final_f06

final_f06 - EE 428 Final Exam 20 December 2006 Last Name...

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Unformatted text preview: EE 428 Final Exam 20 December 2006 Last Name (Print): 80‘ o as. First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem 1 2 3 - INSTRUCTIONS 1. You have 2 hours to complete this exam. This is a closed book exam. You may use three 8.5” x 11” note sheets. Calculators are allowed. P9730 Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6. 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. To receive credit, you must show your work. Problem 1: (25 Points) The feedback system in laboratory #4 that regulates the location of the ball along a beam requires a measurement of the ball velocity, however, the apparatus only provides a signal proportional to position x(t). Passing the position measurement :c(t) through a filter with transfer function . _l/_(§l_ 103 HW‘ (s)‘s+1o provides an estimate of the velocity v(t). 1. (1 point) Express the frequency response function H ( you) in a form for suitable for sketching a Bode magnitude and phase plot. 2. (8 points) Neatly sketch the Bode magnitude and phase plot of the filter H ( 3w) in Figure 1. Your sketch must: 0 Indicate the slope of all straight-line segments. 0 Indicate the location of corner frequencies. Based on your Bode plot: 0 Specify the DC gain. 0 Determine the high frequency gain. 0 In one sentence or two short sentences, discuss why the filter output is not an exact measure of the ball velocity v(t) = :b(t) and under what conditions the filter output provides a good estimate of the ball velocity. I I a. ugh?) I): am; ,Q.‘m Hcs) :0. Note that, the 6.032, mat/mean. ,9 ppm 5’0 ‘0 «513 «A. «42 wpprouhes tern. aé A (a‘b "b ‘ Fn- ‘ Has): to. 1J1:- Bng. ma {ml/pa fl Pr‘g‘tb 19931“? a“ - LOJ‘BJ “1193,?! [ngfl = (o. In con-Erqflb) H'CS) 00L; Wflroxm‘cées 5 whey I 65.;Smafiz CW < (o “HQ/Say.) Tl“— pJéL ~m H19 lengs w ALVA maveny 6611/) b in)“ do aotw‘br maohafl’tcé. w‘\ [00 (0‘ La" 103 Figure 1: Bode magnitude and phase plot of H ( 3w). 3. (8 points) Sketch the polar plot of H ( 30.1) in Figure 2. Your sketch must: 0 Indicate the point corresponding to w = 0. 0 Indicate the points corresponding to corner frequencies. 0 Indicate the point corresponding to u) = 00. Polar Plot of H(s) 1- o 0 7o lac3707 lo “15° ra- I 00 00 i0 lml H0 (0)} -—2-0'2'4 6 8 '10 RelHGGH —10 —8 —6 —4 Figure 2: Polar plot of H (1w). 4. (8 points) Sketch the Nyquist diagram of H (s) in Figure 3, and indicate the direction of travel using arrows. Nyquist Diagram for H(s) lm{ H(s) } ~10 —8 —6 —4 -2 0 Rel H(s) } Figure 3: Nyquist plot of H(s). f ’0 2". 0 ad— 8 a o Problem 2: (25 Points) 1. (15 points) Figure 5 shows the Bode magnitude and phase plot of the loop transfer function for the feedback control system shown in Figure 4. R(S) - G(S) Y(S) H(s) ; Figure 4: Feedback control system. 0 (5 points) Determine the phase margin of the system, and specify the frequency pr at which the phase margin is measured. Indicate the phase margin and pr in Figure 5. .