Table Of ContentInternational Series on
SYSTEMS AND CONTROL, Volume 2
Editorial Board
Professor M G SINGH, UMIST, Manchester, England (Co-ordinating Editor)
Professor H AKASHI, University of Kyoto, Japan
Professor Y C HO, Harvard University, USA
Academician B PETROV, Moscow Aviation Institute, USSR
Other Titles of Interest
ANAND
Introduction to Control Systems
HAASE
Real-Time Programming 1980
HAIMES & KINDLER
Water and Related Land Resource Systems
ISERMANN
System Identification Tutorials
ISERMANN & KALTENECKER
Digital Computer Applications to Process Control
JANSSEN, PAU & STRASZAK
Dynamic Modelling and Control of National Economies
LESKIEWICZ & ZAREMBA
Pneumatic and Hydraulic Components and Instruments in Automatic Control
MUNDAY
Automatic Control in Space
NAJIM & ABDEL-FATTAH
Systems Approach for Development 1980
PATEL & MUNRO
Multivariable Systems Theory and Design
RAUCH
Control Applications of Nonlinear Programming
REMBOLD
Information - Control Problems in Manufacturing Technology 1979
SINGH et al.
Applied Industrial Control: An Introduction
SINGH & TITLI
Systems: Decomposition, Optimisation and Control
SUBRAMANYAM
Computer Applications in Large Scale Power Systems
TITLI & SINGH
Large Scale Systems: Theory and Applications
VAN CAUWENBERGHE
Instrumentation and Automation in the Paper, Rubber, Plastics and
Polymerisation Industries
Pergamon Related Journal (Free specimen copy available on requestj
AUTOMATICA
CLASSICAL AND
MODERN CONTROL
WITH WORKED EXAMPLES
by
JEAN-PIERRE ELLOY
and
JEAN-MARIE PIASCO
Maitres-Assistants at the Ecole Nationale
Suporieure de Mocanique, Nantes, France
Translated from the original French
by Mrs. Barbara Beeby
PERGAMON PRESS
OXFORD · NEW YORK · TORONTO · SYDNEY · PARIS · FRANKFURT
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FEDERAL REPUBLIC Pergamon Press GmbH, 6242 Kronberg-Taunus,
OF GERMANY Hammerweg 6, Federal Republic of Germany
Copyright ©1981 Pergamon Press Ltd.
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted
in any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the
publisher
First edition 1981
British Library Cataloguing in Publication Data
Elloy, Jean-Pierre
Classical and modern control with worked examples.
-(International series on systems and control;
v.2)
1. Control theory
I. Title II. Piasco, Jean-Marie
629.8'312 TJ213
ISBN 0-08-026745-9 (Hardcover)
ISBN 0-08-026746-7 (Flexicover)
LOC 81-81 059
In order to make this volume available as economically
and as rapidly as possible the authors' typescripts have
been reproduced in their original forms. This method un
fortunately has its typographical limitations but it is
hoped that they in no way distract the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
To Jacqueline and Xavier
ff. P. ELLOY)
To Claudine, Laurence and Guillaume
(J. M. PIASCOJ
ACKNOWLEDGEMENTS
This book is the result of ten years teaching in the Automatic
Control laboratory of the Ecole Nationale de Mecanique in Nantes
(France). It contains problems in automatic control which were put to
the students by the authors and their teacher colleagues, R.Mezencev,
M. Hamy, B. Cheneveaux, Ph. de Larminat and J.L. Jeanneau, to whom
I wish to offer my sincere thanks.
I must particularly emphasize that the book would never have seen
the light of day had it not been for the perspicacity of my friend
and erstwhile colleague Madan G. Singh (UMIST, Manchester, England)
who was able to pick out from the pile of papers cluttering up my
desk the manuscript of these problems and who greatly stimulated us
to put them together in a book. To him I offer the expression of my
friendship and heartfelt gratitude.
I should also like to mention the excellent work of my wife,
Jacqueline Elloy, who took care of the typing and pagination of the
book. I am duty bound to admit that if any errors remain they are of
my own making and certainly not of my wife's ; to imply otherwise
would be to risk causing great arguments between us.
In addition I should like to thank Mrs. Barbara Beeby who trans
lated the original French text into English.
I conclude with the hope that Professor M.G. Singh will one day
find time to look once more at the papers lying around on my desk.
Who knows, perhaps he will find there material for a new book I
J.P. ELLOY, NANTES, December 1980
1
PREFACE
This work is devoted to the problems of analysis and control
of continuous time systems. It contains exercises which increase in
difficulty. When a new concept appears it is fully explained and
then taken up again in a shorter form in the exercises which follow.
The book is aimed essentially at students, technicians and qualified
engineers who wish to acquaint themselves in a practical way with
both the traditional and the modern principles of Automatic Control,
and with their application to industrial processes of all kinds.
Each of the three parts to the book is divided into two chap
ters. The first chapter of each part consists of a course abstract ;
the second chapter contains the exercises relevant to the course in
question.
