Controlled and Conditioned Invariants in Linear System Theory

Preface......Page 3
Glossary......Page 5
Contents......Page 9
1.1 Basic Concepts and Terms......Page 13
1.2 Some Examples of Dynamic Systems......Page 15
1.3 General De.nitions and Properties......Page 22
1.4 Controlling and Observing the State......Page 32
1.5.1 Graphic Representations of Interconnected Systems......Page 36
1.5.2 Cascade, Parallel, and Feedback Interconnections......Page 40
1.6 A Review of System and Control Theory Problems......Page 42
1.7 Finite-State Systems......Page 47
1.7.1 Controllability......Page 50
1.7.2 Reduction to the Minimal Form......Page 55
1.7.3 Diagnosis and State Observation......Page 58
1.7.4 Homing and State Reconstruction......Page 61
1.7.5 Finite-Memory Systems......Page 63
References......Page 66
2.1.1 Linear Time-Varying Continuous Systems......Page 69
2.1.2 Linear Time-Varying Discrete Systems......Page 73
2.1.3 Function of a Matrix......Page 74
2.1.4 Linear Time-Invariant Continuous Systems......Page 78
2.1.5 Linear Time-Invariant Discrete Systems......Page 85
2.2.1 Linear Time-Varying Continuous Systems......Page 88
2.2.2 Linear Time-Varying Discrete Systems......Page 91
2.2.3 Linear Time-Invariant Systems......Page 92
2.2.4 Computation of the Matrix Exponential Integral......Page 96
2.2.5 Approximating Continuous with Discrete......Page 99
2.3 IO Representations of Linear Constant Systems......Page 101
2.4 Relations Between IO and ISO Representations......Page 104
2.4.1 The Realization Problem......Page 106
2.5.1 Linear Time-Varying Systems......Page 112
2.5.2 Linear Time-Invariant Systems......Page 116
2.5.3 The Liapunov and Sylvester Equations......Page 119
2.6.1 Linear Time-Varying Systems......Page 123
2.6.2 Linear Time-Invariant Systems......Page 130
References......Page 135
3.1.1 Some Subspace Algebra......Page 137
3.2.1 Invariants and Changes of Basis......Page 140
3.2.2 Lattices of Invariants and Related Algorithms......Page 141
3.2.3 Invariants and System Structure......Page 143
3.2.4 Invariants and State Trajectories......Page 145
3.2.5 Stability and Complementability......Page 146
3.3 Controllability and Observability......Page 149
3.3.1 The Kalman Canonical Decomposition......Page 150
3.3.2 Referring to the Jordan Form......Page 155
3.3.3 SISO Canonical Forms and Realizations......Page 157
3.3.4 Structural Indices and MIMO Canonical Forms......Page 162
3.4 State Feedback and Output Injection......Page 167
3.4.1 Asymptotic State Observers......Page 175
3.4.2 The Separation Property......Page 178
3.5 Some Geometric Aspects of Optimal Control......Page 182
3.5.1 Convex Sets and Convex Functions......Page 184
3.5.2 The Pontryagin Maximum Principle......Page 189
3.5.3 The Linear-Quadratic Regulator......Page 200
3.5.4 The Time-Invariant LQR Problem......Page 201
References......Page 208
4.1 Controlled and Conditioned Invariants......Page 211
4.1.1 Some Speci.c Computational Algorithms......Page 215
4.1.2 Self-Bounded Controlled Invariants and their Duals......Page 217
4.1.3 Constrained Controllability and Observability......Page 222
4.1.4 Stabilizability and Complementability......Page 223
4.2 Disturbance Localization and Unknowninput State Estimation......Page 230
4.3 Unknown-Input Reconstructability, Invertibility, and Functional Controllability......Page 237
4.3.1 A General Unknown-Input Reconstructor......Page 239
4.3.2 System Invertibility and Functional Controllability......Page 242
4.4 Invariant Zeros and the Invariant Zero Structure......Page 244
4.4.1 The Generalized Frequency Response......Page 245
4.4.2 The Role of Zeros in Feedback Systems......Page 249
4.5 Extensions to Quadruples......Page 251
4.5.1 On Zero Assignment......Page 255
References......Page 256
5.1 The Five-Map System......Page 259
5.1.1 Some Properties of the Extended State Space......Page 262
5.1.2 Some Computational Aspects......Page 267
5.1.3 The Dual-Lattice Structures......Page 272
5.2 The Dynamic Disturbance Localization and the Regulator Problem......Page 279
5.2.1 Proof of the Nonconstructive Conditions......Page 282
5.2.2 Proof of the Constructive Conditions......Page 286
5.2.3 General Remarks and Computational Recipes......Page 292
5.2.4 Su.cient Conditions in Terms of Zeros......Page 297
5.3 Reduced-Order Devices......Page 298
5.3.1 Reduced-Order Observers......Page 302
5.3.2 Reduced-Order Compensators and Regulators......Page 304
5.4 Accessible Disturbance Localization and Model-Following Control......Page 307
5.5 Noninteracting Controllers......Page 310
References......Page 315
6.1 The Single-Variable Feedback Regulation Scheme......Page 319
6.2 The Autonomous Regulator: A General Synthesis Procedure......Page 324
6.2.1 On the Separation Property of Regulation......Page 333
6.2.2 The Internal Model Principle......Page 335
6.3 The Robust Regulator: Some Synthesis Procedures......Page 336
6.4 The Minimal-Order Robust Regulator......Page 347
6.5 The Robust Controlled Invariant......Page 350
6.5.1 The Hyper-Robust Disturbance Localization Problem......Page 355
6.5.2 Some Remarks on Hyper-Robust Regulation......Page 357
References......Page 358
A.1 Sets, Relations, Functions......Page 361
A.2 Fields, Vector Spaces, Linear Functions......Page 375
A.3 Inner Product, Orthogonality......Page 392
A.4 Eigenvalues, Eigenvectors......Page 399
A.5 Hermitian Matrices, Quadratic Forms......Page 419
A.6 Metric and Normed Spaces, Norms......Page 422
References......Page 436
B.1 Gauss-Jordan Elimination and LU Factorization......Page 439
B.2 Gram-Schmidt Orthonormalization and QR Factorization......Page 443
B.3 The Singular Value Decomposition......Page 447
B.4 Computational Support with Matlab......Page 448
References......Page 450
Index......Page 451