Bilinear Control Systems (eBook)

Matrices in Action

David Elliott (Autor)

eBook Download: PDF

2009 | 2009
X, 281 Seiten
Springer Netherlands (Verlag)
978-1-4020-9613-6 (ISBN)

Infos

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.

Inhalt

Preface 5
Contents 7
Introduction 10
Matrices in Action 11
Stability: Linear Dynamics 16
Linear Control Systems 17
What Is a Bilinear Control System? 19
Transition Matrices 22
Controllability 29
Stability: Nonlinear Dynamics 33
From Continuous to Discrete 37
Exercises 39
Symmetric Systems: Lie Theory 42
Introduction 42
Lie Algebras 43
Lie Groups 53
Orbits, Transitivity, and Lie Rank 63
Algebraic Geometry Computations 69
Low-Dimensional Examples 77
Groups and Coset Spaces 79
Canonical Coordinates 81
Constructing Transition Matrices 83
Complex Bilinear Systems 86
Generic Generation 88
Exercises 90
Systems with Drift 92
Introduction 92
Stabilization with Constant Control 94
Controllability 98
Accessibility 109
Small Controls 113
Stabilization by State-Dependent Inputs 116
Lie Semigroups 125
Biaffine Systems 128
Exercises 133
Discrete-Time Bilinear Systems 135
Dynamical Systems: Discrete-Time 136
Discrete-Time Control 137
Stabilization by Constant Inputs 139
Controllability 140
A Cautionary Tale 149
Systems with Outputs 151
Compositions of Systems 152
Observability 154
State Observers 159
Identification by Parameter Estimation 161
Realization 162
Volterra Series 169
Approximation with Bilinear Systems 171
Examples 173
Positive Bilinear Systems 173
Compartmental Models 178
Switching 180
Path Construction and Optimization 187
Quantum Systems 192
Linearization 194
Equivalent Dynamical Systems 195
Linearization: Semisimplicity and Transitivity 199
Related Work 205
Input Structures 207
Concatenation and Matrix Semigroups 207
Formal Power Series for Bilinear Systems 210
Stochastic Bilinear Systems 213
Matrix Algebra 220
Definitions 220
Associative Matrix Algebras 222
Kronecker Products 225
Invariants of Matrix Pairs 228
Lie Algebras and Groups 230
Lie Algebras 230
Structure of Lie Algebras 234
Mappings and Manifolds 236
Groups 243
Lie Groups 245
Algebraic Geometry 251
Polynomials 251
Affine Varieties and Ideals 252
Transitive Lie Algebras 255
Introduction 255
The Transitive Lie Algebras 259
References 263
Index 276

Produkt-Details

• Erscheint lt. Verlag: 1.9.2009
• Reihe/Serie: Applied Mathematical Sciences
• Zusatzinfo: X, 281 p. 8 illus., 2 illus. in color.
• Verlagsort: Dordrecht
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik ► Theorie / Studium
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Algebra • brandonwiskunde • control systems • Control Theory • Geometry • Lie algebra • Lie Algebras • Matrix • Matrix groups • matrix theory • nonlinear system • System • Systems Theory
• ISBN-10: 1-4020-9613-5 / 1402096135
• ISBN-13: 978-1-4020-9613-6 / 9781402096136