a
`
`ROBERTO SAUCEDO
`Manager, Command and Control Systems
`International Business Machines, Inc.
`and
`University of California at Los Angeles
`
`EARL E. SCHIRING
`Staff Engineer
`Logicon, Inc.
`and
`University of California at Los Angeles
`
`INTRODUCTION
`TO CONTINUOUS
`AND DIGITAL
`CONTROL
`SYSTEMS
`
`R
`
`THE MACMILLAN COMPANY
`COLLIER· MACMILLAN LIMITED. London
`
`Micro Motion 1029
`
`1
`
`

`
`\
`
`ti
`
`© Copyright, The Macmillan Company, 1968
`All rights reserved. No part of this book may be repro(cid:173)
`duced or transmitted in any form or by any means,
`electronic or mechanical, including photocopying, re(cid:173)
`cording or by any· information storage and retrieval
`system, without permission in writing from the Publisher.
`
`PRINTING 3456789 YEAR
`
`3456789
`
`Library of Congress catalog card number: 68-12075
`
`THE MACMILLAN COMPANY, NEW YORK
`COLLIER-MACMILLAN CANADA, LTD., TORONTO, ONTARIO
`
`Printed in the United States of America
`
`2
`
`

`
`~pts [Ch.l
`
`or extended
`nd noncon(cid:173)
`. production
`tic controls.
`
`n is not in(cid:173)
`;ystem is an
`'er, with his
`arrive at the
`llUman pilot
`~ading at all
`i the actual
`
`nted by the
`sensed and
`
`ller uses an
`ystem being
`system, the
`lfe 1-2. The
`nervous and
`;raft control
`tis change of
`n open-loop
`) the ground
`e magnitude
`
`~
`
`path
`
`Ie end of each
`
`§1.2] Open-Loop and Closed-Loop Systems
`
`3
`
`Reference
`input
`
`: Comparator
`
`Disturbances
`
`Output
`
`Output
`< - - - - - - - -1 measuring 1+ - - - - - - - - - '
`device
`
`FIGURE 1-3. Block diagram of closed-loop system.
`
`and direction of the side winds, the accuracy of the compass instrument, the
`engine thrust magnitude, and the accuracy of the controller all contribute to
`errors between the desired flight path and the actual one.
`A closed-loop cont;rol system is one in which the output quantity is fed back
`for comparison with the input, and the difference, or error quantity, is then
`applied to the amplifier, as shown in Figure 1-3. In such a feedback system, if
`the output is not the desired one, an error is sensed by the comparator and the
`control system will act in a manner to reduce the error to zero, or to maintain
`the output equal to the desired input. Thus the feedback system has the ability
`to correct for load disturbances and inaccuracies in the controller. If in the
`aircraft navigation example the pilot can see the ground, and if he is familiar
`with many landmarks along his flight, then he has the capability of correcting
`his flight-path errors by this visual feedback, as shown in Figure 1-4.
`In general, open-loop control systems are characterized by moderate accuracy,
`high sensitivity to environmental conditions, and slow response. The advantages
`of open-loop control are simplicity and low cost. In contrast, closed-loop or
`feedback control systems are characterized by high accuracy, rapid response, and
`relative independence of environmental factors.
`
`Desired
`flight path
`
`Disturbances
`
`Actual
`: Aircraft 1-....--'-":.:..=.::.:...... ....
`flight path
`
`FIGURE 1-4. Closed-loop aircraft navigation system.
`
`1.3 NOMENCLATURE
`The functional block diagram of a closed-loop system in Figure 1-5 shows the
`nomenclature that is in common use. The command input is transformed by the
`reference input element, or reference selector, into a signal that is calibrated in
`terms of the output. This signal is called the reference input. The actuating signal
`is applied to the feedforward elements, which include the amplifier, controller,
`
`3
`
`

