This book introduces and surveys bilinear control systems theory from a mathematical viewpoint. The exposition is at the first-year graduate level. Topics include an introduction to Lie algebras, matrix groups and semigroups; the controllability, stabilization, and observability of bilinear systems; nonlinear systems analytically equivalent to bilinear systems; series representations for input-outputmappings; and random inputs. To assist the reader there are exercises; Mathematica scripts; and appendices on matrix analysis, differentiable maps and manifolds, Lie algebra, and Lie groups.
Author(s): David Elliott
Series: Applied Mathematical Sciences
Edition: 1
Publisher: Springer
Year: 2009
Language: English
Pages: 290
Tags: Автоматизация;Теория автоматического управления (ТАУ);
Contents......Page 8
1. Introduction......Page 11
1.1 Matrices in Action......Page 12
1.2 Stability: Linear Dynamics......Page 17
1.3 Linear Control Systems......Page 18
1.4 What Is a Bilinear Control System?......Page 20
1.5 Transition Matrices......Page 23
1.6 Controllability......Page 30
1.7 Stability: Nonlinear Dynamics......Page 34
1.8 From Continuous to Discrete......Page 38
1.9 Exercises......Page 40
2.1 Introduction......Page 43
2.2 Lie Algebras......Page 44
2.3 Lie Groups......Page 54
2.4 Orbits, Transitivity, and Lie Rank......Page 64
2.5 Algebraic Geometry Computations......Page 70
2.6 Low-Dimensional Examples......Page 78
2.7 Groups and Coset Spaces......Page 80
2.8 Canonical Coordinates......Page 82
2.9 Constructing Transition Matrices......Page 84
2.10 Complex Bilinear Systems......Page 87
2.11 Generic Generation......Page 89
2.12 Exercises......Page 91
3.1 Introduction......Page 93
3.2 Stabilization with Constant Control......Page 95
3.3 Controllability......Page 99
3.4 Accessibility......Page 110
3.5 Small Controls......Page 114
3.6 Stabilization by State-Dependent Inputs......Page 117
3.7 Lie Semigroups......Page 126
3.8 Biaffine Systems......Page 129
3.9 Exercises......Page 134
4. Discrete-Time Bilinear Systems......Page 137
4.1 Dynamical Systems: Discrete-Time......Page 138
4.2 Discrete-Time Control......Page 139
4.3 Stabilization by Constant Inputs......Page 141
4.4 Controllability......Page 142
4.5 A Cautionary Tale......Page 151
5. Systems with Outputs......Page 153
5.1 Compositions of Systems......Page 154
5.2 Observability......Page 156
5.3 State Observers......Page 161
5.4 Identification by Parameter Estimation......Page 163
5.5 Realization......Page 164
5.6 Volterra Series......Page 171
5.7 Approximation with Bilinear Systems......Page 173
6.1 Positive Bilinear Systems......Page 175
6.2 Compartmental Models......Page 180
6.3 Switching......Page 182
6.4 Path Construction and Optimization......Page 189
6.5 Quantum Systems......Page 194
7. Linearization......Page 197
7.1 Equivalent Dynamical Systems......Page 198
7.2 Linearization: Semisimplicity and Transitivity......Page 202
7.3 Related Work......Page 208
8.1 Concatenation and Matrix Semigroups......Page 211
8.2 Formal Power Series for Bilinear Systems......Page 214
8.3 Stochastic Bilinear Systems......Page 217
A.1 Definitions......Page 225
A.2 Associative Matrix Algebras......Page 227
A.3 Kronecker Products......Page 230
A.4 Invariants of Matrix Pairs......Page 233
B.1 Lie Algebras......Page 235
B.2 Structure of Lie Algebras......Page 239
B.3 Mappings and Manifolds......Page 241
B.4 Groups......Page 248
B.5 Lie Groups......Page 250
C.1 Polynomials......Page 257
C.2 Affine Varieties and Ideals......Page 258
D.1 Introduction......Page 261
D.2 The Transitive Lie Algebras......Page 265
References......Page 269
Index......Page 283
C......Page 284
E......Page 285
I......Page 286
M......Page 287
R......Page 288
S......Page 289
Z......Page 290
