Systems & Control: Foundations & Applications

Series Editor Tamer Ba¸sar, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Editorial Board Karl Johan Åström, Lund University of Technology, Lund, Sweden Han-Fu Chen, Academia Sinica, Beijing, China Bill Helton, University of California, San Diego, CA, USA Alberto Isidori, Sapienza University of Rome, Rome, Italy Miroslav Krstic, University of California, San Diego, CA, USA H. Vincent Poor, Princeton University, Princeton, NJ, USA Mete Soner, ETH Zürich, Zürich, Switzerland; Swiss Finance Institute, Zürich, Switzerland Roberto Tempo, CNR-IEIIT, Politecnico di Torino, Italy

More information about this series at http://www.springer.com/series/4895 Emilia Fridman

Introduction to Time-Delay Systems Analysis and Control Emilia Fridman School of Electrical Engineering Tel Aviv University Tel Aviv, Israel

ISSN 2324-9749 ISSN 2324-9757 (electronic) ISBN 978-3-319-09392-5 ISBN 978-3-319-09393-2 (eBook) DOI 10.1007/978-3-319-09393-2 Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014945739

Mathematics Subject Classification (2010): 34K05, 34K06, 34K20, 34K35, 93B36, 93C23, 93C57, 93D09

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Springer Basel is part of Springer Science+Business Media (www.birkhauser-science.com) To the memory of my parents, Izabella and Moisei. To the memory of my teachers, Vadim V. Strygin and Vladimir B. Kolmanovskii. Preface

Time-delay often appears in many real-world engineering systems either in the state, the control input, or the measurements. Delays are strongly involved in challenging areas of communication and information technologies: in stabilization of networked controlled systems and in high-speed communication networks. Time-delay is, in many cases, a source of instability. However, for some systems, the presence of delay can have a stabilizing effect. The stability analysis and robust control of time- delay systems (TDSs) are, therefore, of theoretical and practical importance. As in systems without delay, an efficient method for stability analysis of TDSs is the Lyapunov method. For TDSs, there exist two main Lyapunov methods: the Krasovskii method of Lyapunov functionals (1956) and the Razumikhin method of Lyapunov functions (1956). The Krasovskii method is applicable to a wider range of problems and it leads usually to less conservative results, than the Razumikhin method. The Lyapunov stability criterion for linear systems without delay can be for- mulated in terms of a linear matrix inequality (LMI). The realization that LMI may be treated as a convex optimization problem and the development of the efficient interior point method led to formulation of many control problems and their solutions in the form of LMIs [17]. The LMI approach to analysis and design of TDSs provides constructive finite-dimensional conditions, in spite of significant model uncertainties. Modeling of continuous-time systems with digital control in the form of continuous-time systems with time-varying delay [166] and the extension of Krasovskii method to TDSs without any constraints on the delay derivative [72] and to discontinuous delays [76] have allowed the development of the time-delay approach to sampled-data and to network-based control. The beginning of the twenty-first century can be characterized as the “time-delay boom” leading to numerous important results. The books that have been published so far have been restricted to detailed presentation of certain specific time-delay topics. The purpose of this book is twofold, to familiarize the non-expert reader

