PID Tuning: A Modern Approach via the Weighted Sensitivity Problem

155 Pages·2020·5.16 MB·English

by  Salvador Alcántara Cano, Ramon Vilanova Arbós, Carles Pedret i Ferré

PID Tuning

PID Tuning
A Modern Approach via the Weighted
Sensitivity Problem
Salvador Alc´antara Cano
Ramon Vilanova Arbo´s
Carles Pedret i Ferr´e
Firsteditionpublished2021
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To my son, the indomitable pirate, Salvita, and wife, Rosi, for riding
the waves of life with me.
—Salvador Alca´ntara Cano
To all who have contributed to my PID pathway, especially those that are not fully
aware of it. Thanks to my wife, Rosa, for filling the space between the lines.
—Ramon Vilanova Arb´os
Thanks to the people that made this book possible, especially those who gave
me guidance in my career path.
—Carles Pedret i Ferr´e

Contents
Foreword xi
Preface xiii
Authors xv
1 Introduction 1
1.1 Servo, regulation, and stability . . . . . . . . . . . . . . . . . . . . . . . .. 1
1.2 Industrial PID control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2
1.3 Internal model and H∞ control . . . . . . . . . . . . . . . . . . . . . . . .. 3
1.3.1 Internal model control . . . . . . . . . . . . . . . . . . . . . . . . .. 3
1.3.2 H∞ control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4
1.3.3 Blending internal model and H∞ control. . . . . . . . . . . . . . .. 5
1.3.4 Vilanova’s (2008) design for robust PID tuning revisited . . . . . .. 6
1.4 Outline of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8
I MODEL-MATCHING APPROACH TO ROBUST
PID DESIGN 17
2 Simple Model-Matching Approach to Robust PID Control 19
2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19
2.1.1 The control framework . . . . . . . . . . . . . . . . . . . . . . . . .. 19
2.1.2 The model-matching problem . . . . . . . . . . . . . . . . . . . . .. 20
2.1.3 The model-matching problem within H∞ control . . . . . . . . . .. 21
2.2 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22
2.2.1 Initial formulation for set-point response . . . . . . . . . . . . . . .. 22
2.2.2 Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . .. 24
2.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
2.3.1 Nominal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26
2.3.2 Robust stability . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
2.4 Automatic PID tuning derivation . . . . . . . . . . . . . . . . . . . . . .. 29
2.4.1 Control effort constraints . . . . . . . . . . . . . . . . . . . . . . .. 32
2.5 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
3 Alternative Design for Load Disturbance Improvement 39
3.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
3.1.1 The control framework . . . . . . . . . . . . . . . . . . . . . . . . .. 40
3.1.2 The model-matching problem formulation . . . . . . . . . . . . . .. 41
3.2 Model-matching solution for PID design . . . . . . . . . . . . . . . . . . .. 41
3.3 Trade-off tuning interval considering load disturbances . . . . . . . . . .. 43
3.3.1 Nominal stability . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45
3.4 Tuning guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
3.5 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49
vii
viii Contents
4 Analysis of the Smooth/Tight–Servo/Regulation Tuning Approaches 57
4.1 Revisiting the model-matching designs . . . . . . . . . . . . . . . . . . . . 57
4.2 Smooth/tight tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Servo/regulation tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Implementation aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
II WEIGHT SELECTION FOR SENSITIVITY SHAPING 71
5 H∞ Design with Application to PI Tuning 73
5.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3 Weight selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Stability and robustness analysis . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Application to PI tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.1 Stable/unstable plants . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5.2 Integrating plant case (τ →∞) . . . . . . . . . . . . . . . . . . . . . 81
5.6 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Generalized IMC Design and H Approach 89
2
6.1 Motivation for the input/output disturbance trade-off . . . . . . . . . . . . 89
6.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.3 Weight selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4.1 Interpretation in terms of alternative IMC filters . . . . . . . . . . . 95
6.4.2 Extension to plants with integrators or complex poles . . . . . . . . 96
6.5 Performance and robustness analysis . . . . . . . . . . . . . . . . . . . . . . 97
6.6 Tuning guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.7 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
III WEIGHTED SENSITIVITY APPROACH FOR ROBUST
PID TUNING 105
7 PID Design as a Weighted Sensitivity Problem 107
7.1 Context, motivation, and objective . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Servo/regulation and robustness/performance trade-offs . . . . . . . . . . . 108
7.3 Unifying tuning rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.4 Special cases and tuning-rule simplifications . . . . . . . . . . . . . . . . . 111
7.4.1 First-order cases (τ =0) . . . . . . . . . . . . . . . . . . . . . . . . 111
2
7.4.2 Second-order cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.5 Applicability: normalized dead time range . . . . . . . . . . . . . . . . . . 113
8 PID Tuning Guidelines for Balanced Operation 115
8.1 Robustness and comparable servo/regulation designs . . . . . . . . . . . . 115
8.2 Servo/regulation performance evaluation: J and J indices . . . . . . 116
max avg
8.3 PI control using first-order models . . . . . . . . . . . . . . . . . . . . . . . 118
8.3.1 Stable and integrating cases . . . . . . . . . . . . . . . . . . . . . . . 118
8.3.1.1 Tuning based on J . . . . . . . . . . . . . . . . . . . . . 119
max
8.3.1.2 Tuning based on J . . . . . . . . . . . . . . . . . . . . . 120
avg
8.3.2 Unstable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8.3.2.1 Tuning based on J and J . . . . . . . . . . . . . . . . 120
max avg
Contents ix
8.4 PID control using second-order models . . . . . . . . . . . . . . . . . . . . 123
8.4.1 Stable and integrating cases . . . . . . . . . . . . . . . . . . . . . . . 123
8.4.1.1 Tuning based on J . . . . . . . . . . . . . . . . . . . . . 123
max
8.4.1.2 Tuning based on J . . . . . . . . . . . . . . . . . . . . . 124
avg
8.4.2 Unstable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.4.2.1 Tuning based on J and J . . . . . . . . . . . . . . . . 125
max avg
Appendix A 127
Bibliography 131
Index 137