Analysis, Modeling and Stability of Fractional Order Differential Systems 2: The Infinite State Approach

Cover......Page 1
Half-Title Page......Page 3
Dedication......Page 4
Title Page......Page 5
Copyright Page......Page 6
Contents......Page 7
Foreword......Page 15
Preface......Page 17
PART 1: Initialization, State Observation and Control......Page 21
  1.1. Introduction......Page 23
   1.2.2. Response of a linear system......Page 24
   1.2.3. Input/output solution......Page 26
   1.2.4. State space solution......Page 27
   1.2.5. First-order system example......Page 28
   1.3.2. Free response of a simple FDE......Page 30
   1.4.2. State space representation......Page 34
   1.4.3. Input/output formulation......Page 35
   1.5.2. Initialization of the fractional integrator......Page 37
   1.5.3. Initialization of the Riemann–Liouville derivative......Page 39
   1.5.4. Initialization of an elementary FDS......Page 41
   1.5.5. Conclusion......Page 53
  2.1. Introduction......Page 55
   2.2.3. Observability and controllability criteria for a linear integer order system......Page 57
   2.2.4. Observability and controllability of FDS......Page 59
   2.3.1. Introduction......Page 60
   2.3.2. Elementary approach......Page 61
   2.3.3. Cayley–Hamilton approach......Page 65
   2.3.4. Gramian approach......Page 69
   2.3.5. Gilbert’s approach......Page 72
   2.3.6. Conclusion......Page 77
   2.3.7. Pseudo-controllability example......Page 78
   2.4.1. Introduction......Page 80
   2.4.2. Observability of the distributed state......Page 82
   2.4.3. Controllability of the distributed state......Page 84
  2.5. Conclusion......Page 85
  3.1. Introduction......Page 87
  3.2. Initialization: problem statement......Page 88
   3.3.1. Fractional observer definition......Page 91
   3.3.2. Stability analysis......Page 92
   3.3.3. Convergence analysis......Page 94
   3.3.4. Numerical example 1: one-derivative system......Page 96
   3.3.5. Numerical example 2: non-commensurate order system......Page 98
   3.4.1. Introduction......Page 101
   3.4.2. Non-commensurate order principle......Page 102
   3.4.3. Gradient algorithm......Page 104
   3.4.4. One-derivative FDE example......Page 107
   3.4.5. Two-derivative FDE example......Page 111
   A.3.1. Convergence of gradient algorithm......Page 115
   A.3.2. Stability and limit value of λ......Page 118
  4.1. Introduction......Page 119
   4.2.1. Introduction......Page 120
   4.2.2. Numerical simulation example......Page 121
   4.3.1. Introduction......Page 123
   4.3.2. State control of a fractional integrator......Page 124
   4.4.1. Introduction......Page 141
   4.4.2. Principle of state control......Page 142
   4.4.3. State control of two integrators in series......Page 144
   4.4.4. Numerical example......Page 146
   4.4.5. State control of a two-derivative FDE......Page 149
   4.4.6. Pseudo-state control of the two-derivative FDE......Page 150
  4.5. Conclusion......Page 151
  5.1. Introduction......Page 153
   5.2.2. An identification algorithm dedicated to fractional models......Page 154
   5.2.3. Simulation of the diffusive RC line......Page 159
   5.2.4. Experimental identification......Page 169
   5.3.1. Introduction......Page 174
   5.3.2. Natural relaxation......Page 175
   5.3.3. Principle of the reset technique......Page 176
   5.3.4. Proposed reset procedure......Page 178
   5.3.5. Experimental results......Page 179
   5.3.6. Comments......Page 184
   5.3.7. Conclusion......Page 185
PART 2: Stability of Fractional Differential Equations and Systems......Page 187
  6.1. Introduction......Page 189
   6.2.1. Simulation of an FDE......Page 191
   6.2.2. Stability of the simulation scheme......Page 192
   6.2.3. Stability analysis of FDEs using the Nyquist criterion......Page 194
   6.3.1. Introduction......Page 195
   6.3.2. Open-loop transfer function......Page 196
   6.3.3. Drawing of HOL (jω) graph in the complex plane......Page 197
   6.3.4. Stability of the third-order ODE......Page 198
   6.4.2. Drawing of HOL (jω) graph in the complex plane......Page 202
   6.4.3. Stability of the one-derivative FDE......Page 204
   6.4.4. Stability of the two-derivative FDE......Page 207
   6.4.5. Stability of the N-derivative FDE......Page 214
