Linear Systems Theory

【实例简介】Linear Systems Theory 2ND EDITION by João P. Hespanha

【实例截图】

【核心代码】

Contents
Preamble xiii
Linear Systems I — Basic Concepts 1
I System Representation 3
1 State-Space Linear Systems 5
1.1 State-Space Linear Systems 5
1.2 Block Diagrams 7
1.3 Exercises 11
2 Linearization 12
2.1 State-Space Nonlinear Systems 12
2.2 Local Linearization Around an Equilibrium Point 12
2.3 Local Linearization Around a Trajectory 15
2.4 Feedback Linearization 16
2.5 Practice Exercises 22
2.6 Exercises 27
3 Causality, Time Invariance, and Linearity 31
3.1 Basic Properties of LTV/LTI Systems 31
3.2 Characterization of All Outputs to a Given Input 33
3.3 Impulse Response 34
3.4 Laplace and Z Transforms (Review) 37
3.5 Transfer Function 38
3.6 Discrete-Time Case 39
3.7 Additional Notes 40
3.8 Exercises 42
4 Impulse Response and Transfer Function of
State-Space Systems 43
4.1 Impulse Response and Transfer Function for LTI Systems 43
4.2 Discrete-Time Case 44
4.3 Elementary Realization Theory 45
4.4 Equivalent State-Space Systems 49
4.5 LTI Systems in MATLAB
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50
4.6 Practice Exercises 52
4.7 Exercises 53
5 Solutions to LTV Systems 56
5.1 Solution to Homogeneous Linear Systems 56
5.2 Solution to Nonhomogeneous Linear Systems 58
viii CONTENTS
5.3 Discrete-Time Case 59
5.4 Practice Exercises 61
5.5 Exercises 62
6 Solutions to LTI Systems 64
6.1 Matrix Exponential 64
6.2 Properties of the Matrix Exponential 65
6.3 Computation of Matrix Exponentials Using
Laplace Transforms 67
6.4 The Importance of the Characteristic Polynomial 68
6.5 Discrete-Time Case 69
6.6 Symbolic Computations in MATLAB
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70
6.7 Practice Exercises 72
6.8 Exercises 74
7 Solutions to LTI Systems: The Jordan Normal Form 76
7.1 Jordan Normal Form 76
7.2 Computation of Matrix Powers using the
Jordan Normal Form 78
7.3 Computation of Matrix Exponentials using the
Jordan Normal Form 80
7.4 Eigenvalues with Multiplicity Larger than 1 81
7.5 Practice Exercise 82
7.6 Exercises 83
II Stability 85
8 Internal or Lyapunov Stability 87
8.1 Lyapunov Stability 87
8.2 Vector and Matrix Norms (Review) 88
8.3 Eigenvalue Conditions for Lyapunov Stability 90
8.4 Positive-Definite Matrices (Review) 91
8.5 Lyapunov Stability Theorem 91
8.6 Discrete-Time Case 95
8.7 Stability of Locally Linearized Systems 98
8.8 Stability Tests with MATLAB
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103
8.9 Practice Exercises 103
8.10 Exercises 105
9 Input-Output Stability 108
9.1 Bounded-Input, Bounded-Output Stability 108
9.2 Time Domain Conditions for BIBO Stability 109
9.3 Frequency Domain Conditions for BIBO Stability 112
9.4 BIBO versus Lyapunov Stability 113
9.5 Discrete-Time Case 114
9.6 Practice Exercises 115
9.7 Exercises 118
CONTENTS ix
10 Preview of Optimal Control 120
10.1 The Linear Quadratic Regulator Problem 120
10.2 Feedback Invariants 121
10.3 Feedback Invariants in Optimal Control 122
10.4 Optimal State Feedback 122
10.5 LQR with MATLAB
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124
10.6 Practice Exercise 124
10.7 Exercise 125
III Controllability and State Feedback 127
11 Controllable and Reachable Subspaces 129
11.1 Controllable and Reachable Subspaces 129
11.2 Physical Examples and System Interconnections 130
11.3 Fundamental Theorem of Linear Equations (Review) 134
11.4 Reachability and Controllability Gramians 135
11.5 Open-Loop Minimum-Energy Control 137
11.6 Controllability Matrix (LTI) 138
11.7 Discrete-Time Case 141
11.8 MATLAB
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Commands 145
11.9 Practice Exercise 146
11.10 Exercises 147
12 Controllable Systems 148
12.1 Controllable Systems 148
12.2 Eigenvector Test for Controllability 150
12.3 Lyapunov Test for Controllability 152
12.4 Feedback Stabilization Based on the Lyapunov Test 155
12.5 Eigenvalue Assignment 156
12.6 Practice Exercises 157
12.7 Exercises 159
13 Controllable Decompositions 162
13.1 Invariance with Respect to Similarity Transformations 162
13.2 Controllable Decomposition 163
13.3 Block Diagram Interpretation 165
13.4 Transfer Function 166
13.5 MATLAB
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Commands 166
13.6 Exercise 167
14 Stabilizability 168
14.1 Stabilizable System 168
14.2 Eigenvector Test for Stabilizability 169
