Continuous Time Dynamical Systems: State Estimation and Optimal Control with Orthogonal Functions

Continuous Time Dynamical Systems: State Estimation and Optimal Control with Orthogonal Functions......Page 2
Contents......Page 6
List of Abbreviations......Page 12
List of Figures......Page 14
Preface......Page 20
Acknowledgements......Page 22
About the Authors......Page 24
1.1 Optimal Control Problem......Page 28
1.2 Historical Perspective......Page 30
1.3 Organisation of the Book......Page 35
2.1 Introduction......Page 38
2.2 Block-Pulse Functions (BPFs)......Page 41
2.2.1 Integration of B(t)......Page 42
2.2.3 Representation of C(t)f(t) in terms of BPFs......Page 43
2.2.4 Representation of a time-delay vector in BPFs......Page 44
2.2.5 Representation of reverse time function vector in BPFs......Page 45
2.3 Legendre Polynomials (LPs)......Page 46
2.4 Shifted Legendre Polynomials (SLPs)......Page 47
2.4.1 Integration of L(t)......Page 49
2.4.2 Product of two SLPs......Page 50
2.4.3 Representation of C(t)f(t) in terms of SLPs......Page 51
2.4.4 Representation of a time-delay vector function in SLPs......Page 52
2.4.5 Derivation of a time-advanced matrix of SLPs......Page 54
2.4.6 Algorithm for evaluating the integral in Eq. (2.75)......Page 55
2.5 Nonlinear Operational Matrix......Page 57
2.6 Rationale for Choosing BPFs and SLPs......Page 59
3.1 Introduction......Page 62
3.2 Inherent Filtering Property of OFs......Page 65
3.3 State Estimation......Page 66
3.3.2 Recursive algorithm via BPFs......Page 70
3.3.3 Recursive algorithm via SLPs......Page 71
3.3.4 Modification of the recursive algorithm of Sinha and Qi-Jie......Page 72
3.4 Illustrative Examples......Page 74
3.5 Conclusion......Page 85
4.1 Introduction......Page 92
4.2 Analysis of Linear Optimal Control Systems Incorporating Observers......Page 95
4.2.1 Kronecker product method......Page 96
4.2.2 Recursive algorithm via BPFs......Page 97
4.2.3 Recursive algorithm via SLPs......Page 98
4.3 Illustrative Example......Page 99
4.4 Conclusion......Page 104
5.1 Introduction......Page 106
5.2 Optimal Control of LTI Systems Described by Integro-Differential Equations......Page 107
5.3 Illustrative Example......Page 111
5.4 Conclusion......Page 113
6.1 Introduction......Page 116
6.2 LQG Control Problem......Page 117
6.3 Unified Approach......Page 120
6.3.1 Illustrative example......Page 123
6.4 Recursive Algorithms......Page 124
6.4.1 Recursive algorithm via BPFs......Page 128
6.4.2 Recursive algorithm via SLPs......Page 129
6.4.3 Illustrative example......Page 131
6.5 Conclusion......Page 132
7.1 Introduction......Page 136
7.2 Recursive Algorithms......Page 138
7.2.1 Recursive algorithm via BPFs......Page 140
7.2.2 Recursive algorithm via SLPs......Page 141
7.3 Unified Approach......Page 142
7.4 Illustrative Examples......Page 144
7.5 Conclusion......Page 148
8 Optimal Control of Time-Delay Systems......Page 152
8.1 Introduction......Page 153
8.2 Optimal Control of Multi-Delay Systems......Page 156
8.2.1 Using BPFs......Page 160
8.2.2 Using SLPs......Page 162
8.2.3 Time-invariant systems......Page 164
8.2.5 Illustrative examples......Page 165
8.3 Optimal Control of Delay Systems with Reverse Time Terms......Page 173
8.3.1 Using BPFs......Page 178
8.3.2 Using SLPs......Page 180
8.3.3 Illustrative example......Page 181
8.4 Conclusion......Page 182
9.1 Introduction......Page 186
9.2 Computation of the Optimal Control Law......Page 187
9.3 Illustrative Examples......Page 189
9.4 Conclusion......Page 193
10.1 Introduction......Page 196
10.2 Hierarchical Control of LTI Systems with Quadratic Cost Functions......Page 197
10.2.1 Partial feedback control......Page 199
10.2.2 Interaction prediction approach......Page 200
10.3 Solution of Hierarchical Control Problem via BPFs......Page 201
10.3.1 State transition matrix......Page 202
10.3.2 Riccati matrix and open-loop compensation vector......Page 204
10.3.3 State vector......Page 205
10.4 Extension to Linear Time-Varying Systems......Page 207
10.5 Computational Algorithm......Page 210
10.6 Illustrative Examples......Page 211
10.7 Conclusion......Page 217
11 Epilogue......Page 226
Bibliography......Page 230
Index......Page 244