   6.5.1. Introduction......Page 215
   6.5.3. Stability analysis......Page 216
   6.5.4. Application to an example......Page 218
   6.6.1. Definitions......Page 220
   6.6.2. Stability......Page 221
   6.6.3. Application to an example......Page 222
  7.1. Introduction......Page 225
  7.2. Pseudo-energy stored in a fractional integrator......Page 226
   7.3.2. Electrical distributed network......Page 231
   7.3.3. Stored energy......Page 234
   7.3.4. Power dissipated in the fractional integrator......Page 235
   7.3.5. Energy storage......Page 236
   7.3.6. Integer order and fractional order integrators......Page 239
   7.3.7. Characterization of fractional energy and its dissipation......Page 246
   7.3.8. Fractional energy invariance......Page 251
   7.4.2. Energy of the closed-loop model......Page 254
   7.4.3. Energy of the open-loop model......Page 257
   7.4.4. Stored energies with a step input excitation......Page 259
  8.1. Introduction......Page 267
   8.2.1. Problem statement......Page 269
   8.2.2. Numerical simulation......Page 271
   8.2.3. Physical interpretation......Page 273
   8.2.4. Theoretical interpretation......Page 274
   8.3.2. The integer order case......Page 278
   8.3.3. Lyapunov function of N-derivative systems......Page 281
   8.3.4. Stability condition......Page 285
   8.4.1. Introduction......Page 289
   8.4.2. State space model of the open-loop representation......Page 290
   8.4.3. State space models of the closed-loop representation......Page 291
   8.4.4. Energy and stability of the open-loop representation......Page 292
   8.4.5. Energy and stability of the closed-loop representation......Page 294
   8.4.6. Definition of a stability test for a > 0......Page 296
   8.5.1. Introduction......Page 301
   8.5.2. Problem statement......Page 302
   8.5.3. LMI generalization for N = 3......Page 303
   8.5.4. Application example......Page 309
   A.8.1. Lemma......Page 310
   A.8.2. Matignon’s criterion......Page 311
  9.1. Introduction......Page 313
   9.2.1. Usual capacitor and inductor devices......Page 315
   9.2.2. Fractional capacitor and inductor......Page 316
   9.2.3. Energy storage and dissipation in fractional devices......Page 319
   9.2.4. Reversibility of energy and energy balance......Page 321
   9.3.2. Analysis of the series RLC circuit......Page 322
   9.3.3. Stability analysis......Page 324
   9.4.2. Analysis of the series RLC* circuit......Page 326
   9.4.3. Experimental stability analysis......Page 327
   9.4.4. Theoretical stability analysis......Page 330
   9.4.5. Conclusion......Page 334
   9.5.1. Circuit modeling......Page 335
   9.5.2. Stability analysis......Page 337
   9.6.2. Analysis of the series RL*C* circuit......Page 340
   9.6.3. Theoretical stability analysis......Page 342
   9.7.2. Analysis of the commensurate order FDE......Page 345
   9.7.3. Application to stability......Page 347
   9.8.1. Introduction......Page 348
   9.8.3. Lyapunov function of a fractional differential system......Page 349
   9.8.4. Stability analysis......Page 351
   9.8.5. Conclusion......Page 354
   A.9.1. The infinite length LG line......Page 355
   A.9.2. Energy storage and dissipation in the fractional capacitor......Page 359
   A.9.3. Some integrals......Page 361
  10.1. Introduction......Page 363
   10.2.2. Linearization......Page 364
   10.2.3. Nonlinear system analysis......Page 365
   10.2.4. Local stability of a one-derivative nonlinear fractional system......Page 369
   10.3.2. The variable gradient method......Page 373
   10.3.3. Nonlinear system with one derivative......Page 374
   10.3.4. Nonlinear system with two fractional derivatives......Page 377
   10.4.1. Electrical nonlinear system......Page 383
   10.4.3. Simulation of the nonlinear system......Page 384
   10.4.4. Limit cycle......Page 385
   10.5.1. Linearization......Page 386
   10.5.2. Local stability......Page 387
   10.5.3. Validation of stability results......Page 389
   10.6.2. Approximation of the first harmonic [MUL 09]......Page 391
   10.6.4. Amplitude of the limit cycle......Page 392
   10.6.5. Prediction of the limit cycle......Page 394
References......Page 397
Index......Page 415
Other titles from iSTE in Systems and Industrial Engineering – Robotics......Page 417
EULA......Page 429