The Dynamics of Automatic Control Systems

752 Pages·1961·40.146 MB·English

by  E. P. Popov and A. D. Booth (Auth.)

THE DYNAMICS OF
AUTOMATIC CONTROL SYSTEMS
by
E. P. POPOV
Translated from the Russian
PERGAMON PRESS
OXFORD • LONDON. NEW YORK • PARIS
1962
PERGAMON PEESS LTD.
Eeadington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W.l
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Copyright
©
1961
Pergamon Press Ltd.
Library of Congress Card No. 60-16485
Printed in Poland to the order of Panstwowe Wydawnictwo Naukowe
by Drukarnia Uniwersytetu Jagiellonskiego, Krakdw
ENGLISH EDITORS INTRODUCTION
PROFESSOR POPOV'S book on the theory of servomechanisms and
control systems will be of interest to English readers in showing the
meticulous detail into which Eussian engineers and students are
led when discussing these subjects.
A few standard and useful examples are taken and form the basis
for discussion at all levels, from the most elementary to the most
complete and advanced, so that a sense of continuity is preserved
throughout the volume.
The translation preserves much of the Eussian sentence structure
and the editor did not find it possible to alter this without a more
or less complete rewriting of the translator's manuscript. Neverthe­
less, it is hoped that a readable result has been obtained and
that the spirit of the whole is faithful to that of the original.
A. D. B.
FOREWORD
THE WIDE study of the theory of automatic control in our insti­
tutions of higher education is a very recent development. Many
engineers therefore have come to study it only when already engaged
in practical work. The present book has the aim of assisting these
broad circles of engineers and students to acquaint themselves in
the most accessible form with the foundations of the theory of
automatic control, in which the main role is played by the dynam­
ics of control systems.
Following the aim of the most accessible development, the author
has consciously tried to reduce to a minimum the use of operational
calculus and the theory of functions of the complex variable and
everywhere, where possible, to confine himself to the use of the
symbolic operational method as a means of simplifying the notation
and manipulation of differential equations.
The book presents those theoretical methods of analysis and
synthesis of automatic control systems common to systems of various
physical natures and designs. The concrete examples presented in
the book, therefore, by no means pretend to reproduce the designs
of contemporary automatic control systems. They will be only the
simplest functional circuits serving to illustrate the principal ideas
in the construction of automatic control systems and the application
of the theoretical methods developed here. By analogy, the reader
may then apply these ideas and methods to the concrete automatic
control systems of interest to him.
Attempting to describe in detail the fundamental results of the
theory of automatic control, the author has been forced to limit
himself to brief remarks on a number of special questions. Statistical
methods and questions of representation of automatic control systems
have been completely neglected here. Each of these important
fields is so specialised that it may constitute the subject of separate
books.
The author expresses his appreciation to Comrades Ya. Z. Tsypkin,
O. K. Sobolev and I. V. Korol'kov for valuable remarks when re­
viewing the manuscript.
CHAPTER I
FORMS OF AUTOMATIC CONTROL SYSTEMS
1. The concept of closed automatic systems
There exist very many forms of automatic systems performing
various functions in the control of the most varied physical pro­
cesses in all branches of engineering. These systems consist of
mechanical, electrical and other devices of widely different designs,
assembled in an overall complex of mutually interacting circuits.
As examples of automatic systems we may list:
(a) automatic light switches including a photocell which reacts
to the intensity of daylight and a special device for turning on the
light, operating on a definite signal from this photocell;
(b) a machine for vending given objects (e.g. tickets, chocolates)
when a given combination of coins is fed into it;
(c) automatic machines, automatic production lines and automatic
factory departments;
(d) remote-control systems in which a defined combination of
heavy or complicated operations in the controlled object are carried
out upon depression of a button or an easy rotation of a handle
on the control panel;
(e) automatic regulation of motor speed, maintaining constant
angular velocity of the motor independently of the external load
(analogously, temperature, pressure, voltage and frequency regu­
lators, etc.);
(f) automatic pilots, maintaining a definite course and altitude
of an aircraft without intervention by the pilot;
(g) servomechanisms in which an arbitrary variation in time of
some quantity applied to the input is exactly copied at the output
to a defined precision;
(h) a tracking system, in which the barrel of an anti-aircraft gun
is automatically trained on a flying aircraft;
(i) a computer carrying out a defined mathematical operation
(differentiation, integration, the solution of equations, etc.);
(j) measuring devices operating on the compensation principle;
(k) synchronous remote transmitters, etc.
