Digital control systems : design, identification and implementation

Communications and Control Engineering

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Ioan D. Landau and Gianluca Zito

Digital Control

Systems

Design, Identification and Implementation

With 238 Figures

123

Ioan D. Landau, PhD

Gianluca Zito, PhD

Lab. d’Automatique de Grenoble (INPG/CNRS)

ENSIEG

BP 46

38402 Saint Martin d’Heres

France

Series Editors

E.D. Sontag · M. Thoma · A. Isidori · J.H. van Schuppen

British Library Cataloguing in Publication Data

Landau, Ioan D., 1938-

Digital control systems : design, identification and

implementation. - (Communications and control engineering)

1. Digital control systems 2. Digital control systems -

Design and construction

I. Title II.Zito, Gianluca

629.8’9

ISBN-10: 1846280559

Library of Congress Control Number: 2005931921

Communications and Control Engineering Series ISSN 0178-5354

ISBN-10: 1-84628-055-9 e-ISBN 1-84628-056-7 Printed on acid-free paper

ISBN-13: 978-1-84628-055-9

© Springer-Verlag London Limited 2006

MATLAB® is the registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-

2098, U.S.A. http://www.mathworks.com

Scilab is Copyright © 1989-2005. INRIA ENPC; Scilab is a trademark of INRIA: www.scilab.org

Digital Control Systems is a revised translation of Commande des systèmes: conception, identification,

mise en oeuvre (2-7462-0478-9) published by Hermes-Lavoisier, Paris, 2002

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as

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The publisher makes no representation, express or implied, with regard to the accuracy of the infor￾mation contained in this book and cannot accept any legal responsibility or liability for any errors or

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Printed in Germany

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To Lina, Vlad, Carla, Maria Luisa, Francesco

“Ce qui est simple est toujours faux

Ce qui ne l’est pas est inutilisable.”

(Paul Valery)

Preface

The extraordinary development of digital computers (microprocessors,

microcontrollers) and their extensive use in control systems in all fields of

applications has brought about important changes in the design of control systems.

Their performance and their low cost make them suitable for use in control systems

of various kinds which demand far better capabilities and performances than those

provided by analog controllers.

However, in order really to take advantage of the capabilities of

microprocessors, it is not enough to reproduce the behavior of analog (PID)

controllers. One needs to implement specific and high-performance model based

control techniques developed for computer-controlled systems (techniques that

have been extensively tested in practice). In this context identification of a plant

dynamic model from data is a fundamental step in the design of the control system.

The book takes into account the fact that the association of books with software

and on-line material is radically changing the teaching methods of the control

discipline. Despite its interactive character, computer-aided control design software

requires the understanding of a number of concepts in order to be used efficiently.

The use of software for illustrating the various concepts and algorithms helps

understanding and rapidly gives a feeling of the various phenomena.

Complementary information and material for teaching and applications can be

found on the book website:

http://landau-bookic.lag.ensieg.inpg.fr

The Aim of the Book

The aim of this book is to give the necessary knowledge for the comprehension and

implementation of digital techniques for system identification and control design.

These techniques are applicable to various types of process. The book has been

written taking into account the needs of the designer and the user of such systems.

Theoretical developments that are not directly relevant to the design have been

omitted. The book also takes into account the availability of dedicated control

software. A number of useful routines have been developed and they can be freely

ix

x Preface

downloaded from the book website. Details concerning effective implementation

and on-site optimization of the control systems designed have been provided.

An important feature of the book, which makes it different from other books on

the subject, is the fact that equal weight has been given to system identification and

control design. This is because both techniques are equally important for design

and optimization of a high-performance control system. A control engineer has to

possess a balance of knowledge in both subjects since identification cannot be

dissociated from control design. The book also emphasizes control robustness

aspects and controller complexity reduction, both very important issues in practice.

The Object of Study

The closed loop control systems studied in this book are characterized by the fact

that the control law is implemented on a digital computer (microprocessor,

microcontroller). This type of system is sketched in Figure 0.1.

The continuous-time plant to be controlled is formed by the set of actuator,

process and sensor. The continuous-time measured output y(t) is converted into a

sequence of numbers {y(k)} by an analog-to-digital converter (ADC), at sampling

instants k defined by the synchronization clock. This sequence is compared with

the reference sequence{r(k)} and the resulting sequence of errors is processed by

the digital computer using a control algorithm that will generate a control

sequence {u(k)}. By means of a digital-to-analog converter (DAC), this sequence

is converted into an analog signal, which is usually maintained constant between

the sampling instants by a zero-order hold (ZOH).