- ° wake this A 6” < "moo FM x 20 1 when \6Hl =o&8 7 Paul— .529 (own 9" o (5 points) Estimate the peak overshoot of the closed-loop system to a unit-step input. garage; ' Pm < O) the, clodeg 'Qoaip fiskm [I Uflj'tab\e..‘) MP area. 0 (5 points) What is the system Type number ? Justify your answer by citing appropriate features of the Bode magnitude and phase plots in Figure 5. 145 w —§O'. (x) ; 'I'ZOO c2) slope if 1M” {S —%OJBA)&0 719%" two obseruw’bum In’otcw‘éz’ aw isdb-km Magnitude (dB) Phase (deg) 200 150 100 50 ——1OO ~150 —200 —180 —225 Bode Diagram 101 102 Frequency (rad/sec) Figure 5: Bode magnitude and phase plot of the loop transfer function. 2. (10 points) Figure 7 shows the Nyquist diagram for the feedback control system shown in Figure 6 when the proportional control gain 06(3) = K is set to unity. Y(s) Figure 6: Feedback control system. 0 (5 points) Determine the phase margin of the system and indicate the phase margin Figure 7. U-‘ma_ av pra'bi‘abv'b') Pm *9 ‘15"0 (see; Five/<17). o (5 points) Determine the gain margin of the system in dB. 6M > == 2’7? 20 QoémléMl :- Sow Imaginary Axis Nyquis1 Diagram —1 ~05 0 0.5 Real Axis Figure 7: Nyquist diagram for K = 1. 1.5 2A5 Problem 3: (50 Points) The feedback control system in Figure 8 has the plant transfer function 1 (1%“)2 Design a compensator Gc(s) for the feedback control system in Figure 8 so that the following design specifications are met 011(3) = o The steady—state error to a unit-step input is 0.01. o The phase margin is 60°. A semilog graph is provided in Figures 9 and 10 for your analysis. In order to receive partial credit, your work must be well-organized and detailed. R(S) Gc(s) (312(8) Y(S) Figure 8: Feedback control system with compensator Gc(s). 7:016 hlan (an .12.. aLJJMQVgi “I? elflep av vfiagfl 0' PW'Q‘a. Cont/afler. However) the two £e5a7ro fi‘u/e. waft? 6]! #‘éro’ré ’tr‘t’lglej‘fé res/Manic cLaruér‘zr‘ér‘cs . 7’06 PMW' 0“? Je‘yfl Fm’waer W ,7) “i the, expense. “- Smaller Pewlc. aver, F1054: ( [~57 ‘70 we r56 -6 o % 17’) c reagefl/ Se/t’t'hi, {SM-b (32° "’5 “Q’S‘é 2”“ S/wo +l lg) We 70 “muse ' Le/KQI" 023?; n 6°C“ : S/a-wo +1 0.. > I choose K to Sw‘blfi‘ca. Sféa-oifibdfi mum? rev/.remen‘és‘. - - s=t<=> a“: -—L—"‘=ol :> ’~p"9?. kp'Q‘mar-P I+Kp " 3-190 7 I e ( w), ' Pave/ll shows flo, (So-E; muvnd'y-fi/ anfl/ phase plo‘l';% C F f l<l ' |K<>91=oéus «:4: 609m: We revil/sec. 14-1; the; emblem; Br K GP (WPM) :. - (630) arg— 5'0 Pm % k— 600 75 'l'l2.°_ ' Set $173 :‘ Pm kiwi ~ p‘m 7" 5—0 E 53". 1t Sn —_ l" $1011,” "Pg: 7- 0.33 oy—‘7.6 0v a 23 6 6mm mwo an .Q/ a. '42th W0 6w; -—- _£9.. _, '57 " ken/2K6 O (‘19; flee. Ffifl______.—-———<V Gocg : ‘7? 5/157 v] M 5/04va 1'( Da5L q 10 {K €p[ 606 25, K6, E «£73 O . . . . . . . . . . . . . . , . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. '9'0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . .. -l80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . ; I :Pm x110 2 10" ‘0‘ 10" to? Figure 9: Semilog graph for Bode magnitude and phase plots. 11 Figure 10: Semilog graph for Bode magnitude and phase plots. 12 ...
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final_f06 - EE 428 Final Exam 20 December 2006 Last Name...

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