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