An Introduction to Fractional Control

✍ Scribed by Duarte Valério, José Sá da Costa

• Publisher: The Institution of Engineering and Technology
• Year: 2013
• Tongue: English
• Leaves: 382
• Series: IET Control Engineering Series 91
• Category: Library

✦ Synopsis

Fractional control techniques provide an effective way to control dynamic behaviours, using fractional differential equations. This can include the control of fractional plants, the control of a plant using a fractional controller, or the control of a plant so that the controlled system will have a fractional behaviour to achieve a performance that would otherwise be hard to come by. An Introduction to Fractional Control outlines the theory, techniques and applications of fractional control. The theoretical background covers fractional calculus with real, complex and variable orders, fractional transfer functions, fractional identification and pseudo-state-space representations, while the control systems explored include: fractional lead control, fractional lag control, first, second and third generation Crone control, fractional PID, PI and PD control, fractional sliding mode control, logarithmic phase Crone control, fractional reset control, fractional H2 and H8 control, fractional predictive control, trajectory planning and fractional time-varying control. Each chapter contains solved examples, where the subject addressed is either expanded or applied to concrete cases, and references for further reading. Common definitions and proofs are included, along with a bibliography, and a discussion of how MATLAB can be used to assist in the design and implementation of fractional control. This is an essential guide for researchers and advanced students of control engineering in academia and industry.

