Lecture Notes in Control and Information Sciences 423

Lecture Notesin Control and Information Sciences 423

Editors: M. Thoma, F. Allgöwer, M. Morari

Rifat Sipahi, Tomáš Vyhlídal,Silviu-Iulian Niculescu,and Pierdomenico Pepe (Eds.)

Time Delay Systems:Methods, Applicationsand New Trends

ABC

Series Advisory BoardP. Fleming, P. Kokotovic,A.B. Kurzhanski, H. Kwakernaak,A. Rantzer, J.N. Tsitsiklis

Editors

Prof. Rifat SipahiNortheastern UniversityDepartment of Mechanical andIndustrial EngineeringBoston, USA

Prof. Tomáš VyhlídalCzech Technical UniversityFaculty of Mechanical EngineeringDepartment of Instrumentation and

Control EngineeringPrague, Czech Republic

Prof. Silviu-Iulian NiculescuL2S (UMR CNRS 8506)CNRSSupélecGif-sur-YvetteFrance

Prof. Pierdomenico PepeUniversity of L’AquilaDepartment of Electrical andInformation EngineeringL’Aquila, Italy

ISBN 978-3-642-25220-4 e-ISBN 978-3-642-25221-1

DOI 10.1007/978-3-642-25221-1

Lecture Notes in Control and Information Sciences ISSN 0170-8643

Library of Congress Control Number: 2011940774

c© 2012 Springer-Verlag Berlin Heidelberg

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Preface

In many problems in engineering, physics, and biology, various processes take effectonly after a certain amount of time elapses after the onset of a stimulus, an input, andany means of cause. This period of time, which is application specific, can arise dueto many reasons including, among others, transmitting information across a wirelessnetwork in network control systems, shipping products from one location to anotherin supply chains, producing a decision in car driving upon receiving a stimulus.

Due to the presence of delays, instantaneous information cannot be available andtherefore many control actions are produced based on the historical evolution ofthe governing dynamics. This represents a major source of instability, especiallyin cases when the controllers are designed by neglecting delays. In other words, acontroller designed in a conventional way by assuming that the system at hand didnot present delays, may not necessarily guarantee the stability of the same systemthat actually includes delays.

Since 1950s, many studies have been devoted to understanding the effects ofdelays and exploring control theoretic approaches to circumvent the detrimental ef-fects of delays to dynamical behavior. One of the first tools to deal with delays isthe celebrated Smith predictor, dated 1957. On the other hand, an interesting phe-nomenon discovered in these studies is the so-called stabilizing effect of delays: in-ducing delays in the control loop may lead to stable closed-loop system which canbe lost when the delay becomes zero. Such a stabilizing feature of delay in dynami-cal systems works against the human intuition and therefore necessitates analyticalstudies. Consequently, this feature has become the motivation of many theoreticaland experimental studies in the literature, especially in the past two decades.

During this period, many results have been presented at the main control confer-ences (IEEE CDC, ACC, IFAC symposia), in specialized workshops (IFAC Time-Delay Systems series), and published in the leading journals of control engineering,manufacturing engineering, operations research, mathematical biology, systems andcontrol theory, applied and numerical mathematics. Furthermore, numerous bookson the topic appeared, and two review articles were published quite recently in IEEEControl Systems Magazine. In the control area, the results are primarily based on theanalysis of systems with delays (such as stability, controllability, performance), and

VI Preface

on the synthesis of controllers (designing an appropriate control law that achievescertain control objectives). For various specific classes of models, including linear,nonlinear, and stochastic systems, many results have been obtained, while in the nu-merical analysis area the research has focused on time-stepping, discretization andsemi-discretization, computation of stability and corresponding stability charts, andbifurcation analysis.

Although many results are mature, the existing work has so far remained validfor stand-alone systems that have relatively small dimensions and a small numberof delays. On the other hand, the challenges faced by the researchers is increasing,in parallel with the rapidly developing technology and higher expectations fromresearchers and control engineer practitioners. The main challenges are due to han-dling large-scale systems, nonlinear systems, discrete-continuous type mixed sys-tems, interconnected systems over networks, multi-input multi-output systems withdelays of various features such as time-varying, uncertain, and stochastic delays.In the analysis and controller synthesis of such systems, there still remains manyuncharted research territories where new results and major impacts are needed.

The arising complexities in the control problems are also in parallel with theversatility of systems aligned with the emergence of multiple disciplines, includingengineering, physics, chemistry, biology, operations research, and economics. Thistrend also calls for researchers from different disciplines to work together, in orderto better understand and address control problems. Therefore, it is of no surprisetoday that control engineers, medical doctors, chemical engineers, and mathemati-cians work together in defining new frameworks for dealing with the complexityof the systems to be handled. In one sense, the research on time delay systemsis expanding and becoming more pervasive, and the publications in this field willgrow even further and will be accessed by many researchers including those outsideengineering.

