For both their theoretical and practical impact, time-delay systems have been an enduring theme in the study of differential equations, stochastic processes, game theory, and systems theory. The subject has broad applications to a number of areas, including mechanical, electrical and chemical engineering, mathematics, biology, and economics.This book is a self-contained, coherent presentation of the background and progress of the stability of time-delay systems. Focusing on techniques, tools, and advances in numerical methods and optimization algorithms, the authors develop material, which up until now, has been scattered in technical journals and conference proceedings. Special emphasis is placed on systems with uncertainty and stability criteria which can be computationally implemented.Features and Topics:* Systematic and comprehensive coverage of robust stability for time-delay systems, including time-domain and frequency-domain approaches* Stability criteria formulated using linear matrix inequalities (LMI), providing a powerful toolbox for practicing engineers* Strong stability conditions developed to provide a solid basis for design of feedback control and filtering* Balance of intuition and rigor, stressing concepts rather than technical details* Emphasis on comparisons and connections among various approaches* Mathematical prerequisites integrated within each chapter, with more elementary material covered in two appendicesRequiring only basic knowledge of linear systems and Lyapunov stability theory, Stability of Time-Delay Systems will be accessible to a broad audience of researchers, professional engineers, and graduate students. It may be used for self-study or as a reference; portions of the text may be used in advanced graduate courses and seminars.

Preface

xiii

Notation

xix

Introduction to Time-Delay Systems

(28)

Introduction

(4)

A simple time-delay system

(3)

Functional differential equations

(2)

Stability of time-delay systems

(5)

Stability concept

(1)

Lyapunov-Krasovskii Theorem

(2)

Razumikhin Theorem

(2)

Linear systems

(2)

Linear time-invariant systems

(3)

Neutral time-delay systems

(2)

Outline of the text

(3)

Notes

(4)

A brief historic note

(1)

Application examples

(1)

Analysis of time-delay systems

(2)

I Frequency Domain Approach

(116)

Systems with Commensurate Delays

(38)

Introduction

(3)

Some classical stability tests

(10)

2-D stability tests

(2)

Pseudo-delay methods

(2)

Direct method

(6)

Frequency-sweeping tests

(11)

Constant matrix tests

(12)

Notes

(2)

Classical results

(1)

Frequency-sweeping and constant matrix tests

(1)

Systems with Incommensurate Delays

(48)

Introduction

(1)

Small gain/ μ theorem

(9)

Small gain theorem

(7)

Structured singular value

(2)

Frequency-sweeping conditions

(8)

Computational complexity analysis

(9)

Basic complexity concepts

(2)

Proof of NP-hardness

(6)

Sufficient stability conditions

(10)

Systems of one delay

(7)

Systems of multiple delays

103

(3)

Neutral delay systems

106

(7)

Summary

113

(1)

Notes

114

(3)

Small gain theorem and μ

114

(1)

Stability of systems with incommensurate delays

114

(1)

Complexity issues

115

(1)

Sufficient conditions and neutral systems

115

(2)

Robust Stability Analysis

117

(28)

Uncertain systems

117

(1)

Characteristic quasipolynomial

117

(2)

Zeros of a quasipolynomial

119

(6)

Exponential diagram

120

(2)

Potential diagram

122

(3)

Uncertain quasipolynomial

125

(4)

Value set

126

(1)

Zero exclusion principle

126

(3)

Edge Theorem

129

(6)

Stability of an edge subfamily

132

(2)

Interval quasipolynomial

134

(1)

Multivariate polynomial approach

135

(9)

Multivariate polynomials

136

(1)

Stable polynomials

137

(3)

Stability of an interval multivariate polynomial

140

(2)

Stability of a diamond family of multivariate polynomials

142

(2)

Notes

144

(1)

II Time Domain Approach

145

(128)

Systems with Single Delay

147

(50)

Introduction

147

(4)

Delay-independent stability criteria based on the Razumikhin Theorem

151

(5)

Single delay case

151

(3)

Distributed delay case

154

(2)

Simple delay-dependent stability criteria based on the Razumikhin Theorem

156

(13)

Model transformation

157

(1)

Simple delay-dependent stability criteria using explicit model transformation

158

(2)

Additional dynamics

160

(5)

Simple delay-dependent stability criteria using implicit model transformation

165

(4)

Delay-independent stability criteria based on the Lyapunov-Krasovskii Stability Theorem

169

(3)

Systems with single delay

169

(2)

Systems with distributed delays

171

(1)

