Stability and Stabilization of Time-Delay Systems An Eigenvalue-Based Approach Wim Michiels Katholieke Universiteit Leuven Leuven, Belgium Silviu-Iulian Niculescu Laboratoire des Signaux et Systemes Gif-sur-Yvette, France SIHJTL Society for Industrial and Applied Mathematics Philadelphia
Preface Symbols Acronyms xiii xix xxi I Stability analysis of linear time-delay Systems 1 1 Spectral properties of linear time-delay Systems 3 1.1 Time-delay Systems of retarded type 4 1.1.1 Initial value problem 4 1.1.2 Spectrum: definitions 7 1.1.3 Asymptotic growth rate of Solutions and stability 8 1.1.4 Spectrum: qualitative properties 9 1.1.5 Spectrum: continuity properties 11 1.1.6 Computation of characteristic roots 13 1.2 Time-delay Systems of neutral type 15 1.2.1 Initial value problem 15 1.2.2 Spectrum: definitions 16 1.2.3 Asymptotic growth rate of Solutions and stability 19 1.2.4 Spectrum: qualitative properties 20 1.2.5 Spectrum: continuity properties 22 1.2.6 Computation of characteristic roots 30 1.3 Notes and references 31 2 Pseudospectra and robust stability analysis 33 2.1 Introduction 33 2.2 Pseudospectra for nonlinear eigenvalue problems 35 2.2.1 Definition and expressions 35 2.2.2 Connection with stability radii 37 2.2.3 Computational issues 38 2.2.4 Application to time-delay Systems 39 2.3 Structured pseudospectra for nonlinear eigenvalue problems 43 v
vi 2.3.1 Exploiting the system's structure 43 2.3.2 Definition and expressions 43 2.3.3 Computational issues and special cases 46 2.3.4 Application to time-delay Systems 49 2.4 Illustrative examples 49 2.4.1 Second-order System 50 2.4.2 Feedback controlled semiconductor laser 51 2.5 Notes and references 54 3 Computation of stability regions in parameter Spaces 57 3.1 Introduction 57 3.2 Basic notions and definitions 58 3.3 From D-subdivision to numerical continuation 59 3.3.1 D-subdivision and stability crossing boundaries 59 3.3.2 T-decomposition and delay stability intervals 64 3.3.3 Numerical continuation 69 3.4 Computing the crossing direction of characteristic roots 71 3.4.1 Simple crossing characteristic roots 71 3.4.2 Semisimple characteristic roots 75 3.4.3 Further analysis: basic ideas 78 3.4.4 Delay interdependence and crossing direction evaluation. 80 3.5 Notes and references 81 4 Stability regions in delay-parameter Spaces 85 4.1 Introduction 85 4.2 Invariance properties 86 4.2.1 Delay shifts and characteristic roots 86 4.2.2 Crossing direction invariance 87 4.3 Algebraic methods 88 4.3.1 Elimination principle: basic ideas 88 4.3.2 Matrix pencil approach and crossing characterization... 90 4.3.3 Particular cases and other elimination techniques 94 4.4 Geometrie methods 98 4.4.1 Identification of crossing points 100 4.4.2 Stability crossing curves 102 4.4.3 Tangents, smoothness, and crossing direction 107 4.5 Notes and references 110 5 Stability of delay rays and delay-interference 113 5.1 Introduction 113 5.2 Preliminary results 115 5.2.1 Definitions and assumptions 115 5.2.2 Introductory example 116 5.3 Properties of some associated matrix-valued funetions 118 5.4 Delay-independent stability and delay-interference phenomena... 120 5.4.1 Delay-independent stability characterization 121
VII 5.4.2 Delay-interferencecharacterization 121 5.5 Illustrative examples 124 5.5.1 Interterence in parameterized scalar delay Systems... 124 5.5.2 Delay rays and second-order delay Systems 125 5.6 Notes and references 127 6 Stability of linear periodic Systems with delays 131 6.1 Introduction 131 6.2 Systems with fast varying coefficients 132 6.2.1 Averaging periodic Systems 132 6.2.2 Computational tools 134 6.2.3 Analytical tools 135 6.3 General case 137 6.3.1 Collocation scheme 137 6.3.2 Computation of stability determining eigenvalues... 138 6.3.3 Computation of stability regions 139 6.3.4 Special cases 140 6.3.5 Comparison with the averaging based approach 141 6.4 Illustrative examples 141 6.4.1 Variable spindle speed cutting machine 141 6.4.2 Forced elastic column 143 6.5 Notes and references 145 II Stabilization and robust stabilization 147 7 The continuous pole placement method 149 7.1 Introduction 149 7.2 Motivation 150 7.2.1 Afinite-dimensional Controller for an infinite-dimensional problem 150 7.2.2 Methods based on prediction 153 7.2.3 Scalar example 154 7.3 Continuous pole placement algorithm 155 7.3.1 Description of the algorithm 155 7.3.2 Theoretical properties 159 7.3.3 Optimization point of view 160 7.4 Illustrative examples 162 7.4.1 Model problem: stabilizing a third-order system 162 7.4.2 General stabilization problems 165 7.5 Extensions of State feedback 170 7.5.1 Multiple input, multiple outpul Systems 170 7.5.2 Observer based Controllers 171 7.5.3 Finite-dimensional dynamic State feedback 173 7.6 Systems of neutral type 173 7.6.1 Algorithm 173