vii viii Preface with TDSs and to provide a systematic treatment of modern ideas and techniques for experts. The book leads the reader from some basic classical results to recent developments on Lyapunov-based analysis and design with applications to the hot topics of sampled-data and network-based control. It should be of interest for researchers working in the field, for graduate students in engineering and applied mathematics, and for practicing engineers. It may also be used as a textbook for a graduate course on TDSs. The book is based on the course “Introduction to time- delay systems” for graduate students in Engineering and Applied Mathematics that I taught in Tel Aviv University in 2011–2012 and 2012–2013 academic years. The sufficient background to follow most of the material are the undergraduate courses in mathematics and an introduction to control. Chapters 1 and 2 are introductory and are aimed at illustrating the new features that are brought by the time-delay. Chapter 1 discusses models with time-delays and gives some mathematical background: solution concept, the step method, and the state of TDS. Chapter 2 presents solution to linear TDSs and treats characteristic equation of LTI systems. Section 2.3 discusses the effects of delay on stability and presents the classical direct frequency domain method for stability analysis. Section 2.4 considers controllability and observability of TDSs. The emphasis in this book is on the Lyapunov-based analysis and design (Chaps. 3–7). Chapters 3 and 4 present stability and performance analysis of continuous-time TDSs, starting from the simple stability conditions and showing the ideas and tools that essentially improve the results. The objective is to provide useful techniques that will allow the reader not only to apply the existing methods but also to develop new ones. Chapter 5 provides solutions to the classical linear quadratic regulator problem for LTI systems in terms of Riccati equations, and to robust control of systems with time-varying delays in terms of LMIs. The LMI-based design conditions are derived via the descriptor method [52], having almost the same form for the continuous and for the discrete-time systems. Therefore, the LMI- based analysis and design for the discrete-time systems in Chap. 6 are presented as a simple extension of the continuous-time results of Chaps. 3–5. Chapter 7 develops a time-delay approach to the hot topic of sampled-data and networked control systems. I would like to strongly encourage readers to send me suggestions, comments, and corrections by e-mail ([email protected]). I wish to acknowledge my great debt to many colleagues and students who helped me in writing this book. My special thanks to former students Kun Liu, Christophe Fiter, Oren Solomon, and Vladimir Suplin for their great help. Of the friends and colleagues with whom I have had worked on problems directly pertinent to this book, it is a pleasure to acknowledge: Uri Shaked, Jean-Pierre Richard, Michel Dambrine, Silviu Niculescu, Yury Orlov, Valery Glizer, Sabine Mondie, Alexander Seuret, Laurentiu Hetel, and Frederic Gouaisbaut. I am happy to acknowledge Leonid Mirkin for fruitful discussions on time-delay and sampled-data systems. Great thanks to Springer Editor Donna Chernyk who invited me to write a book in the correct time (when I was preparing a course on TDSs in Tel Aviv University). Preface ix

Heartfelt thanks to my family, Eugenii and Boris Shustin, for moral and emotional support, and to my brother, Leonid Fridman, for encouragement and inspiring experience. This book is dedicated to my parents, Izabella Goldreich and Moisei Fridman, and to my teachers, Vadim Vasil’evich Strygin and Vladimir Borisovich Kolmanovskii. I am grateful to Tel Aviv University for an environment that allowed me to write this book, and to the Israel Science Foundation for supporting my research on TDSs.

Tel Aviv, Israel Emilia Fridman June 2014 Contents

1 Introduction ...... 1 1.1 Models with Time-Delay ...... 1 1.1.1 ShoweringPerson...... 1 1.1.2 Sampled-DataControlandNCSs ...... 3 1.1.3 Congestion Control in Communication Networks ...... 5 1.1.4 Drilling System Model ...... 6 1.1.5 Long Line with Tunnel Diode and Models of Lasers ...... 9 1.1.6 VehicularTrafficFlows...... 10 1.1.7 Neural Networks, Population Dynamics, and Epidemic Models ...... 11 1.2 Solution Concept, the Step Method and the State of TDS ...... 13 1.2.1 Classification of TDSs and the Step Method for RetardedDifferentialEquations(RDEs)...... 14 1.2.2 The Step Method for Neutral Type Differential Equations(NDEs)...... 16 1.3 General Functional-Differential Equations ...... 17 1.3.1 Initial Value (Cauchy) Problem for General RDEs...... 17 1.3.2 Initial Value Problem for NDEs ...... 19 1.4 TDSs and Infinite-Dimensional Systems ...... 20 1.4.1 TDSsandPDE...... 20 1.4.2 TDSs and Abstract Systems in Hilbert or Banach Spaces ...... 20 1.5 A HistoricalNote...... 22 2 Linear TDSs ...... 23 2.1 Linear TDSs: Fundamental Matrix and Solution...... 23 2.1.1 LTITDSs:TheVariation-of-ConstantsFormula ...... 23 2.1.2 General LTI RDE and the Variation-of-ConstantsFormula...... 25