`
`§lA] ContI
`
`raise the tell
`preset value
`tinuous coni
`and control
`type system
`
`1.4 CON
`In this se(
`It is recogni
`and of the 1
`necessary d,
`nature of c(
`practical aJ:
`exerted inel
`rotational),
`Consider
`The inlet v~
`set at a con:
`of the inlet'
`the float ah
`to raise the
`similar to t
`Another
`output ofa
`electric po,
`pressurized
`inherentne
`reactor dec
`as a result (
`as the elect
`the reactor
`power leve
`for reactor
`
`4
`
`Control Systems Concepts
`
`[Cb. 1
`
`and plant of Figure 1-1. The output of these feedforward elements is the control
`system output. This output is sensed by the feedback element and is fed back to
`the comparator as the feedback signal, where it is compared with the reference
`input to form the actuating error signal. In addition, an output element may exist
`such as an output indicator.(2)
`Common terms for control systems that are widely used in the literature are
`servomechanism or servo, and regulator. A servomechanism is a control system
`in which the output is the position, velocity, or acceleration of a mechanical
`shaft, such as that corresponding to a motor. The term regulator usually applies
`to a control system wherein the output quantity is desired to be maintained
`constant despite unwanted disturbances to the system. An electric generator is
`an example of a regulator, in that it is desired to maintain constant the generator
`voltage regardless of the load fluctuations on the generator.
`If control systems are classified according to the nature of the signals at the
`various points of the feedback loop, then the three kinds of systems are con(cid:173)
`tinuous, sampled-data or pulsed-data systems,t and relay-type systems. A
`continuous control system is one in which all the signals in the control loop may
`be considered as continuous functions of time, and an example of a continuous
`system is the regulation of speed of a motor using tachometric feedback. In such
`a system, the tachometer feeds back a signal proportional to the motor shaft
`speed to be compared with the desired speed.
`A sampled- or pulsed-data system is one in which one or more signals in the
`loop are in the form of a pulse train or a numerical code, such as the output of
`a digital computer. An example of a sampled-data system is that of a radar
`tracking system. The signals sent and received are in pulse form, and are used
`to position an antenna to track some desired object.
`A system in which a relay type of element is present in an otherwise continuous
`type of system is called a relay, or on-off, system. The thermostat home-heating
`system is an example of an on-off system. When the temperature at the thermo(cid:173)
`stat is below a preset value, the full force of the furnace is suddenly applied to
`
`Command. 1 Relere~~ce
`input
`i selector
`
`Reference +
`input
`
`Actuatin
`error
`
`Output
`
`Feedback
`signal
`
`FIGURE 1-5. Functional block diagram 0/ closed-loop system.
`
`t The terms" sampled-data,"" digital," and" discrete" are used interchangeably in many
`instances throughout the text, although fine distinctions are made by many researchers.
`The term" sampled-data" is the most general term, and includes all systems which contain
`signals in pulsed form. The term "digital" implies that a digital component such as a
`digital sensor or a digital computer is used in the system.
`
`4
`
`

`
`§1.4] Control
`
`Numerous c
`occur naturall:
`system compo
`human body r
`more general 51
`The oculomot(
`discussed prev
`object, such as
`the retina fron
`muscles are COl
`in the center 0
`In ecology a
`of Canada. In
`territory under
`constitute the I
`the lynx popul
`population wiU
`rabbits. To co
`overpopulatior
`complete. (9)
`Another nat
`1-12. This pan
`(10), which is
`income is the Sl
`these quantitie:
`
`8
`
`Control Systems Concepts
`
`[Ch. 1
`
`8
`
`FIGURE 1-10. Sampled-data/ire-control system.
`
`or missile launcher is commanded for the gun control system. The error signal
`in pulsed form is converted to a continuous signal and applied to the gunmount
`drive motor, and the gun is positioned accordingly. The angle of the gun is
`sensed and convetted to digital form and fed back for the error comparison.
`While the diagram shows a simple single-axis representation, two separate
`channels of azimuth and elevation are positioned simultaneously in the actual
`equipment. In addition, if the fire-control system is on board a naval vessel,
`the ship is subjected to sea motion that must also be accounted for in the fire(cid:173)
`control computer. (6)
`Another example ofa sampled-data system is the control of an electro(cid:173)
`optical telescope to effect automatic tracking of a star. The telescope must be
`continuously repositioned if the star is to remain in the field of view of the tele(cid:173)
`scope, owing primarily to the rotation of the earth. This system is illustrated in
`Figure 1-11. A television camera is mounted on the telescope and an image is
`generated on the screen. The set is aligned such that the center of the raster is
`coincident with the optical axis of the telescope. Additional tracker electronics
`determine the displacement of the image with respect to the center of the raster
`and furnish a signal proportional to the pointing error of the telescope. Since
`television is by nature a repetitive scanning device, it is also by nature a sampled(cid:173)
`data component. The pointing error is amplified and shaped by stabilization
`electronics, and is then applied to the servomotor, which positions the telescope
`in a direction to decrease the pointing error. While a standard television applied
`to this problem has a sampling rate so high in comparison to earth rate that the
`system appears to operate as a continuous control system, the same system
`applied to automatic tracking of a high speed missile or aircraft could have
`system time constants of nearly the order of magnitude of the television sampling
`rate, so that a sampled-data analysis would be required.(7)
`
`Star
`Line of sight
`
`'--~r----'
`
`r - - - - - - , Optical
`axis
`
`FIGURE 1-11. Electro-optical telescope tracker.
`
`5
`
`

