Bifurcation Control: Theory and Applications

Preface......Page 6
Contents Bifurcation Control......Page 9
1 Introduction......Page 11
2 Local Bifurcations and Power System Behaviors......Page 14
3 Local Bifurcations in Power Systems......Page 16
4 Dynamics after Local Bifurcations......Page 18
4.2 Limit-induced bifurcation......Page 19
4.3 Hopf bifurcation......Page 20
4.5 SAD voltage collapse model......Page 21
4.6 Hopf voltage collapse model......Page 22
5 Computational Tools......Page 24
6 Performance Indices for Assessing Voltage Collapse......Page 29
7 Persistence of Local Bifurcations under Unmodelled Dynamics......Page 30
8 Concluding Remarks......Page 33
References......Page 34
1 Introduction......Page 39
2.1 Basic operation......Page 41
2.2 Period-doubling bifurcation......Page 42
3.1 Design to avoid bifurcation......Page 43
3.2 EAects on dynamical response......Page 44
3.3 Experimental measurements......Page 47
3.4 Variable ramp compensation......Page 49
4.1 Bifurcation analysis......Page 50
4.2 Bifurcation behavior of the pfc boost converter by computer simulations......Page 54
5 Conclusions......Page 56
References......Page 57
1 Introduction......Page 59
2.2 Hypersurfaces of # and their normal vector N......Page 60
2.3 Margin sensitivity......Page 61
3 Closest Bifurcations......Page 63
3.2 Direct methods......Page 64
3.3 Optimization and continuation formulations......Page 65
4.1 Codimension one bifurcations......Page 66
4.3 Constraints......Page 68
4.4 Simplifications for saddle-node bifurcation......Page 69
5 Applications......Page 70
5.1 Large scale electric power systems......Page 71
5.2 Design of Hamiltonian satellites......Page 72
6 Conclusions......Page 73
References......Page 74
1 Introduction......Page 77
2.1 Necessary conditions for static bifurcation......Page 78
2.2 Examples: a first look......Page 80
3.2 Locating static bifurcation points......Page 81
3.3 Nonlinear control system properties......Page 82
4 The Automobile......Page 84
4.1 Directional stability......Page 85
4.2 Cornering stability......Page 87
5 Concluding Remarks......Page 89
References......Page 90
1 Introduction......Page 92
2.1 Continuation on regular solutions......Page 93
3 Homoclinic Bifurcation and the Resonant Conditions......Page 94
4 Second-order System......Page 95
4.1 Bifurcation analysis on equilibria......Page 96
5 Third-order System......Page 99
6 Conclusions......Page 106
1 Introduction......Page 108
2 Double Hopf Bifurcation......Page 111
2.1 An electrical circuit......Page 114
2.2 A mechanical system......Page 116
3 Heteroclinic Bifurcation (PDE)......Page 118
4 Chaos (DDE)......Page 123
5 Anti-Control of Hopf Bifurcation......Page 126
6 Conclusions......Page 131
References......Page 133
1 Introduction......Page 136
2 The System......Page 138
3 Background on the Harmonic Balance (HB) Prediction of Limit Cycles......Page 139
4 HB for Bifurcations of Limit Cycles......Page 142
4.2 Flip and secondary Hopf (Neimark-Sacker) bifurcations......Page 143
5 Limit Cycle Bifurcations in Lur’e Systems......Page 146
6 Stability of Predicted Limit Cycles and Their Bifurcations......Page 147
7.1 Problem formulation......Page 150
7.2 Controller design......Page 152
7.3 A simplified technique: the subharmonic bifurcation control......Page 154
8.1 Control of a period doubling bifurcation......Page 155
8.3 Control of a cyclic fold bifurcation......Page 158
References......Page 161
1 Introduction......Page 164
2.1 Introduction......Page 166
2.2 Hybrid models......Page 167
2.3 Control elements......Page 168
2.4 Bifurcations and global control......Page 169
2.5 Optimal coordination and swarming......Page 171
3.2 Dynamical model of power systems......Page 172
3.3 Local controllers......Page 174
3.4 Global controller......Page 178
4.1 Introduction......Page 182
4.2 System modelling......Page 183
4.3 Control problem formulation......Page 184
4.4 A tree search method......Page 187
4.5 Simulation results......Page 189
5 Conclusions......Page 191
Appendix: System data......Page 192
References......Page 193
1 Introduction......Page 197
2.1 De.nitions......Page 198
3 Controlling Procedure......Page 200
4.1 Controlling spiral attractor of Chua’s circuit......Page 201
4.2 Preserving transient chaos of the Lorenz system......Page 204
4.3 Controlling impact oscillator......Page 205
5 Conclusions......Page 210
1 Introduction......Page 212
2.1 Preliminaries......Page 214
3 Bifurcation Theory in the Stabilization Problem......Page 222
3.1 Review of previous results......Page 223
3.2 Domain of attraction of x00......Page 226
3.3 Qualitative analysis of the dynamical behavior on a two-dimensional manifold......Page 228
4 Conclusions......Page 232
1 Introduction......Page 236
2.1 Bifurcation creation in 1-D systems......Page 240
2.2 Bifurcation creation in 2-D systems......Page 242
2.3 Bifurcation creation in -D systems......Page 243
2.4 Anti-control of bifurcations in thermal convection loop......Page 246
3 Fractional Power Control of Bifurcations......Page 249
3.1 FPC of 1-D systems......Page 250
3.2 2-D systems......Page 252
4 Conclusions......Page 253
References......Page 254
1 Introduction......Page 256
2 Bifurcations and Structural Stability......Page 258
3 Bifurcation Analysis for Model Identification......Page 259
3.1 Experimental bifurcation analysis......Page 260
3.2 Deriving the model......Page 261
4 Bifurcation Tailoring......Page 263
4.1 Statement of the problem......Page 265
4.2 Feedback bifurcation tailoring of equilibria......Page 266
5 PID Tuning......Page 268
6 Conclusions......Page 269
1 Introduction......Page 272
2 Model Description......Page 273
3.1 Continuous control......Page 274
3.2 Discrete control......Page 276
4 Conclusions......Page 279
1 Introduction......Page 307
2.1 Preliminaries......Page 309
2.2 The turbo decoding algorithm as a dynamical system......Page 311
3.1 Basic concepts of dynamical system theory......Page 313
3.2 Indecisive and unequivocal .xed points......Page 314
4.1 Bifurcation diagram......Page 316
4.2 Bifurcations of .xed points......Page 317
4.3 Chaotic behavior......Page 318
5 Control of Transient......Page 321
6 Conclusions......Page 325
1 Introduction......Page 281
2 Assumptions and Definitions......Page 283
3.1 Projection method......Page 285
3.2 Bifurcation stabilization......Page 287
4 Local Robustness of Bifurcation Stabilization......Page 289
4.1 Pitchfork bifurcation......Page 290
4.2 Transcritical bifurcation......Page 293
4.3 Hopf bifurcation......Page 295
5 Robustness of Rotating Stall Control for Axial Flow Compressors......Page 298
5.1 Moore-greitzer model for post-stall dynamics......Page 299
5.2 Rotating stall control laws......Page 300
5.3 Robustness of rotating stall control......Page 301
6 Remarks......Page 304
References......Page 305