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# 157aq2 - UNIVERSITY OF CALIFORNIA DAVIS Department of...

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Unformatted text preview: UNIVERSITY OF CALIFORNIA, DAVIS Department of Electrical and Computer Engineering EEC 157 A CONTROL SYSTEMS QUIZ 2 FALL 2005 80 Minutes; Closed Book; 2 pages of notes allowed 1 NAME: Sgt u-ég'ans —’ ALL PROBLEMS ARE BASED ON THE UNITY—FEEDBACK CONTROL SYSTEM: input: Mt) } output: Mt) R(s) 3(3) m) . -O(S) -P(S) error: 5(16) = ﬁt) _- y“) iHPUtOutput transfer—function: Hm(3) : 2E3 E{s) input—error transfer—function: H643) : 3(3) Problem 1 (40 points): P(s) = —-—U~L , C(s) = Kc. £0011, (0 at; (s + 3X3 + 5) . ammaﬁo‘a a) Sketch the ROOT-LOCUS (for Kc 2 U ). F k _b) Find the range of Kc such that the closed-loop system is stable: K: 0 ' 2 C. c) Using the root—locus, ﬁnd the value of Kc such that the closed—loop poles have a damping ratio C = l/w/ﬁ = 0.707 . For this Kc, estimate the percent overshoot and the settling time (to 2 % of the/ﬁnal value) of the unit—step response. a) /P-* .__1_ fdaﬁvtdgras2) weafmqrnpbfuzi ‘3‘? / (5+3 ){95} centroid ﬁgs—3:3? s _4 2 . . '2... 64mm, zomkmve A ._.:- . . ‘ 1.1“) at brmk aw‘UYix- 5) CAM 0 [5+3)(%§)+0»Z Kc: gagsﬁﬂeo. liar Mew; reﬁt-Va PM ”161-0. 2 KC *2 o a K: >- 75/ 0) gr cg; I/Q nematode fem“ *4'55’4 ._ ,3; . Kzl . WWW-=17 ._.> “’7‘ W/ é? MP; £27 («{2; 0'3?ng :4. 32?: . _ 2:? Sec// s + 4 M K > U (5+5)((8'+1)2+9)( * l a) Sketch the ROOT LOCUS (indicate the value of K at important points). b) Find the range of K (non-negative) such that the closed-loop system is stable. (3) Find the value of K such that the steady—state error 653 due to a unit step input 3“ : 1(t) is 633 : 0.05 . Is the system stable for this K ? Problem 2 (50 points): Let 0(8) 2 P(5)C(s) = K 0L) . ,1. :3— Arra" 3w? .. 3 2 1”) chm Poll (5V5 )(sﬁzw (0% ““402 5445+ (20+k)\$+ sorz’fk' F0135 if)” Orwgt fig/”ﬂaw {Ear 7(20yk) 7 5011 4K 1:7 K7-30 n 5411,1926 5% a}! nomngijMK. a) 835: “Kim '5 H “lg-'3 K‘s/(0)3 I 0x05" 11 (s + 7) 82 Problem 3 (40 points): Let (3(3) = P(s)C(s) = K ( K 2 U l a) Sketch the ROOT LOCUS (ﬁnd the value of K at all important points). b) Find the range of K (nondnegative) such that the closed-loop system is stable. Find the value of K such that the closed—loop system is critically damped. :3) Find the steadywstate error 833 due to a unit step input 1'" = 1(t). Find the steady—state error 833 due to a unit ramp input 2" = t (t) Problem 4 (50 points): Let G(s) : P(s)C(s) = m . (show important frequencies 3 and points in the following plots) ' 3) Sketch the POLAR PLOT for the frequency response of 63(3). 1)) Sketch the BODE PLOT for the frequency response of 0(3). (3) Calculate the BANDWIDTH. (:1) Find the steady-state response (of the open—loop system 0(8)) if the input is 10 COS t, t 2 O. Problem 5 {20 points}: The Bode Plot for a system, whose transfer—function is H(s), is given below. Suppose that the input to this system is 5 C08 ( 1015 + 450) 5 t 2 0. Find the steady- st-ate output (approximately). B ad 8 Diagram 0 ‘ ,,,.,,,F l l!!!"'I_—'—_'"'EEII-—!—'.'EIIE_T- _w -30 -30 ﬂm ‘50 Magnitude (dB] -50 _m _Bﬂ .....'....'..' -gg ...
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157aq2 - UNIVERSITY OF CALIFORNIA DAVIS Department of...

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