`
`CHAPTER 8
`
`ROOT lOCUS
`
`One must learn by doing the thing: for though you think you know it, you
`have no certainty until you try.
`SOPHOCLES
`
`INTRODUCTION
`8.1
`This chapter introduces the root locus method, a graphical means for deter(cid:173)
`mining the migration of a system's closed-loop poles as a function of the loop
`gain. Obviously, the closed-loop poles are the roots (or the zeros) of the charac(cid:173)
`teristic equation, and the loci of these roots in a complex plane as the loop gain
`parameter takes on different values constitute the migration mentioned. If
`continuous systems are involved, the complex plane is the s plane; if the system
`is discrete, it is the z plane. The theory and mechanics of the root locus method
`are not affected by the resident complex domain; only the interpretation of the
`effect on the system response is altered.
`The root locus method provides a correlation between the feedback system's
`open-loop poles and zeros and the system's stability characteristics, its frequency
`and transient respons~d its steady-state behavior. It complements the other
`classical techniques very m~ely and is extremely useful to the system analyst. As
`with any technique, its use will be dictated by the particulars of the situation and
`personal preference.
`
`8.2 ROOT LOCUS METHOD FOR CONTINUOUS SYSTEMS
`It has been mentioned that the root locus method provides a means for chart(cid:173)
`ing the migration of the closed-loop poles for a feedback control system as the
`loop gain varies. The starting point is the open-loop pole-zero configuration for
`a feedback control system as depicted in Figure 8-1, where attention is restricted
`
`R(s) +
`
`CIs)
`
`' - - -.. H(s) k - - - '
`
`FIGURE 8-1. Feedback control system.
`
`324
`
`§8.2:
`
`to s
`fune
`
`whe
`and
`fune
`
`Inse
`
`This
`feed
`man
`give
`
`or
`
`The:
`eval
`dete
`thro
`T
`(8.5:
`in tl
`
`To~
`
`Poil
`At 1
`s pi
`sho]
`B
`wh2
`trail
`
`6
`
`