This book presents the most recent trends as well as new directions in the fieldof control and dynamics of time delay systems. The field is extremely active andthe book captures the most recent snapshot of the research results. The book iscollected under five parts: (i) Methodology: From Retarded to Neutral ContinuousDelay Models, (ii) Systems, Signals and Applications, (iii) Numerical Methods, (iv)Predictor-based Control and Compensation, and (v) Networked Control Systemsand Multi-agent Systems. The themes of these chapters closely tie with the discus-sions and motivations above. The contributions in the chapters are a well-balancedcombination of two main resources; invited papers, and the work presented at 2009IFAC Workshop on Time Delay Systems held in Sinaia, Romania and 2010 IFACWorkshop on Time Delay Systems held in Prague, Czech Republic. This workshoprepresents the main specialized meeting venue in the field, and thus captures themost updated research trends. In the selection of the topics and the contributors, theeditors have not only aimed at maintaining the highest technical quality of the pre-sentations, but also at achieving an appropriate balance across the chapters (i)-(v).It is worthy to note that the book proposal does not have significant overlap withthe contents of 2009-2010 IFAC Workshops on Time Delay Systems (IFAC TDS).Attendees of the workshop contributing to this book proposal also improved their

Preface VII

contributions beyond what they presented at the workshop and the editors comfort-ably claim that at least 50% of the book content-wise comprises new contributionsthat are closely relevant to the most recent trends discussed above.

Structure of the Volume

As mentioned earlier, this volume is divided into five parts, where each part is com-posed of five to eight chapters. They are respectively devoted to Part I: Methodology:From Retarded to Neutral Continuous Delay Models, Part II: Systems, Signals andApplications, Part III: Numerical Methods, Part IV: Predictor-based Control andCompensation, and Part V: Networked Control Systems and Multi-agent Systems.The first part is concerned with the new results on retarded and neutral type delaydifferential equations. The second part is devoted to problems with flows, signalsand the corresponding real-world applications. The third part is related to numeri-cal methods, which are indispensable part of the research momentum in this field.The fourth and the fifth parts are concerned with the recent trends, which coverrespectively predictor-based control/compensation and networked control systemswith multi agents. In what follows, we present the different chapters and streams ofthe different parts.

Part I Methodology: From Retarded to Neutral Continuous Delay Models

Almost all continuous dynamical systems with delays are modeled by either re-tarded or neutral-type delay differential equations. It is therefore extremely impor-tant to develop appropriate analysis tools and methods, and synthesis schemes forsuch differential equations. In this research direction, many problems are still open,including, among others, improvements on computational efficiency, control of non-linear systems using observers, new ideas for control system design, connectionsbetween frequency and time-domain tools, new ways in constructing Lyapunovquadratic functionals that are not necessarily continuous. In this part of the book,some of these important problems are discussed and addressed.

The first chapter in this part is contributed by VLADIMIR L. KHARITONOV andpresents new results in computing quadratic functionals for linear neutral-type sys-tems. Particular emphasis here is on the Lyapunov matrices, which are defined byspecial matrix valued functions. The focus on such matrices is important as thiseffort ties with stability analysis and robust stability analysis of time delay systems.

The second chapter is contributed by MICHAEL DI LORETO and JEAN JACQUES

LOISEAU and it discusses the stability of positive difference equations, which arisein many stability problems associated with neutral-type delay differential equations.The chapter shows that the stability of linear difference equations with positive co-efficients is robust with respect to delays. This result is crucial since otherwise thestability analysis of difference equations with multiple delays can be computation-ally cumbersome. The result supplements a critical stability analysis step in neutraltype systems.

The third chapter contributed by KEQIN GU, YASHUN ZHANG, and MATTHEW

PEET concerns the positivity of quadratic functionals, which are key components

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in time-domain based stability analysis. Under single and double integral positivityconditions, the chapter utilizes appropriate notions from operator theory to showthe positivity of quadratic functionals, and studies coupled differential-differenceequations using efficient sum-of-squares tools.

The fourth chapter, which is contributed by MILENA ANGUELOVA andMIROSLAV HALAS, presents a class of retarded-type nonlinear systems that ad-mit an input-output representation of neutral type. Interestingly, the representationbecomes possible only with nonlinearities present in the dynamical system, and theauthors present conditions under which such a representation exists.

The fifth chapter is contributed by GILBERTO OCHOA, SABINE MONDIE, andVLADIMIR L. KHARITONOV. The chapter studies an important problem in linearneutral-type systems, where the computation of critical eigenvalues has direct linkswith system stability. Different from existing results, the authors consider Lyapunovmatrix in order to compute such eigenvalues, which are then used to find the delayintervals in which the system maintains its asymptotic stability.

The last chapter, contributed by ROB H. GIELEN, MIRCEA LAZAR and SORIN

OLARU, focusses on the relation between stability of delay difference equations(DDEs) and the existence of D-contractive sets, which provide a region of attrac-tion. First, it is established that a DDE admits a D-contractive set if and only ifit admits a Lyapunov-Razumikhin function. This and subsequently derived neces-sary conditions provide a first step towards the derivation of a notion of asymptoticstability for DDEs which is equivalent to the existence of a D-contractive set.

Part II Systems, Signals and Applications

This part presents an array of contributions that encompass system-level applica-tions to various real-world control problems, in which time delays affect the in-volved signals. The organization of this chapter is as follows.

The first chapter in this part is contributed by SERGEI AVDONIN and LUCIANO

PANDOLFI. The authors focus on the relation between temperature and flux for heatequations with memory. First, it is proved that the relation between temperature andheat governing equations are strict for small times, but these quantities are essen-tially independent for large times. In the problem analysis, the observation that theindependence can be interpreted as a kind of controllability is utilized.

The second chapter, contributed by LOTFI BELKOURA, investigates the identifi-ability and algebraic identification of time-delay systems. It is shown that the iden-tifiability property of a general class of systems described by convolution equationscan be formulated in terms of approximate controllability or weak controllability,depending on the available models. Consequently, an algebraic method for the iden-tification of delay systems based on both structured inputs and arbitrary input-outputtrajectories is presented. The proposed on-line algorithms for both parameters anddelay estimation are validated on both the simulation and experimental studies withnoisy data.

