Lecture Notes in Control and Information Sciences : Lecture Notes in Control And Iinformation Sciences - Fritz Colonius

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Lecture Notes in Control and Information Sciences

By: Fritz Colonius (Editor), Lars GrÃ¼ne (Editor)

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Paperback | 11 January 2002

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This volume originates from the Third Nonlinear Control Workshop "- namics, Bifurcations and Control", held in Kloster Irsee, April 1-3 2001. As the preceding workshops held in Paris (2000) and in Ghent (1999), it was organized within the framework of Nonlinear Control Network funded by the European Union (http://www.supelec.fr/lss/NCN). The papers in this volume center around those control problems where phenomena and methods from dynamical systems theory play a dominant role. Despite the large variety of techniques and methods present in the c- tributions, a rough subdivision can be given into three areas: Bifurcation problems, stabilization and robustness, and global dynamics of control s- tems. A large part of the fascination in nonlinear control stems from the fact that is deeply rooted in engineering and mathematics alike. The contributions to this volume reflect this double nature of nonlinear control. We would like to take this opportunity to thank all the contributors and the referees for their careful work. Furthermore, it is our pleasure to thank Franchise Lamnabhi-Lagarrigue, the coordinator of our network, for her s- port in organizing the workshop and the proceedings and for the tremendous efforts she puts into this network bringing the cooperation between the d- ferent groups to a new level. In particular, the exchange and the active p- ticipation of young scientists, also reflected in the Pedagogical Schools within the Network, is an asset for the field of nonlinear control.

