`
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`Control Systems Engineering
`
`William J. Palm III
`University of Rhode Island
`
`
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`
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`New York
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`Chichester
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`Brisbane
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`John Wiley 8: Sons
`Toronto
`Singapore
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`Copyright It"
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`I986 by John Wiley & Sons. Inc.
`
`All rights reserved. Published simultaneously Ill Canada.
`
`Reproduction or translation of any part of this work beyond that permitted by Sections 10" and lOX
`ol the l976 United States Copyright Act without the permisston of the copyright owner 1.x unlawful.
`Requests for permission or further information should be addressed to the Permissions Depurltnenl.
`John Wiley & Solis.
`
`Library of (.‘ongrei'x Cataloging-in-I’ublicarinn [)qu
`
`Palm. William J. (William John), 1944
`Control systems engineering.
`
`Bibliography: p.
`Includes index.
`
`1. Automatic control.
`I. Title
`629R 85-26590
`TJZI3.P225 I986
`ISBN 0-471-81086-X
`
`2. Control theory.
`
`Printed in the United States of America
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`l098765432|
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`6.5 CONTROL LAWS
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`335
`
`can be used to drive a final control element
`
`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-
`sure). The other type of pneumatic actuator
`is the force balance type. in which both the
`input and output are pressures. The pres-
`sures are made to act across diaphragms
`whose resulting motion activates a pneu-
`matic relay. Details can be found in the
`specialized literature te.g.. Reference 7).
`
`pneumatic
`
`”(mum
`
`Diaphragm
`
`'4.
`4..
`.‘r:
`'2!
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`r _
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`i
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`'1
`
`Flow—>
`
`3
`
`——->
`
`FIGURE 6.25 Pneumatic flow control
`Vah'c-
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`6.5 CONTROL LAWS
`
`to
`logic elements are designed to act on the actuating (error) signal
`The control
`produce the control signal. The algorithm that is physi 'ally implemented for this
`purpose is the control law or control action. 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:
`
`v'1\'i
`x .\
`t
`‘
`l
`:-
`
`I. Minimize the steady—state error.
`
`2. Minimize the settling time.
`
`3. Achieve other
`overshoot.
`
`transient
`
`specifications.
`
`such as minimizing the maximum
`
`In practice. the design specifications 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. Willi
`the on-ofl
`controller. the controller output is either on or ofl"(fully open or fully closed). Such is
`the case with the thermostat furnace system. The controller output is determined by
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`FORD 1883
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`Flapper
`displacement
`at nozzle
`(inches)
`
`.l'
`)-
`
`\ i
`
`00‘
`u”
`
`artistic curve.
`
`'the back pressure curve.
`:r displacement from the
`unal condition. Then the
`
`mem)
`
`.1. From the geometry of
`
`(6.4-1 I)
`
`(6.442)
`
`the nozzle
`King region.
`assure is well below the
`lmore output pressure is
`antic relay or amplilier
`are 6.24 illustrates this
`belt pressure increases.
`[the supply line. and the
`pproaches atmospheric
`tack pressure decreases.
`lthe atmospheric bleed.
`ressure approaches the
`The relay is said to be
`ause an increase in back
`
`. a decrease in output.
`tressure from the relay
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`336
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`FEEDBACK CONTROL SYSTEMS
`
`I
`
`,.
`
`Neutra. zone
`
`l
`
`FlClIRI-I 6.26 Transfer characteristics
`
`of the on—oll' controller. The actuating
`error is e=r—e. where r=set point.
`c=controlled variable, and I: control
`signal.
`
`the magnitude of the error signal. The
`switching diagram for the onwolT con—
`troller with hysteresis is shown in l-‘igure
`6.26.
`
`An example of an application of an
`on olTeontroller to a liquid level system is
`shown in Figure 6.27“. 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 /.one is provided to prevent
`frequent on oil" switching. or cliarrering,
`which can shorten the life of the device.
`
`The cycling frequency also depends on the time constant of the controlled process and
`the magnitude of the control signal.
`the sensor and control valve will not
`In a real system. as opposed to ideal.
`respond instantaneously. btit 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 6271). The opposite occurs when the valve is turned on. This unwanted elleet
`can be reduced by decreasing the neutral zone. but the cycling frequency increases if
`this is done.
