Linear Systems and Optimal Control
C. K. Chui and Guanrong
Chen
Springer-Verlag, New York, 1989 (ISBN: 3-540-18737-5)
1.
State Space Descriptions ..............................................................
1
1.1 Introduction
...........................................................................
1
1.2 An Example
of Input-Output Relations ................................. 3
1.3 An Example
of State-Space Descriptions ................................ 4
1.4 State-Space
Models ................................................................
5
Exercises
.....................................................................................
6
2.
State Transition Equations and Matrices .......................................
8
2.1 Continuous-Time
Linear Systems ........................................... 8
2.2 Picard's
Iteration .....................................................................
9
2.3 Discrete-Time
Linear Systems ............................................... 12
2.4 Discretization
........................................................................
13
Exercises .....................................................................................
14
3.
Controllability .............................................................................
16
3.1 Control
and Observation Equations ....................................... 16
3.2 Controllability
of Continuous-Time linear Systems ................ 17
3.3 Complete
Controllability of Continuous-Time Linear Systems 19
3.4 Controllability
and Complete Controllability of Discrete-Time
Linear Systems .......................................................................
21
Exercises ......................................................................................
24
4.
Observability and Dual Systems ..................................................
26
4.1 Observability
of Continuous-Time Linear Systems ................ 26
4.2 Observability
of Discrete-Time Linear Systems ...................... 29
4.3 Duality
of Linear Systems ......................................................
31
4.4 Dual Time-Varying
Discrete-Time Linear Systems ................. 33
Exercises ......................................................................................
34
5.
Time-Invariant Linear Systems ....................................................
36
5.1 Preliminary Remarks
.............................................................. 36
5.2 The Kalman Canonical
Decomposition ................................... 37
5.3 Transfer Functions
.................................................................. 43
5.4 Pole-Zero Cancellation
of Transfer Functions ......................... 44
Exercises .......................................................................................
47
6.
Stability ........................................................................................
49
6.1 Free Systems and
Equilibrium Points ....................................... 49
6.2 State-Stability
of Continuous-Time Linear Systems .................. 50
6.3 State-Stability
of Discrete-Time Linear Systems ....................... 56
6.4 Input-Output Stability
of Continuous-Time Linear Systems ..... 61
6.5 Input-Output Stability
of Discrete-Time Linear Systems .......... 65
Exercises .......................................................................................
68
7.
Optimal Control Problems and Variational Methods .................... 70
7.1 The Lagrange,
Bolza, and Mayer Problems ............................. 70
7.2 A Variational
Method for Continuous-Time Systems .............. 72
7.3 Two Examples ........................................................................
76
7.4 A Variational
Method for Discrete-Time Systems ................... 78
Exercises .......................................................................................
79
8.
Dynamic Programming ................................................................
81
8.1 The Optimality
Principle ......................................................... 81
8.2 Continuous-Time
Dynamic Programming ............................... 83
8.3 Discrete-Time
Dynamic Programming .................................... 86
8.4 The Minimum Principle
of Pontryagin .................................... 90
Exercises .......................................................................................
92
9.
Minimum-Time Optimal Control Problems ..................................
94
9.1 Existence of the Optimal
Control Function ............................... 94
9.2 The Bang-Bang Principle
.......................................................... 96
9.3 The Minimum Principle
of Pontryagin for Minimum-Time
Optimal Control Problems .......................................................
98
9.4 Normal Systems .....................................................................
101
Exercises ......................................................................................
103
10.
Notes and References ...............................................................
106
10.1 Reachability and Constructibility
.......................................... 106
10.2 Differential Controllability
.................................................... 107
10.3 State Reconstruction
and Observers ...................................... 107
10.4 The Kalman Canonical
decomposition .................................. 108
10.5 Minimum Realization
............................................................ 110
10.6 Stability of Nonlinear
Systems .............................................. 110
10.7 Stabilization ..........................................................................
112
10.8 Matrix Riccati Equations
....................................................... 112
10.9 Pontryagin's Maximum
Principle .......................................... 113
10.10 Optimal Control of
Distributed Parameter Systems ............. 115
10.11 Stochastic Optimal
Control ................................................. 117
References ......................................................................................
119
Answers and Hints to Exercises
....................................................... 149
Subject Index ..................................................................................
153 |