Linear Dynamical Systems

Front Cover......Page 1
Linear Dynamical Systems......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface to the Revised Edition......Page 12
Preface to the First Edition......Page 14
1.1 Dynamical Systems, Inputs, and Outputs......Page 18
1.2 Internal Description of Σ......Page 20
1.3 Realizations......Page 23
1.4 Controllability and Observability......Page 24
1.5 Stability and Feedback......Page 28
1.6 Optimality......Page 30
1.7 Stochastic Disturbances......Page 34
Notes and References......Page 36
2.2 Dynamical Systems......Page 38
2.3 External Description......Page 44
2.4 Frequency-Domain Analysis......Page 45
2.5 Transfer Functions......Page 47
2.6 Impulse-Response Function......Page 48
Notes and References......Page 50
3.1 Introduction......Page 52
3.2 Basic Definitions......Page 53
3.3 Time-Dependent Linear Systems......Page 56
3.4 Discrete-Time Systems......Page 60
3.5 Constant Systems......Page 64
3.6 Positive Controllability......Page 69
3.7 Relative Controllability......Page 72
3.8 Conditional Controllability......Page 74
3.9 Structural Controllability......Page 75
3.10 Controllability and Transfer Functions......Page 78
3.11 Systems with a Delay......Page 79
Miscellaneous Exercises......Page 81
Notes and References......Page 85
4.1 Introduction......Page 89
4.2 Basic Definitions......Page 90
4.3 Basic Theorems......Page 92
4.4 Duality......Page 98
4.5 Functional Analytic Approach to Observability......Page 99
4.6 The Problem of Moments......Page 100
Miscellaneous Exercises......Page 101
Notes and References......Page 102
5.1 Introduction......Page 105
5.2 State Variable Transformations......Page 107
5.3 Control Canonical Forms......Page 108
5.4 Observer Canonical Forms......Page 114
5.5 Invariance of Transfer Functions......Page 116
5.6 Canonical Forms and the Bezoutiant Matrix......Page 118
5.7 The Feedback Group and Invariant Theory......Page 121
Miscellaneous Exercises......Page 128
Notes and References......Page 131
6.1 Introduction......Page 134
6.2 Algebraic Equivalence and Minimal Realizability......Page 135
6.3 Construction of Realizations......Page 141
6.4 Minimal Realization Algorithm......Page 144
6.5 Examples......Page 145
6.6 Realization of Transfer Functions......Page 148
6.7 Uniqueness of Minimal Realizations......Page 149
6.8 Partial Realizations......Page 150
6.9 Reduced Order Models and Balanced Realizations......Page 155
Miscellaneous Exercises......Page 157
Notes and References......Page 162
7.1 Introduction......Page 164
7.2 Some Examples and Basic Concepts......Page 166
7.3 Routh-Hurwicz Methods......Page 169
7.4 Lyapunov Method......Page 173
7.5 Frequency-Domain Techniques......Page 179
7.6 Feedback Control Systems and Stability......Page 181
7.7 Modal Control......Page 186
7.8 Observers......Page 190
7.9 Structural Stability......Page 191
Miscellaneous Exercises......Page 194
Notes and References......Page 196
8.1 Motivation and Examples......Page 199
8.2 Open-Loop Solutions......Page 202
8.3 The Maximum Principle......Page 204
8.4 Some Computational Considerations......Page 207
8.5 Feedback Solutions......Page 209
8.6 Generalized X–Y Functions......Page 213
8.7 Optimality versus Stability......Page 221
8.8 A Low-Dimensional Alternative to the Algebraic Riccati Equation......Page 231
8.9 Computational Approaches for Riccati Equations......Page 233
8.10 Structural Stability of the Optimal Closed-Loop System......Page 236
8.11 Inverse Problems......Page 238
8.12 Linear Filtering Theory and Duality......Page 244
8.13 The Separation Principle and Stochastic Control Theory......Page 248
8.14 Discrete-Time Problems......Page 250
8.15 Generalized X–Y Functions Revisited......Page 251
Miscellaneous Exercises......Page 252
Notes and References......Page 257
9.1 Algebra, Geometry, and Linear Systems......Page 263
9.2 Mathematical Description of a Linear System......Page 264
9.3 The Module Structure of Ω, Γ, and X......Page 266
9.4 Some System-Theoretic Consequences......Page 270
9.5 Transfer Functions......Page 274
9.6 Realization of Transfer Functions......Page 277
9.7 The Construction of Canonical Realizations......Page 280
9.8 Partial Realizations......Page 288
9.9 Pole-Shifting and Stability......Page 290
9.10 Systems over Rings......Page 291
9.11 Some Geometric Aspects of Linear Systems......Page 295
9.12 Feedback, the McMillan Degree, and Kronecker Indices......Page 300
9.13 Some Additional Ideas from Algebraic Geometry......Page 302
9.14 Pole Placement for Linear Regulators......Page 305
9.15 Multivariable Nyquist Criteria......Page 308
9.16 Algebraic Topology and Simplicia1 Complex of Σ......Page 309
Miscellaneous Exercises......Page 315
Notes and References......Page 327
10.1 Finiteness as a System Property......Page 334
10.2 Reachability and Controllability......Page 336
10.3 Observability and Duality......Page 340
10.4 Stability Theory......Page 342
10.5 Realization Theory......Page 344
10.6 The LQG Problem......Page 347
10.7 Operator Riccati Equations and Generalized X–Y Functions......Page 349
Miscellaneous Exercises......Page 352
Notes and References......Page 362
Index......Page 364