`
`The overshoot and undershoot in on rolleontrol 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. If the control valve in Figure 6.27” is far upstream from the tank. a
`significant lag can exist between the time of control action and its cllect 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 6.3 is appreciable.
`the
`heater will continue to deliver energy to the oven ex en after it has been turned oil.
`An example close to home demonstrates how the capacitance of the plant allects
`
`‘i t‘
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`“5 volts
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`Solenoid
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`Overshoor
`Rear
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`twtiavrm’n/
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`7’,
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`Em[)lylng\ \/ Undorshm"
`C'
`,IV:
`Curve
`
`Zl—
`*
`("l
`
`FIGURE 6.27
`
`(0) Liquid level control with on-ofl' action. (b) Time response.
`
`th)
`
`llfnlf
`
`the suitability of Olle‘OlTCOIl
`obviously be unsuitable hr
`capacitance is greater.
`Another type of Nope:
`mg diagram is shown in Fig
`control by the fact that the di
`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
`are not too severe, In the hon
`
`hardly detectable by the occ
`situations require finer conti
`Consider the tank systt
`controller, we might try sctti
`balances the system at the dl
`this setting in proportion to
`proportional control.
`the ah
`proportional to the error. R1
`are drawn in terms of the
`
`Applying this convention I
`proportional control is desc
`
`where l’ts) is the deviation in
`total valve displacement is y
`
`(.1)
`
`FIGURE 6.28 Transfer Chara
`a dead zone. The control signal
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`the on—olT con-
`tgram for
`ysterests is shown in Figure
`
`tple of an application of an
`ller to a liquid level system is
`ire 6.27u. The time response
`Ire 6.37!) with a solid line is
`ystem in which the control
`intaneously. The controlled
`'3 with an amplitude that
`c width of the neutral zone
`one is provided to prevent
`ff switching. 0r chartering.
`men the life of the device.
`the controlled process and
`
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`6.5 CONTROL LAWS
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`337
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`the suitability of on—ofl‘ control. OnAofi' control of the hot water valve in a shower will
`obviously be unsuitable. bttt
`it
`is acceptable for a bath. because the thermal
`capacitance is greater.
`Another type of two-position control is the hung-hang controller whose switch-
`ing diagram is shown in Figure 628(1. This controller is distinguished from onwofl‘
`control by the fact that the direction or sign of the control signal cart have two values.
`A motor with constant
`torque that can reverse quickly might be modeled as it
`bangirbang device. Because such perfect switching is impossible. a more accurate
`model would include a dead zone (Figure 6.22%). When the error is within the zone.
`the controller output is zero.
`
`Proportional Control
`Tw0~position control is acceptable for many applications in which the requirements
`are not too severe. In the home heating application. the typical 2" F temperature gap is
`hardly detectable by the occupants. Thus. the system is acceptable. However. many
`situations require liner control.
`Consider the tank system shown in Figure 6.27o. 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 controller that adjusts
`this setting in proportion to the deviation of the level from the desired value. This is
`proportional r-omrol.
`the algorithm in which the change in the control signal
`is
`proportional to the error. Recall the convention that block diagrams for controllers
`are drawn in terms of the deviations from a zero-error equilibrium condition.
`Applying this convention to the general
`terminology in Figure 6.7. we see that
`proportional control is described by
`HS): KpEts)
`
`(6.5-1)
`
`ation in the control signal and Kr, is the proportional gain. If the
`where His) is the devi
`and the manually created displacement is x. then
`total valve displacement is vvtt)
`
`ytt) = ert!) + x
`
`(.1!
`
`lb)
`
`FIG UR F. 6.28 Transfer characteristics. (a) Ideal bang-bang control. (b) Bang-bang control with
`a dead tone. The control signal is f; the error signal is 0.
`
` le of the error signal. The
`
`nd control valve will not
`Is. The valve will not close
`will be some delay during
`an by the dotted line in
`| on. This unwanted ellect
`ling frequency increases if
`
`acceptable only when the
`tg ofthe control elements.
`: well as to their distance
`pstream from the tank. a
`ind its elTect on the plant.
`r itself. For example. if the
`'n-6.2 is appreciable. the
`r it has been turned off.
`name of the plant affects
`
`Overshoot
`
`/ Ideal
`,_\_ _b£halior
`\
`\
`
`z
`
`
`
`\: — 7l— — _
`1‘1an \l Undershoot
`I'VE
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`“(“0
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`'l
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`9 response.
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