The third chapter is contributed by RUDY CEPEDA-GOMEZ and NEJAT OLGAC.The study addresses the consensus problem for a group of autonomous agents withsecond order dynamics and time-delayed communications. A consensus protocol for

Preface IX

a system of agents with second-order dynamics is proposed, under the assumptionsthat all agents can communicate with each other and there is a constant communica-tion delay for all. Using the Cluster Treatment of Characteristic Roots (CTCR) pro-cedure, a complete stability picture is obtained, taking into account the variations inthe control parameters and the communication delay. Case studies and simulationsare presented to illustrate the analytical derivations.

The fourth chapter, contributed by ERIK I. VERRIEST, focuses on the analysisof state-space construction for systems with time-varying delay. The fundamentalcondition for the existence of the state-space is that the delay derivative should bebounded by one. Besides, using a discretization approach, simple derivation of thespectral reachability condition for linear time-invariant (LTI) delay systems is pro-vided. It is also shown that when a system with fixed delay is modeled as one in aclass with larger delay, reachability can no longer be preserved.

The fifth chapter of the part is contributed by VLADIMIR RASVAN, which dis-cusses some dynamical models in automatic control that are connected with dis-tributed parameters in one dimension. These models are described by boundaryvalue problems for hyperbolic partial differential equations (PDEs). The functionalequations associated to these problems are considered by using the integration of theRiemann invariants along the characteristics. The chapter contains some illustratingapplications from various fields: nuclear reactors with circulating fuel, canal flowscontrol, overhead crane, drilling devices, without forgetting the standard classicalexample of the nonhomogeneous transmission lines for distortionless and losslesspropagation. Specific features of the control models are discussed in connectionwith the control approach wherever it applies.

The sixth chapter is contributed by TAMAS INSPERGER, RICHARD WOHLFART,JANOS TURI and GABOR STEPAN. The authors study the dynamics of the stickbalancing control where the output for the feedback controller is provided by anaccelerometer attached to the stick. In the analysis, different models are consideredin the loop with proportional-derivative controller. The stability of the feedback sys-tem is studied for the cases with and without feedback delays. Besides, the effect ofcontroller discretization is studied. It is shown that if the accelerometer dynamicsis considered in the model with feedback delay, the advanced closed loop dynamics(with infinitely many unstable poles) is obtaiend. Once the controller is discretized,the system can be stabilized despite its advanced nature.

The seventh chapter, contributed by LOUAY SALEH, PHILIPPE CHEVREL andJEAN-FRANCOIS LAFAY, is dedicated to the study the characteristics of the op-timal preview control for lateral steering of a passenger vehicle, provided the nearfuture curvature of the road is known. The synthesis is performed in continuous timeand leads to a two-degrees of freedom feedback and feedforward controller, whosefeedforward part is a finite impulse response filter. Both the theoretical and experi-mental results show that the preview control action enables the vehicle to track thecenter of the lane with a smaller tracking error.

In the last, the eighth chapter of this part, HITAY OZBAY, CATHERINE BON-NET, HOUDA BENJELLOUN and JEAN CLAIRAMBAULT study local asymptoticstability conditions for the positive equilibrium of a system modeling cell

X Preface

dynamics in leukemia. Local asymptotic stability conditions are derived for the pos-itive equilibrium point of this nonlinear model. As the main result, guidelines fordevelopment of therapeutic actions are proposed based on the stability conditionsderived in terms of inequalities.

Part III Numerical Methods

Roughly speaking, due to the many computational challenges involved in time delaysystems, it is of no surprise today that the need and the demand for reliable numer-ical tools is always a focal point, with more and more versatility expected fromthese tools. In this line of research, this part presents optimal control design tech-niques, discretization of solution operators, applications to distributed delays, toolsfor delay-independent stability test, analysis of non-uniformly sampled systems, nu-merical methods for constructing Lyapunov matrices and analyzing the stability ofsystems found in biology. The details of the chapters are as follows.

The first chapter of this part contributed by WIM MICHIELS provides an overviewof the eigenvalue-based robust control design methods for linear time delay systemswith the fixed-order controllers. The analysis concerns the computation of stabil-ity determining characteristic roots and the computation of H2 and H∞ type costfunctions. The control synthesis is performed using direct optimization algorithmsapplied to minimizing either the spectral abscissa (stabilization problem), or H2,or H∞ norms (robust control design), which are, in general, non-convex and evennon-smooth objective functions. As mentioned in the chapter, the author and hiscolleagues have recently implemented the methods into the freely available soft-ware tools.

In the second chapter, contributed by DIMITRI BREDA, STEFANO MASET andROSSANA VERMIGLIO, a numerical scheme to discretize the solution operators oflinear time-invariant time-delay systems in Hilbert spaces is proposed and analyzed.Combining polynomial collocation and Fourier projection, a numerical scheme isproposed for discretizing the solution operator of the system under consideration.Next, step-by-step analysis of the discretization approaches results in the matri-ces necessary for code implementations, and detailed convergence analysis of themethod is performed.

The third chapter of this part is contributed by ELIAS JARLEBRING, WIM

MICHIELS and KARL MEERBERGEN. The authors show that the Arnoldi method,which is a well-established numerical method for standard and generalized eigen-value problems, can also be used for infinite-dimensional problems. First, the au-thors introduce the infinite Arnoldi method and provide a step-by-step algorithm forimplementation. Consequently, the method is adapted for time-delay system withdistributed delays. After outlining the connection with the Fourier cosine transform,the convergence properties in the eigenvalue computation are illustrated on two ex-amples.