14.3 Popov-Belevitch-Hautus (PBH) Test for Stabilizability 171
14.4 Lyapunov Test for Stabilizability 171
14.5 Feedback Stabilization Based on the Lyapunov Test 173
14.6 MATLAB
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Commands 174
14.7 Exercises 174
x CONTENTS
IV Observability and Output Feedback 177
15 Observability 179
15.1 Motivation: Output Feedback 179
15.2 Unobservable Subspace 180
15.3 Unconstructible Subspace 182
15.4 Physical Examples 182
15.5 Observability and Constructibility Gramians 184
15.6 Gramian-Based Reconstruction 185
15.7 Discrete-Time Case 187
15.8 Duality for LTI Systems 188
15.9 Observability Tests 190
15.10 MATLAB
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Commands 193
15.11 Practice Exercises 193
15.12 Exercises 195
16 Output Feedback 198
16.1 Observable Decomposition 198
16.2 Kalman Decomposition Theorem 200
16.3 Detectability 202
16.4 Detectability Tests 204
16.5 State Estimation 205
16.6 Eigenvalue Assignment by Output Injection 206
16.7 Stabilization through Output Feedback 207
16.8 MATLAB
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Commands 208
16.9 Exercises 208
17 Minimal Realizations 210
17.1 Minimal Realizations 210
17.2 Markov Parameters 211
17.3 Similarity of Minimal Realizations 213
17.4 Order of a Minimal SISO Realization 215
17.5 MATLAB
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Commands 217
17.6 Practice Exercises 217
17.7 Exercises 219
Linear Systems II — Advanced Material 221
V Poles and Zeros of MIMO Systems 223
18 Smith-McMillan Form 225
18.1 Informal Definition of Poles and Zeros 225
18.2 Polynomial Matrices: Smith Form 226
18.3 Rational Matrices: Smith-McMillan Form 229
18.4 McMillan Degree, Poles, and Zeros 230
CONTENTS xi
18.5 Blocking Property of Transmission Zeros 232
18.6 MATLAB
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Commands 233
18.7 Exercises 233
19 State-Space Poles, Zeros, and Minimality 235
19.1 Poles of Transfer Functions versus Eigenvalues of State-Space
Realizations 235
19.2 Transmission Zeros of Transfer Functions versus Invariant Zeros
of State-Space Realizations 236
19.3 Order of Minimal Realizations 239
19.4 Practice Exercises 241
19.5 Exercise 242
20 System Inverses 244
20.1 System Inverse 244
20.2 Existence of an Inverse 245
20.3 Poles and Zeros of an Inverse 246
20.4 Feedback Control of Invertible Stable Systems
with Stable Inverses 248
20.5 MATLAB
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Commands 249
20.6 Exercises 250
VI LQR/LQG Optimal Control 251
21 Linear Quadratic Regulation (LQR) 253
21.1 Deterministic Linear Quadratic Regulation (LQR) 253
21.2 Optimal Regulation 254
21.3 Feedback Invariants 255
21.4 Feedback Invariants in Optimal Control 256
21.5 Optimal State Feedback 256
21.6 LQR in MATLAB
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258
21.7 Additional Notes 258
21.8 Exercises 259
22 The Algebraic Riccati Equation (ARE) 260
22.1 The Hamiltonian Matrix 260
22.2 Domain of the Riccati Operator 261
22.3 Stable Subspaces 262
22.4 Stable Subspace of the Hamiltonian Matrix 262
22.5 Exercises 266
23 Frequency Domain and Asymptotic Properties
of LQR 268
23.1 Kalman’s Equality 268
23.2 Frequency Domain Properties: Single-Input Case 270
23.3 Loop Shaping Using LQR: Single-Input Case 272
23.4 LQR Design Example 275
xii CONTENTS
23.5 Cheap Control Case 278
23.6 MATLAB
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Commands 281
23.7 Additional Notes 282
23.8 The Loop-Shaping Design Method (Review) 283
23.9 Exercises 288
24 Output Feedback 289
24.1 Certainty Equivalence 289
24.2 Deterministic Minimum-Energy Estimation (MEE) 290
24.3 Stochastic Linear Quadratic Gaussian (LQG) Estimation 295
24.4 LQR/LQG Output Feedback 295
24.5 Loop Transfer Recovery (LTR) 296
24.6 Optimal Set-Point Control 297
24.7 LQR/LQG with MATLAB
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302
24.8 LTR Design Example 303
24.9 Exercises 304
25 LQG/LQR and the Q Parameterization 305
25.1 Q-Augmented LQG/LQR Controller 305
25.2 Properties 306
25.3 Q Parameterization 309
25.4 Exercise 309
26 Q Design 310
26.1 Control Specifications for Q Design 310
26.2 The Q Design Feasibility Problem 313
26.3 Finite-Dimensional Optimization: Ritz Approximation 314
26.4 Q Design Using MATLAB
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and CVX 316
26.5 Q Design Example 321
26.6 Exercise 323
Bibliography 325
Index 327