Stability and Stabilization of Time-Delay Systems (Advances in Design & Control) (Advances in Design and Control)

401 Pages·2007·29.55 MB·English

by  Wim Michiels

Stability and
Stabilization of
Time-Delay Systems
Advances in Design and Control
SIAM's Advances in Design and Control series consists of texts and monographs dealing with all
areas of design and control and their applications. Topics of interest include shape optimization,
multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses
on the mathematical and computational aspects of engineering design and control that are usable
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Editor-in-Chief
Ralph C. Smith, North Carolina State University
Editorial Board
Athanasios C. Antoulas, Rice University
Siva Banda, Air Force Research Laboratory
Belinda A. Batten, Oregon State University
John Betts, The Boeing Company
Stephen L. Campbell, North Carolina State University
Eugene M. Cliff, Virginia Polytechnic Institute and State University
Michel C. Delfour, University of Montreal
Max D. Gunzburger, Florida State University
J. William Helton, University of California, San Diego
Arthur J. Krener, University of California, Davis
Kirsten Morris, University of Waterloo
Richard Murray, California Institute of Technology
Ekkehard Sachs, University of Trier
Series Volumes
Michiels, Wim and Niculescu, Silviu-lulian, Stability and Stabilization of Time-Delay Systems: An
Eigenvalue-Based Approach
loannou, Petros, and Fidan, Baris, Adaptive Control Tutorial
Bhaya, Amit, and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and
Matrix Problems
Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic
Programming for Optimization of Dynamical Systems
Huang, J., Nonlinear Output Regulation: Theory and Applications
Haslinger, J. and Makinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation,
and Computation
Antoulas, Athanasios C, Approximation of Large-Scale Dynamical Systems
Gunzburger, Max D., Perspectives in Flow Control and Optimization
Delfour, M. C. and Zolesio, J.-P, Shapes and Geometries: Analysis, Differential Calculus, and
Optimization
Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming
El Ghaoui, Laurent and Niculescu, Silviu-lulian, eds., Advances in Linear Matrix Ineguality Methods
in Control
Helton, J. William and James, Matthew R., Extending H°° Control to Nonlinear Systems: Control
of Nonlinear Systems to Achieve Performance Objectives
Stability and
Stabilization of
Time-Delay Systems
An Eigenvalue-Based Approach
W im Michiels
Katholieke Universiteit Leuven
Leuven, Belgium
Silviu-lulian Niculescu
Laboratoire des Signaux et Systemes
Gif-sur-Yvette, France
siam
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2007 by the Society for Industrial and Applied Mathematics.
10 987654321
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Library of Congress Cataloging-in-Publication Data
Michiels, W„ (Wim)
Stability and stabilization of time-delay systems : an Eigenvalue-based approach / Wim
Michiels, K.U. Leuven, Silviu-lulian Niculescu.
p. cm. -- (Advances in design and control ; 12)
Includes bibliographical references and index.
ISBN 978-0-898716-32-0 (alk. paper)
1. Automatic control. 2. Time delay systems. 3. Stability. 4. Eigenvalues. I. Leuven, K. U.
II. Niculescu, Silviu-lulian. III. Title.
TJ213.M485 2007
629.8'3-dc22
2007061746
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Contents
Preface xiii
Symbols xix
Acronyms 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
Contents
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
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
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 Geometric 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
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 functions 118
5.4 Delay-independent stability and delay-interference phenomena . . .. 120
5.4.1 Delay-independent stability characterization 121
VII
5.4.2 Delay-interference characterization 121
5.5 Illustrative examples 124
5.5.1 Interference in parameterized scalar delay systems . . .. 124
5.5.2 Delay rays and second-order delay systems 125
5.6 Notes and references 127
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
Stabilization and robust stabilization 147
The continuous pole placement method 149
7.1 Introduction 149
7.2 Motivation 150
7.2.1 A finite-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 output 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 Contents
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 Characterization of 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 problem 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
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 back
ward differences 249
12.3.2 Control laws based on exactpole placement and low-gain
design 252
12.4 Illustrative example 255
12.5 Notes and references 255
Congestion control algorithms in networks 257
13.1 Algorithms for single connection models 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 models 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
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 direc
tions 285
14.5 Illustrative example 286
14.6 Multivariable case 287
14.6.1 Practical stability condition 288
14.6.2 Stability domain 289