An Introduction to Fractional Control

382 Pages·2013·13.163 MB·English

by  Duarte Valério, José Sá da Costa

Control Engineering Series 91
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An Introduction to Fractional Control
n An Introduction to

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Fractional Control
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Fractional control techniques provide an effective way to control dynamic behaviours, Duarte Valério is a research
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using fractional differential equations. This can include the control of fractional plants, assistant at the Technical University t
the control of a plant using a fractional controller, or the control of a plant so that the of Lisbon. He works in the field of io
controlled system will have a fractional behaviour to achieve a performance that would control engineering, studying the n
control of dynamic systems using
otherwise be hard to come by. fractional order derivatives; the use of t
An Introduction to Fractional Control outlines the theory, techniques and applications autonomous, pro-active, interacting o

of fractional control. After an initial introduction to fractional calculus, the book explores systems applied to distributed plants;
many fractional control systems including fractional lead control, fra ctional lag control, and the control of plants to produce F
first, second and third generation Crone control, fractional PID, PI and PD control, electricity from sea wave energy. ra
fractional sliding mode control, logarithmic phase Crone control, fractional reset control, José Sá da Costa is Professor of c
fractional H2 and H∞ control, fractional predictive control, and fractional time-varying CUonnivterorsl itEyn ogifn Leisebrionng aatn dth eH eTeadch onfi ctahle tio
control.
Centre of Intelligent Systems of the n
Each chapter contains solved examples and references for further reading. Common Institute of Mechanical Engineering, a
definitions and proofs are included, along with a discussion of how MATLAB can be used Lisbon. His research interests include l
to assist in the design and implementation of fractional control. fractional systems, fault tolerant C
control, surgery robots, active noise
This book is an essential guide for researchers and advanced students of control and vibration control, and control o
engineering in academia and industry. in health and ocean wave energy n
devices. t
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Cover image: fractional derivatives of a sinusoid.
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The Institution of Engineering and Technology tad
www.theiet.org
978-1-84919-545-4
Fractional Control.indd 1 24/10/2012 15:55:35
IETCONTROL ENGINEERING SERIES 91
An Introduction to
Fractional Control
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An Introduction to
Fractional Control
Duarte Vale´rio and Jose´ Sa´ da Costa
The Institution ofEngineeringand Technology
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‘I just take the train from platform nine and
three-quarters at eleven o’clock,’ he read.
His aunt and uncle stared.
‘Platform what?’
‘Nine and three-quarters.’
‘Don’t talk rubbish,’ said Uncle Vernon, ‘there is
no platform nine and three-quarters.’
J. K. Rowling, Harry Potter and the Philosopher’s Stone, Chapter 6

Contents
Acknowledgements xiii
Introduction xv
Notation xx
1 Preliminaries 1
1.1 The Gamma function 1
1.1.1 Definition and basic properties 1
1.1.2 Combinations 4
1.2 Calculus: integer orders 5
1.3 Laplace transforms 7
1.4 Continued fractions 9
1.4.1 Continued fractionexpansions of real numbers 10
1.4.2 Evaluation of continued fractions 10
1.4.3 Continued fractionexpansions of functions 11
Further reading 14
Appendix to Chapter 1 14
Part I Fractional derivatives withreal orders 31
2 Fractional calculus: real orders 33
2.1 Simple examples 33
2.1.1 Derivatives of elt 33
2.1.2 Derivatives of tl 34
2.1.3 Derivatives of sinðltÞ and cosðltÞ 35
2.2 Definitions 36
2.2.1 Non-local operator 38
2.2.2 Terminals 39
2.2.3 Linear operator 39
2.2.4 Geometrical interpretation 40
2.2.5 Alternative notations and names 40
2.2.6 Relations between the different definitions 40
2.2.7 Law of exponents 41
2.3 Laplace transforms 41
2.3.1 Riemann–Liouville and Gru¨nwald–Letnikoff definitions 41
2.3.2 Caputo definition 42
2.3.3 Some Laplace transforms 43
viii An introduction to fractional control
2.4 Some fractional derivatives 46
2.5 Numerical evaluation 51
2.5.1 Approximations based upon the definitions 51
2.5.2 Short memory principle 53
Further reading 54
Appendix to Chapter 2 56
3 Fractional transfer functions 79
3.1 Definitions 79
3.1.1 SISOtransfer functions 79
3.1.2 Fractional MIMOtransfer function matrixes 80
3.2 Time responses 81
3.3 Stability 84
3.4 Frequency response 86
3.4.1 Frequencyresponse of a generic transfer function 86
3.4.2 Study of sa 89
3.4.3 Study of (cid:1)s(cid:3)1a 90
þ1
a
3.4.4 Study of 1 94
ðsÞ2aþ2zðsÞaþ1
a a
3.4.5 Steady-state errors 97
3.4.6 Some irrational fractional transfer functions 99
3.5 Stability, continued 102
Further reading 105
Appendix to Chapter 3 106
4 Approximationsof fractional transfer functions 121
4.1 Continuousapproximations 121
4.1.1 Crone approximation 121
4.1.2 Carlson approximation 124
4.1.3 Matsuda approximation 126
4.2 Discrete-time approximations of sa 128
4.2.1 Gru¨nwald–Letnikoff approximation 128
4.2.2 Approximations based on truncated series 129
4.2.3 FIRapproximations based on time responses 132
4.2.4 Impulse response approximation 132
4.2.5 Step responseapproximation 133
4.2.6 Poles and zeros 134
4.3 General considerations 135
4.3.1 Approximations of sa 135
4.3.2 Approximations of general fractional transfer functions 135
4.3.3 Approximations of irrational fractional transfer functions 136
Further reading 136
Appendix to Chapter 4 137
Contents ix
5 Fractional identification 141
5.1 Identification from a time response 141
5.1.1 Knowncommensurability order 141
5.1.2 Non-commensurable model 142
5.1.3 Filtering 143
5.1.4 Order optimisation 143
5.2 Identification from a frequency response 144
5.2.1 Levy’s method 144
5.2.2 Levy’s method, first formulation 145
5.2.3 Levy’s method, second formulation 147
5.2.4 Making useof data from many frequencies 149
5.2.5 Levy’s method: summing matrixes 149
5.2.6 Levy’s method: stacking matrixes 150
5.2.7 Weighted Levy’s method 151
5.2.8 Iterations of Sanathanan and Koerner 154
5.3 Identification from the phase of a frequency response 156
5.3.1 Integer transfer function 157
5.3.2 Discrete-time transfer function without zeros 161
5.3.3 Discrete-time transfer function without poles 163
5.3.4 Fractional transfer function without zeros 164
5.3.5 Fractional transfer function without poles 165
5.3.6 Solving the four cases above 166
5.3.7 Fractional and discrete-time transfer functions with
both poles and zeros 168
5.3.8 The effect of noise 170
Further reading 175
Appendix to Chapter 5 175
6 First andsecond-generationCronecontrol 181
6.1 Robustnesstowardsplantgain variations 181
6.2 First-generation Crone controller 181
6.3 Second-generation Crone controller 182
6.4 Filters 185
Further reading 186
Appendix to Chapter 6 186
7 Fractional PIDs 189
7.1 Definition 189
7.1.1 Integer PID 189
7.1.2 Fractional PID 190
7.1.3 Frequency behaviour 191
7.1.4 Tuning methods 191
7.2 Analytical tuning: frequency response 192
7.3 Analytical tuning: internal model control 194