The fourth chapter, contributed by ALI FUAT ERGENC, presents an originalmethod for determining delay-independent stability zones of the general LTI dy-namics with multiple delays against parametric uncertainties. The method utilizes

Preface XI

the extended Kronecker summation and unique properties of self-inversive polyno-mials. The main result of the chapter is represented by the sufficient condition fordelay-independent stability. Several examples demonstrate the applicability of thetheoretical results.

In the fifth chapter, LAURENTIU HETEL, ALEXANDRE KRUSZEWSKI and JEAN-PIERRE RICHARD propose a method for computing the Lyapunov exponent of sam-pled data systems with sampling jitter. The proposed method is hybrid, in the sensethat it combines continuous-time models (based on time-delay systems) with poly-topic embedding methods (specific to discrete-time approaches). A lower boundof the Lyapunov exponent can be expressed as a generalized eigenvalue problem.Numerical examples complete the presentation and illustrate the improvement incomparison with other classical approaches.

In the sixth chapter, OLGA N. LETYAGINA and ALEXEY P. ZHABKO propose anumerical procedure for the construction of Lyapunov functionals with a prescribedtime-derivative for the case of delay systems with periodic coefficients. Similar tothe case of time-invariant systems, the functionals are defined by using special Lya-punov matrices. Next, the issues of existence, exponential estimation, robust stabil-ity and computational issues are studied.

The seventh chapter by WARODY LOMBARDI, SORIN OLARU and SILVIU-IULIAN NICULESCU proposes a generic approach to obtain polytopic models forsystems with inputs affected by time-delays. Three different cases are studied thatarise during the discretization process with respect to the structure of the continuous-state transition matrix. The goal is to model the variable input delays as a polytopicuncertainty in order to preserve a linear difference inclusion framework. The pro-posed approach guarantees a fixed complexity simplex-type global embedding.

Part IV Predictor-Based Control and Compensation

Predictor-based control goes back to 1950s with the results of Smith predictor,which has become one of the major contributions in control systems field in terms ofcontrolling a class of systems with delay. Many other predictive-based approachescurrently exist in the literature, including a popular one called model predictive con-trol, which has found broad application in industry. Presence of delay poses majorproblems in designing predictors and using them as compensation schemes.

The first chapter by MIROSLAV KRSTIC provides a tutorial introduction to meth-ods for stabilization of systems with long input delays, the so-called predictor feed-back techniques. The methods are based on techniques originally developed forboundary control of partial differential equations using the backstepping approach.Several adaptive and compensation schemes are considered for both linear and non-linear systems with delays of different nature. Primarily, the chapter demonstratesthat the construction of backstepping transformations allow one to deal with de-lays and PDE dynamics at the input, as well as in the main line of applying controlaction. It is shown that using direct and inverse backstepping transformations, Lya-punov functionals and explicit stability estimates can be constructed.

The second chapter, contributed by ION NECOARA, IOAN DUMITRACHE andJOHAN A.K. SUYKENS, proposes two methods for solving distributively separable

XII Preface

convex problems - the proximal center method and the interior-point Lagrangianmethod. Next, the methods are extended to the case of separable non-convex prob-lems but with a convex objective function using a sequential convex programmingframework. It is also proven that some relevant centralized model predictive control(MPC) problems for a network of coupling linear (non-linear) dynamical systemscan be recast as separable convex (non-convex) problems for which the proposeddistributed methods can be applied.

The third chapter in this part, contributed by SERGIO TRIMBOLI, STEFANO

DI CAIRANO, ALBERTO BEMPORAD and ILYA V. KOLMANOVSKY focuses ona model predictive control method with delay compensation for controlling air-to-fuel ratio and oxygen storage in spark ignited engines. The control architecture isbased on a delay-free model predictive controller that enforces constraints on theactuators and on the operating range of the variables. Consequently, the architecturecomprises a delay compensation strategy based on a state predictor that counter-acts the time-varying delay. Simulations of the closed-loop system with a detailednonlinear model have been shown.

The fourth chapter, contributed by ALFREDO GERMANI, COSTANZO MANES

and PIERDOMENICO PEPE, concerns the control of a class of nonlinear retardedsystems via an observer-based stabilization scheme. The contributions include bothlocal and global stability, as well as ways to separately design the observer and thecontrollers, in order to achieve stabilization.

The fifth chapter, contributed by PAVEL ZITEK, VLADIMIR KUCERA and TOMAS

VYHLIDAL, focuses on developing an appropriate cascade control architecture fortime-delayed plants based on affine parameterization. The work makes use of quasi-integrating meromorphic functions in order to prescribe the desired open-loop be-havior. The arising cascade control has several advantages including its superiorityover standard control schemes.

The last chapter, contributed by MARCUS REBLE and FRANK ALLGOWER,presents results on model predictive control (MPC) for nonlinear time-delay sys-tems. Two schemes for calculating stabilizing design parameters based on the Ja-cobi linearization of the nonlinear time-delay system are presented. In the first part,an arbitrary stabilizing linear local control law is considered. It is shown that a sta-bilizable Jacobi linearization implies the existence of a suitable quadratic terminalcost functional and terminal region. In the second part, a simpler terminal region isderived using additional Lyapunov-Razumikhin conditions.