2 3
4 The Dynamics of Automatic Control Systems
All these and similar automatic systems may be divided into two
main classes:
(1) automatic machines carrying out definite types of single or
repeated operations; among these are, for example, lighting switches,
ticket vendors, automatic machines, machine guns, velocity switching
mechanisms, etc.;
(2) automatic systems which over a fairly long period of time
vary some physical quantity in a necessary manner (or maintain
it constant) (the coordinates of a moving object, velocity, voltage,
frequency, temperature, pressure, acoustic level, etc.) in some con­
trolled process. Among these are automatic controls, servo-
mechanisms, automatic pilots, certain computing devices, certain
measuring equipment, remote control systems, etc.
In the present book we shall consider only automatic systems
of the second class. These are divided into open and closed loop
automatic systems.
The general structural circuit of an open loop system is shown
in two forms (a and b) in Fig. 1. This is the simplest control system:
semi-automatic when the source of commands is the human being,
ISource Inter* Con.
of medi ate rolled!
Source Inter- [commands device obj ec t|
(a) of [♦-I mediate Ou tpu t (b)
commands device organ Meassuu rree -] fI In ter-
meen t U-j roedia te
de vice I [ device
FIG. 1
and automatic if the source of commands is the variation of certain
external conditions in which the given system operates (temperature
or pressure of the surrounding medium, electric current, illumina­
tion, range of frequency, etc.).
The second structural circuit shown in Fig. 1 differs from the
first in that besides control organs there are also measurement
devices which permit the course of the process in the controlled
object to be observed.
A characteristic of open loop systems is that the functioning of
the system is not directly dependent on the result of its action.
A natural further improvement of automatic systems is the con­
nection of the output (measuring instruments) back to the input
(source of commands) so that the measuring instruments, measuring
a certain quantity characterising a definite process in the control
object should themselves be simultaneously a source of action on
the system, where the magnitude of this action will depend on the
Forms of Automatic Control Systems 5
magnitude by which the measured quantities in the controlled
object deviate from their required values.
Closed loop automatic systems thus arise. It is easy to conceive
that in the closed loop automatic system there is a complete inter­
action of operation of all circuits with each other. The courses of
all processes in a closed loop system are radically different from
processes in open loop systems. The closed loop system reacts com­
pletely differently to external perturbing forces. Various valuable
properties of closed loop automatic systems make them irreplaceable
in all cases where precise and high-speed automatic systems are
required for control, measurement or for carrying out mathematical
calculations. Dynamic calculations take on special importance in
the design of any closed loop automatic system.
Closed loop automatic systems exist in engineering in the form
of automatic regulators, servo mechanisms, computers, compen­
sation-type measuring systems, automatic pilots, stabilisation
systems, remote control systems, etc.
All these forms of closed loop automatic systems may be reduced
to the single general circuit represented in Fig. 2. If this represents
Measurement of control action effect
—1
Con t rolled Con t ro1
(regu1 a ted) s ys tern
obj ec t ( regu 1 ator!
L J
Control action
FIG. 2
a servo mechanism the external driving force y(t) may be arbitrary
in time and it is required that the controlled object should repeat
as exactly as possible this arbitrarily-given time variation in the
form of a mechanical motion or displacement or in the form of
variation of an arbitrary electrical or other physical quantity. At
the same time it should be possible to eliminate as far as possible
interference of all external perturbing forces f(t) on the controlled
object (and possibly on the control system itself).
If it represents an automatic regulation system the role of the
control system (Fig. 2) is played by a regulator, the role of the external
driving force y(t) by the setting of the regulator or programme
control. In this case it is always assumed that substantial external
perturbing forces f{t) act on the regulated object (for example,
variation of load). The automatic regulation system should operate
so that with actual external perturbing forces, regardless of their
magnitude, the output should be as nearly as possible that to which
6 Thy Dynamics of Automatic Control Systems
the regulator is adjusted, or else this quantity should vary according
to a definite programme given by the programme control.
All other forms of closed automatic systems (automatic controls,
automatic pilots, computers, compensation-type instruments, stabili­
sation systems, remote-control systems, etc.) reduce to these two:
either they are designed on the principle of a servomechanism or
on the principle of the automatic regulation system. We observe
that the difference between these, as is evident from the above,
consists more in function than in the general principle of operation.
Therefore, the methods of dynamic calculation are essentially
the same for automatic regulation systems and for servomechanisms
(and for all closed-loop automatic systems in general).
2. Servomechanisms and control systems
Let us consider the operating principle of closed-loop automatic
systems in greater detail, using examples.
Servomechanisms. The general structural circuit of a servomech­
anism is shown in Fig. 3.
x
Sou ree
of
commands pbjectU.-/ Pertur -\
\f(t) J
FIG. 3
It consists of a forward circuit (from the source to the controlled
object) and feedback from the "output" of the system to its "input".
