9 781292 024059

ISBN 978-1-29202-405-9

Modern Control Systems

Richard C. Dorf Robert H. BishopTwelfth Edition

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Pearson Education LimitedEdinburgh GateHarlowEssex CM20 2JEEngland and Associated Companies throughout the world

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Printed in the United States of America

ISBN 10: 1-292-02405-4ISBN 13: 978-1-292-02405-9

ISBN 10: 1-292-02405-4ISBN 13: 978-1-292-02405-9

Feedback Control System Characteristics

CP1 Consider a unity feedback system with

Obtain the step response and determine the percentovershoot. What is the steady-state error?

CP2 Consider the transfer function (without feedback)

When the input is a unit step, the desired steady-statevalue of the output is one. Using the step function, showthat the steady-state error to a unit step input is 0.8.

G(s) =

4

s2+ 2s + 20

.

G(s) =

12

s2+ 2s + 10

.

CP3 Consider the closed-loop transfer function

Obtain the family of step responses for and 500. Co-plot the responses and develop a table ofresults that includes the percent overshoot, settlingtime, and steady-state error.

<CP Consider the feedback system in Figure CP Sup-pose that the controller is

.Gc(s) = K = 10

K = 10, 200,

T(s) =

5 K

s2+ 15s + K

.

COMPUTER PROBLEMS

represented by the system in Figure DP8(b), wherethe nominal values are ms and ms.(a) Compute the sensitivity and the sensitivity .ST

t2STt1

t2 = 2t1 = 20(b) Design the controller gain K such that the steady-state tracking error to a unit step disturbance is lessthan 0.05.

(b)

(a)

K

�

��

�

Td(s)

Y(s)R(s)1

t1s � 1

1

s (t2s � 1)

Videocamera

Pulley

FIGURE DP8Remote-controlledTV camera.

Controller�

�

��

R(s) K1

s(s�1.91)Y(s)

Td(s)

Plant

Ea(s)

FIGURE CP4Unity feedbacksystem withcontroller gain K.

310

Feedback Control System Characteristics

�

�

Controller Process

1s � a

KR(s) Y(s)

�

� ��

K0

Td(s)

ud(s)r(t)

u (s)1�J

s2 � bJ

skJ

�

Controller

Mechanicalsystem

(a) (b)

Elasticshaft td(t), Disturbance

torque

Brakingdevice

u

r(t), Inputtorque

FIGURE CP7(a) A torsionalmechanical system.(b) The torsionalmechanical systemfeedback controlsystem.

(a) Develop an m-file to compute the closed-looptransfer function and plot the unitstep response. (b) In the same m-file, compute thetransfer function from the disturbance to theoutput and plot the unit step disturbance re-sponse. (c) From the plots in (a) and (b) above, esti-mate the steady-state tracking error to the unit stepinput and the steady-state tracking error to the unitstep disturbance input. (d) From the plots in (a) and(b) above, estimate the maximum tracking error to theunit step input and the maximum tracking error to theunit step disturbance input. At approximately whattimes do the maximum errors occur?

CP5 Consider the closed-loop control system shown inFigure CP5. Develop an m-file script to assist in thesearch for a value of k so that the percent overshootto a unit step input is greater than 1%, but less than10%. The script should compute the closed-looptransfer function and generate thestep response. Verify graphically that the steady-stateerror to a unit step input is zero.

CP6 Consider the closed-loop control system shown inFigure CP6.The controller gain is The nominalvalue of the plant parameter is The nominala = 1.

K = 2.

T(s) = Y(s)>R(s)

Y(s)Td(s)

T(s) = Y(s)>R(s)value is used for design purposes only, since in realitythe value is not precisely known. The objective of ouranalysis is to investigate the sensitivity of the closed-loop system to the parameter a.

(a) When show analytically that the steady-state value of y(t) is equal to 2 when r(t) is a unitstep. Verify that the unit step response is within2% of the final value after 4 seconds.

(b) The sensitivity of the system to changes in the para-meter a can be investigated by studying the effectsof parameter changes on the transient response.Plot the unit step response for and 5.Discuss the results.

CP7 Consider the torsional mechanical system in FigureCP7(a). The torque due to the twisting of the shaft is

the damping torque due to the braking deviceis the disturbance torque is td(t); the inputtorque is r(t); and the moment of inertia of the me-chanical system is J. The transfer function of the tor-sional mechanical system is

G(s) =

1>J

s2+ (b>J)s + k>J

.

- bu;- ku;

a = 0.5, 2,

a = 1,

�

�

Controller

10s

1s � k

R(s) Y(s)

Process

FIGURE CP5A closed-loopnegative feedbackcontrol system.

FIGURE CP6A closed-loopcontrol systemwith uncertainparameter a.

311

Feedback Control System Characteristics

�

�

R(s) Y(s)

N(s)

G(s)

�

�H(s)

FIGURE CP9 Closed-loop system with nonunityfeedback and measurement noise.

A closed-loop control system for the system is shownin Figure CP7(b). Suppose the desired angle

and (a) Determine the open-loop response of the

system for a unit step disturbance (b) With the controller gain determine the

closed-loop response, to a unit step distur-bance.

