Analog Automation and Digital Feedback Control Techniques

259 Pages·2018·7.7 MB·English

by  Jean Mbihi

Analog Automation and Digital Feedback Control Techniques

Series Editor
Jean-Paul Bourrières
Analog Automation
and Digital Feedback
Control Techniques

Jean Mbihi
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Part 1. Analog Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . 1
Chapter 1. Models of Dynamic Processes . . . . . . . . . . . . . . . . . . . . . 3
1.1. Introduction to dynamic processes . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1. Definition, hypotheses and notations . . . . . . . . . . . . . . . . . . . . . 3
1.1.2. Implications of hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.3. Dynamic model: an automation perspective . . . . . . . . . . . . . . . . . 5
1.2. Transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1. Existence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2. Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3. General structure of a transfer function . . . . . . . . . . . . . . . . . . . . 8
1.2.4. Tools for the analysis of the properties
of transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5. First- and second-order transfer functions . . . . . . . . . . . . . . . . . . 8
1.3. State models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2. Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3. General structure of the state model . . . . . . . . . . . . . . . . . . . . . . 14
1.4. Linear state models with constant parameters . . . . . . . . . . . . . . . . . . . 15
1.4.1. Linearization-based construction . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2. Structure of a linear state model with constant parameters . . . . . . . . . 16
1.4.3. Properties of a model without pure input delay (τ = 0) . . . . . . . . . . 18
0
1.5. Similarity transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6. Exercises and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
vi Analog Automation and Digital Feedback Control Techniques
Chapter 2. Experimental Modeling Approach
of Dynamic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1. Introduction to experimental modeling . . . . . . . . . . . . . . . . . . . . . . 39
2.1.1. Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.2. Principle of experimental modeling . . . . . . . . . . . . . . . . . . . . . . 39
2.1.3. Experimental modeling methodology . . . . . . . . . . . . . . . . . . . . . 40
2.2. Step response-based modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.1. Model of order 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.2. Under-damped model of order 2 (ξ < 1) . . . . . . . . . . . . . . . . . . 44
2.2.3. Damped model of order ≥ 2 (Strejc method) . . . . . . . . . . . . . . . . 46
2.3. Frequency response-based modeling . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4. Modeling based on ARMA model . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.1. ARMA model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.2. Parameter estimation of an ARMA model . . . . . . . . . . . . . . . . . . 54
2.5. Matlab-aided experimental modeling . . . . . . . . . . . . . . . . . . . . . . . 56
2.6. Exercises and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Chapter 3. Review of Analog Feedback Control Systems . . . . . . . . . . . 73
3.1. Open-loop analog control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1.1. Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.1.2. Open-loop control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2. Analog control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3. Performances of an analog control system . . . . . . . . . . . . . . . . . . . . 75
3.3.1. Closed-loop transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.2. Performance quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4. Simple analog controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5. PID/PIDF controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.1. Structure and role of the parameters of
a PID/PIDF controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.2. Ziegler–Nichols methods for parameter calculation . . . . . . . . . . . . . 79
3.5.3. Calculation of parameters by pole placement . . . . . . . . . . . . . . . . 79
3.5.4. Direct calculation of optimal PID parameters . . . . . . . . . . . . . . . . 81
3.5.5. LQR-based indirect calculation of optimal
PID parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.6. Implementation of analog controllers . . . . . . . . . . . . . . . . . . . . . 85
3.6. Controllers described in the state space . . . . . . . . . . . . . . . . . . . . . . 86
3.6.1. Principle and block diagram of a linear state feedback . . . . . . . . . . . 86
3.6.2. Techniques for calculating the state feedback gain . . . . . . . . . . . . . 87
3.6.3. Integral action state feedback . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6.4. State feedback with integral action and observer . . . . . . . . . . . . . . . 90
