Control Systems Engineering
`
`William J. Palm III
`University of Rhode Island
`
`New York
`
`Chichester
`
`Brisbane
`
`John Wiley & Sons
`Toronto
`Singapore
`
`I
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`l
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`BMW1082
`Page 1 of 5
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`Copyright© 1986 by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Canada.
`
`Reproduction or translation of any part of this work beyond that permitted by Sections 107 and 108
`of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
`Requests for permission or further information should be addressed 10 the Permissions Dcpartmen1,
`John Wiley & Sons.
`
`Library of Con,:ress Catalo,:in,:-in-Pub/ication Data
`
`Palm, William J. (William John), 1944-
`Control systems engineering.
`
`Bibliography: p.
`Includes index.
`I. Automatic control. 2. Control theory.
`I. Title
`TJ213.P225 1986 629.8 85-26590
`ISBN 0-471-81086-X
`
`Printed in the United States of America
`
`10 9 8 7 6 5 4 3 2 I
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`Pneumat;c
`pressure
`
`FIGURE 6.25 Pneumatic flow control
`valve.
`
`6.5 CONTROL LAWS 335
`
`can be used to drive a final control clement
`like the pneumatic actuating valve shown
`in Figure 6.25. The pneumatic pressure
`acts on the upper side of the diaphragm
`and is opposed by the return spring.
`The nozzle flapper is a force distance
`type of actuator, because its input is a
`displacement and its output a force (pres(cid:173)
`sure). The other type of pneumatic actuator
`is the force balance type, in which both the
`input and output are pressures. The pres(cid:173)
`sures arc made to act across diaphragms
`whose resulting motion activates a pneu(cid:173)
`matic relay. Details can be found in the
`specialized literature (e.g., Reference 7).
`
`6.5 CONTROL LAWS
`The control logic elements arc designed to act on the actuating (error) signal to
`produce the control signal. The algorithm that is physically implemented for this
`purpose is the control fall' or control act ion. A non zero error signal results from either
`a change in command or a disturbance. The general function of the controller is to
`keep the controlled variable near its desired value when these occur. More specifically,
`the control objectives might be stated as follows:
`
`1. Minimize the steady-state error.
`2. Minimize the settling time.
`J\.chic,e other transient specifications, such as minimizing the maximum
`3.
`overshoot .
`
`ln practice, the design spccification:s for a controller are more detailed. For example,
`the bandwidth might also be specified along with a safety margin for stability. We
`never know the numerical values of the system's parameters with true certainty, and
`some controller designs can be more sensitive to such parameter uncertainties than
`other designs. So a parameter sensitivity specification might also be included. We will
`return to this topic later, but for now. a general understanding of control objectives is
`sufficient for our purpose.
`The following two control laws form the basis of many control systems.
`
`Two-Position Control
`
`Two-position control is the most familiar type perhaps because of its use in home
`thermostats. The control output takes on one of two values. With the on- off
`controller, the controller output is either on or off (fully open or fully closed). Such is
`the case with the thermostat furnace system. The controller output is determined by
`
`X -
`
`. Flapper
`I
`_...._--'--l.,.__ d 1sptacement
`O.O 1
`at noute
`( inches)
`lb l
`
`:an-istic curve.
`
`"lhe back pressu re curve.
`er displacement from the
`ainaJ condition. Then the
`
`(6.4-10)
`
`:-r. From the geometry of
`
`(6.4- 11)
`
`(6.4-12)
`
`• :mg region, the nozzle
`cssurc is well below the
`f :nore output pressure is
`catic relay or amplifier
`i:::re 6.24 illustrates this
`IL~ pressure increases
`r ;he supply line, and th:
`1pproaches atmospheric
`iack pressure decreases,
`f the atmospheric bleed,
`ressure approaches the
`fhe relay is said to be
`a use an increase in back
`a decrease in output.
`,ressure from the relay
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`336
`
`FEEDBACK CONTROL SYSTEMS
`
`I
`
`___ __. ____ __, ___ ,,
`
`Neutral zone
`
`FIG URE 6.26 Transfer characteris tics
`of the on- off controller. T he actuating
`error is e = r - c, where r = set point,
`c = controlled variable, and /=control
`s ignal.
`
`the magnitude of the error signal. The
`switching diagram for the on- off con(cid:173)
`troller with hysteresis is shown 111 Figure
`6.26.
`An example of an application of an
`on- olTcontroller to a liquid level system is
`shown in Figure 6.27a. The time response
`shown in Figure 6.27h with a solid line is
`for an ideal system in which the control
`valve acts instantaneously. The controlled
`variable cycles with an amplitude that
`depends on the width of the neutral zone
`or gap. This zone is provided to prevent
`frequent on off switching, or chaueri11g,
`which can shorten the life of the device.
`The cycli ng frequency also depends on the time constant of the controlled process and
`the magnitude of the control signal.