Part V Networked Control Systems and Multi-Agent Systems

Versatile control systems can be created by the synergy of the combination ofmany subsystems that effectively and synchronously work together to deliverhigh-performance behavior that cannot be otherwise obtained from individual sub-systems. Many engineering as well as biological systems work with this principle,bringing together an ensemble of dynamic systems. In engineering systems, the sub-systems can be physically distanced from each other yet can communicate via var-ious communication media, such as internet. The emergence of wireless networks,advancements in robotics, control systems, as well as the need to understand how

Preface XIII

a group of animals coordinate their group behavior have led to many new trends,especially in the presence of communication delays among the group members. Inthis part, most recent results in networked control systems and multi-agent systemsunder the influence of delays are presented.

The first chapter, contributed by ALEXANDRE SEURET and KARL H. JOHANS-SON, derives a robust controller for networked control systems with uncertain plantdynamics. The link between the nodes is disturbed by time-varying communica-tion delays, samplings and time-synchronization. A stability criterion for a robustcontrol is presented in terms of LMIs based on Lyapunov-Krasovskii techniques. Asecond-order system example is considered and the relation between the admissiblebounds of the synchronization error and the size of the uncertainties is computed.

The second chapter, contributed by RAFAEL C. MELO, JEAN-MARIE FARINES

and JULIO E. NORMEY-RICO, presents the modelling and congestion control ofTCP protocols. A nonlinear and a simple linear model are used to represent TCPincluding comparative results with NS-2 network simulator. Two control schemesare derived based on simple first-order plus dead-time model: i) general predictivecontroller; ii) proportional-integral plus Smith Predictor. These are compared with anonlinear based predictive controller. The results obtained using NS-2 demonstratethat the PI plus Smith Predictor offers the best trade-off between complexity andperformance to cope with the process dead-time and network disturbances.

The third chapter of the part, contributed by WEI QIAO and RIFAT SIPAHI, stud-ies the indirect relationship between the delay margin of coupled systems and dif-ferent graphs these systems form via their different topologies. A four-agent lineartime-invariant (LTI) consensus dynamics is taken as a benchmark problem with asingle delay and second-order agent dynamics. First, Cluster Treatment of Charac-teristic Roots (CTCR) is applied to reveal the delay margin for all possible topolo-gies of the agents. Next, the stability analysis is extended to the case when graphstransform from one to another as the coupling strengths of some links between theagents weaken and vanish.

In the fourth chapter by FATIHCAN M. ATAY, the author studies the consensusproblem on directed and weighted networks in the presence of time-delays. The con-nection structure of the network with information transmission delays is describedby a normalized Laplacian matrix. It is shown that consensus is achieved if and onlyif the underlying graph contains a spanning tree, independently of the value of thedelay. Next, the consensus value is calculated and it is shown that the consensusvalue is determined not just by the initial states of the nodes at time zero, but alsoon their past history over an interval of time.

The fifth chapter is contributed by IRINEL-CONSTANTIN MORARESCU, SILVIU-IULIAN NICULESCU, and ANTOINE GIRARD. It discusses consensus problems fornetworks of dynamic agents with fixed and switching topologies in presence of de-lay in the communication channels. The study provides sufficient agreement con-ditions in terms of delay and the second largest eigenvalue of the Perron matricesdefining the collective dynamics. An exact delay bound is determined assuring thepreservation of the initial network topology. Besides, the authors present an analysis

XIV Preface

of the agreement speed when the asymptotic consensus is achieved. Some numericalexamples complete the presentation.

The last chapter of this part is contributed by KUN LIU and EMILIA FRIDMAN

on the H∞ control of networked control systems. The chapter presents a new stabilityand L2-gain analysis of such systems inspired by discontinuous Lyapunov functionsthat were introduced in the literature. The novelty in this chapter is represented bythe extensions of time-dependent Lyapunov functional approach to networked con-trol systems, where variable sampling intervals, data packet dropouts and variablenetwork-induced delays are taken into account.

Last but not least, we would like to thank the editors of the LNCIS book series forprofessionally handling the volume and the reviewers for their careful suggestionswhich improved the overall quality of the volume.

Boston, Rifat SipahiPrague, Tomas VyhlıdalGif-sur-Yvette, Silviu-Iulian NiculescuL’Aquila, Pierdomenico Pepe

August 2011

Contents

Part I Methodology: From Retarded to Neutral Continuous Delay Models

Lyapunov Functionals and Matrices for Neutral Type Time DelaySystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Vladimir L. Kharitonov

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Lyapunov Matrices and Functionals . . . . . . . . . . . . . . . . . . . . . . . . . 63.1 Existence and Uniqueness Issues . . . . . . . . . . . . . . . . . . . . 73.2 Computational Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Lyapunov Functionals: A New Form . . . . . . . . . . . . . . . . . . . . . . . . 95 Quadratic Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

6.1 Exponential Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Quadratic Performance Index . . . . . . . . . . . . . . . . . . . . . . 136.3 Robustness Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 The H2 Norm of a Transfer Matrix . . . . . . . . . . . . . . . . . . 15

7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

On the Stability of Positive Difference Equations . . . . . . . . . . . . . . . . . . . . 19Michael Di Loreto and Jean Jacques Loiseau

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Stability of Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Zeros Location of Difference Equations . . . . . . . . . . . . . . 222.3 Stability and Robust Stability of Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 Positive Difference Equations . . . . . . . . . . . . . . . . . . . . . . 283.2 Stability of Monovariable Positive Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Stability of Multivariable Positive Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Asymptotic Behavior of Positive Difference