viii 7.6.2 Illustrative example 175 7.7 Notes and references 175 8 Stabilizability with delayed feedback: a numerical case study 177 8.1 Introduction 177 8.2 Characterizationof stabilizable Systems 178 8.2.1 System representation 178 8.2.2 Class of stabilizable Systems for the unit delay 179 8.2.3 Class of stabilizable Systems for arbitrary delay values.. 184 8.2.4 Discussion 186 8.2.5 Noncyclic System matrix 187 8.3 Simultaneous stabilization over a delay interval 188 8.4 Notes and references 190 9 The robust stabilization problein 193 9.1 Introduction 193 9.2 Stability radii as robustness measures 194 9.3 Stabilization versus robust stabilization 195 9.4 Robust stabilization procedure 196 9.4.1 Continuity properties 196 9.4.2 Algorithm 198 9.5 Illustrative example 201 9.6 Notes and references 203 10 Stabilization using a direct eigenvalue optimization approach 205 10.1 Introduction 205 10.2 Eigenvalue optimization approach 206 10.2.1 Smoothness properties of spectral abscissa function...206 10.2.2 Gradient sampling algorithm 207 10.2.3 Application to linear time-delay Systems 211 10.2.4 Extension to nonlinear time-delay Systems 212 10.3 Illustrative examples 213 10.3.1 Model problem: a third-order System 213 10.3.2 Semiconductor laser 213 10.4 Notes and references 216 III Applications 217 11 Stabilization by delayed Output feedback: single delay case 219 11.1 Introduction 219 11.2 Characterization of all stabilizable second-order Systems 220 11.3 Necessary conditions for stabilizability 226 11.4 Controller construction 227 11.4.1 Prerequisites 227 11.4.2 Stabilization using the delay parameter 229
ix 11.4.3 Stabilization using the delay and gain parameter 234 11.5 Geometry of stability regions 235 11.5.1 Identification of crossing points 235 11.5.2 Classification of stability crossing curves, smoothness, and crossing directions 238 11.6 Illustrative examples 241 11.6.1 Second-order System 241 11.6.2 Sixth-order System 242 11.7 Notes and references 245 12 Stabilization by delayed Output feedback: multiple delay case 247 12.1 Introduction 247 12.2 Necessary conditions for stabilizability 248 12.3 Stabilization of multiple integrators 249 12.3.1 Control laws based on numerical differentiation with backward differences 249 12.3.2 Control laws based on exact pole placement and low-gain design 252 12.4 Illustrative example 255 12.5 Notes and references 255 13 Congestion control algorithms in networks 257 13.1 Algorithms for Single connection modeis with two delays 258 13.1.1 Model and related remarks 258 13.1.2 Linear stability analysis 260 13.1.3 Interpretations and discussions 263 13.2 TCP/AQM congestion avoidance modeis with one delay 266 13.2.1 Model and related remarks 266 13.2.2 Transformation 267 13.2.3 Stability analysis 268 13.3 Notes and references 273 14 Smith predictor for stable Systems: delay sensitivity analysis 275 14.1 Introduction 275 14.2 Sensitivity of stability w.r.t. infinitesimal delay mismatches 277 14.2.1 Instability mechanism 277 14.2.2 Conditions for practical stability 278 14.3 Stability analysis and critical delay mismatches 282 14.4 Geometry of stability regions 284 14.4.1 Identification of crossing points 284 14.4.2 Stability crossing curves: smoothness and crossing directions 285 14.5 Illustrative example 286 14.6 Multivariable case 287 14.6.1 Practical stability condition 288 14.6.2 Stability domain 289
X 14.7 Notes and referenees 291 15 Controlling unstable Systems using finite spectrum assignment 293 15.1 Introduction 293 15.2 Preliminaries 294 15.3 Implementation of the integral 296 15.3.1 Instability mechanism 296 15.3.2 Stability conditions 297 15.3.3 Removing restrictions 301 15.4 Delay mismatch 304 15.5 Output feedback 304 15.5.1 Static output feedback 305 15.5.2 Dynamic output feedback and relations with Smith Predictors 305 15.6 Notes and referenees 307 16 Consensus problems in traffic flow applications 309 16.1 Introduction 309 16.2 Extension of stability Iheory to Systems with distributed delays....312 16.3 Conditions for the realizalion ofa consensus 316 16.3.1 Prerequisites 316 16.3.2 Computation of stability regions 317 16.4 Examples 320 16.5 Other modeis 322 16.6 Notes and referenees 324 17 Stability analysis of delay modeis in biosciences 325 17.1 Introduction 325 17.2 Delay effects on stability in some human respiration modeis 326 17.2.1 Delay model and its linearization 326 17.2.2 Stability analysis and delay intervals 327 17.2.3 Discussions and interpretations 330 17.3 Delays in immune dynamics model of leukemia 331 17.3.1 Delay model and its linearization 331 17.3.2 Stability analysis in the delay-parameter space 332 17.3.3 Illustrative example and discussions 338 17.4 Notes and referenees 342 Appendix 345 A. 1 Rouche's theorem 345 A.2 Structured Singular value (ssv) 345 A.3 Continuity properties 347 A.4 Interdependency of numbers 347 A.5 Software 348
XI Bibliography 351 Index 375