1. Getting Started
1
1.1 Using the Application for the First Time
1
1.2 The Structure of the Application
2
1.3 The Control Objects
3
1.4 Traditional Notations
5
1.5 The Control Format
7
1.6 The Notation for the Imaginary Unit
9
1.7 Numericalizing for Speed
9
1.8 Evaluating Examples in This Guide
10
2. Introduction: Extending Mathematica to Solve Control Problems
11
3. Description of Dynamic Systems
27
3.1 Transfer Function Representations
27
3.2 State-Space Representations
33
3.3 Continuous-Time versus Discrete-Time Systems
39
3.4 The Traditional Notations
41
3.5 Discrete-Time Models of Continuous-Time Systems
44
3.5.1 The Conversion Methods
48
3.6 Discrete-Time Models of Systems with Delay
53
3.7 Continuous-Time Models of Discrete-Time Systems
54
4. Time-Domain Response
57
4.1 Symbolic Approach
57
4.2 Simulating System Behavior
65
4.3 Step, Impulse, and Other Responses
74
5. Classical Methods of Control Theory
80
5.1 Root Loci
80
5.2 The Bode Plot
85
5.2.1 The Basic Function
85
5.2.2 Gain and Phase Margins
89
5.3 The Nyquist Plot
93
5.4 The Nichols Plot
95
5.5 The Singular-Value Plot
96
6. System Interconnections
98
6.1 Elementary Interconnections
98
6.1.1 Connecting in Series
98
6.1.2 Connecting in Parallel
103
6.1.3 Closing Feedback Loop
105
6.2 Arbitrary Interconnections
110
6.3 State Feedback
114
6.4 Manipulating a System's Contents
115
6.5 Using Interconnecting Functions for Controller Design
119
7. Controllability and Observability
122
7.1 Tests for Controllability and Observability
122
7.2 Controllability and Observability Constructs
126
7.3 Dual System
130
8. Realizations
132
8.1 Irreducible (Minimal) Realizations
133
8.2 Kalman Canonical Forms
136
8.3 Jordan Canonical (Modal) Form
138
8.4 Internally Balanced Realizations
140
8.5 Dominant Subsystem
142
8.6 Pole-Zero Cancellation
144
8.7 Similarity Transformation
146
8.8 Recovering the Transformation Matrix
148
9. Feedback Control Systems Design
150
9.1 Pole Assignment with State Feedback
150
9.1.1 Ackermann's Formula
154
9.1.2 Robust Pole Assignment
158
9.2 State Reconstruction
161
10. Optimal Control Systems Design
165
10.1 Linear Quadratic Regulator
166
10.2 Optimal Output Regulator
171
10.3 Riccati Equations
172
10.4 Discrete Regulator by Emulation of Continuous Design
178
10.5 Optimal Estimation
181
10.6 Discrete Estimator by Emulation of Continuous Design
184
10.7 Kalman Estimator
185
10.8 Optimal Controller
192
11. Nonlinear Control Systems
197
11.1 Local Linearization of Nonlinear Systems
197
11.2 Rational Polynomial Approximations
202
12. Miscellaneous
204
12.1 Ordered Schur Decomposition
204
12.2 Lyapunov Equations
206
12.3 Rank of Matrix
210
12.4 Part Count and Consistency Check
210
12.5 Displaying Graphics Array Objects Together
211
12.6 Systems with Random Elements
211
References
214
Index
215