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Conclusions and Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Positivity of Complete Quadratic Lyapunov-Krasovskii Functionals inTime-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Keqin Gu, Yashun Zhang, and Matthew Peet

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Positive Quadratic Integral Expressions . . . . . . . . . . . . . . . . . . . . . . 394 Stability of Coupled Differential-Difference Equations . . . . . . . . . 415 SOS Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 Numerical Example and Observation . . . . . . . . . . . . . . . . . . . . . . . . 467 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

On Retarded Nonlinear Time-Delay Systems That Generate NeutralInput-Output Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Milena Anguelova and Miroslav Halas

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 Algebraic Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 Input-Output Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1 State Elimination Algorithm . . . . . . . . . . . . . . . . . . . . . . . 513.2 Neutral Input-Output Equations . . . . . . . . . . . . . . . . . . . . . 53

4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1 Input-Output Equations for Linear Delay Systems . . . . . 544.2 Input-Output Equations for Nonlinear Delay

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 Retarded Input-Output Representation . . . . . . . . . . . . . . . . . . . . . . . 577 Conclusions, Discussion and Open Problems . . . . . . . . . . . . . . . . . 58References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Computation of Imaginary Axis Eigenvalues and Critical Parametersfor Neutral Time Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Gilberto Ochoa, Sabine Mondie, and Vladimir L. Kharitonov

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Contents XVII

2.2 Lyapunov-Krasovskii Functionals . . . . . . . . . . . . . . . . . . . 632.3 Properties of the Lyapunov Matrix . . . . . . . . . . . . . . . . . . 63

3 Computation of the Lyapunov Matrix . . . . . . . . . . . . . . . . . . . . . . . 644 Frequency Domain Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . 665 Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Set-Induced Stability Results for Delay Difference Equations . . . . . . . . . . 73Rob H. Gielen, Mircea Lazar, and Sorin Olaru


2.1 Delay Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . 753 D-Contractive Sets and K L -Stability . . . . . . . . . . . . . . . . . . . . . . 764 Necessary Conditions for D-Contractive Sets . . . . . . . . . . . . . . . . . 795 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Part II Systems, Signals and Applications

Temperature and Heat Flux Dependence/Independence for HeatEquations with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Sergei Avdonin and Luciano Pandolfi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871.1 Further Applications and References . . . . . . . . . . . . . . . . 88

2 The Interpretation of Dependence/Independence of Heat andFlux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3 Strict Dependence at Small Times . . . . . . . . . . . . . . . . . . . . . . . . . . 904 Independence at Large Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.1 Preliminary Transformations and AsymptoticEstimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Riesz Bases and Riesz Sequences . . . . . . . . . . . . . . . . . . . 924.3 Temperature and Flux Regularity . . . . . . . . . . . . . . . . . . . 944.4 The Proof of Independence . . . . . . . . . . . . . . . . . . . . . . . . 98

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Identifiability and Algebraic Identification of Time Delay Systems . . . . . 103Lotfi Belkoura

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1031.1 The Distribution Framework . . . . . . . . . . . . . . . . . . . . . . . 104

2 Identifiability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.1 Sufficiently Rich Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3 Algebraic Identification: The Structured Case . . . . . . . . . . . . . . . . . 1083.1 Structured Signals and Their Annihilation . . . . . . . . . . . . 108

XVIII Contents

3.2 Application to a Single Delay Identification . . . . . . . . . . . 1083.3 Application to a Simultaneous Parameters and Delay

Identification: Experimental Example . . . . . . . . . . . . . . . . 1104 Algebraic Identification: The Unstructured Case . . . . . . . . . . . . . . 112

4.1 The Cross Convolution Approach . . . . . . . . . . . . . . . . . . . 1124.2 Application to a Delay Identification . . . . . . . . . . . . . . . . 1134.3 Experimental Results (Continued) . . . . . . . . . . . . . . . . . . . 115

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Stability Analysis for a Consensus System of a Group of AutonomousAgents with Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Rudy Cepeda-Gomez and Nejat Olgac

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.1 First Factor of (9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.2 Second Factor of (9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3 Complete Stability Picture . . . . . . . . . . . . . . . . . . . . . . . . . 126

4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

State Space for Time Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Erik I. Verriest

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1352 State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363 Functional Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374 Non-differential Functional Equation . . . . . . . . . . . . . . . . . . . . . . . 1415 Reachability of Systems with Fixed Point Delay . . . . . . . . . . . . . . 142

5.1 PBH Test for Delay Systems . . . . . . . . . . . . . . . . . . . . . . . 1425.2 State Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Delays. Propagation. Conservation Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . 147Vladimir Rasvan

1 Introduction and Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1472 Lossless and Distortionless Propagation . . . . . . . . . . . . . . . . . . . . . 1493 The Multi-wave Case. Application to the Circulating Fuel

Nuclear Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 A Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535 Dynamics and Control for Systems of Conservation Laws . . . . . . 1566 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Contents XIX

Equations with Advanced Arguments in Stick Balancing Models . . . . . . . 161Tamas Insperger, Richard Wohlfart, Janos Turi, and Gabor Stepan

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1612 Different Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623 Analysis of the Different Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Optimal Control with Preview for Lateral Steering of a PassengerCar: Design and Test on a Driving Simulator . . . . . . . . . . . . . . . . . . . . . . . 173Louay Saleh, Philippe Chevrel, and Jean-Francois Lafay

