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Stability of Time-Delay Systems (Gu KeQin).pdf
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Stability of Time-Delay Systems
Keqin Gu, Vladimir L. Kharitonov and Jie Chen
ii
ABSTRACT abstract
Contents
1 Introduction to Time-Delay Systems
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 A simple time-delay system . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
1.3 Functional differential equations
1.4 Stability of time-delay systems
. . . . . . . . . . . . . . . .
1.4.1
Stability concept . . . . . . . . . . . . . . . . . . . .
1.4.2 Lyapunov-Krasovskii theorem . . . . . . . . . . . . .
1.4.3 Razumikhin theorem . . . . . . . . . . . . . . . . . .
1.5 Linear systems
. . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Linear time-invariant systems . . . . . . . . . . . . . . . . .
1.7 Neutral time delay systems
. . . . . . . . . . . . . . . . . .
1.8 Outline of the text . . . . . . . . . . . . . . . . . . . . . . .
1.9 Notes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9.1 A brief historic note . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
1.9.2 Application examples
1.9.3 Analysis of time-delay systems
. . . . . . . . . . . .
2 Systems with Commensurate Delays
2-D stability tests
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Some classical stability tests . . . . . . . . . . . . . . . . . .
2.2.1
. . . . . . . . . . . . . . . . . . .
2.2.2 Pseudo-delay methods . . . . . . . . . . . . . . . . .
2.2.3 Direct method . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Frequency sweeping tests
2.4 Constant matrix tests
2.5 Notes
2.5.1 Classical results . . . . . . . . . . . . . . . . . . . . .
. . .
2.5.2 Frequency-sweeping and constant matrix tests
3 Systems with Incommensurate Delays
3.2.1
3.2.2
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Small gain/ theorem . . . . . . . . . . . . . . . . . . . . .
Small gain theorem . . . . . . . . . . . . . . . . . . .
Structured singular value . . . . . . . . . . . . . . .
3.3 Frequency-sweeping conditions
. . . . . . . . . . . . . . . .
3.4 Computational complexity analysis . . . . . . . . . . . . . .
3.4.1 Basic complexity concepts . . . . . . . . . . . . . . .
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3.5.1
3.5.2
3.4.2 Proof of NP-hardness
3.5 Sufficient stability conditions
89
. . . . . . . . . . . . . . . . .
95
. . . . . . . . . . . . . . . . .
Systems of one delay . . . . . . . . . . . . . . . . . .
96
Systems of multiple delays . . . . . . . . . . . . . . . 102
3.6 Neutral delay systems
. . . . . . . . . . . . . . . . . . . . . 105
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Small gain theorem and . . . . . . . . . . . . . . . 114
Stability of systems with incommensurate delays . . 114
. . . . . . . . . . . . . . . . . . . 115
. . . . . . 115
3.7.1
3.7.2
3.7.3 Complexity issues
3.7.4
Sufficient conditions and neutral systems
4 Frequency domain robust stability analysis
117
. . . . . . . . . . . . . . . . . . . . . . . 117
4.1 Uncertain systems
4.2 Characteristic quasipolynomial
. . . . . . . . . . . . . . . . 117
4.3 Zeros of a quasipolynomial . . . . . . . . . . . . . . . . . . . 119
4.3.1 Exponential diagram . . . . . . . . . . . . . . . . . . 120
4.3.2 Potential diagram . . . . . . . . . . . . . . . . . . . 122
4.4 Uncertain quasipolynomial . . . . . . . . . . . . . . . . . . . 125
4.4.1 Value set
. . . . . . . . . . . . . . . . . . . . . . . . 126
4.4.2 Zero exclusion principle . . . . . . . . . . . . . . . . 126
4.5 Edge theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Stability of an edge subfamily . . . . . . . . . . . . . 132
Interval quasipolynomial . . . . . . . . . . . . . . . . 133
4.6 Multivariate polynomial approach . . . . . . . . . . . . . . . 135
. . . . . . . . . . . . . . . 136
Stable polynomials . . . . . . . . . . . . . . . . . . . 136