Delay-dependent stability criteria using a simple Lyapunov-Krasovskii functional

172

(3)

Stability criteria using explicit model transformation

172

(1)

Stability criteria using implicit model transformation

173

(2)

Complete quadratic Lyapunov-Krasovskii functional

175

(7)

Introduction

175

(2)

Fundamental solution and matrix Uw(τ)

177

(2)

Lyapunov-Krasovskii functionals

179

(3)

Discretized Lyapunov functional method for systems with single delay

182

(11)

Introduction

182

(1)

Discretization

183

(2)

Lyapunov-Krasovskii functional condition

185

(3)

Lyapunov-Krasovskii derivative condition

188

(3)

Stability criterion and examples

191

(2)

Notes

193

(4)

Results based on the Razumikhin Theorem

193

(1)

Model transformation and additional dynamics

194

(1)

Lyapunov-Krasovskii method

194

(3)

Robust Stability Analysis

197

(36)

Introduction

197

(1)

Uncertainty characterization

197

(10)

Polytopic uncertainty

198

(1)

Subpolytopic uncertainty

199

(2)

Norm-bounded uncertainty

201

(4)

Block-diagonal uncertainty

205

(2)

Robust stability criteria based on the Razumikhin Theorem

207

(6)

Delay-independent stability for systems with single delay

207

(2)

Delay-independent stability criteria for systems with distributed delays

209

(2)

Delay-dependent stability criteria with explicit model transformation

211

(2)

Delay-independent stability criteria using the Lyapunov-Krasovskii functional

213

(6)

Systems with single delay

213

(5)

Systems with distributed delays

218

(1)

Delay-dependent stability criteria using simple Lyapunov-Krasovskii functional

219

(5)

Complete quadratic Lyapunov-Krasovskii functional approach

224

(3)

Discretized Lyapunov functional method for systems with single delay

227

(4)

General case

227

(2)

Norm-bounded uncertainty

229

(2)

Notes

231

(2)

Uncertainty characterization

231

(1)

Stability results based on the Razumikhin Theorem and Lyapunov-Krasovskii functional

232

(1)

Systems with Multiple and Distributed Delays

233

(40)

Introduction

233

(1)

Delay-independent stability criteria of systems with multiple Delays

234

(1)

Simple delay-dependent stability criteria of systems with multiple delays

235

(3)

Complete quadratic functional for general linear systems

238

(3)

Discretized Lyapunov functional method for systems with multiple delays

241

(19)

Problem setup

242

(2)

Discretization

244

(2)

Lyapunov-Krasovskii functional condition

246

(5)

Lyapunov-Krasovskii derivative condition

251

(7)

Stability condition and examples

258

(2)

Discretized Lyapunov functional method for systems with distributed delays

260

(11)

Problem statement

261

(1)

Discretization

262

(1)

Lyapunov-Krasovskii functional condition

263

(2)

Lyapunov-Krasovskii derivative condition

265

(5)

Stability criterion and examples

270

(1)

Notes

271

(2)

III Input-Output Approach

273

(36)

Input-Output Stability

275

(34)

Introduction

275

(1)

Input-output stability

276

(3)

Method of comparison systems

279

(8)

Problem setup

279

(2)

Frequency domain approach

281

(3)

Time domain approach

284

(3)

Scaled small gain problem

287

(5)

Robust stability under dynamical uncertainty

292

(4)

Problem setup

292

(1)

Uncertainty characterization

293

(2)

Robust small gain problem

295

(1)

Approximation of delay elements

296

(11)

Approximation of time-varying delay

296

(3)

Approximation of distributed delays

299

(4)

Approximation by multiple delays

303

(4)

Notes

307

(2)

A Matrix Facts

309

(6)

Notation

309

(1)

Determinant

310

(1)

Eigenvalue problems

310

(2)

Singular value decomposition

312

(1)

Norms

312

(2)

Matrix measure

314

(1)

Kronecker product and sum

314

(1)

Notes

314

(1)

B Linear Matrix Inequalities and Quadratic Integral Inequalities

315

(12)

Basic LMI problem

315

(1)

GEVP

316

(2)

Transforming nonlinear matrix inequalities to LMI form

318

(1)

S-procedure

319

(1)

Elimination of matrix variables

319

(3)

Quadratic integral inequalities

322

(2)

Notes

324

(3)

Bibliography

327

(24)

Index

351

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Stability of Time-Delay Systems

9780817642129

0817642129

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