Optimization and control of bilinear systems : theory, algorithms, and applications

398 Pages·2008·3.21 MB·English

by  P M Pardalos, Vitaliy Yatsenko

OPTIMIZATION AND CONTROL
OF BILINEAR SYSTEMS
Springer Optimization and Its Applications
VOLUME 11

Managing Editor
Panos M. Pardalos (University of Florida)
Editor—Combinatorial Optimization
Ding-Zhu Du (University of Texas at Dallas)
Advisory Board
J. Birge (University of Chicago)
C.A. Floudas (Princeton University)
F. Giannessi (University of Pisa)
H.D. Sherali (Virginia Polytechnic and State University)
T. Terlaky (McMaster University)
Y. Ye (Stanford University)
Aims and Scope
Optimization has been expanding in all directions at an astonishing rate
during the last few decades. New algorithmic and theoretical techniques have
been developed, the diffusion into other disciplines has proceeded at a rapid
pace, and our knowledge of all aspects of the field has grown even more
profound. At the same time, one of the most striking trends in optimization is
the constantly increasing emphasis on the interdisciplinary nature of the
field. Optimization has been a basic tool in all areas of applied mathematics,
engineering, medicine, economics and other sciences.
The series Optimization and Its Applications publishes undergraduate
and graduate textbooks, monographs and state-of-the-art expository works
that focus on algorithms for solving optimization problems and also study
applications involving such problems. Some of the topics covered include
nonlinear optimization (convex and nonconvex), network flow problems,
stochastic optimization, optimal control, discrete optimization, multi-
objective programming, description of software packages, approximation
techniques and heuristic approaches.
OPTIMIZATION AND CONTROL
OF BILINEAR SYSTEMS
Theory, Algorithms, and Applications
By
PANOS M. PARDALOS
University of Florida, Gainesville, FL
VITALIY YATSENKO
Space Research Institute of NASU-NASU, Kyiv, Ukraine
ABC
Panos M. Pardalos Vitaliy Yatsenko
Department of Industrial Space Research Institute
and Systems Engineering of NASU-NASU
University of Florida Glushkov Avenue, 40
303 Weil Hall Kyiv 03680
P.O.Box 116595 Ukraine
Gainesville FL 32611-6595 [email protected]
USA
[email protected]
ISBN:978-0-387-73668-6 e-ISBN:978-0-387-73669-3
DOI:10.1007/978-0-387-73669-3
LibraryofCongressControlNumber:2007934549
AMS SubjectClassifications:93C10, 93B29, 93B52, 93EE10, 81P68
(cid:1)c 2008SpringerScience+BusinessMedia,LLC
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Contents
Preface xiii
Acknowledgments xvii
Foreword xix
Notation xxi
Introduction xxiii
1. SYSTEM-THEORETICAL DESCRIPTION
OF OPEN PHYSICAL PROCESSES 1
1 Reduction of Nonlinear Control Systems
to Bilinear Realization 3
1.1 Equivalence of Control Systems 3
1.2 Lie Algebras, Lie Groups,
and Representations 4
1.3 Selection of Mathematical Models 6
1.4 BilinearLogic-DynamicalRealizationofNonlinear
Control Systems 10
2 Global Bilinearization of Nonlinear Systems 12
3 Identification of Bilinear Control Systems 19
4 Bilinear and Nonlinear Realizations
of Input-Output Maps 20
4.1 Systems on Lie Groups 20
4.2 Bilinear Realization of Nonlinear Systems 22
4.3 Approximation of Nonlinear Systems by Bilinear
Systems 23
5 Controllability of Bilinear Systems 25
6 Observability of Systems on Lie Groups 27
6.1 Observability and Lie Groups 27
6.2 Algorithms of Observability 33
vii
viii Contents
6.3 Examples 36
6.4 Decoupling Problems 38
7 Invertibility of Control Systems 40
7.1 Right-Invariant Control Systems 40
7.2 Invertibility of Right-Invariant Systems 43
7.3 Left-Inverses for Bilinear Systems 49
8 Invertibility of Discrete Bilinear Systems 56
8.1 Discrete Bilinear Systems and Invertability 56
8.2 Construction of Inverse Systems 57
8.3 Controllability of Inverse Systems 58
9 Versal Models and Bilinear Systems 59
9.1 General Characteristics of Versal Models 59
9.2 Algorithms 60
10 Notes and Sources 64
2. CONTROL OF BILINEAR SYSTEMS 65
1 Optimal Control of Bilinear Systems 66
1.1 Optimal Control Problem 66
1.2 Reduction of Control Problem to Equivalent
Problem for Bilinear Systems 67
1.3 Optimal Control of Bilinear Systems 70
1.4 On the Solution of the Euler–Lagrange Equation 72
2 Stability of Bilinear Systems 74
2.1 Normed Vector Space 75
2.2 Continuous Bilinear Systems 77
2.3 Discrete Bilinear Systems 79
3 Adaptive Control of Bilinear Systems 82
3.1 Control of Fixed Points 82
3.2 Control of Limit Cycles 88
3.3 Variations in the Control Dynamics 89
4 Notes and Sources 91
3. BILINEARSYSTEMSANDNONLINEARESTIMATION
THEORY 93
1 Nonlinear Dynamical Systems and Adaptive Filters 94
1.1 Filtration Problems 94
1.2 Problem Statement 97
1.3 Preliminaries on Nonlinear and Bilinear Lattice
Models 98
Contents ix
1.4 Adaptive Filter for Lattice Systems 100
1.5 Identification of Bilinear Lattice Models 103
1.6 A Generalization for Nonlinear Lattice Models 109
1.7 Estimation of the State Vector of CA3 Region 111
1.8 Detection and Prediction of Epileptic Seizures 115
2 Optimal Estimation of Signal Parameters Using Bilinear
Observations 118
2.1 Estimation Problem 118
2.2 Invertibility of Continuous MS and Estimation
of Signal Parameters 119
2.3 Estimation of Parameters of an Almost Periodic
Signal Under Discrete Measurements 124
2.4 Neural Network Estimation of Signal Parameters 127
2.5 Finite-Dimensional Bilinear Adaptive Estimation 129
2.6 Example 130
3 Bilinear Lattices and Nonlinear Estimation Theory 131
3.1 Lattice Systems and DMZ Equations 131
3.2 Structure of Estimation Algebra 135
4 Notes and Sources 138
4. CONTROL OF DYNAMICAL PROCESSES
AND GEOMETRICAL STRUCTURES 139
1 Geometric Structures 141
1.1 Metric Spaces 142
1.2 Optimal Control 143
1.3 IdentificationofNonlinearAgentsandYang–Mills
Fields 145
1.4 The Estimation Algebra of Nonlinear Filtering
Systems 146
1.5 Estimation Algebra and Identification Problems 147
2 Lie Groups and Yang–Mills Fields 149
3 Control of Multiagent Systems and Yang–Mills
Representation 152
4 Dynamic Systems, Information, and Fiber Bundles 154
5 Fiber Bundles, Multiple Agents,
and Observability 164
5.1 Smooth Nonlinear Systems 166
5.2 Minimality and Observability 168
6 Notes and Sources 176