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1732 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1743 The H2/LQ-Preview Problem Solution . . . . . . . . . . . . . . . . . . . . . . 1754 Solution Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775 Application to Car Lateral Steering Control . . . . . . . . . . . . . . . . . . 178

5.1 Simplified Lateral Model of the Vehicle . . . . . . . . . . . . . . 1785.2 Experimental Setup and Results . . . . . . . . . . . . . . . . . . . . 180

6 Preview Horizon and Weighting Matrix Impact . . . . . . . . . . . . . . . 1837 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Local Asymptotic Stability Conditions for the Positive Equilibrium ofa System Modeling Cell Dynamics in Leukemia . . . . . . . . . . . . . . . . . . . . . 187Hitay Ozbay, Catherine Bonnet, Houda Benjelloun, and Jean Clairambault

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1872 Mathematical Model of Cell Dynamics in Leukemia . . . . . . . . . . . 1883 Stability Analysis for the Positive Equilibrium . . . . . . . . . . . . . . . . 190

3.1 Local Asymptotic Stability for μ > 0 . . . . . . . . . . . . . . . . 1913.2 Local Asymptotic Stability for μ < 0 . . . . . . . . . . . . . . . . 191


Part III Numerical Methods

Design of Fixed-Order Stabilizing and H2 - H∞ Optimal Controllers:An Eigenvalue Optimization Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Wim Michiels

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2012 Solving Analysis Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

2.1 Computation of Characteristic Roots and the SpectralAbscissa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

2.2 Computation of H∞ Norms . . . . . . . . . . . . . . . . . . . . . . . . 2052.3 Computation of H2 Norms . . . . . . . . . . . . . . . . . . . . . . . . 206

3 Solving Synthesis Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2083.1 Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2093.2 H∞ and H2 Optimization Problems . . . . . . . . . . . . . . . . . 210

XX Contents

4 Software and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Discretization of Solution Operators for Linear Time Invariant - TimeDelay Systems in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Dimitri Breda, Stefano Maset and Rossana Vermiglio

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2172 Solution Operators and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2193 Projection and Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

3.1 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.2 The Matrix Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

The Infinite Arnoldi Method and an Application to Time-DelaySystems with Distributed Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229Elias Jarlebring, Wim Michiels, and Karl Meerbergen

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2292 Operator Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2303 The Infinite Arnoldi Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2314 Adaption for Time-Delay Systems with Distributed Delays . . . . . 233

4.1 Computing y0 for Distributed Delays . . . . . . . . . . . . . . . . 2334.2 Connection with the Fourier Cosine Transform . . . . . . . . 235

5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2365.1 Example 1: A Rectangular Function . . . . . . . . . . . . . . . . . 2365.2 Example 2: A Gaussian Distribution . . . . . . . . . . . . . . . . . 237


A New Method for Delay-Independent Stability of Time-DelayedSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241Ali Fuat Ergenc

1 Introduction and the Problem Statement . . . . . . . . . . . . . . . . . . . . . 2412 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2423 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2444 Example Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

A Hybrid Method for the Analysis of Non-uniformlySampled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253Laurentiu Hetel, Alexandre Kruszewski, and Jean-Pierre Richard

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2532 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Contents XXI

3 Lower Estimate of the Lyapunov Exponent . . . . . . . . . . . . . . . . . . . 2563.1 Time-Delay Model of the System . . . . . . . . . . . . . . . . . . . 2563.2 Theoretical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2573.3 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2583.4 Numerical Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 260

4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2625 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

A Numerical Method for the Construction of Lyapunov Matrices forLinear Periodic Systems with Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . 265Olga N. Letyagina and Alexey P. Zhabko

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2652 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2663 Complete Type Lyapunov Functionals . . . . . . . . . . . . . . . . . . . . . . . 2674 Existence Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2695 Exponential Estimation and Robust Stability . . . . . . . . . . . . . . . . . . 272

5.1 Exponential Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2725.2 Robust Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

6 Computation Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Polytopic Discrete-Time Models for Systems with Time-VaryingDelays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Warody Lombardi, Sorin Olaru, and Silviu-Iulian Niculescu

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2772 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2783 Polytopic Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

3.1 Non-defective System Matrix with Real Eigenvalues . . . 2803.2 Defective System Matrix with Real Eigenvalues . . . . . . . 2813.3 Non-defective System Matrix with Complex-

Conjugated Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 2834 Towards a Less Conservative Simplex Embedding . . . . . . . . . . . . . 2835 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Part IV Predictor-Based Control and Compensation

Predictor Feedback: Time-Varying, Adaptive, and Nonlinear . . . . . . . . . . 291Miroslav Krstic

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912 Lyapunov Functional and Its Immediate Benefits . . . . . . . . . . . . . . 2923 Delay-Robustness, Delay-Adaptivity, and Time-Varying

Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2943.1 Robustness to Delay Mismatch . . . . . . . . . . . . . . . . . . . . . 2943.2 Delay-Adaptive Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . 295

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3.3 Time-Varying Input Delay . . . . . . . . . . . . . . . . . . . . . . . . . 2974 Predictor Feedback for Nonlinear Systems . . . . . . . . . . . . . . . . . . . 2995 Non-holonomic Unicycle Controlled over a Network . . . . . . . . . . 3026 State-Dependent Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Smoothing Techniques-Based Distributed Model Predictive ControlAlgorithms for Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307Ion Necoara, Ioan Dumitrache, and Johan A.K. Suykens

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3072 Application of Smoothing Techniques to Separable Convex

