Dynamics, Bifurcations and Control

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This book contains a selection of the papers presented at the 3rd NCN Workshop which was focused on "Dynamics, Bifurcations and Control". The peer-reviewed papers describe a number of ways how dynamical systems techniques can be applied for analysis and design problems in control with topics ranging from bifurcation control via stability and stabilizaton to the global dynamical behaviour of control systems. The book gives an overview of the current status of the field.

Author(s): Fritz Colonius, Lars GrĂ¼ne
Series: Lecture Notes in Control and Information Sciences
Edition: 1
Publisher: Springer
Year: 2002

Language: English
Pages: 299

front-matter.pdf......Page 1
Dynamics, Bifurcations, and Control
......Page 3
Preface
......Page 5
Table of Contents......Page 6
Introduction......Page 11
Description of the System......Page 12
Equilibrium points......Page 13
Stability of the remaining equilibria......Page 14
Hopf bifurcation at the equlibrium point......Page 15
Numerical analysis of the global dynamical behavior......Page 16
Desired operating behaviour......Page 22
Acknowledgments......Page 23
References......Page 24
Introduction......Page 25
Neural network model and preliminaries......Page 27
Limit cycles in a competivite neural network......Page 31
Hopf bifurcations in sigmoidal neural networks......Page 34
Period-doubling bifurcation in a third-order neural network......Page 38
References......Page 40
Introduction......Page 44
Behaviour of rate limiters......Page 45
Describing function of rate limiters......Page 48
Limit cycle analysis of systems with rate limiters......Page 49
Saddle-node bifurcation of periodic orbits......Page 50
Supercritical Hopf-like bifurcation......Page 53
Acknowledgments......Page 56
References......Page 57
Introduction......Page 58
Hopf bifurcation......Page 59
Averaged model......Page 61
Calculation of shift in the critical value of the bifurcation parameter......Page 62
Numerical example......Page 64
Combined Stability Monitoring and Control......Page 65
Detection of Impending Bifurcation in a Power System Model......Page 67
References......Page 71
Introduction......Page 73
Problem formulation......Page 74
Normal form and invariants......Page 76
Normal form......Page 77
Invariants......Page 78
Resonant terms......Page 79
Systems with a double-zero uncontrollable mode......Page 81
Systems with single-zero uncontrollable modes......Page 82
Bifurcation control using state feedback......Page 83
Bifurcation with quadratic degeneracy......Page 84
Bifurcation with cubie degeneracy......Page 85
The cusp bifurcation and hysteresis......Page 87
Other related issues......Page 89
Conclusions......Page 90
References......Page 91
Abnormal trajectories......Page 94
Single-input affine systems......Page 95
Asymptotic of the reachable sets......Page 96
Applications th the optimal status of an abnormal trajectory......Page 99
Application to the sub-Riemannian case......Page 101
References......Page 103
Introduction......Page 105
Perturbations of linear retarded equations......Page 107
The harmonic oscillator under delayed feedback......Page 108
The linear equation......Page 109
The reduced equation and averaging......Page 112
Controlling the amplitude and frequency of oscillations......Page 113
References......Page 117
Introduction......Page 119
Conic analysis of uncertain friction......Page 123
Darmonic balance......Page 126
Frequencial synthesis using QFT......Page 129
Discussion......Page 130
References......Page 131
Introduction......Page 133
Closed loop time-optimal stabilization for a third-order integrator......Page 135
Sliding-mode implementation of the time-optimal controller......Page 139
Simulation result......Page 143
Acknowledgements......Page 145
References......Page 146
Introduction......Page 147
A moving example......Page 148
Problem formulation and preliminary results......Page 150
Sufficient conditions for stability of periodic solutions......Page 153
Application example......Page 156
References......Page 158
Introduction......Page 160
Controller design......Page 161
Evolutionary basics......Page 163
Fitness evaluation......Page 165
Evolutionary formulation......Page 166
Case study - ball & beam system......Page 168
Conclusions......Page 170
References......Page 171
Introduction......Page 174
Differential games approaches to nonlinear H control......Page 176
Other stability questions......Page 182
Building a feedback solution for nonlinear H control......Page 183
References......Page 189
Introduction......Page 192
Preliminaries......Page 193
The discrete time case......Page 196
Continuous time......Page 198
Conclusion......Page 200
References......Page 201
Introduction: Constrained Hamiltonian systems......Page 202
Generalities......Page 204
The algebroid structure of an integrable subbundle of a tangent bundle......Page 206
Dirac structures......Page 207
Application: Mechanical system with constraints......Page 209
Port controlled Hamiltonian systems......Page 210
Constrained mechanical systems and algebroids......Page 212
Control of constrained mechanical systems......Page 213
References......Page 215
Introduction......Page 216
Basic definition......Page 217
Strong inner pairs......Page 218
The dynamic index......Page 220
The index of a control set near a periodic orbit......Page 223
References......Page 229
Introduction......Page 231
Nonautonomous Hamiltonian systems......Page 233
Generalization of Yakubovich's theorem......Page 236
References......Page 238
Introduction......Page 239
Assumptions and preliminaries......Page 240
Localization of the global attractor......Page 243
Longtime behavior and estimates of the Hausdor dimension of the global attractor......Page 246
References......Page 251
Introduction......Page 253
Peak-to-peak dynamics......Page 254
The control problem......Page 258
Lorenz system......Page 259
Delay-differential systems......Page 261
Concluding remarks......Page 263
References......Page 265
Introduction......Page 267
Definitions and notations......Page 269
Feedforward normal form......Page 272
m-inveriants......Page 273
Main results......Page 274
Feedforward form: the general step......Page 275
Strict feedforward form......Page 277
Nice feedforward form......Page 278
Examples......Page 279
Feedforward system in R4......Page 281
References......Page 283
Introduction......Page 285
The Maximum Principle......Page 287
A Necessaty and Sufficient Condition......Page 289
Noether Theorem for Optimal Control......Page 290
Examples......Page 292
References......Page 293
Participants of the 3rd NCN Workshop "Dynamics, Bifurcations and Control" Kloster Irsee, Germany, April 1-3, 2001......Page 295