We also call servomechanisms such automatic systems in which
the output quantity x(t) reproduces the variations of the input
quantity y(t) where the automatic system reacts to the difference x
x
between the output and input quantities. We shall term the quanti­
ties x and y the regulated and desired quantities respectively.
The term "reproduces" employed here loosely denotes the equality
of regulated and desired quantities at each moment of time, i.e. we
are roughly concerned with the equality *
Frequently the regulated quantity x differs from the desired quan­
tity y in scale or in physical character. We then require
co{t) ™h y(t), (2.2)
0
* The sign of approximate equality is used in the formula since a real system
cannot operate absolutely exactly; there is always a certain error (deviation).
Forms of Automatic Control Systems 7
where Tc is a given constant coefficient. The same relates to the
0
compensation-type measurement systems.
If we are considering a computer, operating on the servo principle,
by the above term "reproduction" should be understood the satis­
faction of a definite mathematical relationship, for example, for
differentiating devices we require
x{t) * fc ^ , (2.3)
0
and for integrating
t
' x(t) **Jc f y{t)M. (2.4)
0
o
Consequently, "reproduction" of an input quantity y(t) at the
output of the system x{t) may occur in various forms.
A servomechanism has feedback from output to input (Fig. 3)
which, in essence, serves for measurement of the result of the action
of the system. At the input to the system subtraction occurs:
x = y — x . The device carrying out this subtraction will be termed
x 2
the error detector (Fig. 3). The error magnitude x acts on the in­
x
termediate circuit, and through it on the controlled object. The
system operates so as constantly to reduce to zero the error x and
x
thus to satisfy the required relationship (2.1), (2.2), (2.3) or (2.4)
with arbitrary form of variation of y with time.
Obviously the magnitude x (Fig. 3) applied by the feedback
2
should have the same physical character and the same scale as the
driving quantity y. Therefore, the feedback loop should be able to
transform the regulated quantity x, inversely to that which is carried
out in the forward circuit of the system, namely, for (2.1), (2.2),,
(2.3) and (2.4) there should be
x C dx{t)
— , x = x(t)dt, x = dt
2 2
respectively.
The source of commands for the input device (Fig. 3) may be
either a special device or the variation of the external conditions
in which the system operates.
For remote control of artillery and machine guns, in the circuit
of Fig. 3 they are the controlled objects, and the input device is
a handle on the control panel or sight, turned, for example, by
a human hand according to an arbitrary function y(t). The controlled
object under the action of the intermediate circuits of the servo-
mechanism must reproduce this motion in an appropriate manner
(e.g. the circuit on p. 258 in reference 36).
8 The Dynamics of Automatic Control Systems
For automatic stabilisation of antenna position (on an aircraft
or vessel) the input device is a gyroscope measuring the angle of
deviation of the aircraft or vessel from some given direction. This
angle will be given by a function y(t) (Fig. 3). For the controlled
object (antenna) to retain a constant position in space, its servo-
mechanism must carry out a motion x(t) with respect to the body
of the aircraft which exactly repeats the motion of the aircraft y(t)
measured by the gyroscope.
Contemporary electromechanical, electrical and electronic measure­
ments of non-electrical quantities (distance, velocity, temperature,
volume) are frequently based on the servomechanism principle.
An example is a device for measuring the airspeed of an aircraft,
the block diagram of which is shown in Fig. 4. A Pitot tube, consti­
tuting the input device, establishes a pressure p = y(t) dependent
c Turbopump
Pi
Pi »ot |l-9i0—lD|txi afl fL mearenno« m Pneumatic Ainidrsipceaetdo r
tube meter relay (tacho*
meter)
At r speed
#41
Am p 1 i - ** Elect ric
f ier motor
FIG. 4
on the airspeed of the aircraft. An electric motor at the output of
the servomechanism gives an angular velocity co = x{t) proportional
to the pressure p. The forward circuit of the system carries out-
transformation of pressure into angular velocity of the electric
motor, while the feedback (turbopump) effects the transformation
of the angular velocity into pressure, which must be applied to the
error detector (differential manometer). The scale of the tachometer,
measuring the angular velocity of the electric motor, is calibrated
in airspeed.
As an example of a computing system operating on the servo-
mechanism principle we present the simplest structural scheme of
an electromechanical integrator (Fig. 5). Here the angular velocity
of the electric motor is proportional to an arbitrary voltage V = y{t)
and consequently the angle of rotation 9 = x(t) is proportional
to the integral of the given function y(t). The forward loop (amplifier
and motor) integrates the input quantity. Therefore, the feedback
(tachometer) should differentiate the output quantity of the system
so that the input to the amplifier is a difference between similar
quantities.