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3082.1 Proximal Center Decomposition Method . . . . . . . . . . . . . 3102.2 Interior-Point Lagrangian Decomposition Method . . . . . 3112.3 Application of Smoothing Techniques to Separable

Non-convex Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123 Distributed Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . 313

3.1 Distributed MPC for Coupling Non-linear Dynamics . . . 3133.2 Distributed MPC for Coupling Linear Dynamics . . . . . . 3153.3 Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 316


Model Predictive Control with Delay Compensation for Air-to-FuelRatio Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319Sergio Trimboli, Stefano Di Cairano, Alberto Bemporad,and Ilya V. Kolmanovsky

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3202 Model of AFR and Oxygen Storage . . . . . . . . . . . . . . . . . . . . . . . . . 322

2.1 Air-to-Fuel Ratio Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 3222.2 Oxygen Storage Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 323

3 Model Predictive Control with Delay Compensation . . . . . . . . . . . 3243.1 State Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3243.2 Model Predictive Controller . . . . . . . . . . . . . . . . . . . . . . . . 325

4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3275 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

Observer-Based Stabilizing Control for a Class of Nonlinear RetardedSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331Alfredo Germani, Costanzo Manes, and Pierdomenico Pepe

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3323 Separation Theorem: Global Results . . . . . . . . . . . . . . . . . . . . . . . . 3334 Separation Theorem: Local Results . . . . . . . . . . . . . . . . . . . . . . . . . 336

Contents XXIII

5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Cascade Control for Time Delay Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Pavel Zıtek, Vladimır Kucera, and Tomas Vyhlıdal

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3432 Meromorphic Quasi-integration Transfer Function . . . . . . . . . . . . . 3453 Parameterization Based Design of the Secondary

Control Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3474 Master Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3505 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

Design of Terminal Cost Functionals and Terminal Regions for ModelPredictive Control of Nonlinear Time-Delay Systems . . . . . . . . . . . . . . . . . 355Marcus Reble and Frank Allgower

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3552 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3573 Model Predictive Control for Nonlinear Time-Delay Systems . . . 3584 General Linearization-Based Design . . . . . . . . . . . . . . . . . . . . . . . . 3605 Design by Combination of Lyapunov-Krasovskii and

Lyapunov-Razumikhin Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . 3626 Brief Discussion of Both Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 3647 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Part V Networked Control Systemsand Multi-agent Systems

Networked Control under Time-Synchronization Errors . . . . . . . . . . . . . . 369Alexandre Seuret and Karl H. Johansson


2.1 Definition of the Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.2 Synchronization and Delays Models . . . . . . . . . . . . . . . . . 371

3 Observer-Based Networked Control . . . . . . . . . . . . . . . . . . . . . . . . . 3724 Stabilization under Synchronization Error . . . . . . . . . . . . . . . . . . . . 373

4.1 Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3734.2 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

5 Application to a Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3776 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3777 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

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Modelling and Predictive Congestion Control of TCP Protocols . . . . . . . . 383Rafael C. Melo, Jean-Marie Farines, and Julio E. Normey-Rico

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3832 Modeling TCP Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

2.1 Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3852.2 Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3862.3 Simple Model for Control Purposes . . . . . . . . . . . . . . . . . 3872.4 Comparative Dynamical Models Simulation Results . . . 388

3 Congestion Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3893.1 Case Study 1: Controller Performance Comparison . . . . 3903.2 Case Study 2: Sample Time Effects . . . . . . . . . . . . . . . . . 3903.3 Case Study 3: RTT Variation Effects . . . . . . . . . . . . . . . . . 391


Dependence of Delay Margin on Network Topology: SingleDelay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395Wei Qiao and Rifat Sipahi


2.1 Linear-Time Invariant Model with Delays . . . . . . . . . . . . 3972.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3983.1 Dependence of Delay Margin on Topology - Weak

Damping Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3983.2 Effects of Topology Transition on Stability . . . . . . . . . . . 4003.3 Stability Analysis with Respect to Damping . . . . . . . . . . 401


Consensus in Networks under Transmission Delays and theNormalized Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407Fatihcan M. Atay

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4072 The Normalized Laplacian and Consensus in the Absence of

Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4093 Consensus under Transmission Delays . . . . . . . . . . . . . . . . . . . . . . . 4104 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

4.1 Zero Eigenvalue and Spanning Trees . . . . . . . . . . . . . . . . 4124.2 Undirected Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4134.3 Normalized versus Non-normalized Laplacian . . . . . . . . 4134.4 Transmission versus Processing Delays . . . . . . . . . . . . . . 4144.5 Distributed Delays, Discrete-Time Systems . . . . . . . . . . . 415

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

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Consensus with Constrained Convergence Rate and Time-Delays . . . . . . 417Irinel-Constantin Morarescu, Silviu-Iulian Niculescu, and Antoine Girard


2.1 Algebraic Graph Theory Elements . . . . . . . . . . . . . . . . . . 4182.2 Consensus Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

3 Consensus Problem for Networks with Fixed Topology . . . . . . . . . 4213.1 Convergence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.2 Delay Margin Assuring a Fixed Network Topology . . . . 423

4 Agreement Speed in Networks with Dynamic Topology . . . . . . . . 4255 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4276 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

H∞ Control of Networked Control Systems via DiscontinuousLyapunov Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429Kun Liu and Emilia Fridman

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4292 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

3.1 Exponential Stability and L2-Gain Analysis . . . . . . . . . . . 4333.2 Application to Network-Based

Static Output-Feedback Design . . . . . . . . . . . . . . . . . . . . . 4364 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4375 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

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