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AME 50652; Spring 2016; Homework 3; Due February 17th, 2016

Read: FPEN Chapter 4.

Problems:

• Problem 1: FPEN 4.4

• Problem 2: FPEN 4.17

• Problem 3: FPEN 4.18

• Problem 4: FPEN 4.20

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Problems for Section 4.1: The BasicEquations of Control

4.1 If S is the sensitivity of the unity feedback system to changes in the planttransfer function and T is the transfer function from reference to output, showthat S + T = 1.4.2 We define the sensitivity of a transfer function G to one of its parameters kas the ratio of percent change in G to percent change in k.

The purpose of this problem is to examine the effect of feedback on sensitivity.In particular, we would like to compare the topologies shown in Fig. 4.24 forconnecting three amplifier stages with a gain of ­K into a single amplifier with again of ­10.

a. For each topology in Fig. 4.24 , compute β so that if K = 10, Y = ­10R.b. For each topology, compute when . (Use the respective βvalues found in part (a).) Which case is the least sensitive?

c. Compute the sensitivities of the systems in Fig. 4.24(b,c) to β andβ . Using your results, comment on the relative need for precision insensors and actuators.

SkG=dG/Gdk/k=d ln Gd ln k=kGdGdk.

i

SkG G=YR i

2

3

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Figure 4.24 Three­amplifier topologies for Problem 4.2

4.3 Compare the two structures shown in Fig. 4.25 with respect to sensitivityto changes in the overall gain due to changes in the amplifier gain. Use therelation

as the measure. Select H and H so that the nominal system outputs satisfy F= F , and assume KH > 0.

Figure 4.25 Block diagrams for Problem 4.3

S=d ln Fd ln K=KFdFdK,

1 2 1

2 1

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4.4 A unity feedback control system has the open­loop transfer function

a. Compute the sensitivity of the closed­loop transfer function to changes inthe parameter A.

b. Compute the sensitivity of the closed­loop transfer function to changes inthe parameter a.

c. If the unity gain in the feedback changes to a value of β ≠ 1, compute thesensitivity of the closed­loop transfer function with respect to β.

4.5 Compute the equation for the system error for the feedback system shownin Fig. 4.5 .

G(s)=As(s+a).

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Problems for Section 4.2: Control of Steady­State Error

4.6 Consider the DC motor control system with rate (tachometer) feedbackshown in Fig. 4.26(a) .

a. Find values for K′ and k so that the system of Fig. 4.26(b) has thesame transfer function as the system of Fig. 4.26(a) .

b. Determine the system type with respect to tracking θ and compute thesystem K in terms of parameters Kand k .

c. Does the addition of tachometer feedback with positive k increase ordecrease K ?

Figure 4.26 Control system for Problem 4.6

4.7 A block diagram of a control system is shown in Fig. 4.27 .a. If r is a step function and the system is closed­loop stable, what is thesteady­state tracking error?

b. What is the system type?c. What is the steady­state error to a ramp velocity 5.0 if K = 2 and K isadjusted to give a system step overshoot of 17%?

t′

r

v ′ t′

t

v

2 1

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Figure 4.27 Closed­loop system for Problem 4.7

4.8 A standard feedback control block diagram is shown in Fig. 4.5 with thevalues

a. Let W = 0 and compute the transfer function from R to Y.b. Let R = 0 and compute the transfer function from W to Y.c. What is the tracking error if R is a unit­step input and W ≡ 0?d. What is the tracking error if R is a unit­ramp input and W ≡ 0?e. What is the system type and the corresponding error coefficient?

G(s)=1s; Dc(s)=2(s+1)s; H(s)=100(s+100).

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a. Let W = 0 and compute the transfer function from R to Y.b. Let R = 0 and compute the transfer function from W to Y.c. What is the tracking error if R is a unit­step input and W ≡ 0?d. What is the tracking error if R is a unit­ramp input and W ≡ 0?e. What is the system type and the corresponding error coefficient?

Figure 4.28 Control system for Problem 4.10

4.9 A generic negative feedback system with non­unity transfer function in the feedbackpath is shown in Fig. 4.5 .

a. Find the steady­state tracking error for this system to a ramp reference input.b. If G(s) has a single pole at the origin in the s­plane, what is the requirement on H(s)such that the system will remain a Type 1 system?

c. Suppose,

showing a lead compensation in the feedback path. What is the value of the velocityerror coefficient, K ?

4.10 Consider the system shown in Fig. 4.28 , where

a. Prove that if the system is stable, it is capable of tracking a sinusoidal reference inputr = sin ω t with zero steady­state error. (Hint: Look at the transfer function from R to

d. G(s)=1s(s+1)2; Dcl(s)=0.73; H(s)=2.75s+10.36s+1,

v

Dc(s)=K(s+α)2s2+ωo2.

o

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E and consider the gain at ω .)b. Use Routh’s criterion to find the range of K such that the closed­loop system remainsstable if ω = 1 and α = 0.25.