xi xii Contents

2.1.3 Adjoint Equation and the Variation-of-Constants Formula for LTVRDE...... 26 2.2 LTI TDSs: Characteristic Equation and Transfer Function ...... 29 2.2.1 A SimpleRDEandItsCharacteristicRoots...... 29 2.2.2 A General LTI RDE and Its Characteristic Roots ...... 29 2.2.3 A SimpleNeutralDifferentialEquation ...... 31 2.2.4 LTI NDE: Characteristic Roots and Exponential Stability...... 32 2.2.5 On Robustness of Stability of NDE with Respect to Small Delays ...... 33 2.2.6 NDE with Strongly Stable Difference Operators ...... 35 2.2.7 Transfer Function Matrix of a TDS ...... 36 2.3 Effects of the Delay on Stability: A Frequency Domain Analysis...... 36 2.3.1 Destabilizing and Stabilizing Effects of the Delay ...... 36 2.3.2 Direct Method for Stability of Single Delay CharacteristicEquation...... 37 2.4 On Controllability and Observability of TDSs ...... 43 2.4.1 Controllability ...... 43 2.4.2 Observability ...... 46 2.4.3 Stabilizability and Detectability ...... 48 3 Lyapunov-Based Stability Analysis ...... 51 3.1 General TDS and the Direct Lyapunov Method...... 51 3.1.1 The Stability Notions and Preliminaries ...... 52 3.1.2 Lyapunov–Krasovskii Approach: RDE ...... 53 3.1.3 Lyapunov–Razumikhin Approach: RDE ...... 55 3.1.4 Lyapunov–Krasovskii Approach: NDE ...... 57 3.2 A Linear Matrix Inequality Approach to Stability ...... 59 3.2.1 Stability of Linear Uncertain Systems andSimpleLMIs...... 59 3.2.2 Standard LMI Problems ...... 61 3.2.3 Schur Complement Lemma and S-Procedure ...... 62 3.3 Delay-Independent Conditions for Linear TDSs ...... 63 3.3.1 LMIs Via the Krasovskii Approach: RDE ...... 63 3.3.2 LMIs Via the Razumikhin Approach: RDE ...... 64 3.3.3 Robust Stability of Linear RDE with Polytopic Type or Norm-Bounded Uncertainties ...... 67 3.3.4 LMIs Via the Krasovskii Approach: NDE ...... 69 3.4 Lyapunov–Krasovskii Method for Linear Descriptor TDSs ...... 70 3.4.1 PreliminariesonDescriptorSystems...... 71 3.4.2 DescriptorSystemswithDiscreteDelays...... 73 3.4.3 The Direct Lyapunov–Krasovskii Method forDescriptorDelaySystems ...... 74 Contents xiii

3.4.4 LMIs for the Delay-Independent Stability ...... 76 3.4.5 LMI for the Strong Stability of the Difference Operator ... 77 3.5 Delay-Dependent Criteria: Preliminaries ...... 79 3.5.1 Model Transformations and the Descriptor Method ...... 80 3.5.2 The Descriptor Method for Uncertain Systems...... 82 3.5.3 The Descriptor Method: Neutral Type Systems ...... 83 3.5.4 Singularly Perturbed Systems and the DescriptorMethod...... 84 3.5.5 Free-Weighting Matrices Technique and Integral Inequalities ...... 86 3.6 Delay-Dependent Conditions: The Krasovskii Method...... 88 3.6.1 Simple Delay-Dependent Conditions ...... 88 3.6.2 Improved Delay-Dependent Conditions ...... 90 3.6.3 Further Improvement: A Reciprocally Convex Approach ...... 95 3.6.4 About the Reduced-Order LMI Conditions...... 99 3.7 IntervalorNon-smallTime-VaryingDelay ...... 101 3.8 Stability of Linear Systems with Distributed Delays ...... 103 3.9 General Lyapunov Functionals for LTI Retarded Systems ...... 108 3.9.1 Necessary Stability Conditions and General Lyapunov Functionals ...... 109 3.9.2 About the Discretized Lyapunov Functional Method ...... 114 3.9.3 Simple, Augmented, and General Lyapunov Functionals.. 115 3.10 Wirtinger’s Inequality, Its Extensions, and Augmented Lyapunov Functionals ...... 118 3.10.1 Wirtinger-Based Integral Inequalities ...... 119 3.10.2 Stability of Linear Systems with Constant Delays Via Augmented Lyapunov Functionals ...... 121 3.11 About the Stability Analysis of Nonlinear TDSs...... 125 3.11.1 Stability in the First Approximation ...... 125 3.12 Notes ...... 133 4 Performance Analysis of TDSs ...... 135 4.1 Exponential and ISS ...... 135 4.1.1 Exponential Stability: The Lyapunov–Krasovskii Approach ...... 136 4.1.2 Halanay’s Inequality and the Exponential Stability ...... 138 4.1.3 Input-to-State Stability ...... 141 4.2 PassivityandPositiveRealness...... 144 4.2.1 PassivityAnalysisofLTVTDSs...... 144 4.2.2 PassivityandPositiveRealnessofLTITDSs...... 145 4.3 L2-GainAnalysisofTDSs ...... 147 4.3.1 H1 -NormofLTISystems ...... 148 4.3.2 Lyapunov-Based L2-GainAnalysis...... 150 xiv Contents

