Dynamics, Bifurcations, and Control

Dynamics, Bifurcations, and Control
......Page 4
Preface
......Page 6
Table of Contents......Page 7
Participants of the 3rd NCN Workshop \"Dynamics, Bifurcations and Control\" Kloster Irsee, Germany, April 1-3, 2001......Page 12
Introduction......Page 17
Description of the system......Page 18
Equilibrium points
......Page 19
Stability of the remaining equilibria
......Page 20
Hopf bifurcation at the equilibrium point x00
......Page 21
Numerical analysis of the global dynamical behavior
......Page 22
Desired operating behaviour
......Page 28
Conclusions
......Page 29
Introduction......Page 31
Neural network model and preliminaries......Page 33
Limit cycles in a competitive neural network......Page 37
Hopf bifurcations in sigmoidal neural networks......Page 40
Period-doubling bifurcations in a third-order neural network......Page 44
Conclusion......Page 46
Introduction......Page 50
Behaviour of rate limiters......Page 51
Describing function of rate limiters......Page 54
Limit cycle analysis of systems with rate limiters......Page 55
Saddle-node bifurcation of periodic orbits......Page 56
Supercritical Hopf-like bifurcation......Page 59
Acknowledgments......Page 62
References......Page 63
Introduction......Page 64
Hopf bifurcation......Page 65
Averaged model......Page 67
Calculation of shift in the critical value of the bifurcation parameter......Page 68
Numerical example......Page 70
Combined Stability Monitoring and Control......Page 71
Detection of Impending Bifurcation in a Power System Model......Page 73
References......Page 77
Introduction......Page 79
Problem formulation......Page 80
Normal from and invariants 
......Page 82
Normal from
......Page 83
Invariants
......Page 84
Resonant terms
......Page 85
Systems with a double-zero uncontrollable mode
......Page 87
Systems with simple-zero uncontrollable modes
......Page 88
Bifurcation control using state feedback
......Page 89
Bifurcations with quadratic degeneracy
......Page 90
Bifurcations with cubic degeneracy
......Page 91
The cusp bifurcation and hysteresis
......Page 93
Other related issues
......Page 95
Conclusions
......Page 96
Abnormal trajectories
......Page 100
Single-input affine systems
......Page 101
Asymptotics of the reachable sets
......Page 102
Application to the optimality status of an abnormal trajectory
......Page 105
Application to the sub-Riemannian case
......Page 107
Introdection
......Page 111
Perturbations of linear retarded equations
......Page 113
The harmonic oscillator under delayed feedback
......Page 114
The linear equation
......Page 115
The reduced equation and averaging
......Page 118
Controlling the amplitude and frequency of oscillations
......Page 119
Conclusion
......Page 123
Introduction
......Page 125
Conic analysis of uncertain friction
......Page 129
Harmonic balance
......Page 132
Frequencial synthesis using QFT
......Page 135
Discussion
......Page 136
Introduction......Page 139
Closed loop time-optimal stabilization for a third-order intergrator......Page 141
Sliding-mode implementation of the time-optimal controller......Page 145
Simulation results......Page 149
Conclusions......Page 151
Introduction......Page 153
A motivating example......Page 154
Problem formulation and preliminary results......Page 156
Sufficient conditions for stability of periodic solutions......Page 159
Application example......Page 162
Conclusions......Page 164
Introduction......Page 166
Controller design
......Page 167
Evolutionary basics
......Page 169
Fitness evaluation
......Page 171
Evolutionary formulation
......Page 172
Case study - ball & beam system
......Page 174
Conclusions
......Page 176
References......Page 177
Introduction......Page 180
Differential games approach to nonlinear H∞ control......Page 182
Other stability questions......Page 188
Building a feedback solution for nonlinear H∞ control......Page 189
Introduction......Page 198
Preliminaries......Page 199
The discrete time case......Page 202
Continuous time......Page 204
Conclusion......Page 206
References......Page 207
Introduction: Constrained Hamiltonian systems......Page 208
Generalities......Page 210
The algebroid structure of an integrable subbundle of a tangent bundle......Page 212
Dirac structures......Page 213
Application: Mechanical system with constraints......Page 215
Port controlled Hamiltonian systems......Page 216
Constrained mechanical systems and algebroids......Page 218
Control of constrained mechanical systems......Page 219
References......Page 221
Introduction......Page 222
Basic definitions......Page 223
Strong inner pairs......Page 224
The dynamic index......Page 226
The index of a control set near a periodic orbit......Page 229
Introduction......Page 237
Nonautonomous Hamiltonian systems......Page 239
Generalization of Yakubovich\'s theorem......Page 242
Introduction......Page 245
Assumptions and preliminaries......Page 246
Localization of the global attractor......Page 249
Longtime behavior and estimates of the Hausdor  dimension of the global attractor......Page 252
References......Page 257
Introduction......Page 259
Peak-to-peak dynamics......Page 260
The control problem......Page 264
Lorenz system......Page 265
Delay-differential systems......Page 267
Concluding remarks......Page 269
References......Page 271
Introduction......Page 273
Definitions and notations......Page 275
Feedforward normal from......Page 278
m-invariants......Page 279
Main results
......Page 280
Feedforward form: the general step
......Page 281
Strict feedforward form
......Page 283
Nice feedforward form
......Page 284
Examples
......Page 285
Feedforward systems in R4
......Page 287
Introduction
......Page 291
The Maximum Principle
......Page 293
A Necessary and Sufficient Condition
......Page 295
Noether Theorem for Optimal Control
......Page 296
Examples
......Page 298