(c) Plot the open-loop versus the closed-loop responseto the disturbance input. Discuss your results andmake an argument for using closed-loop feedbackcontrol to improve the disturbance rejection prop-erties of the system.

CP8 A negative feedback control system is depicted inFigure CP8. Suppose that our design objective is tofind a controller of minimal complexity suchthat our closed-loop system can track a unit step inputwith a steady-state error of zero.(a) As a first try, consider a simple proportional

controller

where K is a fixed gain. Let Plot the unitstep response and determine the steady-stateerror from the plot.

(b) Now consider a more complex controller

where and This controller isknown as a proportional, integral (PI) controller.Plot the unit step response, and determine thesteady-state error from the plot.

(c) Compare the results from parts (a) and (b), anddiscuss the trade-off between controller complex-ity and steady-state tracking error performance.

CP9 Consider the closed-loop system in Figure CP9,whose transfer function is

(a) Obtain the closed-loop transfer function and the unit step response; that is, let

and assume that N(s) = 0.R(s) = 1>sY(s)>R(s)

T(s) =

G(s) =

10s

s + 100 and H(s) =

5s + 50

.

K1 = 20.K0 = 2

Gc(s) = K0 +

K1

s,

K = 2.Gc(s) = K,

Gc(s)

u(t)K0 = 50,

(set r(t) = 0).u(t)

J = 1.ud = 0°, k = 5, b = 0.9,

(b) Obtain the disturbance response when

is a sinusoidal input of frequency Assume that

(c) In the steady-state, what is the frequency andpeak magnitude of the disturbance response frompart (b)?

CP10 Consider the closed-loop system is depicted inFigure CP10. The controller gain K can be modified tomeet the design specifications.

(a) Determine the closed-loop transfer function

(b) Plot the response of the closed-loop system for10, and 50.

(c) When the controller gain is determinethe steady-state value of y(t) when the distur-bance is a unit step, that is, when and

CP11 Consider the non-unity feedback system is depictedin Figure CP11.

(a) Determine the closed-loop transfer function

(b) For 12,and 15,plot the unit step responses.Determine the steady-state errors and the settlingtimes from the plots.

For parts (a) and (b), develop an m-file that computesthe closed-loop transfer function and generates theplots for varying K.

K = 10,T(s) = Y(s)>R(s).

R(s) = 0.Td(s) = 1>s

K = 10,K = 5,

T(s) = Y(s)>R(s).

R(s) = 0.v = 10 rad>s.

N(s) =

100

s2+ 100

�

�

Controller Process

10s � 10

R(s) Y(s)Gc(s)FIGURE CP8A simple single-loop feedbackcontrol system.

312

Feedback Control System Characteristics

Loss of gain A reduction in the amplitude of the ratio ofthe output signal to the input signal through a system,usually measured in decibels.

Open-loop system A system without feedback that directlygenerates the output in response to an input signal.

Steady-state error The error when the time period islarge and the transient response has decayed, leavingthe continuous response.

System sensitivity The ratio of the change in the systemtransfer function to the change of a process transferfunction (or parameter) for a small incrementalchange.

Tracking error See error signal.

Transient response The response of a system as a func-tion of time.

Closed-loop system A system with a measurement of theoutput signal and a comparison with the desired out-put to generate an error signal that is applied to theactuator.

Complexity A measure of the structure, intricateness, orbehavior of a system that characterizes the relation-ships and interactions between various components.

Components The parts, subsystems, or subassembliesthat comprise a total system.

Disturbance signal An unwanted input signal that affectsthe system’s output signal.

Error signal The difference between the desired outputR(s) and the actual output Y(s). Therefore,

Instability An attribute of a system that describes a ten-dency of the system to depart from the equilibriumcondition when initially displaced.

Loop gain The ratio of the feedback signal to the con-troller actuating signal. For a unity feedback systemwe have L(s) = Gc(s)G(s).

E(s) = R(s) - Y(s).

TERMS AND CONCEPTS

Controller�

�

��

R(s) 1s2�s�6.5

Ks�1s�15

Y(s)

Td(s)

Process

FIGURE CP10Closed-loopfeedback systemwith externaldisturbances.

Controller

�

�

R(s) 20s2�4.5s�64

1s�1

K

Sensor

Y(s)

Process

FIGURE CP11Closed-loop systemwith a sensor in thefeedback loop.

ANSWERS TO SKILLS CHECK

True or False: (1) True; (2) True; (3) False; (4) False;(5) True

Multiple Choice: (6) a; (7) b; (8) a; (9) b; (10) c;(11) a; (12) b; (13) b; (14) c; (15) c

Word Match (in order, top to bottom): e, h, k, b, c, f,i, g, d, a, j

313

Feedback Control System Characteristics

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18. M. Eslami, Theory of Sensitivity in Dynamic Sys-tems, Springer-Verlag, New York, 1994.

19. Y. M. Pulyer, Electromagnetic Devices for MotionControl, Springer-Verlag, New York, 1992.

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CHAPTER REFERENCES

314