Stability of an interval multivariate polynomial
. . . 139
Stability of a diamond family of multivariate polyno-
mials . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.7 Notes and References . . . . . . . . . . . . . . . . . . . . . . 143
4.6.1 Multivariate polynomials
4.6.2
4.6.3
4.6.4
4.5.1
4.5.2
5 Systems with Single Delay
145
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.1
5.2 Delay-independent stability criteria based on Razumikhin
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.2.1
Single delay case . . . . . . . . . . . . . . . . . . . . 148
5.2.2 Distributed delay case . . . . . . . . . . . . . . . . . 152
5.3 Simple delay-dependent stability criteria based on Razu-
mikhin Theorem . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3.1 Model transformation . . . . . . . . . . . . . . . . . 155
5.3.2
Simple delay-dependent stability using explicit model
transformation . . . . . . . . . . . . . . . . . . . . . 156
5.3.3 Additional dynamics . . . . . . . . . . . . . . . . . . 158
5.3.4
Simple delay-dependent stability using implicit model
transformation . . . . . . . . . . . . . . . . . . . . . 163
v
5.4 Delay-independent stability based on Lyapunov-Krasovskii
stability theorem . . . . . . . . . . . . . . . . . . . . . . . . 166
Systems with single delay . . . . . . . . . . . . . . . 166
5.4.1
5.4.2
Systems with distributed delays . . . . . . . . . . . . 168
5.5 Delay-dependent stability criteria using simple Lyapunov-
5.6 Complete quadratic Lyapunov-Krasovskii functional
Krasovskii functional . . . . . . . . . . . . . . . . . . . . . . 170
Stability criteria using explicit model transformation 170
5.5.1
Stability criteria using implicit model transformation 171
5.5.2
. . . . 173
Introduction . . . . . . . . . . . . . . . . . . . . . . 173
5.6.1
. . . . . . 175
5.6.2 Fundamental solution and matrix UW (τ )
5.6.3 Lyapunov-Krasovskii functionals . . . . . . . . . . . 177
5.7 Discretized Lyapunov functional method for systems with
single delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . 179
5.7.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 181
5.7.3 Lyapunov-Krasovskii functional condition . . . . . . 182
5.7.4 Lyapunov-Krasovskii derivative condition . . . . . . 186
5.7.5
Stability criterion and examples . . . . . . . . . . . . 189
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.8 Notes
5.8.1 Results based on Razumikhin Theorem . . . . . . . 191
5.8.2 Model transformation and additional dynamics . . . 191
5.8.3 Lyapunov-Krasovskii method . . . . . . . . . . . . . 192
6 Robust Stability Analysis
193
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.2 Uncertainty characterization . . . . . . . . . . . . . . . . . . 193
6.2.1 Polytopic uncertainty . . . . . . . . . . . . . . . . . 194
6.2.2
Subpolytopic uncertainty . . . . . . . . . . . . . . . 195
6.2.3 Norm bounded uncertainty . . . . . . . . . . . . . . 197
6.2.4 Block-diagonal uncertainty . . . . . . . . . . . . . . 201
6.3 Robust stability based on Razumikhin Theorem . . . . . . . 203
6.3.1 Delay-independent stability for systems with single
delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.3.2 Delay independent stability for systems with distrib-
uted delays . . . . . . . . . . . . . . . . . . . . . . . 204
6.3.3 Delay-dependent stability with explicit model trans-
formation . . . . . . . . . . . . . . . . . . . . . . . . 207
6.4 Delay independent stability using Lyapunov-Krasovskii func-
tional
6.4.1
6.4.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Systems with single delay . . . . . . . . . . . . . . . 209
Systems with distributed delays . . . . . . . . . . . . 214
6.5 Delay-dependent stability using simple Lyapunov-Krasovskii
functional
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
6.6 Complete quadratic Lyapunov-Krasovskii functional approach220
vi
6.7 Discretized Lyapunov functional method for systems with
6.8 Notes
single delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.7.1 General case
. . . . . . . . . . . . . . . . . . . . . . 223
6.7.2 Norm bounded uncertainty . . . . . . . . . . . . . . 225
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.8.1 Uncertainty characterization . . . . . . . . . . . . . 227
6.8.2
Stability results based on Razumikhin Theorem and
Lyapunov-Krasovskii functional . . . . . . . . . . . . 228
7 Systems with Multiple and distributed delays
229
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2 Delay-independent stability of systems with multiple delays 230
7.3 Simple delay-dependent stability of systems with multiple
delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.4 Complete quadratic functional for general linear systems . . 234
7.5 Discretized Lyapunov functional method for systems with
multiple delays . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.5.1 Problem setup . . . . . . . . . . . . . . . . . . . . . 238
7.5.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 240
7.5.3 Lyapunov-Krasovskii functional condition . . . . . . 242
7.5.4 Lyapunov-Krasovskii derivative condition . . . . . . 248
7.5.5
Stability condition and examples . . . . . . . . . . . 255
7.6 Discretized Lyapunov functional method for systems with
distributed delays . . . . . . . . . . . . . . . . . . . . . . . . 257
7.6.1 Problem statement . . . . . . . . . . . . . . . . . . . 257
7.6.2 Discretization . . . . . . . . . . . . . . . . . . . . . . 259
7.6.3 Lyapunov-Krasovskii functional condition . . . . . . 260
7.6.4 Lyapunov-Krasovskii derivative condition . . . . . . 262
Stability criterion and examples . . . . . . . . . . . . 266
7.6.5
7.7 Notes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8 Stability under dynamic uncertainty
8.1
8.2
8.3 Method of comparison systems
269
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Input-output stability . . . . . . . . . . . . . . . . . . . . . 270
. . . . . . . . . . . . . . . . 273
8.3.1 Frequency domain approach . . . . . . . . . . . . . . 273
8.3.2 Time domain approach . . . . . . . . . . . . . . . . 277
8.4 Scaled small gain problem . . . . . . . . . . . . . . . . . . . 279
8.5 Robust stability under dynamic uncertainty . . . . . . . . . 284
8.5.1 Problem setup . . . . . . . . . . . . . . . . . . . . . 284
8.5.2 Uncertainty characterization . . . . . . . . . . . . . 285
8.5.3 Robust small gain problem . . . . . . . . . . . . . . 287
8.6 Approximation approach . . . . . . . . . . . . . . . . . . . . 288
8.6.1 Approximation of time-varying delay . . . . . . . . . 288
8.6.2 Approximation of distributed delay coefficient matrix 291
vii
8.6.3 Approximation by multiple delays . . . . . . . . . . 296
8.7 Passivity and generalization . . . . . . . . . . . . . . . . . . 299
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
8.8 Notes
A Matrix facts
303
A.1 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
A.2 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . 304
A.3 Singular value decomposition . . . . . . . . . . . . . . . . . 306
A.4 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
A.5 Kronecker product and sum . . . . . . . . . . . . . . . . . . 308
A.6 Notes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
B LMI & Quadratic Integral Inequalities
309
B.1 Basic LMI problem . . . . . . . . . . . . . . . . . . . . . . . 309
B.2 Generalized Eigenvalue Problem (GEVP)
. . . . . . . . . . 310
B.3 Transforming nonlinear matrix inequalities to LMI form . . 312
B.4 S-procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
B.5 Elimination of matrix variables . . . . . . . . . . . . . . . . 313
B.6 Quadratic Integral Inequalities
. . . . . . . . . . . . . . . . 316
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
B.7 Notes
Bibliography
319
B.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
viii
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