Control Systems Engineering
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`
`William Palm III
`University of Rhode Island
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`New York
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`Chichester
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`Brisbane
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`John Wiley & Sons
`Toronto
`Singapore
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`Copyright ((3) I986 by John Wiley & Sons, Inc.
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`All rights reserved. Published simultaneously in Canada.
`
`Reproduction or translation of any part of this work beyond that permitted by Sections 107 and I08
`of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful.
`Requests for permission or further inl'orm-ation should be addressed to the Permissions Department,
`John Wiley & Sons.
`
`Library of Congress CataI0ging-in-Publication Dam
`
`.,,..
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`Palm, William J. (William John), 1944-
`Control systems engineering.
`
`Bibliography: p.
`Includes index.
`
`1. Automatic control.
`I. Title
`TJ213.P225 I986
`629.8
`ISBN 0-471-81086-X
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`2. Control theory.
`85-26590
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`Printed in the United States of America
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`""ev'"atic
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`pressure
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`Diaphragm
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`Flow '—>
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`3 —>
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`FIGURE 6.25 Pneumatic flow control
`V3'V¢-
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`6.5 CONTROL LAWS
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`335
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`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 (e.g., Reference 7).
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`6.5 CONTROL LAWS
`
`The control logic elements are 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 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:
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`1. Minimize the steady—state error.
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`2. Minimize the settling time.
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`3. Achieve other
`overshoot.
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`transient
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`specifications,
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`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
`sufiieient for our purpose.
`The following two control laws form the basis of many control systems.
`
`Two-Position Control
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`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~ojf
`controller, the controller output is either on or oiT(fully open or fully closed). Such is
`the case with the thermostat furnace system. The controller output is determined by
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`->-
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`Kb,
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`Flapper
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`d.':*:‘::::‘.:"‘
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`(inches)
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`:Ieristic curve.
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`‘the. back pressure curve.
`dlsplacement from the
`amal condition. Then the
`
`(6.4-10)
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`rr. From the geometry of
`
`(6.4- I l)
`
`(6.4-12)
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`the nozzle
`‘W38 TCgi0n.
`ware is well below the
`{more output pressure is
`mric relay or amplifier
`titre 6.24 illustrates this
`
`traick pressure increases,
`{the supply line, and the
`approaches atmospheric
`rack pressure decreases,
`fthe atmospheric bleed,
`ressure approaches the
`The relay is said to be
`ause an increase in back
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`i a decrease in output.
`Iressure from the relay
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`the suitability of on—offcon¥tti
`obviously be unsuitable. E
`capacitance is greater.
`Another type of two-pm
`ing diagram is shown in Fig
`control by the fact that the (ii
`A motor with constant ton
`
`bang—bang device. Because
`model would include a dead
`the controller output is zero
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`Proportional Control
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`Two-position control is aecc
`are not too severe. In the hon
`
`hardly detectable by the occ
`situations require finer conu
`Consider the tank systt
`controller, we might try setti:
`balances the system at the dl
`this setting in proportion to
`proportional control,
`the alt
`proportional to the error. R-
`are drawn in tertns of the
`
`Applying this convention t
`proportional control is desc:
`
`where F (s) is the deviation in
`total valve displacement is _\
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`(0)
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`FIGURE 6.28 Transfer chart:
`a dead zone. The control signal
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`'1
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`FIGURE 6.27
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`(a) Liquid level control with on—olf action. (b) Time response.
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`336
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`FEEDBACK CONTROL SYSTEMS
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`f
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`Neutral zone
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`,.
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`FIGURE 6.26 Transfer characteristics
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`of the on—olf controller. The actuating
`error
`is e=r-c, where r=set point.
`c=controlled variable, and f: control
`signal.
`
`the magnitude of the error signal. The
`switching diagram for the on—oFf con-
`troller with hysteresis is shown in Figure
`6.26.
`
`An example of an application of an
`on~olTeontroller to a liquid level system is
`shown in Figure 6.27a. The time response
`shown in Figure 6.27}: 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 roll" switching, or chattering,
`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, 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 elleet
`can be reduced by decreasing the neutral zone. but the cycling frequency increases if
`this is done.
`
`The overshoot and undershoot in on—otTcontrol 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 6.27a is far upstream from the tank, a
`significant lag can exist between the time of control action and its ellect on the plant.
`Another source oftimc lag is the capacitance ofthe controller itself. For example, if the
`heater capacitance in the temperature controller of Section 6.2 is appreciable, the
`heater will continue to deliver energy to the oven even after it has been turned ofi”.
`An example close to home demonstrates how the capacitance of the plant allects
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`R a]
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`Overshoot
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`beh:vior / Ideal
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`\~—-7I———
`E mmying‘ \ 1 Undorshoot
`CUYVG
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`/Filling
`I curve
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`(b)
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`time
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`Solenoid
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`S“°°"'
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`wet
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`6.5 CONTROL LAWS
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`the suitability of on—ofl' control. ()n—ofi‘ 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—b(mg controller whose switch-
`ing diagram is shown in Figure 6.28a. This controller is distinguished from on—oiT
`control by the fact that the direction or sign of the control signal can have two values.
`A inoior 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 6.281)). When the error is within the Zone,
`the controller output is zero.
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`Proportional Control
`Two-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 6.2711. 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 mritrol.
`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
`Fts) = K,,E(s)
`
`(6-5-I)
`
`where F(s) is the deviation in the control signal and K’, is the proportional gain. lf the
`total valve displacement is _\'(t) and the manually created displacement is x, then
`
`ytt) -— Kpettl + .\'
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`ie of the error signal. The
`igram for
`the on—olT con-
`Ystcresis is shown in Figure
`
`‘Pie of an application of an
`ller to a liquid level system is
`ire 6.27a. The time response
`ire 6.27b with a solid line is
`ystem in which the control
`mtaneously. The controlled
`-8 with an amplitude that
`e width of the neutral zone
`one is provided to prevent
`H Swltchlflg, Or chattering,
`irten the life of the device_
`the controlled process and
`
`nd control valve will not
`IS._ The valve will not close
`Will be some delay during
`"" by the dotted line in
`Ion. This unwanted effect
`["18 frequency increases if
`
`’“°°¢Ptable only when the
`Ig of the control elements,
`: well as to their distance
`pstream from the tank, a
`Ind its eliect on the plant,
`r itself. For example, ifthe
`'n.6.2 is appreciable, the
`it has been turned oil".
`ltance of the plant affects
`
`Overshooi
`
`
`
`.\:‘7’-~-
`Kvmg \/ Undershoot
`We
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`I
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`time
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`it
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`e response.
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`FIGURE 6.28 Transfer characteristics. (a) [deal bang-bang control. (b) Bang-bang control with
`a dead zone. The control signal is f; the error signal is 0.
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