4.4 The Input–Output Approach: Stability and L2-GainAnalysis ...... 152 4.4.1 The Small-Gain Theorem and Simple Stability Conditions...... 152 4.4.2 Stability of Systems with Non-small Delays ...... 158 4.4.3 L2-Gain Analysis of Linear TDSs with Norm-Bounded Uncertainties ...... 164 4.4.4 Relation Between Input–Output and Exponential Stability...... 166 4.5 Systems with Infinite Delays: Stability and L2-GainAnalysis...... 167 4.5.1 PreliminariesandProblemFormulation ...... 168 4.5.2 Integrable Kernels: Exponential Stability and L2-GainAnalysis...... 172 4.5.3 Gamma-Distributed Delay with a Gap: Exponential Stability...... 177 4.5.4 Examples: Traffic Flow Models on the Ring ...... 184 4.6 Exponential Stability of Singularly Perturbed TDSs...... 185 4.7 Exponential Stability of Diffusion Time-Delay PDEs ...... 191 4.7.1 Diffusion PDE Under Study and Preliminaries...... 191 4.7.2 Delay-Independent Stability Conditions: Halanay’s Inequality ...... 193 4.7.3 Delay-Dependent Stability Conditions: Krasovskii Method Via the Descriptor Approach ...... 195 5 Control Design for TDSs ...... 199 5.1 ThePredictor-BasedDesignandLQR ...... 199 5.1.1 Predictor-Based Control: Constant Input Delay ...... 199 5.1.2 The Predictor-Based Design and the Reduction Approach ...... 200 5.1.3 Infinite Horizon LQR: Constant Input Delay ...... 202 5.1.4 InfiniteHorizonLQR:ConstantStateDelay ...... 203 5.1.5 Extension to H1 Control:ConstantStateDelay...... 207 5.2 LMI-BasedDesign:Time-VaryingDelays ...... 209 5.2.1 State-Feedback Control: An Uncertain Input Delay...... 209 5.2.2 H1 Filtering: A Time-Varying Measurement Delay ...... 212 5.3 H1 Control Via Descriptor Discretized Lyapunov Functional Method ...... 215 5.3.1 L2-GainAnalysis ...... 216 5.3.2 H1 Control...... 222 5.4 Control of TDSs Under Actuator Saturation ...... 225 5.4.1 Delay-Dependent Methods and the First Delay Interval ... 227 5.4.2 Solution Bounds Via the Lyapunov–Krasovskii Method... 227 5.4.3 Control Under Actuator Saturation: Regional Stabilization...... 232 5.4.4 Generalized Sector Condition ...... 238 Contents xv

5.5 Notes ...... 241 5.5.1 Stability and Control of Systems with State-Dependent Delay ...... 241 6 Discrete-Time Delay Systems ...... 243 6.1 Stability and Performance Analysis of Discrete-Time TDSs ...... 243 6.1.1 Analysis of Discrete-Time Delay Systems Via Augmentation ...... 244 6.1.2 Transfer Function of Discrete-Time TDS...... 245 6.1.3 LMI Stability Conditions: The Direct Lyapunov Method.. 246 6.1.4 l2-Gain Analysis Via the Krasovskii Approach...... 251 6.1.5 Exponential and Input-to-State Stability ...... 252 6.1.6 The Input–Output Approach to Stability ofDiscrete-TimeTDSs...... 255 6.2 ControlofDiscrete-TimeDelaySystems ...... 259 6.2.1 Infinite Horizon LQR for LTI Discrete-Time DelaySystems...... 259 6.2.2 The Predictor-Based Design and the Reduction Approach ...... 261 6.2.3 LMI-BasedDesign:Time-VaryingDelays ...... 263 6.3 Control of Discrete-Time Delay Systems with Input Saturation .... 266 6.3.1 Solution Bounds Via Delay-Dependent Lyapunov–Krasovskii Methods ...... 266 6.3.2 State-Feedback Control with Input Saturation...... 270 6.4 Notes ...... 272 7 Sampled-Data and NCSs: A Time-Delay Approach ...... 273 7.1 PreliminariesonSampled-DataControl...... 273 7.1.1 DiscretizationofLTISampled-DataSystems ...... 274 7.1.2 Effects of Sampling on Stability ...... 275 7.1.3 Three Main Approaches to Sampled-Data Control...... 278 7.2 Stability and L2-GainAnalysisofSampled-DataSystems...... 279 7.2.1 Time-Dependent Lyapunov Functional Method ...... 281 7.2.2 Simple Stability Conditions: Variable Sampling ...... 282 7.2.3 AnImprovedSampled-DataAnalysis ...... 286 7.3 Sampled-Data Control of Switched Affine Systems ...... 292 7.3.1 Stabilization by Continuous State-Dependent Switching .. 292 7.3.2 Practical Stabilization by Sampled-Data Switching ...... 294 7.4 Wirtinger’s Inequality and Sampled-Data Control ...... 299 7.4.1 Input–Output Approach to Stability Via Wirtinger’s Inequality...... 301 7.4.2 Wirtinger-Based Lyapunov Functionals and LMIs: Variable Sampling and Constant Input/OutputDelay ...... 302 7.5 A Time-Delay Model of an NCS: Sampling, Packet Dropouts, and Communication Delays ...... 309 xvi Contents

