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Unless
specified otherwise, problems come from:
Feedback Control of Dynamic Systems, 4th Ed. By
Note: Numbers refer to problems at the
end of the chapter, Ex refers to exercises
within the chapter. Homework is due by
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#4: 3.3, 3.7(b,g), 3.8(a,c) use matlab to compare all answers |
1. (a) C11=C21=1, (b)C11=C21=0, C12=1, C21=-1, (c) C11=0, C21=1/2, (d) C11=1/4j, C12=1/2-1/2j, 2. (a) y(t)=1-t+t^3/2-e^-t (b) 1, t^2e^-2tsin(2t+phi), tsin(2t+phi) all non-zero, 3.(a)1, e^-100tcos(33t+phi), t, te^-100tcos(33t+phi),(b)all of (a)+cos(33t+phi) 4. 3.3(a) 3s/s^2+36, (b) (2+2s)/(s^2+4)+2/(s+1)^2+4, (c) 2/s^3+3/(s+2)^2+9, 3.7(b)1, -10/9, 1/9, (g) 1,1, 3.8(a)1/4, -1/4, -1/2, (c) 2, 0 -2 |
Mon, Jan.24 |
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Problem Set #2 Solution 2 Updated 1/30 |
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1.(a) Yes, 0 (b) Yes, -1/2 (c) no (d) yes, 5 (e) yes, none (f) no (g,h,i) yes, 0 2.(a) no, (b) 40ms (c) yes, no 3.(a,c) stable 4. (a) (k/JL)/(s^2+R/Ls+k^2/JL), poles=>-R/2L +-sqrt(R/2L-k^2/JL) (b) k/RJ/(s+k^2/RJ), pole=-k^2/RJ, no zero (c) L(not equal to)=0 poles = -140+-120j, L=0 poles = -123 |
Wed., Jan. 26 |
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1.(a) yes, (b) no (c) not bounded (d) yes (e) bounded, zero 2. {H1(1-H3)+H2+H3(1-H1)}/{(1-H1)(1-H3)} 3.(a)2 (b) -1 3.24 0<K<=2.86 @K=2 ->settling time = 4sec. 3.27 0.6 < zeta < 0.9 (b)K=2, KI=13/4 3.29 (a) 0.2/s+0.104 (b) 19.23 (c) 0.2/s(s+0.104) (d) 0.2K/s^2+0.104s+0.2K (e) K<6.50x10^-2 (f) K>=1.01 |
Feb. 2 |
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Do not do 2.f. |
1.a.A=[-4 0 0;1 0 -1;1 2 0], B=[1; 0;0], C=[1 20], D=0. poles=4, +-sqrt(2)j; b&c.H(s)=(s^2+2s+1)/(s+4)*(s^2+2); d.non-zero:e^-4t, cos(t+phi), cos(sqrt(2)t+phi); Not BIBO stable, bounded 2.a.[-1/RC -1/C 1/C;1/L 0 0;-1/L 0 0], [0;0;1/L],[-1 0 0], [1]; b.[-R/(Ls^2RC+sL+2R)] + 1, c.Y(s)=5(Ls^2C+RCs+1)/(s(Ls^2C+RCs+1+R)), d. 3rd order, e. No conditions 3.a.s^2+[1/(RC)]s+(1/LC), b.-(1/RC)<+-sqrt[(1/RC)^2-4(1/LC)], c.yzi(t)=-8e^-1/2t; 4.3.9e.2+1/3e^t-7/3e^-2t; 3.16a. -1, b. Can not do -> unstable pole.; 3.23a. Ldi/dt+Ri+1/Cint(i(t))dt, b.1/Cint(i(t))dt, c. 1/(s^2LC+sRC+1), d. R=40ohm;3.30 a.1.667e^-6/s(s+1/30), b. 1.667Ke^-6/(s^2+(1/30)s+1.667ke^-6), c.K<477 d. K>=304 |
Feb. 9 |
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1.a.Y(s)/R(s) = (kk_p)/(s+a+kk_p), Y(s)/D(s) = (k)/(s+a+kk_p), BIBO when a+kk_p>0, Gains: r->y, (kk_p)/(a+kk_p); d->y, (k)/(a+kk_p); No.; b. yss=193.5rad/s, ess=6.5rad/s; c. yss=.98rad/sec, ess=.1rad/s; d. Y/R = (kk_p)/(s^2+as+kk_p), Y/D = (ks)/(s^2+as+kk_p), a>0, kk_p>0, Gains: r->y 1; d->y 0, Yes to both; e. Y/R=(kk_p)/(s^2+as+kk_p); Y/D=(k)/(s^2+as+kk_p), Gains: r->y 1, d->y 1/kp, perfect tracking – not disturbance rej.; (2) a. Yes, b. No, c. No; (3) a<1, 0<b<a(1-a); (4) a+kk_D>0, 0<k_I<k_p(a+kk_D) |
Feb. 25 |
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(1) b. centroid=-8, c. centroid = (2+a-b)/2, a>0 and (b-a)>2; (2) asymptote: +-90, centroid=1, 1<k<2/3, -36degrees; (3) breakaway point at -3, k>1/2; (4) centroid=-5/3, asymptotes +-60, 180, 9<k<24, breakaway => -3, -1/3, crossing at -5, +-sqrt(3)j; (5) angles of departure = 180, +-60, angle of arrival = 0, stable all k except k=2. (6) c. natural freq. = 1, damping=0.05, peak=20dB, f. natural freq. = 10, damping=1/10, peak=14dB |
March 11 |
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4. (a) Yes, (b)GM=2, PM=90, (c)4 (d) 4, 3cos(t-90), -2cos(5t), (e) 4/3, 3sqrt(2)/2cos(t-45), -4cos(5t) |
April 1 |
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April 15 |
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April 22 |