midterm2_soln

# midterm2_soln - Name: :UFID Problem 1. A [10 points] Sketch...

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Unformatted text preview: Name: :UFID Problem 1. A [10 points] Sketch the Bode plot of the following transfer function: 2 G“) : s2 + 225 + 40 Label the :c— and y—axis of both the magnitude and phase plots carefully. _L W9= ~3—¢—-~ = a up): Wife enema) > (A Wmme r £j95+1ngl7§+Q -w 32:“! , 2 Zolca‘o—E‘B - 20‘oglakcolli4'lj ~— 201031 LCM»): +12% Problem 2. I - [10 points] Suppose the system 17(3) = is driven by the input signal \$(t) = oos(5t) + 10 sin(10t + 77/ What will be the output signal y(t) at steady state? co 3 Bl; ~w—3-l }—-> 3+an>xa S—lzclﬂ WW)“ l 5.“ 47.6 h ‘ - '71 L l V l _. +3 l0 Si 4(“2 & S'l’ﬁxle OVJYPM ‘ l l’ . - ‘ s "n ’ . :10 \Sm \ot+—+ L.- +) 310+ l < 3 ~ J10 l > : IO 9m (‘OJH 1% ~ Ma“) r) 1 (JO 3 50.35% +10 ﬁloQO'l—l—ﬂb __> g—fﬂ—ﬁ‘ Stewﬂa 54142 owﬁwl ! a 1m 7%? Dir/73' 255217 . :. 805(‘3Jpgr— r393) —l— 12. Slvx (\UJE— 0’4139) Es {TOT Problem ‘3. 1” r‘ A u. [5 + 5 = 10 points] 1 W hat is the natural fre uencv and dam in ratio of G(s) — —s:-20—7 ' q u p g ‘2s2+6s+20' 2. Let G1(s) = ———k—— and (279(5) = ‘Determine the value of 59 + 25 + 10 ‘ (5+O.O2s + 0.5)(32 + 25 + 50) I: so that the steady state outputs of G1 (8) and 02(3) in response to a unit step input are the same? . p - : n “isz , h —L +7, , 9 i. Z'CS 0),. 2) 92:10 => wﬁ:ﬁ—O ' 6““A ‘ sv—l- 3\$+10 ' 3 n I gyro“: '5 2 ~--— = O°L13L43 2W0 V x . I 4m LAer s+éxr {Mm/f" :1. S+£aolna git:le om'l‘pwl‘gl Q‘Csl In Yi/SQDOWEL I {g 1.6“03 =3 Problem 4. [10 pt] A second order transfer function H (s) has the following transient response speciﬁcations: rise time if? S 0.1 sec7 peak overshoot 1%, S 0.16, and settling time is 3 0.511 sec. Sketch (accurately) the region in the complex plane where the poles of H (5) may lie to meet these speciﬁcations. 526.0»«6‘ order Mud Problem 6. A proportional controller is to be designed (for the feedback loop shown in the ﬁgure ) to meet the following transient performance speciﬁcations: rise time t, g 0.2 sec, (ii) settling time ts S 10 sec, and (iii) peak overshoot NI?) 3 0.1 in response to a unit step command in the reference 'r(t). The‘plant is P(s) = 1 32+.2s+7' 1. Determine the expressions for the damping ratio and natural frequency of the closed loop transfer function from R(s) to Y(s) as a function of the proportional gain kp. 2. Among the three parameters: rise time, settling time, and peak overshoot of the closed loop, which can be changed by varying lap? ' 3. Determine the minimum value of the proportional gain kp so that the closed loop rise time speciﬁcation is met. ' 4. For this value of lap, What are the values of the peak overshoot and the settling time of the closed loop? \ , 14 r .. H PC _ SV+0.234 9 Pro? oy‘l‘ibMi 1 -—v —— M h. x 60‘ I K I + x ' k? “4Y0 Y- \$V+o‘zs + 9 C (5;: 1x1, ) SVA— 0:19 i- E/Jl" hﬁclwwl +veclurﬂxneg) wn == 4+“? . “1 l _ - l \$01MA ‘I wd'lb f _ ' 7' 7"9‘ _ M‘— AW)- 07% 51 {NY} IOWM l©\[,7_+u,p W96 film “El” E“ C ’H can be WW4? Rom gel/gloss cowpkax Win FAQ/a bﬁ QMO'DS‘WVJ k? >_é.99) \$3 W :1 tr: QPPHCCKAQ) + 9 M n; ‘ £5: ﬁg 4!" M : La:— ‘ﬁT—T’ ﬁnd ? is 0\ ’S'MKC-‘l‘ib‘ﬁ <3» or 0 n? 7 r _ ‘ H 80 {ﬁn iv and M8g cam lea. aangrLél lag Var/am?) \DWl' ’tg eanvw't L32, i' 1 1‘8 ~ “Nam. {raw <02 4J+ﬁ 1—?» kg: 2 #4) 7 _. Lo 11:5. _\ -' _ "“ {F}; _ ‘9 MP — UQ __ mﬂd fit-5:- J; 2> \bmai'? 2> 1%) 5“: W§n§mvxm véde Problem 7. [10 points] Consider the closed loop system shown in the : :1: Plot the (i) real axis part and (ii) the asymptotes of the root locus of the closed loop with respect to ﬁgure, where 0(5) 2 the parameter 2. v Y PC __ 5+7- , 5’0 “r 72: : Mae. ‘ - WW7) .\ + >+% ‘ "Sb E472.) (SV—l- log + Q32)?“ 5+9» 55440) ' 73/0 (5+?) 2) [H : 70(541) : 533’ by”? #05 53+lzsv+203+ 505+Z~70 ' "50 ’1 4— Z» 93+ ugh?“ \_, N Y? C.l/\0\Y‘0xC+IZ\r\SJH=L £1 35 \+ 2G. [10 points] l’(s) Problem 8. Consider the closed loop control system shown in the ﬁgure (similar to the cruise control system in the mini project), Where C(s) = 1 + is a PI controller. Sketch the 3 root locus of the closed loop poles with respect to the integral gain ki. H“ Y 34 k] Z > "’ —* —- 2 5 ‘ M ‘ R (HPC : E+ s+kq ‘7: 59+ €£+ 23+ 219 s ‘ 9+: a 5+k: 2:") H: 563+?) 1 + 1‘R: a 35+ 3}) CkaVRC‘l—Qr‘S-l-fc 1 k l —. 0 5(5+7) W .‘ 0 L“) 2 A, 22W» . YL M SCSJF?) Palﬁa‘. O and ~¥ ...
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## This note was uploaded on 06/06/2011 for the course EML 4312 taught by Professor Dixon during the Fall '07 term at University of Florida.

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midterm2_soln - Name: :UFID Problem 1. A [10 points] Sketch...

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