Theory of Control Systems Described by Differential Inclusions (eBook)

Preface 6
Contents 10
1 Convex Sets and Convex Functions 13
1.1 Normed Spaces and Inner Product Spaces 13
1.1.1 Sets and Mappings 13
1.1.2 Normed Spaces 15
1.1.3 Elementary Topology 19
1.1.4 Limitation Theorems 21
1.1.5 Inner Product Spaces 24
1.2 Convex Sets 26
1.2.1 Convex Sets and Their Properties 27
1.2.2 Construction of Convex Sets 29
1.2.3 Separation Theorems 35
1.3 Convex Functions 40
1.3.1 Convex Functions and Their Elementary Properties 40
1.3.2 Examples of Convex Functions 43
1.3.3 Derivative of Convex Functions 45
1.4 Semi-continuous Functions 53
1.4.1 Semi-continuous Functions and Their ElementaryProperties 53
1.4.2 Examples of Semi-continuous Functions 57
References 63
2 Set-Valued Mappings and Differential Inclusions 64
2.1 Set-Valued Mappings 64
2.1.1 Definition of the Set-Valued Mappings 64
2.1.2 Continuities of Set-Valued Mappings 66
2.1.3 Tangent Cones and Normal Cones 73
2.1.4 Derivative of Set-Valued Mappings 78
2.2 Selection of Set-Valued Mappings 83
2.2.1 The Minimal Selection 84
2.2.2 Michael Selection Theorem 87
2.2.3 Lipschitzian Approximation 91
2.2.4 Theorems for Fixed Points 98
2.3 Differential Inclusions and Existence Theorems 100
2.3.1 Differential Equations and Differential Inclusions 100
2.3.2 Why Do We Propose Differential Inclusions? 102
2.3.3 Existence Theorems of Solution of Differential Inclusions 108
2.3.4 The Existence of Solutions of Time-delayed Differential Inclusions 117
2.4 Qualitative Analysis for Differential Inclusions 119
2.4.1 Qualitative Analysis for Lipschitzian DifferentialInclusions 119
2.4.2 Qualitative Analysis for Upper Semi-continuous Differential Inclusions 123
2.4.3 Convexification of Differential Inclusions and Relaxed Theorem 129
2.5 Stability of Differential Inclusions 135
2.5.1 Dini Derivatives 135
2.5.2 Definitions of Stability of Differential Inclusions 139
2.5.3 Lyapunov-like Criteria for Stability of DifferentialInclusions 142
2.6 Monotonous Differential Inclusions 151
2.6.1 Monotonous Set-valued Mappings and Their Properties 152
2.6.2 Minty Theorem 154
2.6.3 Yosida Approximation 159
2.6.4 Maximally Monotonous Differential Inclusions 163
References 167
3 Convex Processes 168
3.1 Convex Processes in Linear Normed Spaces 168
3.1.1 Convex Processes and Their Adjoint Processes 169
3.1.2 The Norm of Convex Processes 174
3.1.3 Fundamental Properties of Convex Processes 175
3.2 Convex Processes in Spaces with Finite Dimensions 179
3.2.1 Adjoint Processes in n-Dimensional Space 179
3.2.2 Structure of Convex Processes 184
3.3 Controllability of Convex Processes 196
3.3.1 T-Controllability 196
3.3.2 Controllability 203
3.4 Stability of Convex Process Differential Inclusions 206
3.4.1 Stability of Convex Processes 207
3.4.2 Construction of Lyapunov Functions 214
Reference 219
4 Linear Polytope Control Systems 220
4.1 Polytope Systems 220
4.1.1 Linear Control Systems and Matrix Inequalities 220
4.1.2 Linear Polytope Systems 225
4.2 Convex Hull Lyapunov Functions 229
4.2.1 Convex Hull Quadratic Lyapunov Functions 229
4.2.2 Layer Sets for the Convex Hull Quadratic Function 232
4.3 Control of Linear Polytope Systems 243
4.3.1 Feedback Stabilizability for Linear Polytope Systems 244
4.3.2 Feedback Stabilization for Linear Polytope Systems with Time-delay 250
4.3.3 Disturbance Rejection for Linear Polytope Systems 253
4.4 Saturated Control for Linear Control Systems 260
4.4.1 Saturated Control Described by Set-Valued Mappings 260
4.4.2 Stabilization by the Saturated Control 262
4.4.3 Disturbance Rejection by the Saturated Control 267
References 273
5 Luré Differential Inclusion Systems 274
5.1 Luré Systems 274
5.1.1 Luré Systems and Absolute Stability 274
5.1.2 Positive Realness and the Positive Realness Lemma 277
5.1.3 Criterion for Absolute Stability 281
5.2 Stabilization of Luré Differential Inclusion Systems 283
5.2.1 An Example of the Luré Differential Inclusion System 284
5.2.2 Stabilization of Luré Differential Inclusion Systems 286
5.2.3 Zeroes and Relative Degree of Control Systems 287
5.2.3.1 The Zeroes of System 287
5.2.3.2 Relative Degree of System 288
5.2.3.3 Hurwitz Vectors 291
5.2.4 Feedback Positive Realness 292
5.2.5 Feedback Stabilization – Single-Variable Systems 297
5.2.6 Feedback Stabilization – Multivariable Systems 299
5.3 Luenberger Observers and Separated Design 301
5.3.1 Well-Posedness 302
5.3.2 The Luenberger State Observer 304
5.3.3 State Feedback Based on Observer 309
5.3.4 Reduce-Order Luenberger Observer 311
5.4 Linear Observers of Luré Differential Inclusion Systems 315
5.4.1 Single-Variable Systems 316
5.4.2 Multivariable Systems 323
5.5 Adaptive Luenberger Observers 325
5.5.1 Adaptive Luenberger Observers 326
5.5.2 Reduced-Order Adaptive Observers 330
5.5.3 An Example of Adaptive Observer 332
5.6 Adaptive Linear Observers 335
5.6.1 Persistent Excitation 336
5.6.2 Linear Adaptive Observers 343
References 350
Index 351