3.6.5. State feedback with output error compensator . . . . . . . . . . . . . . . . 92
Contents vii
3.7. Principle of equivalence between PID and LQR controllers . . . . . . . . . . . 92
3.7.1. Proof of the equivalence principle . . . . . . . . . . . . . . . . . . . . . . . 93
3.7.2. Equivalence relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.7.3. Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.8. Exercises and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Part 2. Synthesis and Computer-aided Simulation
of Digital Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . . 123
Chapter 4. Synthesis of Digital Feedback Control
Systems in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.1. Synthesis methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2. Transfer function G(z) of a dynamic process . . . . . . . . . . . . . . . . . . . 125
4.2.1. Sampled dynamic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2.2. Discretization of G(p) if input delay τ = 0 . . . . . . . . . . . . . . . . . 126
c 0
4.2.3. Discretization of G(s) if input delay τ # 0 . . . . . . . . . . . . . . . . . . 128
c 0
4.2.4. Examples of calculation of G(z) by
discretization of G(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
c
4.3. Transfer function D(z): discretization method . . . . . . . . . . . . . . . . . . . 136
4.3.1. Interest of discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.3.2. Discretization of D(s) by invariance methods . . . . . . . . . . . . . . . . 137
c
4.3.3. Discretization of D(s) by transformation methods . . . . . . . . . . . . . 139
c
4.3.4. z-Transfer functions of simple controllers . . . . . . . . . . . . . . . . . . 142
4.3.5. General structure of D(z) and recurrence equation . . . . . . . . . . . . . . 144
4.3.6. Discretization of transfer functions with Matlab . . . . . . . . . . . . . . . 145
4.4. Transfer function D(z): model method . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.1. Principle of the model method . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.4.2. Examples of direct design of digital controllers . . . . . . . . . . . . . . . 146
4.4.3. Conditions for the use of model approach . . . . . . . . . . . . . . . . . . 148
4.4.4. Practical rules for using the model approach . . . . . . . . . . . . . . . . . 149
4.5. Discrete block diagram of digital control . . . . . . . . . . . . . . . . . . . . . 150
4.5.1. Closed-loop characteristic transfer functions . . . . . . . . . . . . . . . . . 151
4.5.2. Sampling frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.6. Exercises and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Chapter 5. Computer-aided Simulation of
Digital Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.1. Approaches to computer-aided simulation . . . . . . . . . . . . . . . . . . . . 177
5.2. Programming of joint recurrence equations . . . . . . . . . . . . . . . . . . . . 178
5.2.1. Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.2.2. Example of Matlab® programming . . . . . . . . . . . . . . . . . . . . . . 179
5.3. Simulation using Matlab macro programming . . . . . . . . . . . . . . . . . . 183
viii Analog Automation and Digital Feedback Control Techniques
5.4. Graphic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.5. Case study: simulation of servomechanisms . . . . . . . . . . . . . . . . . . . 187
5.5.1. Simulation of a speed servomechanism . . . . . . . . . . . . . . . . . . . . 187
5.5.2. Simulation of a position servomechanism . . . . . . . . . . . . . . . . . . 191
5.6. Exercises and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
Chapter 6. Discrete State Models of Dynamic Processes . . . . . . . . . . . . 199
6.1. Discretization of the state model of a dynamic process . . . . . . . . . . . . . 199
6.1.1. Discretization of a state model . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.1.2. Discretization of a state model with input delay . . . . . . . . . . . . . . . 201
6.2. Calculation of {A, B, C, D} parameters
of a discrete state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.2.1. Calculation of A = eAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.2.2. Calculation of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.2.3. Calculation of C and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.3. Properties of a discrete state model {A, B, C, D} . . . . . . . . . . . . . . . . . 208
6.3.1. Infinity of state models of one dynamic process . . . . . . . . . . . . . . . 208
6.3.2. Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.3.3. Controllability and stabilizability . . . . . . . . . . . . . . . . . . . . . . . 209
6.3.4. Observability and detectability . . . . . . . . . . . . . . . . . . . . . . . . 210
6.4. Exercises and solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Appendix 1. Table of Z-transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Appendix 2. Matlab® Elements Used in This Book . . . . . . . . . . . . . . . . 219
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227