7.6 NCSs Under Scheduling Protocols: Round Robin Protocol ...... 314 7.6.1 NCSs Under RR Protocol and the Switched System Model ...... 315 7.6.2 Stability and L2-Gain Analysis of NCSs: Variable k ...... 320 7.6.3 Stability and L2-Gain Analysis of NCSs: Constant  ...... 324 7.6.4 Examples...... 327 7.7 NCSs Under TOD Protocol ...... 330 7.7.1 NCS Under TOD Protocol and the Hybrid Time-Delay Model ...... 331 7.7.2 ISS Under TOD and Quantization ...... 335 7.7.3 Examples...... 340 7.8 Discrete-TimeNCSs...... 342 7.8.1 NCSs Under TOD Protocol and a Hybrid Time-Delay Model ...... 342 7.8.2 Partial Exponential Stability of the Discrete-Time Hybrid Delayed System ...... 345 7.9 Notes ...... 349

Erratum ...... E1

References...... 351

Index ...... 361 Notation

Sets and Spaces

C The set of complex numbers R The set of real numbers Rn The Euclidean space with the norm jj Rnm The space of real n  m matrices with the induced norm jj RC The set of nonnegative real numbers Z The set of integers ZC The set of nonnegative integers N The set of natural numbers x.t/;P x.t/R The first and the second derivatives (or right-hand derivative) of x with respect to time t n xt W Œh;0 ! R xt ./ D x.t C /;  2 Œh;0 CŒa;b;C.Œa;b;Rn/ The space of continuous functions  W Œa;b ! Rn with the norm kkC D max2Œa;b j./j C mŒa;b The space of m times continuously differentiable functions  W Œa;b ! Rn with the norm d m m kkC DkkC CCkdm kC n Lp.a;b/; p 2 N The space of functions  W .a;b/ ! R with the norm hR i 1 b p p kkLp D a j./j d n L2Œ0;1/ The space of functions  W RC ! R with the norm R  1 1 2 2 kkL2 D 0 j./j d n L1.a;b/ The space of essentially bounded functions  W .a;b/ ! R

with the norm kk1 D esssup2.a;b/ j./j

xvii xviii Notation

WŒa;b The space of absolutely continuous functions  W Œa;b ! Rn d d with d 2 L2.a;b/ and with the norm kkW DkkC Ckd kL2 H 1.0;l/ The Sobolev space of absolutely continuous functions Rn d  W Œ0;l ! with d 2 L2.0;l/ H 2.0;l/ The Sobolev space of functions  W Œ0;l ! Rn d d 2 with absolutely continuous d and with d2 2 L2.0;l/P l Œ0;1/ The space of  W ZC ! Rn with the norm kk2 D 1 j.k/j2 2 l2 kD0

Vectors and Matrices colfa;bg Column vector ŒaT bT T M T ;M Transpose, component-wise complex conjugate P>0.P 0/ The symmetric matrix P is positive (semi-positive) definite max.P /;min.P / The maximum and the minimum eigenvalue of P D P T 2 Rnn  The symmetric elements of the symmetric matrix .M / An eigenvalue of the quadratic matrix A .M/ Spectral radius of a square matrix M ,maxi ji .M /j I;In Identity matrix j Imaginary unit with j 2 D1

Abbreviations

ISS Input-to-state stability LMI Linear matrix inequality LTI Linear time-invariant MAD Maximum allowable delay MATI Maximum allowable transmission interval NCS Networked control system NDE Neutral type differential equation ODE Ordinary differential equation PDE Partial differential equation RDE Retarded type differential equation RR Round-robin TDS Time-delay system TOD Try-once-discard ZOH Zero-order-hold

The original version of this book was revised. An erratum to this book can be found at http://dx.doi.org/10.1007/978-3-319-09393-2_8