4.11 Consider the system shown in Fig. 4.29 , which represents control of the angle ofa pendulum that has no damping.

a. What condition must D (s) satisfy so that the system can track a ramp reference inputwith constant steady­state error?

Figure 4.29 Control system for Problem 4.11

b. For a transfer function D (s) that stabilizes the system and satisfies the condition inpart(a), find the class of disturbances w(t) that the system can reject with zerosteady­state error.

4.12 A unity feedback system has the overall transfer function

Give the system type and corresponding error constant for tracking polynomial referenceinputs in terms of ζ and ω .

4.13 Consider the second­order system

o

o

c

c

Y(s)R(s)=T(s)=ωn2s2+2ζωns+ωn2.

n

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We would like to add a transfer function of the form in series with G(s) ina unity feedback structure.

a. Ignoring stability for the moment, what are the constraints on K, a, and b so that thesystem is Type 1?

b. What are the constraints placed on K, a, and b so that the system is both stable andType 1?

c. What are the constraints on a and b so that the system is both Type 1 and remainsstable for every positive value for K?

4.14 Consider the system shown in Fig. 4.30(a) .

a. What is the system type? Compute the steady­state tracking error due to a rampinput r(t) = r t1(t).

b. For the modified system with a feed forward path shown in Fig. 4.30(b) , give thevalue of H so the system is Type 2 for reference inputs and compute the K in thiscase.

c. Is the resulting Type 2 property of this system robust with respect to changes in H ,that is, will the system remain Type 2 if H changes slightly?

G(s)=1s2+2ζs+1.

Dc(s)=K(s+a)(s+b)

o

f a

f

f

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Figure 4.30 Control system for Problem 4.14

4.15 A controller for a satellite attitude control with transfer function has beendesigned with a unity feedback structure and has the transfer function .

G = 1/s2Dc(s)=10(s+2)s+5

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PRINTED BY: dhoelzle@nd.edu. Printing is for personal, private use only. No part of thisbook may be reproduced or transmitted without publisher's prior permission. Violators willbe prosecuted.4.15 A controller for a satellite attitude control with transfer function has beendesigned with a unity feedback structure and has the transfer function .

a. Find the system type for reference tracking and the corresponding error constant forthis system.

b. If a disturbance torque adds to the control so that the input to the process is u + w,what is the system type and corresponding error constant with respect to disturbancerejection?

4.16 A compensated motor position control system is shown in Fig. 4.31 . Assume thatthe sensor dynamics are H(s) = 1.

Figure 4.31 Control system for Problem 4.16

a. Can the system track a step reference input r with zero steady­state error? If yes,give the value of the velocity constant.

b. Can the system reject a step disturbance w with zero steady­state error? If yes, givethe value of the velocity constant.

c. Compute the sensitivity of the closed­loop transfer function to changes in the plantpole at ­2.

G = 1/s2Dc(s)=10(s+2)s+5

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d. In some instances there are dynamics in the sensor. Repeat parts (a) to (c) for and compare the corresponding velocity constants.

4.17 The general unity feedback system shown in Fig. 4.32 has disturbance inputs w ,w , and w and is asymptotically stable. Also,

Show that the system is of Type 0, Type l , and Type (l + l ) with respect to disturbanceinputs w , w , and w , respectively.

Figure 4.32 Single input–single output unity feedback system with disturbance inputs

4.18 One possible representation of an automobile speed­control system with integralcontrol is shown in Fig. 4.33 .

a. With a zero reference velocity input v = 0, find the transfer function relating theoutput speed v to the wind disturbance w.

b. What is the steady­state response of v if w is a unit­ramp function?c. What type is this system in relation to reference inputs? What is the value of thecorresponding error constant?

d. What is the type and corresponding error constant of this system in relation totracking the disturbance w?

H(s)=20s+20

1

2 3

G1(s)=K1∏i=1m1(s+z1i)sl1∏i=1m1(s+p1i),  G2(s)=K2∏i=1m1(s+z2i)sl2∏i=1m1(s+p2i).

1 1 2

1 2 3

c

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Figure 4.33 System using integral control

4.19 For the feedback system shown in Fig. 4.34 , find the value of α that will make thesystem Type 1 for K = 5. Give the corresponding velocity constant. Show that the system isnot robust by using this value of α and computing the tracking error e = r – y to a stepreference for K = 4 and K = 6.

Figure 4.34 Control system for Problem 4.19

4.20 Suppose you are given the system depicted in Fig. 4.35(a) where the plantparameter a is subject to variations.

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Figure 4.35 Control system for Problem 4.20

a. Find G(s) so that the system shown in Fig. 4.35(b) has the same transfer functionfrom r to y as the system in Fig. 4.35(a) .

b. Assume that a = 1 is the nominal value of the plant parameter. What are the systemtype and the error constant in this case?

c. Now assume that a = 1+ δa, where δa is some perturbation to the plant parameter.What are the system type and the error constant for the perturbed system?