`T n a real system, as opposed to ideal, the sensor and control valve will not
`respond instantaneously, but have their own time constants. The valve will not close
`at the instant the height reaches the desired level. There will be some delay during
`which flow continues into the tank. The result is shown by the dotted line in
`Figure 6.27b. The opposite occurs when the valve is turned on. This unwanted effect
`can be reduced by decreasing the neutral zone. but the cycling frequency increases if
`this is done.
`The overshoot and undershoot in on- off control will be acceptable only when the
`time constant of the process is large compared to the time lag of the control elements.
`This lag is related to the time constants of the elements as well as to their distance
`from the plant. rf the control valve in Figure 6.27a is far upstream from the tank, a
`significant lag can exist between the time of control action and its effect on the plant.
`Another source of time lag is the capacitance of the controller itself. for example, if the
`heater capacitance in the temperature controller of Section 6.2 is appreciable, the
`heater will con tinue 10 deliver energy to the oven even after it has been turned off.
`An example close to home demonstrates how the capacitance of the plant affects
`
`h
`
`~T
`.c n
`u"'
`,: a,
`Jj_
`
`Solenoid
`
`Toggle
`switch
`
`115 volts
`
`Supply
`
`(a)
`
`curve
`
`(b )
`
`11mc
`
`FIGURE 6.27
`
`(a) Liquid level control with on-off action. (h) Time response.
`
`the suitability of on-off coc:.=a
`obviously be unsuitable. ..
`capacitance is greater.
`Another type of two-p,...'1!
`ing diagram is shown in fi!
`control by the fact that the o
`A motor with constant ton
`bang- bang device. Because
`model would include a dead
`the controller output is zero
`
`Proportional Control
`
`Two-position control is acce
`arc not too severe. Tn the hoo
`hardly detectable by the occ
`situations require finer conu
`Consider the tank sym
`controller, we might try setti
`balances the system at the ct,
`this setting in proportion to
`proportional control, the alJ
`proportional to the error. R,
`are drawn in terms of the
`Applying this convention t
`proportional control is desc
`
`where F(s) is the deviation ir
`total valve displacement is )
`
`f
`
`(a )
`
`F IGU RE 6.28 Trans fer chant
`a dead 7,0ne. The control signal
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`je of the error signal. The
`igram for the on- ofT con(cid:173)
`ysteresis is shown in Figure
`
`iple of an application of an
`lier to a liquid level system is
`ire 6.27a. The time response
`ire 6.27b with a solid line is
`ystcm in which the control
`in taneously. The cont rolled
`·s with an ampl itude that
`e width of the neutral zone
`o ne is provided to prevent
`ff switching, or chattering,
`1rten the life of the device.
`the controlled process and
`
`nd control valve will not
`:s. The valve will not close
`;viii be some delay during
`rn by the dotted line in
`I on. This unwanted etTect
`ling frequency increases if
`
`acceptable only when the
`1g of the control elements.
`; well as to their distance
`pstream from the tank, a
`1~d its effect on the plant.
`r ttself. For example. if the
`•n. 6.2 is appreciable, the
`r it has been turned off.
`itance of the plant affects
`
`Overshoot
`Ideal
`
`/
`
`·-\~7\---
`~-7~~
`=
`
`Kymg' , / Undershoot
`
`time
`
`, )
`
`e response.
`
`6.5 CONTR OL LAWS 337
`
`the suitabil ity of on-off control. On- off control of the hot water valve in a shower will
`obviously be unsuitable. but it is acceptable for a bath, because the thermal
`capacitance is greater.
`Another type of two-position control is the bang- ban}{ controller whose switch-
`ing diagram is shown in Figure 6.28a. This controller is distinguished from on- off
`control by the fact that the direction or sign of the control signal can have two values.
`A motor with constant torque that can reverse quickly might be modeled as a
`bang- bang device. Because such perfect switching is impossible, a more accurate
`model would include a dead zone (figure 6.28b). When the error is within the zone,
`the controller output is 7ero.
`
`Proportional Control
`Two-position control is acceptable for many applications in which the requirements
`arc not too severe. ln the home heat ing application, the typical 2°F temperature gap is
`hardly detectable by the occupants. Thus, the system is acceptable. However, many
`situations require finer control.
`Consider the tank system shown in Figure 6.27a. To replace the two-position
`controller, we might try setting the control valve manually to achieve a flow rate that
`balances the system at the desired level. We might then add a contro11er that adjusts
`this setting in proportion to the deviation of the level from the desired value. This is
`proportional control. the algorithm in which the change in the control signal is
`proportional to the error. Recall the convention that block diagrams for controllers
`arc drawn in terms of the deviations from a zero-error equilibrium condit ion.
`Applying this convention to the general terminology in Figure 6.7, we see that
`proportional control is described by
`F(s) = K vE(s)
`
`(6.5-1)
`
`where F(s) is the deviation in the control signal and KP is the proportional gain. 1f the
`total valve displacement is y(r) and the manually created displacement is x, then
`
`f
`
`l,11
`
`f
`
`(bl
`
`FIGURE 6.28 Transfer characteristics. (a) Ideal bang-bang control. (b) Bang-bang control with
`a dead ione. The control signal is/; the error signal is e.
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