PLANT

Figure 0.1. Digital control system

The Main Stream

Figure 0.2 summarizes the general principles for controller design, implementation

and validation.

For design and tuning of a good controller one needs:

1. To specify the desired control loop performance and robustness

2. To know the dynamic model of the plant to be controlled

3. To possess a suitable controller design method making it possible to

achieve the desired performance and robustness specifications for the

corresponding plant model

Actuator Sensor ADC

DIGITAL

COMPUTER

CLOCK

r(k)

e(k) u(k) u(t) y(t) y(k)

DAC + +

ZOH -

Process

Preface xi

4. To implement the resulting controller taking into account practical

constraints

5. To validate the controller performance on site and, if necessary, to re￾tune it

Reference u y

+

+

DESIGN

METHOD MODEL(S)

Performance

specifications

PLANT

IDENTIFICATION

Robustness

specifications

CONTROLLER

2

1

Figure 0.2. Principle of controller design and validation

In order to obtain a relevant dynamic plant model for design, system

identification techniques using input/output measurements (switch 1 is off, switch

2 is on) should be considered. The methodology for system identification is

presented in the book together with dedicated algorithms implemented as software

tools.

Once the system model is available, the book provides a set of methods (and

the corresponding software tools) for the design of an appropriate controller.

The implementation of the controller should take into account aspects related to

data acquisition, switching from open loop to closed loop, and saturation of the

actuator as well as constraints on the complexity of the controller. These aspects

are examined in detail in the book.

Expected Audience

The book represents a course reference for Universities and Engineering Schools

offering courses on applied computer-controlled systems and system identification.

In addition to its academic audience, Digital Control Systems is aimed at

practising engineers wishing to acquire the concepts and techniques of system

identification, control design and implementation using a digital computer. The

industrial references for the techniques presented in the book and the various

applications described provide useful information for those directly involved in the

real-world uses of control.

Readers who are already familiar with the basics of computer-controlled

systems will find in this book a clear, application oriented, methodology for system

identification and the design of various types of controllers for single-input, single￾output (SISO) systems.

xii Preface

The Content

Chapter 1 briefly reviews the continuous-time control techniques which will be

used later on as a reference for the introduction of basic concepts for computer

control.

Chapter 2 provides a concise overview of computer-controlled systems: the

structure of these systems, the sampling process, discrete-time dynamic models,

the principles of design of discrete-time two-degrees-of-freedom controllers (RST),

and robustness analysis of the control loops.

Chapter 3 presents several pertinent model-based design methods for discrete￾time controllers operating in a deterministic environment. After the design of

digital PID controllers, more general design methods allowing systems of any

order, with or without delay, to be controlled are presented. The robustness of the

closed loop with respect to plant model uncertainties or variations is examined in

detail and appropriate control design methods that take into account robustness

specifications are provided.

The design of discrete-time controllers operating in the presence of random

disturbances is discussed in Chapter 4. The chapter begins with a review of random

disturbances and of models and predictors for random disturbances. Connections

with design in deterministic environments are emphasized.

The basics of system identification using a digital computer are presented in

Chapter 5. Methods that are used for the identification of discrete-time models, and

model validation techniques as well as techniques for order estimation from

input/output data are described in Chapter 6.

Chapter 7 discusses the practical aspects of system identification using data

from several applications: air heater, distillation column, DC motor, and flexible

transmission.

The main goal of this work, the use of control design methods and system

identification techniques in the implementation of a digital controller for a specific

application, is discussed in Chapter 8. Implementation aspects are reviewed and

several applications presented (air heater, speed and position control of a DC

motor, flexible transmission, flexible arm, and hot-dip galvanizing).

For on-site optimization and controller re-tuning a plant model should be

obtained by identification in closed loop (switches 1 and 2 are on in Figure 0.2).

The techniques for identification in closed loop are presented in Chapter 9.

In many situations constraints on the complexity of the controller are imposed

so Chapter 10 presents techniques for controller order reduction.

Appendix A reviews some basic concepts.

Appendix B offers an alternative time-domain approach to the design of

RST digital controllers using one-step-ahead and long-range-predictive control

strategies. Links and equivalence with the design methods presented in Chapter 3

are emphasized.

