Robust Control and Observation of Linear Parameter Varying Time-Delay Systems

Remerciements......Page 3
Summary......Page 5
Introduction and Structure of the Thesis......Page 7
Table of Contents......Page 13
List of Figures......Page 17
List of Tables......Page 19
List of Publications......Page 21
Notations and Acronyms......Page 23
1.1.1 Systèmes à Retards......Page 25
1.1.2 Systèmes Linéaires à Paramètres Variants......Page 27
1.1.3 Les systèmes LPV à retards......Page 28
1.2.1 Contributions Méthodologiques......Page 29
1.2.2 Contributions Théoriques......Page 30
1.3.1 Observation......Page 31
1.3.2 Filtrage......Page 34
1.4.1 Contrôleurs par retour d\'état......Page 36
1.4.2 Contrôleurs séquencés par le retard......Page 38
1.4.3 Contrôleurs par retour dynamique de sortie......Page 39
1.5 Conclusion......Page 41
2 Overview of LPV Systems......Page 43
2.1.1.1 Parameters as functions of states......Page 49
2.1.1.3 External parameters......Page 50
2.1.2.1 Discrete vs. Continuous Valued Parameters......Page 51
2.1.2.3 Differentiable vs. Non-Differentiable Parameters......Page 52
2.2.1.1 Affine and Multi-Affine Systems......Page 54
2.2.2.1 Polytopic Formulation......Page 55
2.2.2.2 Parameter Dependent Formulation......Page 59
2.2.2.3 \'LFT\' Formulation......Page 60
2.3.1 Notions of stability for LTI and LPV systems......Page 62
2.3.2 Stability of Polytopic Systems......Page 74
2.3.3 Stability of Polynomially Parameter Dependent Systems......Page 79
2.3.3.1 Relaxation of matrix functions......Page 80
2.3.3.2 Relaxation of parametrized LMIs by discretization (gridding)......Page 81
2.3.3.3 Relaxation of Parametrized LMIs using methods based on Sum-of-Squares (SoS)......Page 82
2.3.3.4 Global Polynomial Optimization and the Problem of Moments......Page 88
2.3.4 Stability of \'LFT\' systems......Page 93
2.3.4.1 Passivity......Page 94
2.3.4.2 Small-Gain Theorem......Page 96
2.3.4.3 Scaled-Small Gain Theorem......Page 101
2.3.4.4 Full-Block S-procedure and Well-Posedness of Feedback Systems......Page 103
2.3.4.5 Frequency-Dependent D-Scalings......Page 119
2.3.4.6 Analysis via Integral Quadratic Constraints (IQC)......Page 120
2.4 Chapter Conclusion......Page 123
3 Overview of Time-Delay Systems......Page 125
3.1 Representation of Time-Delay Systems......Page 126
3.1.1 Functional Differential Equations......Page 127
3.2 Stability Analysis of Time-Delay Systems......Page 130
3.2.1 Time-Domain Stability Analysis......Page 133
3.2.1.1 On the extension of Lyapunov Theory......Page 134
3.2.1.2 About model transformations......Page 137
3.2.1.3 Additional Dynamics......Page 139
3.2.1.4 Stability Analysis: Lyapunov-Razumikhin Functions......Page 141
3.2.1.5 Stability Analysis: Lyapunov-Krasovskii Functionals......Page 142
3.2.1.6 Stability Analysis: (Scaled) Small-Gain Theorem......Page 150
3.2.1.7 Stability Analysis: Padé Approximants......Page 154
3.2.1.8 Stability Analysis: Integral Quadratic Constraints......Page 156
3.2.1.9 Stability Analysis: Well-Posedness Approach......Page 158
3.2.2 Robustness with respect to delay uncertainty......Page 160
3.2.2.1 Frequency domain: Matrix Pencil approach......Page 161
3.2.2.2 Frequency domain: Rouché\'s Theorem......Page 162
3.3 Chapter Conclusion......Page 163
4 Definitions and Preliminary Results......Page 165
4.1.1 Delay Spaces......Page 166
4.1.2 Parameter Spaces......Page 169
4.1.3 Class of LPV Time-Delay Systems......Page 172
4.2 Relaxation of Polynomially Parameter Dependent LMIs......Page 173
4.3 Relaxation of Concave Nonlinearity......Page 177
4.4 Polytopic Systems and Bounded-Parameter Variation Rates......Page 183
4.5.1 Simple Lyapunov-Krasovskii functional......Page 186
4.5.2 Associated Relaxation......Page 189
4.5.3 Reduced Simple Lyapunov-Krasovskii functional......Page 191
4.5.4 Associated Relaxation......Page 192