`
`374
`
`Root Locus
`
`[Ch. 8
`
`8.6 ROOT LOCUS METHOD FOR NEGATIVE LOOP GAIN
`It will be recalled that for a feedback control system as depicted in Figure 8-1,
`the loop gain K was considered always positive. It is possible for the loop gain
`to be negative, in which case the system is depicted as in Figure 8-40. In this
`figure the feedback signal is positively fed back and the loop gain is once again
`assumed positive. Feedback systems of this type are called regenerative or
`positive feedback sy~tems and most often appear as inner loops in a multiloop
`system.
`
`R(s) +
`+
`
`CIs)
`
`FIGURE 8-40. Regenerative feedback control system.
`
`The closed-loop transfer function for the system portrayed in Figure 8-40 is
`
`KG(s}
`KG(s)H(s)
`
`(8.142)
`
`where G(s) and H(s) are rational algebraic functions. The characteristic equation
`is
`
`1 - KG(s)H(s) = 0
`
`(8.143)
`
`The root locus method, just as before, provides a graphical means for evaluating
`the roots of (8.143) as the loop gain varies from 0 to 00. The difference between
`(8.143) and the characteristic equation for negative feedback systems is in the
`sign. As a consequence, the root locus for positive feedback systems consists of
`points in the s plane which satisfy
`
`r
`r
`
`KIG(s)H(s)1
`
`1
`
`/G(s)H(s) = k· 360°
`
`k any integer
`
`(8.144)
`
`(8.145)
`
`As a matter of construction; points are found in the s plane which satisfy
`(8.145) and the resulting locus is then calibrated by use of (8.144).
`
`Construction Rules
`The construction of the root loci for positive feedback systems is similar to
`that for negative feedback systems. Hence only these construction rules which
`differ appreciably will be stated and these without proof. The proofs will be
`left to the reader. The construction rules involved are:
`
`5. The root loci are asymptotic to straight lines, for large values of s, with
`angles given by
`
`k = 0,1, ... , n - m
`
`1
`
`t I
`
`7
`
`

`
`400
`
`30
`
`I
`
`l5 20
`13
`-l2
`'""0
`
`10
`
`0
`
`0..
`.Q
`"10
`.;
`-0
`:>±:
`10
`c:
`ao
`'" :z; -20
`-30
`
`0
`
`-30
`
`-60
`
`-90
`
`120
`
`0>
`
`"10
`
`'"
`~ 0
`"'­
`-;;'0
`c:
`'"
`" , ­
`"'~
`'" 0
`'" 0..
`..c::. a..
`
`Frequency Response
`
`[eh. 9
`
`•
`
`I
`
`1 1 I I ~l~ ~: ci
`
`I
`~:o~
`T«~: O.IQ
`1~:'0.20
`~J.-:: ~~:O'YO
`~ : 0.40
`!
`[-I­ ~~-1 ~I: 0'1
`57
`I ~:1;~ ~ I I
`~
`
`~: 0.707/ i
`
`Asymptote
`
`,'"
`
`t =0.10
`~ =0.20
`
`- 30
`
`i ­ 20
`
`I- 10 ~
`o u
`.E
`e
`'" N
`
`.10
`I
`o
`11
`o
`I :
`o
`+'0
`
`+ 30
`
`+60
`
`+90
`
`+120
`
`5..­
`0 '"..­
`2.,
`
`N
`
`+150
`
`!
`.1 !
`
`'.' ., ..
`
`:
`
`",'(1
`
`, I>~, ~
`
`/",.
`
`§9.7]
`
`1 E
`14
`12
`lC
`E
`E
`4
`
`.0
`"0
`C
`0
`c:;
`~
`0
`0
`
`~ C
`0 a..
`-;:
`
`-4
`-6
`-E
`
`FJ
`
`For ~
`stitutir
`as
`
`A plot
`0.001 :
`occurs
`The
`
`For W
`when,
`functi(
`values
`pair oj
`widely
`factor.
`
`Exa,
`respon
`transfe
`angle c
`
`-150
`-180 L-.lJ-"-lUJ11Ull-l::l:t£i~~§~ + 180
`
`0.1
`
`1.0
`
`10
`
`FIGURE 9--16.
`
`(iW)2] ',H
`2~w
`Bode ploto! 1 + -:;;- + ';'n
`.•
`[
`
`magnitude correction curves shown in Figure 9-17. In both Figures 9-16 and
`9-17, the scale on the right is to be used when the quadratic factor represents a
`zero term.
`In many applications, particularly those involving transfer functions of
`aircraft and missiles, the natural damping ratio of the airframe is normally
`< 0.1. The peak value of the db magnitude for any value of damping ratio may
`be derived by differentiation of (9.75). Assuming that i = -1, or the factor is a
`pair of complex conjugate poles, the magnitude is
`1
`
`1
`
`(9.78)
`
`If this result is differentiated and set equal to zero, the ratio of w/wn where the
`peak magnitude occurs is found to be
`
`~ < 0.707
`
`(9.79)
`
`8
`
`8