Appendix C presents a state space approach to the design of RST digital

controllers. The equivalence with the design approach presented in Chapter 3 is

emphasized. The linear quadratic control is also discussed.

Appendix D presents some important concepts in robustness.

Appendix E demonstrates the Youla–Kucera parametrization of digital

controllers which is useful for a number of developments.

Preface xiii

Appendix F describes a numerically robust algorithm for recursive

identification.

Appendix G is dedicated to the presentation of suggested laboratory sessions

that use data files and functions which can be downloaded from the book website.

Appendix H gives a list and a brief description of the MATLAB®- and Scilab￾based functions and C++ programs implementing algorithms presented in the book.

These functions and programs can also be downloaded from the book website.

The book website gives access, to the various functions and programs as well

as to data files. It contains descriptions of additional laboratory sessions and slides

for a number of chapters, tutorials and courses related to the material included in

the book that can be downloaded; all the MATLAB® files used for generating the

examples and figures in the text can also be found on the website.

How to Read the Book

The book can be read in different ways after the basic control concepts presented in

Chapters 1 and 2 have been assimilated. If the reader is mainly interested in control

algorithms, it would be useful for him/her to read Chapters 3 and 4 and then

Chapters 5, 6, 7 and 8. If the reader is mainly interested in identification

techniques, he or she can jump straight to Chapters 5, 6 and 7 and then return to

Chapters 3, 4 and 8. Those who are familiar with the basics of computer-controlled

systems can even start with Section 2.5. Chapters 9 and 10 follow dependently

from Chapter 8. Figure 0.3 shows the interdependence between the various

chapters.

Course Configurations

A complete basic course on digital control should cover most of the material

presented in Chapters 2, 3, 5, 8 and Section 4.1. For an advanced course, all

chapters might be included. For an introductory course in digital control one can

use Chapters 2, 3 and 8. For an introductory course on system identification one

can use Chapters 5, 6 and 7.

Why this Book?

The book reflects the first author’s more than twenty-five years of experience in

teaching, design and implementation of digital control systems. Involvement in

many industrial projects and feedback from an industrial and academic audience

from various countries in Europe, North and South America and Asia have played

a major role in the selection, organization and presentation of the material.

Experience from writing the book System Identification and Control Design,

Prentice Hall, 19901 (Information and System Sciences Series) has been also very

useful.

The present book is a revised translation of a book (Commande des systèmes –

conception, identification et mise en oeuvre) published in 2002 by Hermes￾Lavoisier, Paris.

1 Revised taranslation of a book published by Hermes Paris, 1988 (second edition 1993).

xiv Preface

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Figure 0.3. Logical dependence of the various chapters

The most recent academic courses based on the material in the present book

include PhD courses delivered in 2004 at Universita Technologica de Valencia,

Spain (robust discrete time controller design) and Escuela Superior de Ingenerios

de Sevilla, Spain (system identification in open and closed loop).

Acknowledgments

We wish to acknowledge the large number of contributors on whose work our

presentation is partly based as well as the discussions we have had with many of

the experts in the field. In particular we wish to mention: G. Zames, V.M. Popov,

B.D.O. Anderson, K.J. Aström, R. Bitmead, D. Clarke, G. Franklin, M. Gevers, G.

Goodwin, T. Kailath, L. Ljung, M. Morari, M. Tomizuka and G. Vinnicombe.

We would like to thank M. M'Saad, N. M'Sirdi, M. Samaan, H.V. Duong, F.

Rolland, A. Voda – Besançon, J. Langer, A. Franco and H. Bourlés for their

collaboration in the development and implementation of various techniques for

system identification and control design.

We would like to thank A. Karimi, H. Prochazka, A. Constantinescu, D. Rey,

A. Rousset, P. Rival, F. Bouziani and G. Stroian who have contributed to this

project.

The support of the Laboratoire d’Automatique de Grenoble (INPG/CNRS/UJF)

and Adaptech for the experimental aspects has been essential.

We also would like to thank the participants in various courses. Their remarks

and suggestion have been very valuable.

