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Unformatted text preview: Final Exam, EE3530, Spring 2002 PyMe/rl; 1 (10 pts). Design .Qntroller using quarter ratio method for syétem with the
following step response. RT '.' (I ' \J,‘ . 1B ‘ . > ‘ ' , A V x V ; . i ‘ Figure 1: Problem 1 —— Déaign PI Controller ' , 5 gr ' ‘ ' k.  M:
W 2 (15 pts) Find the range of which all the rodts of the followipg polynomial are
in the ogen'lefthaliplanem; h ‘ I " I 3/"
.\ "'x‘ , ,'\ sﬂwﬁ+mﬁ+mﬁ+a+f=a‘g
,.  ‘ > , ‘ . V..,..'.,.. _.. .._~...  :_. I
L117 CE a? a QLf 71) w""' H Pym: 3 Consgder a. systemlshown in Figure 2. 1'Ii‘hei transfer, funﬂiqufjcﬁe plant is 9(8) =
I) ' _ , . m. ..._. , . "1 v .m V... _ ‘ (I s; $1.156 . The transfer function of theEélifmlier (s) = .K‘, Le” the Cami0115; is at”?
M“ proportional controller with gain K. . ‘ _ W Q a) (15 pts) Detergﬁne the range of K so that; the 3313mm is stable. L b) (15 pts) Determine the range of. v 90 the' steadystate error MessShir: 2101 for
‘9 reference sipal 9‘0?) = t1 ‘ I giggleﬁi'é'Coﬁsider a system shown in'Figm:e ZMagfer funcfgioxx of tﬂe plagg is G (a ..—...._V .—~ .. ~ ~  )
. _  p: r _...,. a: The tramsferfgﬂgtion of the controller is _ ' ‘ ' .  V. V . , a) (10 pts) Determine the range~oﬁgaipggandpole location a so that the system is stable.
w v  . . . v . __— ———— ~ , / b) C 15 P138) Speciﬁ' the gain K and pole location a so that overall cloaedioop response to a 'a— 4.2..‘__ unitstep input haa'an overshoot of do more {him 25%, ade a 1% Settling time of no more than 0.1 see. > « RM Figure 2: Problem 3 29nd Problem 4 1/," ; ‘ .
' Richly/é 10 ] Causideza_ system with transfer function g
I [/f ‘ ’ . V '
6(5) = (ad—gOOO) Compute the the overshoot andf\se£ﬂir!1g time of the step response of. the system. (Hint:
Truncate the system 33 a aegﬁnd order system by éﬁminaﬁng nondomjnwé poles and zeros.) ‘ Problem 6 (1 pts) Show that a. systém with transfer function 6(5) I I > K an ampliﬁer with conﬁrm W __._.... . x ‘ w " Final Emm, EE3530, 173112003 '/I:;oblem 1 (15/ pts) Let. G be a linear ntabla system with frequency response shown by Figure
/
\\ input be
yﬁpsinﬁ) + 3cos(10t)]1(t). Determine the steady state. response Mt). Bo'du Dingmmc From U11) qumﬂmdhﬂt)
Figure: 1: Problem 1 — Frequency Response Problem 2 (20 ' 5) Consider the feedback system shown by Figure 2. Derive the aarwitivitjr function with
gmﬁect to the variation of plant. model G. What. is the signiﬁcanne of atudyizig the 3W function? Problem 3 (15 pts] Consider a systém with transfer function x" a  1
a = ———~——___._
V’m 85+34+33+23,2+75+K '
where K is a parameter. Is it. possible to find parameter K so thaw the system in stable? Justify your answer or determine the range of K that. guarantees ahabﬂity. V ji
Problem 4 (20 pie) Consider a. feedback system shown in Figure 2 T he transfer ﬁmcim 0f
thep13, 13 0(3) = F—i; Where p is a negative uncertain parameter. The transfer function of the controller is 0(a) = if . Detormine the condltion for K so that the feedhadc system. is stable for ANY value of the uncertain parameter 3:. GI!) YM Figure 2: Problem 2. Problem 4 midProblem 5 r' Problem 5 .1
Consider a feedback system shown in Figure 2. The transfer fundan of His plant is G'(s) = m . '
V . 1:;1'3117 ‘ (a) (15 pts) Suppose that; the reference input signal in r(t) = 1(t‘)iand that C(o) = K is a.
proportional controller with gain K. Can the steady state error be arbitrarily decreased by increasing the controller gain K? Justify your answer. (13) (15 pta) Suppose that the reference. input signal is r(t) = t 105) and that 0(3) = § is an integral controller with. gain X: What is the smallest: error (an be whiéved by increasing the controller gain K Justify yoiir mower. Zl+=i_=0. (a) (15 pts) Ugﬁhismndeuo ﬁnd the transfer function T(s) = the ampliﬁcation circuit shown by Figure 1. A: ~—— — / @15 pts) Compute the sensWioa Si. (Hint: 3'}; = limAqu‘ I?! = jig) Figure 1': Problem 1  Ampliﬁcation Circuit Problem 2 (15 pts) Consider 5, system with transfer function
. J 6(8); 3+1 r” .
‘55+‘s4+s3+252+73+K / K/where K is a parameter. Is it. possible to ﬁnd parameter K 50 that (he system is stableY Justify. your answer or detemline the range of X that. guarantees stability.
' ' I I I ’ 'I i 1
Proplem 3 (20 pts) Consider a feedback system ‘shown in Figue 2._ The tramsfer function of the plant is' G'(s) = where daisan: uncertain parameter, The 'trailsfer funC'ﬁOn 0f the f,
I  > _ M ‘ V _ I
controller is C(s) = éffshow that. the feedback system is stable for positive a (Hint: \r /
If
Apply Rout}: stability criterion). Problem 4 Consider a "system shown in Figure 2, The'tmﬁsfer functioﬁ o:£ the plant is 9(3) = $4 The transfer functioﬁef like, Egntroller is 0(a) = {1%. ,( 15 pts)‘ Determine the range of gain. K endjpole location}: so thatithe s§retem is stable. .1' ‘ Figure 2: Problem 3 and Problem 4 b) (20 1)sz Specify the gain K and pole logation 32 so that overall closedloop réépbnse to a.
14612 input has an' overshoot of no more than 25%, and a 1% settling {time of no more
\j ' than 0.1 sec. ...
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 Fall '07
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