Elliott Bilinear Control Systems

Matrices in Action
1. Auflage 2009
ISBN: 978-1-4020-9613-6
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 169, 283 Seiten

Reihe: Applied Mathematical Sciences

The mathematical theory of control became a ?eld of study half a century ago in attempts to clarify and organize some challenging practical problems and the methods used to solve them. It is known for the breadth of the mathematics it uses and its cross-disciplinary vigor. Its literature, which can befoundinSection93ofMathematicalReviews,wasatonetimedominatedby the theory of linear control systems, which mathematically are described by linear di?erential equations forced by additive control inputs. That theory led to well-regarded numerical and symbolic computational packages for control analysis and design. Nonlinear control problems are also important; in these either the - derlying dynamical system is nonlinear or the controls are applied in a n- additiveway.Thelastfourdecadeshaveseenthedevelopmentoftheoretical work on nonlinear control problems based on di?erential manifold theory, nonlinear analysis, and several other mathematical disciplines. Many of the problems that had been solved in linear control theory, plus others that are new and distinctly nonlinear, have been addressed; some resulting general de?nitions and theorems are adapted in this book to the bilinear case.

Autoren/Hrsg.

Elliott, David

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;Introduction;10
3.1;Matrices in Action;11
3.2;Stability: Linear Dynamics;16
3.3;Linear Control Systems;17
3.4;What Is a Bilinear Control System?;19
3.5;Transition Matrices;22
3.6;Controllability;29
3.7;Stability: Nonlinear Dynamics;33
3.8;From Continuous to Discrete;37
3.9;Exercises;39
4;Symmetric Systems: Lie Theory;42
4.1;Introduction;42
4.2;Lie Algebras;43
4.3;Lie Groups;53
4.4;Orbits, Transitivity, and Lie Rank;63
4.5;Algebraic Geometry Computations;69
4.6;Low-Dimensional Examples;77
4.7;Groups and Coset Spaces;79
4.8;Canonical Coordinates;81
4.9;Constructing Transition Matrices;83
4.10;Complex Bilinear Systems;86
4.11;Generic Generation;88
4.12;Exercises;90
5;Systems with Drift;92
5.1;Introduction;92
5.2;Stabilization with Constant Control;94
5.3;Controllability;98
5.4;Accessibility;109
5.5;Small Controls;113
5.6;Stabilization by State-Dependent Inputs;116
5.7;Lie Semigroups;125
5.8;Biaffine Systems;128
5.9;Exercises;133
6;Discrete-Time Bilinear Systems;135
6.1;Dynamical Systems: Discrete-Time;136
6.2;Discrete-Time Control;137
6.3;Stabilization by Constant Inputs;139
6.4;Controllability;140
6.5;A Cautionary Tale;149
7;Systems with Outputs;151
7.1;Compositions of Systems;152
7.2;Observability;154
7.3;State Observers;159
7.4;Identification by Parameter Estimation;161
7.5;Realization;162
7.6;Volterra Series;169
7.7;Approximation with Bilinear Systems;171
8;Examples;173
8.1;Positive Bilinear Systems;173
8.2;Compartmental Models;178
8.3;Switching;180
8.4;Path Construction and Optimization;187
8.5;Quantum Systems;192
9;Linearization;194
9.1;Equivalent Dynamical Systems;195
9.2;Linearization: Semisimplicity and Transitivity;199
9.3;Related Work;205
10;Input Structures;207
10.1;Concatenation and Matrix Semigroups;207
10.2;Formal Power Series for Bilinear Systems;210
10.3;Stochastic Bilinear Systems;213
11;Matrix Algebra;220
11.1;Definitions;220
11.2;Associative Matrix Algebras;222
11.3;Kronecker Products;225
11.4;Invariants of Matrix Pairs;228
12;Lie Algebras and Groups;230
12.1;Lie Algebras;230
12.2;Structure of Lie Algebras;234
12.3;Mappings and Manifolds;236
12.4;Groups;243
12.5;Lie Groups;245
13;Algebraic Geometry;251
13.1;Polynomials;251
13.2;Affine Varieties and Ideals;252
14;Transitive Lie Algebras;255
14.1;Introduction;255
14.2;The Transitive Lie Algebras;259
15;References;263
16;Index;276