Grenoble, Ioan Doré Landau

May 2005 Gianluca Zito

Contents

List of Principal Notation .................................................................................xxiii

1 Continuous Control Systems: A Review ......................................................... 1

1.1 Continuous-time Models............................................................................ 1

1.1.1 Time Domain.................................................................................. 1

1.1.2 Frequency Domain ......................................................................... 2

1.1.3 Stability .......................................................................................... 4

1.1.4 Time Response ............................................................................... 6

1.1.5 Frequency Response....................................................................... 7

1.1.6 Study of the Second-order System ............................................... 10

1.1.7 Systems with Time Delay............................................................. 14

1.1.8 Non-minimum Phase Systems...................................................... 15

1.2 Closed-loop Systems................................................................................ 16

1.2.1 Cascaded Systems ........................................................................ 16

1.2.2 Transfer Function of Closed-loop Systems .................................. 17

1.2.3 Steady-state Error......................................................................... 18

1.2.4 Rejection of Disturbances............................................................. 19

1.2.5 Analysis of Closed-loop Systems in the Frequency Domain:

Nyquist Plot and Stability Criterion ............................................ 20

1.3 PI and PID Controllers............................................................................. 22

1.3.1 PI Controller................................................................................. 22

1.3.2 PID Controller .............................................................................. 23

1.4 Concluding Remarks................................................................................ 24

1.5 Notes and References............................................................................... 24

2 Computer Control Systems............................................................................ 25

2.1 Introduction to Computer Control ........................................................... 25

2.2 Discretization and Overview of Sampled-data Systems.......................... 28

2.2.1 Discretization and Choice of Sampling Frequency ...................... 28

2.2.2 Choice of the Sampling Frequency for Control Systems.............. 31

2.3 Discrete-time Models............................................................................... 34

2.3.1 Time Domain................................................................................ 34

xv

xvi Contents

2.3.2 Frequency Domain ....................................................................... 38

2.3.3 General Forms of Linear Discrete-time Models ........................... 42

2.3.4 Stability of Discrete-time Systems ............................................... 44

2.3.5 Steady-state Gain.......................................................................... 46

2.3.6 Models for Sampled-data Systems with Hold .............................. 47

2.3.7 Analysis of First-order Systems with Time Delay........................ 49

2.3.8 Analysis of Second-order Systems ............................................... 52

2.4 Closed Loop Discrete-time Systems........................................................ 55

2.4.1 Closed Loop System Transfer Function ....................................... 55

2.4.2 Steady-state Error ......................................................................... 56

2.4.3 Rejection of Disturbances............................................................. 57

2.5 Basic Principles of Modern Methods for Design of Digital

Controllers ............................................................................................... 58

2.5.1 Structure of Digital Controllers .................................................... 58

2.5.2 Digital Controller Canonical Structure......................................... 61

2.5.3 Control System with PI Digital Controller ................................... 64

2.6 Analysis of the Closed Loop Sampled-Data Systems in the Frequency

Domain .................................................................................................... 66

2.6.1 Closed Loop Systems Stability..................................................... 66

2.6.2 Closed Loop System Robustness.................................................. 69

2.7 Concluding Remarks................................................................................ 81

2.8 Notes and References............................................................................... 84

3 Robust Digital Controller Design Methods .................................................. 85

3.1 Introduction.............................................................................................. 85

3.2 Digital PID Controller ............................................................................. 86

3.2.1 Structure of the Digital PID 1 Controller...................................... 87

3.2.2 Design of the Digital PID 1 Controller......................................... 90

3.2.3 Digital PID 1 Controller: Examples ............................................. 95

3.2.4 Digital PID 2 Controller ............................................................... 99

3.2.5 Effect of Auxiliary Poles ........................................................... 102

3.2.6 Digital PID Controller: Conclusions .......................................... 104

3.3 Pole Placement....................................................................................... 105

3.3.1 Structure ..................................................................................... 105

3.3.2 Choice of the Closed Loop Poles (P(q-1))................................... 107

3.3.3 Regulation (Computation of R(q-1) and S(q-1))........................... 108

3.3.4 Tracking (Computation of T(q-1))............................................... 113

3.3.5 Pole Placement: Examples.......................................................... 116

3.4 Tracking and Regulation with Independent Objectives ......................... 117

3.4.1 Structure .................................................................................... 120

3.4.2 Regulation (Computation of R(q-1) and S(q-1))........................... 121

3.4.3 Tracking (Computation of T(q-1))............................................... 124

3.4.4 Tracking and Regulation with Independent Objectives:

Examples .................................................................................... 125

3.5 Internal Model Control (Tracking and Regulation) ............................... 129

3.5.1 Regulation .................................................................................. 129

3.5.2 Tracking...................................................................................... 131