedded SystemsInstitute (ESI) as a Research Fellow. Since 2006, hehas been with the Department of Mechanical Engi-neering, TU/e, where he is currently a Full Professor

in the Hybrid and Networked Systems Group. He held visiting research posi-tions at the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland(2001) and at the University of California at Santa Barbara (2008). In 2004, hewas also at the Research & Development laboratory, Océ, Venlo, The Nether-lands. Currently, he is an Associate Editor for AUTOMATICA and Nonlinear Anal-ysis: Hybrid Systems. His current research interests include hybrid and non-smooth dynamical systems, networked control systems and constrained systemsincluding model predictive control.

Dr. Heemels serves in the international program committees of variousconferences.

Andrew R. Teel (S’91–M’92–SM’99–F’02) re-ceived the A.B. degree in Engineering Sciences fromDartmouth College, Hanover, NH, in 1987, and theM.S. and Ph.D. degrees in electrical engineeringfrom the University of California, Berkeley, in 1989and 1992, respectively.

He was a Postdoctoral Fellow at the Ecole desMines de Paris, Fontainebleau, France. In Septemberof 1992 he joined the faculty of the ElectricalEngineering Department, University of Minnesota,Minneapolis, where he was an Assistant Professor

until September of 1997. In 1997, he joined the faculty of the Electrical andComputer Engineering Department, University of California, Santa Barbara,where he is currently a Professor.

Dr. Teel received NSF Research Initiation and CAREER Awards, the 1998IEEE Leon K. Kirchmayer Prize Paper Award, the 1998 George S. Axelby Out-standing Paper Award, the first SIAM Control and Systems Theory Prize in1998, and the 1999 Donald P. Eckman Award and the 2001 O. Hugo SchuckBest Paper Award from the American Automatic Control Council.

Nathan van de Wouw (M’08) received the M.Sc.degree (with honors) and Ph.D. degree in mechan-ical engineering from the Eindhoven University ofTechnology, Eindhoven, the Netherlands, in 1994 and1999, respectively.

From 1999 until 2010, he was with the Dynamicsand Control Group, Department of MechanicalEngineering, Eindhoven University of Technologyas an Assistant/Associate Professor. In 2000, hasbeen working at Philips Applied Technologies,Eindhoven, The Netherlands, and, in 2001, he has

been working at the Netherlands Organisation for Applied Scientific Research(TNO), Delft, The Netherlands. He has held positions as a Visiting Professorat the University of California Santa Barbara, in 2006 and 2007 and at theUniversity of Melbourne, Australia, in 2009 and 2010. He has published a largenumber of journal and conference papers and co-authored the books UniformOutput Regulation of Nonlinear Systems: A convergent Dynamics Approach(Boston, MA: Birkhauser, 2005) and Stability and Convergence of MechanicalSystems with Unilateral Constraints (New York: Springer-Verlag, 2008). Hiscurrent research interests are the analysis and control of nonlinear/non-smoothsystems and networked control systems.

Dragan Nesic (SM’01–F’08) received the B.E. de-gree from The University of Belgrade, Yugoslavia,in 1990, and the Ph.D. degree from Systems Engi-neering, RSISE, Australian National University, Can-berra, Australia in 1997.

He is currently a Professor in the Department ofElectrical and Electronic Engineering (DEEE), Uni-versity of Melbourne, Australia. From 1997 to 1999,he held postdoctoral positions at the University ofMelbourne, Australia, the Université Catholique deLouvain, Louvain la Neuve, Belgium and the Uni-

versity of California, Santa Barbara, CA. Since February 1999 he has been withThe University of Melbourne. He is an Associate Editor for Automatica, Systemsand Control Letters, and the European Journal of Control. His research inter-ests include networked control systems, discrete-time, sampled-data and contin-uous-time nonlinear control systems, input-to-state stability, extremum seekingcontrol, applications of symbolic computation in control theory, and so on.

Dr. Nesic received the Humboldt Research Fellowship funded by theAlexander von Humboldt Foundation, Germany, in 2003. He was an AustralianProfessorial Fellow from 2004 to 2009 (which is a research position funded bythe Australian Research Council). He is a Fellow of IEAust and a DistinguishedLecturer of CSS of the IEEE. He served as an Associate Editor for the IEEETRANSACTIONS ON AUTOMATIC CONTROL.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 8

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