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Unformatted text preview: BB 428 EXAM II 7 December 1998 Name: 3 olu'lzigas
ID#: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO 50 “Weight
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II This test consists of four problems. Answer each problem on the exam itself; if you use additional
paper, repeat the identifying information above, and staple it to the rest of your exam when you
hand it in. The quality of your analysis and evaluation is as important as your answers. Your
reasoning must be precise and clear; your complete English sentences should convey what you are doing. Problem 1: (25 points) The closedloop system in Figure 1 contains the plant 32—23+2
s(s+1) U G43) 2 Kg. Gal”) = and proportional controller controller ulant Y(s) Figure 1: Closedloop system with cascade compensation. 1. (4 points) Express the characteristic equation of the closedwloop system in a form suitable for con
structing the root locus of the closedloop system as the proportional gain K0 is varied from zero to
inﬁnity. 2. (4 points) Calculate the location of the break away point. 3. (4 points) Determine the value of the control gain Kc for which the closedloop poles cross the go) axis.
Specify the location of the crossing point on the imaginary axis. 4. (4 points) Determine the angles of arrival at the complex zeros. 5. (4 points) Sketch the complete root locus using the graph in Figure 2. Appropriately label the break
away point, 3w axis crossings, and angles of arrival. Use arrows to indicate the direction of travel along U
the loci as K9 increases towards inﬁnity. Root Locus of Gp(s) lmag Axis Figure 2: Root locus of the closedloop system as the proportional gain K0 is varied from zero
towards inﬁnity. 3 From Part C0.) tha. ohmEzrta'mo 9%. Vodka can 59 eﬁf’reSSQcEL (:5
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“(04")51 1" (I ‘LKAS 4. Zko _... 0 TL ROV'HN a.“ I“? 1.5 5"" K0 + I Zko
5‘ [2 ko .
s" 2 k. $76!: ko = J 55 the]: the, 52mg— row LJ 1:271). 2 The Ow x“ I out? ebvccbon 1‘s Problem 2: (25 points) The root locus of the closedsystem in Figure 3 is shown in Figure 4 for 0 < Kg < so. The numerator and
denominator polynomials of the plant transfer function Gp(s) are monic. Answer the following questions
using the root locus in Figure 4 and the ruler provided on the last page of the exam. controller  lam Y(s) Figure 3: Closedloop system with cascade compensation. 1. (8 points) For what range of K, is the system BIBO stable ? 2. (8 points) Determine the gain K0 for which the dominant pole pair of the closedloop system has
C = 0.8. 3. (9 points) For the gain determined in part (2), what is the steadystate error of the closedloop system
in response to a unitstep input ? l. Wham k0 .. 0’ the. gloseaQoop poles gnue, In the. 19,“: le‘F plane, A; Ko :ocrmesJ the. Pale; cram 'HIQ. dw ”L: at Peiﬂ'i'ﬁ S. qua Sf ( walkedAg cm Flawq. '1). Use. ﬂ'lg "Imam‘lDWle— Conﬂl'lzign £9 Jz'EerMIPIa. i512. .29.ng # k, for “5“ch
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° ‘ (3.8)(55) T' system is 8180 .siaaﬂe ﬁf o<K. (0.39 it 2.. 711.. ruled 52...,5; gmﬂhgglm (7") an on she.
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cs+ LX544) 6, (53 = Real Axis Figure 4: Root locus of the closedloop system as the proportional gain K0 is varied from zero towards inﬁnity.
(o3)Cos
a». acts) 6,6.) = k. ”"L 3. as  kP= S6, (o+2.)(o+l) —m S: (0.0'1‘0 (int—:9 : 0‘33 I
1+ KP Problem 3: (25 points) Figure 6 shows the Bode magnitude and phase plot of the plant Gp(s) in Figure 5. controller lant U Figure 5: Closedloop system with cascade compensation. 1. (6 points) Based on the Bode magnitude and phase plots, write an expression for the plant transfer
function G,(s). 2. (6 points) What is the gain margin and phase crossover frequency of the uncompensated open—loop
transfer function B(s)/E(s) : Gp(s) ? 3. (6 points) What is the phase margin and gain crossover frequency of the uncompensated openloop
transfer function B(s)/E(s) = G901) ? 4. (7 points) Suppose proportional control Gc(s) = K9 is used. What is the steadystate error due
to a ramp input r(t) 2: tu,(t) if the gain margin of the compensated open 100p transfer function
B(s)/E(s) = Koapp) is 6 dB ? l. Fraun 'H'c. muvnpbﬂﬂl. put, 'For' w < 10% the. Slope. ls
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GP(3) : 0.5.9 _1.
1. (8 points) Neatly sketch the Bode magnitude and phase plots of Gp(’) 2. (8 points) Sketch the Nyquist plot of the openloop transfer function Gp(s). Indicate the direction of
travel along the contour [‘9' using arrows. 3. (4 points) Based on the Nyquist plot obtained in part (2), determine the range of proportional control
Gc(s) z K,J for which the closedloop system in Figure 7 is BIBO stable. Y(s) Figure 7: Closedloop system with proportioanl control. 4. (5 points) Verify your answer in part (3) by applying the Routh—Hurwitz stability criterion. d‘” "1 ‘, r2 onoawldel
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