Thus, in the first part, which is devoted to linear servo-
systems, chapter A defines the basic principles of continuous auto
matic control and the mechanisms which make it possible to control
a system. Chapter 1 then offers, in the form of exercises, the study
of simple systems, of performance of a servo-system, of the stability
precision and compensation of a servo-system.
The second part is given over to non-linear servo-systems.
Chapter B gives the principle of linearisation using the method of
the 1st harmonic and then describes the study in the phase plane of
simple non-linear systems. Chapter 2 offers a set of problems, some
of which are dealt with using both methods and which are essentially
directed towards the study of auto-oscillations due to a non-
linearity.
3
4 PREFACE
The third part introduces representation in the form of equa
tions of state of linear systems. Chapter C describes different
techniques which make it possible to obtain such representations
and their use in the study of the stability of systems and the
calculation of control laws. Chapter 3 applies these techniques to
multi-variable processes.
This book does not claim to deal with all classical and
modern control. It merely hopes to enable the reader to grasp the
basic principles through the study of concrete problems. He can
then turn to more specialised works if he wishes to extend his
knowledge in this field.
Nantes, December 1980
J.P. ELLOY J.M. PIASCO
NOTATION
dx
x
dt
|v| modulus of v
phase shift of v, conventionally expressed in degrees
Arg(v)
R (x) real part of the complex variable x
I (x) imaginary part of the complex variable x
j U2 = - 1)
s Laplace variable
c£, Laplace transform : ,£(x(t)) = X(s)
«£"" inversion of a Laplace function : x(t) =cC (X(s))
db decibels (20 log)
[x(x)]J x(b) - x(a)
xC[a, b] a < x < b
{::! if x > 0
sign x if x < 0
n
ΣΣ α. a- + a- ... + a
i=l 1 12 n
x column matrix (vector)
T
x row matrix (vector)
T
A transpose of the matrix A
identity matrix
det(A)
determinant of the matrix A
|A|
CMCWE-B
CHAPTER A
LINEAR SERVOMECHANISMS
course abstract
A.l STUDY OF MONOVARIABLE LINEAR SYSTEMS
A.1.1. TRANSFER FUNCTION
A monovariable system with input u(t) and output y(t) is said to be
linear if the relationship between u(t) and y(t) is a linear diffe
rential equation with constant coefficients (a. and b.) :
n , n 1 dt oJ m ,.m 1 dt o
Using the Laplace transforms, this equation gives the transfer
function :
V/N bsm+...+b1s+b
XlsL = _jn 1 g = (
U(s) an sn + ... + a.1s + ao
which can also be represented in the form of a block diagram :
U(s) Y(s)
H(s)
The manipulation of these block diagrams leads to simplification
termed "block diagram algebra".
A.1.2 STABILITY OF LINEAR SYSTEMS
Given a linear system with input u(t) and output y(t), it is said to
be stable if y(t) has a limit of zero for t infinite, whatever may be
the initial conditions, when u(t) is zero.
7
8 CLASSICAL AND MODERN CONTROL WITH WORKED EXAMPLES
From this, we deduce the following condition on the transfer function
of the system : a linear system is stable if all the poles of H(s)
have a real negative part.
Note : the Routh mathematical test makes it possible to calculate
the number of zeros of a polynomial with real negative parts.
A.1.3 RESPONSE OF A LINEAR SYSTEM TO A SINUSOIDAL EXCITATION
In the case in which u(t) = U- sin cot, if the system is stable and
in steady state,
y(t) = Y1 sin (ißt + <j>)
γ
With ^ = |H(jü>) | and φ = Arg [H(jw)] = /H(JO))
Y1/U1 is called the gain and φ the phase.
Depending on the applications, we use one of three graphic repre
sentations of the gain and of the phase :
- representation in the complex plane : we draw the curve H(ju))
(Nyquist curve) when ω varies from zero to infinity.
- representation in the Bode diagram : we draw two curves, the
gain in decibels and the phase in terms of ω (the ω axis being
graduated on a logarithmic scale).
- representation in the Black diagram : we plot the gain vertically
in decibels and the phase horizontally. The curve obtained when ω
varies from zero to infinity is called the Black curve.
A.1.4 1ST DEGREE SYSTEM
The transfer function of a 1st degree system is written as :
γ (s) K
\jis\ = H(s) = ^ + sT in which K is the static gain and T
the time constant.
The impulse response h(t) is the original of H(s):
h(» '^[iHhi] =M
We call "step response" the response to a step input of unit ampli
tude.
For u(t) unit step U(s) =J?[u(t)] = ~ whence Y(s) = 1 ,K m - ^
>*^L J s 1 + sT s
and y(t) = ο£_1 [Y(S)] = K (1-e"^)
The time which is necessary at y(t) to pass from zero to 95 % of
the final value is called "response time" tD : t_ - 3 T ; the "rise
R R
time" (to pass from 10 % to 90 % of the final value)is t - 2.2 T.
m