✦ Table of Contents

An Introduction to Fractional Control......Page 4
Contents......Page 8
Acknowledgements......Page 14
Fractional Calculus......Page 16
Readership......Page 17
Organisation of the book......Page 18
Further reading......Page 19
Notation......Page 21
1.1.1 Definition and basic properties......Page 24
1.1.2 Combinations......Page 27
1.2 Calculus: integer orders......Page 28
1.3 Laplace transforms......Page 30
1.4 Continued fractions......Page 32
1.4.2 Evaluation of continued fractions......Page 33
1.4.3 Continued fraction expansions of functions......Page 34
Appendix to Chapter 1......Page 37
Part I: Fractional derivatives with real orders......Page 54
2.1.1 Derivatives of e(λt)......Page 56
2.1.2 Derivatives of tλ......Page 57
2.1.3 Derivatives of sin(λt) and cos(λt)......Page 58
2.2 Definitions......Page 59
2.2.1 Non-local operator......Page 61
2.2.3 Linear operator......Page 62
2.2.6 Relations between the different definitions......Page 63
2.3.1 Riemann–Liouville and Gru¨nwald–Letnikoff definitions......Page 64
2.3.2 Caputo definition......Page 65
2.3.3 Some Laplace transforms......Page 66
2.4 Some fractional derivatives......Page 69
2.5.1 Approximations based upon the definitions......Page 74
2.5.2 Short memory principle......Page 76
Further reading......Page 77
Relations between the different definitions......Page 79
Proof of theorem 2.4......Page 88
Further proofs from section 2.4......Page 92
Initialisation of fractional derivatives......Page 95
The tautochrone curve......Page 96
The heat equation and the age of the Earth......Page 98
3.1.1 SISO transfer functions......Page 102
3.1.2 Fractional MIMO transfer function matrixes......Page 103
3.2 Time responses......Page 104
3.3 Stability......Page 107
3.4.1 Frequency response of a generic transfer function......Page 109
3.4.2 Study of sα......Page 112
3.4.3 Study of 1/[(s/α)α+1]......Page 113
3.4.4 Study of 1/[(s/α)2α+2ζ(s/α)α+1]......Page 117
3.4.5 Steady-state errors......Page 120
3.4.6 Some irrational fractional transfer functions......Page 122
3.5 Stability, continued......Page 125
Further reading......Page 128
Proof of theorem 3.1......Page 129
Two curiosities......Page 143
4.1.1 Crone approximation......Page 144
4.1.2 Carlson approximation......Page 147
4.1.3 Matsuda approximation......Page 149
4.2.1 Grünwald–Letnikoff approximation......Page 151
4.2.2 Approximations based on truncated series......Page 152
4.2.4 Impulse response approximation......Page 155
4.2.5 Step response approximation......Page 156
4.2.6 Poles and zeros......Page 157
4.3.2 Approximations of general fractional transfer functions......Page 158
Further reading......Page 159
The Carlson approximation......Page 160
The truncated MacLaurin series of the Tustin approximation......Page 162
5.1.1 Known commensurability order......Page 164
5.1.2 Non-commensurable model......Page 165
5.1.4 Order optimisation......Page 166
5.2.1 Levy’s method......Page 167
5.2.2 Levy’s method, first formulation......Page 168
5.2.3 Levy’s method, second formulation......Page 170
5.2.5 Levy’s method: summing matrixes......Page 172
5.2.6 Levy’s method: stacking matrixes......Page 173
5.2.7 Weighted Levy’s method......Page 174
5.2.8 Iterations of Sanathanan and Koerner......Page 177
5.3 Identification from the phase of a frequency response......Page 179
5.3.1 Integer transfer function......Page 180
5.3.2 Discrete-time transfer function without zeros......Page 184
5.3.3 Discrete-time transfer function without poles......Page 186
5.3.4 Fractional transfer function without zeros......Page 187
5.3.5 Fractional transfer function without poles......Page 188
5.3.6 Solving the four cases above......Page 189
5.3.7 Fractional and discrete-time transfer functions with both poles and zeros......Page 191
5.3.8 The effect of noise......Page 193
Complex conjugate zeros and poles, for the integer case......Page 198
Proof of Lemma 5.5......Page 199
Complex conjugate zeros and poles, for the discrete-time case......Page 201
Complex conjugate zeros and poles, for the fractional case......Page 202
6.2 First-generation Crone controller......Page 204
6.3 Second-generation Crone controller......Page 205
6.4 Filters......Page 208
Phase margin and step response overshoot......Page 209
7.1.1 Integer PID......Page 212
7.1.2 Fractional PID......Page 213
7.1.4 Tuning methods......Page 214
7.2 Analytical tuning: frequency response......Page 215
7.3 Analytical tuning: internal model control......Page 217
7.4 Numerical tuning of fractional PIDs......Page 219
7.5 Tuning rules for fractional PIDs......Page 221
7.5.1 S-shaped step response......Page 222
7.5.2 Critical gain......Page 224
Fractional irrational PID......Page 226
Proof of Theorem 7.1......Page 227
Invariance in face of time unit changes......Page 231
8 Fractional reset control......Page 232
Describing functions for non-linearities......Page 239
9.1.1 Definition......Page 242
9.1.2 H2 norm of 1/(sβ)......Page 249
9.1.3 H2 norm of K/(sα+α)......Page 250
9.2.2 Numerical computation......Page 252
9.3 H2 and H∞ controllers......Page 253
Further reading......Page 254
10.1.1 Linear system......Page 256
10.1.3 Fractional MIMO transfer function matrix......Page 257
10.1.4 Non-linear fractional systems......Page 258
10.2 Pseudo-state-space representations of SISO systems......Page 260
10.3 Discretisation......Page 265
The Hadamard product and the Hadamard power......Page 268
11.1 The commensurable SISO case......Page 270
11.1.1 Sliding surface......Page 271
11.1.2 Upper bound for the error......Page 272
11.1.4 Uncertainty in fA(x(t))......Page 275
11.1.5 Uncertainty in fA(x(t)) and in fB(x(t))......Page 276
11.1.6 Avoiding chattering......Page 278
11.2 The more general SISO case......Page 282
11.3 The commensurable MIMO case......Page 286
11.3.1 Sliding surface......Page 287
11.3.2 Uncertainty in f(x(t)) and in B(t)......Page 288
11.4 The more general MIMO case......Page 290
Fractional Lyapunov stability theory......Page 291
12.1 Avoiding obstacles......Page 292
12.2 Reaching the target......Page 293
12.2.1 Maximum acceptable danger level......Page 294
12.2.2 Fractional repulsive force......Page 295
Discrete geometry......Page 297
Part II: Fractional derivatives with complex orders......Page 300
13.1 Preliminaries......Page 302
13.2 Definitions of complex derivatives......Page 303
Complex calculus formulas......Page 305
14.1.2 Time and frequency responses......Page 308
14.1.3 Frequency response of sʒ......Page 309
14.2.2 Frequency response of GR, GI, ḠR and ḠI......Page 310
14.2.3 Non-linearities in the frequency response of GR, GI, ḠR and ḠI......Page 313
14.2.4 Integer approximations of complex order transfer functions......Page 314
14.2.5 Crone approximation of a linear phase......Page 317
14.2.6 Logarithmic phase Crone controller......Page 318
15.1.1 Uncertainties in the Nichols chart......Page 320
15.1.2 Multiple orders......Page 322
15.2 For MIMO plants......Page 324
Closed loop behaviour in the Nichols diagram......Page 325
Part III: Fractional derivatives with variable orders......Page 328
16.1 First definition......Page 330
16.2 Second definition......Page 331
16.3 Third definition......Page 333
16.4 Linear operator......Page 334
16.8 Evaluation of fractional derivatives......Page 335
Further reading......Page 336
Generalised definitions of variable order derivatives......Page 340
17.1 Time-varying transfer functions......Page 342
17.2.1 Approximation for real time-varying orders......Page 343
17.2.2 Approximation for complex time-varying orders......Page 344
17.2.4 Approximations with memory......Page 346
17.3.1 Plant with a constant phase......Page 347
17.3.3 Plant with a varying slope linear phase......Page 348
Further reading......Page 349
Boolean logic......Page 350
Fuzzy logic......Page 351
Approximations of fractional time-varying transfer functions with memory......Page 353
18.1 Types of fractional control......Page 358
18.4 Controllers for MIMO plants......Page 359
Further reading......Page 360
The NINTEGER toolbox......Page 362
Examples......Page 363
References......Page 366
Index......Page 374