4.6 Discretized Lyapunov-Krasovskii Functional for systems with time varying delay and Associated Relaxation......Page 193
4.6.1 Discretized Lyapunov-Krasovskii functional......Page 195
4.6.2 Associated Relaxation......Page 199
4.7 Simple Lyapunov-Krasovskii functional for systems with delay uncertainty......Page 200
4.7.1 Lyapunov-Krasovskii functional......Page 201
4.7.2 Associated Relaxation......Page 204
4.8 Chapter Conclusion......Page 205
5 Observation and Filtering of LPV time-delay systems......Page 207
5.1 Observation of Unperturbed LPV Time-Delay Systems......Page 209
5.1.1 Observer with exact delay value - simple Lyapunov-Krasovskii functional case......Page 210
5.1.2 Observer with approximate delay value......Page 220
5.1.3 Memoryless Observer......Page 223
5.2 Filtering of uncertain LPV Time-Delay Systems......Page 226
5.2.1 Design of robust filters with exact delay-value - simple Lyapunov-Krasovskii functional......Page 227
5.2.2 Design of robust memoryless filters......Page 231
5.2.3 Example......Page 232
5.3 Chapter Conclusion......Page 234
6 Control of LPV Time-Delay Systems......Page 237
6.1 State-Feedback Control laws......Page 239
6.1.1 Memoryless State-Feedback Design - Relaxed Simple Lyapunov-Krasovskii functional......Page 240
6.1.2 Memoryless State-Feedback Design - Relaxed Discretized Lyapunov-Krasovskii functional......Page 244
6.1.3 Memoryless State-Feedback Design - Simple Lyapunov-Krasovskii functional......Page 245
6.1.3.1 About adjoint systems of LPV systems......Page 246
6.1.3.2 LPV Control of LPV time-delay systems using adjoint......Page 248
6.1.4 Memoryless state-feedback - Polytopic approach......Page 259
6.1.5 Hereditary State-Feedback Controller Design - exact delay value case......Page 262
6.1.6 Hereditary State-Feedback Controller Design - approximate delay value case......Page 263
6.1.7 Delay-Scheduled State-Feedback Controllers......Page 265
6.1.7.1 Stability and L2 performances analysis......Page 268
6.1.7.2 Delay-Scheduled state-feedback design......Page 269
6.2 Dynamic Output Feedback Control laws......Page 272
6.2.1 Memoryless observer based control laws......Page 273
6.2.2 Dynamic Output Feedback with memory design - exact delay case......Page 278
6.3 Chapter Conclusion......Page 282
Conclusion and Future Works......Page 283
A.1 Determinant Formulae......Page 285
A.2 Eigenvalues of Matrices......Page 286
A.4 Generalities on Block-Matrices......Page 288
A.5 Kronecker operators and Matrix Tensor Sum and Product......Page 289
A.6 Singular-Values Decomposition......Page 290
A.8 Solving AX=B......Page 291
B Lq and Hq Spaces......Page 292
B.1 Norms for Signals......Page 293
B.2 Norms for Systems......Page 294
C.1 Story......Page 296
C.2 Definitions......Page 297
C.3 How to solve them ?......Page 298
D.1 Dissipative Systems and Supply Rates......Page 300
D.2 Linear Dissipative Systems and Quadratic Supply Rates......Page 301
D.3 Kalman-Yakubovich-Popov Lemma......Page 302
D.4 Schur complement......Page 304
D.5 Positive real lemma......Page 305
D.6 H2 Performances......Page 306
D.7 Generalized H2 performances......Page 307
D.8 Bounded-Real Lemma - H Performances......Page 308
D.9 L-L Performances......Page 309
D.10 S-procedure......Page 310
D.11 Small Gain Theorem......Page 311
D.12 Scalings and Scaled-Small Gain theorem......Page 312
D.13 Full-Block S-procedure......Page 314
D.15 Bounding (Petersen\'s) Lemma......Page 315
D.16 Finsler\'s Lemma......Page 320
D.17 Generalization of Finsler\'s lemma......Page 321
D.18 Projection Lemma......Page 322
D.19 Completion Lemma......Page 323
D.20 Application of the Projection Lemma......Page 326
D.21 Parseval\'s Theorem......Page 329
E.2 Bounding of cross-terms......Page 331
E.3 Padé Approximants......Page 333
E.4 Maximum Modulus Principle......Page 334
E.5 Argument principle......Page 335
E.6 Proof of Rouché\'s theorem......Page 336
Bibliography......Page 339