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HWCE4_Sol

# HWCE4_Sol - HW/CE 4 Due Thursday 1 Suppose we have the...

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Unformatted text preview: HW/CE# 4 Due Thursday, October 20, 2009 1. Suppose we have the following speciﬁcations: MP 3 10% settling time t5 5 4 sec rise time tr 5 0.8 sec Choose phase margin PM and crossover frequency we so that we meet the above. Use the “more exact” tabulations on the second page of the reference sheets. ”’9: ”70 _+ g: 0.4 We PH :2 5'7»? @Ké’: O.é ’ ézbg=\$633 t s {’35 £— 0.23”- vathEASC r. 2.3/3 r ”ff 0'2? an, I (5) i=0: é 331: = 0-7L e-i' 91:042. (2579-: M?- we 3.. M S71M'1’ ”1+4, P7¢4i£ M «JCT; L“? M4:— 2. Make hand sketches of the Bode plots for the following controllers: 100(s + 20) b) D (s) = 0.16(s + 500) D = a) ‘m s+500 s+20 Do both magnitude and angle plots. \$9 warm“ [23163.45 63!? 35/5): Auriga-:79) = 00(29)(T:”_f_) .; _ (54—4719) fad ( E's?” +,> U) l.l.l LL! I: 0 LL! D Q6) 2/!) = 0 {a (ii—FIN) .. amKrao) (1,530+.«j ' (saggy-:0 2‘0 S _\ )5 (EB—J— I) 3. Consider the second order prototype system we have been discussing having closed loop transfer function: wz T s = " ( ) 32 + Zgwns+ a): This system has an equivalent open-loop gain for a unity feedback system of: (02 0(3) = s(s + 29w”) Assume (= 0.4 and w" = 1 176.5 radfs a) Use the tabulated values on the second page of the reference sheets and ﬁnd phase margin PM and crossover frequency wt . ‘1: 9,4,4 ——--a P17=¢3JZ° M 2—31.:- .85- ?) we? [600 rM’AC— ___—_—-I-‘"__'_. v—F——--—-— b) Make a hand sketch of the Bode plot for 6(3). (Magnitude and angle plots) 6/: t: 3.35" a ‘4 #10. HS” 1. C 3..— 1" I 5 9:01)” ) -"""' 4" 1' 4.- : 5,2 (#29,): 5;) 490.2. 1:. 3 --“-—;:---__ n.- -u--— ---n “;---—iu---—-- - -u---—MM- -— -—-—— -—-—--—- 1/900 {wavef/S" L, Um luv-ft .' 4:” PM!» c) Verify sketch results by submitting a MATLAB produced Bode plot. d) Use MATLAB to determine phase margin PM and crossover frequency we and compare results to what you obtained in part a. Bode Diagram Grn - Int {at Int realises] , Pm . 43.1 deg [at 19003 radfsec) _L Q .— h; D M '6‘ 3 ﬁ 10” g I: 3' E 10 4. Assume 6(5) =KL(S), shown below, is the open-loop gain in a unity feedback system. Make hand sketches showing the Nyquist plots for: K :- G(s)=KL(s)=m .4, 4/14? L _ I 2" ' WM“) 7‘ 49“” t 43 3 K = 200, 272, and 300. (Note that this is the same transfer function you did the root locus and Bode plot for in the last homework exercise.) Show critical points (crossover values on the real axis) for all three cases. Determine closed loop system stability for each case. Verify your sketches by submitting MATLAB generated Nyquist plots. 5* \. 69”) = f— = "- 6i” (éﬂaﬁééﬁﬂ _¢u‘+a.;w 68—03:) g; {—4J1—J'w"(48-4«Tz 3 ll xé M¢+ ,arzféa' — 4J2):- 1M;é(;~)f: 0 = —£f-¢o—C"3“W“) .‘i wee M cr=i/Z§_,= 5:95” 4(48) 97—?— 5 in z= 272— we 7"" ” k:- ‘200 “ ﬂ Fo'?§5ﬂ Fall K '-'-'r 2.12.. -—--9 Mug/Zita .SM/t. L I 390 *3 Z can/9:4 A”) Ell/J Fol/OS) duff/“4’; K? Zap ——-—; S‘éeé/g For K = 200 0.3 (15 [3.4 0.2 Imaginary Axis D ~El .8 Nyquist Diagram -E|.E Real AXIS -U4‘ {L2 For K = 272 Nyquist Diagram Imaginary Axis -1 .2 -1 {1.8 ~05 43.4 -E|.2 El For K= 300 Imaginary Axis ...
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