Linear System Theory and Design

Contents......Page 3
Preface......Page 7
1.1 Introduction......Page 10
1.2 Overview......Page 11
2.1 Introduction......Page 14
2.1.1 Causality and Lumpedness......Page 15
2.2 Linear Systems......Page 16
2.3 Linear Time-Invariant (LTI) Systems......Page 20
2.3.1 Op-Amp Circuit Implementation......Page 25
2.4 Linearization......Page 26
2.5 Examples......Page 27
2.6 Discrete-Time Systems......Page 40
2.7 Concluding Remarks......Page 46
3.1 Introduction......Page 53
3.2 Basis, Representation, and Orthonormalization......Page 54
3.3 Linear Algebraic Equations......Page 57
3.4 Similarity Transformation......Page 62
3.5 Diagonal Form and Jordan Form......Page 64
3.6 Functions of a Square Matrix......Page 70
3.7 Lyapunov Equation......Page 79
3.8 Some Useful Formulas......Page 80
3.9 Quadratic Form and Positive Definiteness......Page 82
3.10 Singular-Value Decomposition......Page 85
3.11 Norms of Matrices......Page 87
4.1 Introduction......Page 95
4.2 Solution of LTI State Equations......Page 96
4.2.1 Discretization......Page 99
4.2.2 Solution of Discrete-Time Equations......Page 101
4.3 Equivalent State Equations......Page 103
4.3.1 Canonical Forms......Page 106
4.3.2 Magnitude Scaling in Op-Amp Circuits......Page 107
4.4 Realizations......Page 109
4.5 Solution of Linear Time-Varying (LTV) Equations......Page 115
4.5.1 Discrete-Time Case......Page 119
4.6 Equivalent Time-Varying Equations......Page 120
4.7 Time-Varying Realizations......Page 124
5.2 Input–Output Stability of LTI Systems......Page 130
5.2.1 Discrete-Time Case......Page 135
5.3 Internal Stability......Page 138
5.3.1 Discrete-Time Case......Page 140
5.4 Lyapunov Theorem......Page 141
5.4.1 Discrete-Time Case......Page 144
5.5 Stability of LTV Systems......Page 146
6.1 Introduction......Page 152
6.2 Controllability......Page 153
6.2.1 Controllability Indices......Page 159
6.3 Observability......Page 162
6.3.1 Observability Indices......Page 166
6.4 Canonical Decomposition......Page 168
6.5 Conditions in Jordan-Form Equations......Page 173
6.6 Discrete-Time State Equations......Page 178
6.7 Controllability After Sampling......Page 181
6.8 LTV State Equations......Page 185
7.1 Introduction......Page 193
7.2 Implications of Coprimeness......Page 194
7.2.1 Minimal Realizations......Page 198
7.3 Computing Coprime Fractions......Page 201
7.3.1 QR Decomposition......Page 204
7.4 Balanced Realization......Page 206
7.5 Realizations from Markov Parameters......Page 209
7.6 Degree of Transfer Matrices......Page 214
7.7 Minimal Realizations—Matrix Case......Page 216
7.8 Matrix Polynomial Fractions......Page 218
7.8.1 Column and Row Reducedness......Page 221
7.8.2 Computing Matrix Coprime Fractions......Page 223
7.9 Realizations from Matrix Coprime Fractions......Page 230
7.10 Realizations from Matrix Markov Parameters......Page 234
7.11 Concluding Remarks......Page 236
8.1 Introduction......Page 240
8.2 State Feedback......Page 241
8.2.1 Solving the Lyapunov Equation......Page 248
8.3 Regulation and Tracking......Page 251
8.3.1 Robust Tracking and Disturbance Rejection......Page 252
8.3.2 Stabilization......Page 256
8.4 State Estimator......Page 257
8.4.1 Reduced-Dimensional State Estimator......Page 260
8.5 Feedback from Estimated States......Page 262
8.6 State Feedback—Multivariable Case......Page 264
8.6.1 Cyclic Design......Page 265
8.6.2 Lyapunov-Equation Method......Page 268
8.6.3 Canonical-Form Method......Page 269
8.6.4 Effect on Transfer Matrices......Page 271
8.7 State Estimators—Multivariable Case......Page 272
8.8 Feedback from Estimated States–Multivariable Case......Page 274
9.1 Introduction......Page 278
9.1.1 Compensator Equations—Classical Method......Page 280
9.2 Unity-Feedback Configuration—Pole Placement......Page 282
9.2.1 Regulation and Tracking......Page 284
9.2.2 Robust Tracking and Disturbance Rejection......Page 286
9.3 Implementable Transfer Functions......Page 292
9.3.1 Model Matching—Two-Parameter Configuration......Page 295
9.3.2 Implementation of Two-Parameter Compensators......Page 300
9.4 Multivariable Unity-Feedback Systems......Page 301
9.4.1 Regulation and Tracking......Page 311
9.4.2 Robust Tracking and Disturbance Rejection......Page 312
9.5 Multivariable Model Matching—Two-Parameter Configuration......Page 315
9.6 Concluding Remarks......Page 323
References......Page 328
CHAPTER 2......Page 329
CHAPTER 3......Page 330
CHAPTER 4......Page 332
CHAPTER 5......Page 333
CHAPTER 6......Page 334
CHAPTER 7......Page 335
CHAPTER 8......Page 336
CHAPTER 9......Page 337
INDEX......Page 339