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Bifurcation Problems	p. 1
Controlling an Inverted Pendulum with Bounded Controls	p. 3
Introduction	p. 3
Description of the system	p. 4
Bounded control law	p. 5
Local nonlinear analysis	p. 7
Numerical analysis of the global dynamical behavior	p. 8
Desired operating behaviour	p. 14
Conclusions	p. 15
References	p. 16
Bifurcations of Neural Networks with Almost Symmetric Interconnection Matrices	p. 17
Introduction	p. 17
Neural network model and preliminaries	p. 19
Limit cycles in a competitive neural network	p. 23
Hopf bifurcations in sigmoidal neural networks	p. 26
Period-doubling bifurcations in a third-order neural network	p. 30
Conclusion	p. 32
References	p. 32
Bifurcations in Systems with a Rate Limiter	p. 37
Introduction	p. 37
Behaviour of rate limiters	p. 38
Describing function of rate limiters	p. 41
Limit cycle analysis of systems with rate limiters	p. 42
Bifurcations in systems with a rate limiter	p. 43
Conclusions	p. 49
References	p. 50
Monitoring and Control of Bifurcations Using Probe Signals	p. 51
Introduction	p. 51
Hopf bifurcation	p. 52
Analysis of the effects of near-resonant forcing	p. 54
Numerical example	p. 57
Combined Stability Monitoring and Control	p. 58
Detection of Impending Bifurcation in a Power System Model	p. 60
Conclusions	p. 64
References	p. 64
Normal Form, Invariants, and Bifurcations of Nonlinear Control Systems in the Particle Deflection Plane	p. 67
Introduction	p. 67
Problem formulation	p. 68
Normal form and invariants	p. 70
Bifurcation of control systems	p. 75
Bifurcation control using state feedback	p. 77
The cusp bifurcation and hysteresis	p. 81
Other related issues	p. 83
Conclusions	p. 84
References	p. 85
Bifurcations of Reachable Sets Near an Abnormal Direction and Consequences	p. 89
Setup and definitions	p. 89
Asymptotics of the reachable sets	p. 91
Applications	p. 94
References	p. 98
Stabilization and Robustness101
Oscillation Control in Delayed Feedback Systems	p. 103
Introduction	p. 103
Perturbations of linear retarded equations	p. 105
The harmonic oscillator under delayed feedback	p. 106
Controlling the amplitude and frequency of oscillations	p. 111
Conclusion	p. 115
References	p. 115
Nonlinear Problems in Friction Compensation	p. 117
Introduction	p. 117
Conic analysis of uncertain friction	p. 121
Harmonic balance	p. 124
Frequencial synthesis using QFT	p. 127
Discussion	p. 128
References	p. 129
Time-Optimal Stabilization for a Third-Order Integrator: a Robust State-Feedback Implementation	p. 131
Introduction	p. 131
Closed loop time-optimal stabilization for a third-order integrator	p. 133
Sliding-mode implementation of the time-optimal controller	p. 137
Simulation results	p. 141
Conclusions	p. 143
References	p. 144
Stability Analysis of Periodic Solutions via Integral Quadratic Constraints	p. 145
Introduction	p. 145
A motivating example	p. 146
Problem formulation and preliminary results	p. 148
Sufficient conditions for stability of periodic solutions	p. 151
Application example	p. 154
Conclusions	p. 156
References	p. 156
Port Controller Hamiltonian Synthesis Using Evolution Strategies	p. 159
Introduction	p. 159
Port controlled Hamiltonian systems	p. 160
Controller design	p. 160
Preliminaries on evolution strategies	p. 162
Evolutionary formulation	p. 165
Case study - ball & beam system	p. 167
Conclusions	p. 169
References	p. 170
Feedback Stabilization and HOQ Control of Nonlinear Systems Affected by Disturbances: the Differential Games Approach	p. 173
Introduction	p. 173
Differential games approach to nonlinear Hoo control	p. 175
Other stability questions	p. 181
Building a feedback solution for nonlinear Hoo control	p. 182
References	p. 188
A Linearization Principle for Robustness with Respect to Time-Varying Perturbations	p. 191
Introduction	p. 191
Preliminaries192
The discrete time case	p. 195
Continuous time	p. 197
Conclusion199
References	p. 200
Global Dynamics of Control Systems	p. 201
On Constrained Dynamical Systems and Algebroids	p. 203
Introduction: Constrained Hamiltonian systems	p. 203
What is a Lie algebroid?	p. 205
Dirac structures and Port Controlled Hamiltonian systems	p. 208
Constrained mechanical systems and algebroids	p. 213
Control of constrained mechanical systems	p. 214
References	p. 216
On the Classification of Control Sets	p. 217
Introduction	p. 217
Basic definitions	p. 218
Strong inner pairs	p. 219
The dynamic index	p. 221
The index of a control set near a periodic orbit	p. 224
References	p. 230
On the Frequency Theorem for Nonperiodic Systems	p. 233
Introduction	p. 233
Nonautonomous Hamiltonian systems	p. 235
Generalization of Yakubovich's theorem	p. 238
References	p. 240
Longtime Dynamics in Adaptive Gain Control Systems	p. 241
Introduction	p. 241
Assumptions and preliminaries	p. 242
Localization of the global attractor	p. 245
Longtime behavior and estimates of the Hausdorff dimension of the global attractor	p. 248
References	p. 253
Model Reduction for Systems with Low-Dimensional Chaos	p. 255
Introduction	p. 255
Peak-to-peak dynamics	p. 256
The control problem	p. 260
Examples of application	p. 261
Delay-differential systems	p. 263
Concluding remarks	p. 265
References	p. 267
Feedback Equivalence to Feedforward Forms for Nonlinear Single-Input Control Systems	p. 269
Introduction	p. 269
Definitions and notations	p. 271
Feedforward normal form	p. 274
m-invariants	p. 275
Main results	p. 276
Examples	p. 281
Feedforward systems in E4	p. 283
References	p. 285
Conservation Laws in Optimal Control	p. 287
Introduction	p. 287
Preliminaries	p. 289
Main results	p. 291
Examples	p. 294
References	p. 295
List of Participants	p. 297
Table of Contents provided by Publisher. All Rights Reserved.

Lecture Notes in Control and Information Sciences

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