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CONTROLS FINAL

CONTROLS FINAL - Final Exam EE353U Fall 2002 Problem 1 A...

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Unformatted text preview: Final Exam, EE353U, Fall 2002 Problem 1 A realistic model of an op-amp is given by the equations below A 3+1 V01”: [V+ — V_], if =1._ 2 (a) (15 pts) Use this model to find the transfer function T(s) = of the ampliﬁcation circuit shown by Figure 1. (b) (15 pts) Compute the sensitivity function 53;. (Hint: 83,: : limAnOL—AE = Afg) {a 2R Figure 1: Problem 1 — Ampliﬁcation Circuit Problem 2 (15 pts) Consider a system with transfer function 8+1 0(3): 55+34+53+282+7S+K where K is a parameter. Is it possible to ﬁnd parameter K so that the system is stable? Justify your answer or determine the range of If that guarantees stability. Problem 3 (20 pts) Consider a feedback System shown in Figure 2. The transfer function of the plant is 0(5) 2 “1: where a is an uncertain parameter. The transfer function of the %. Show that the feedback System is stable for any positive a (Hint: ll controller is C(s) Apply Routh stability criterion). Problem 4 Consider a system shown in Figure 2. The transfer function of the plant is 0(5) = L 3—p' ll] - - _ H25. The transfer function of the controller 15 (3(3) — a) (15 pts) Determine the range of gain K and pole location 30 so that the system is stable. 1 l _L Rm + ‘3 5M Ylsl em 'A Figure 2: Problem 3 and Problem 4 b) (20 pts) Specify the gain K and pole location p so that overall closed—loop response to a unit~step input has an overshoot of no more than 25%, and a 1% settling time of no more than 0.1 sec. Final Exam, EE3530, Spring 2003 Problem 1 (15 pts) Let G be a linear stable system with frequency response shown by Figure 1. Let the input be r(t) 2 [23mm + 33in(10t)]1(t). - a -, 13-; Determine the steady state response (5107.3 Fro) i F lg /0 :2 t 0"": PM) 1 ’{ Egg/a " I" Bode Diagrams ( ‘i - d g a (3.”! KW “' l9 /0 ——r r - . I, '. gull: o a qum (and. ,( “7,1, «a ( We < I211 -5 I nr "6 : l— Fim 1! 5') .x, -10 E E g -15 2 in s. '2“ m 3’ o E 0 3 -2o E g: 40 3.5 ’9 -6o 40 .3030” 19‘ Frequency (mdlsec) Figure 1: Problem Problem 2 (20 pts) 1 - Frezquency Response Consider the feedback system shown by Figure 2. Derive the sensitivity function with respect to the variation of plant model G. What is the signiﬁcance of studying the sensitivity function? wlem 3 (15 pts) Consider a system with transfer function 32—5—1 =—,——-——T—_‘- 0(3) s5+s4+s3+2s2+73+K where K is a parameter. Is it possible to ﬁnd parameter K so that the system is stable? Justify your answer or determine the range of X that guarantees stability. ﬂoblem 4 (20 pts) Consider a feedback system shown in Figure 2. The transfer function of the plant is G (3) = ﬁ Where a is an uncertain parameter varying in (0, 00). The transfer function of the controller is C(s) = % . Determine the condition for K so that the feedback System is stable for ANY value of uncertain parameter 0,. “5) ﬁts) Figure 2: Problem 2 and Problem 4 Pro% 5 (30 pts) Consider an electronic circuit system shown in Figure 3. Assume that all resistors are of equal resistance R and that the capacitor constant C’ = %. Assume that NO input voltage (i.e., r(t) = 0) is applied to the circuit and that the switch is initially at OFF state. Predict the consequence when you close up the switch. Justify your prediction. Figure 3: Problem 5 — Circuit System ...
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