Analysis Of Engineering Systems - Homework 8

David Clark MAE 488 Homework #8

10.26) Input I=[2] C=[2] s1=-10; s2=-8; A=[s1 1 (-1)*(2*(s1)^2+2*(s1));s2 1 (-1)*(2*(s2)^2+2*(s2))] Arref=rref(A) Kp1=Arref(1,3) Ki1=Arref(2,3) s1=-10; s2=-20; B=[s1 1 (-1)*(2*(s1)^2+2*(s1));s2 1 (-1)*(2*(s2)^2+2*(s2))] Brref=rref(B) Kp2=Brref(1,3) Ki2=Brref(2,3) s1=-10; s2=-50; C=[s1 1 (-1)*(2*(s1)^2+2*(s1));s2 1 (-1)*(2*(s2)^2+2*(s2))] Crref=rref(C) Kp3=Crref(1,3) Ki3=Crref(2,3) %part B sys1=tf([Kp1, Ki1],[I, (2+(Kp1)), Ki1]) step(sys1) sys2=tf([Kp2, Ki2],[I, (2+(Kp2)), Ki2]) step(sys2) sys3=tf([Kp3, Ki3],[I, (2+(Kp3)), Ki3]) step(sys3) Output I= 2 C= 2 A= -10 -8

1 -180 1 -112

Arref = 1 0 34 0 1 160

David Clark MAE 488 Homework #8

Kp1 = 34 Ki1 = 160 B= -10 1 -20 1 Brref = 1 0 0 1 Kp2 = 58 Ki2 = 400 C= -10 -50 Crref = 1 0 Kp3 = 118 Ki3 = 1000

-180 -760 58 400

1 1 0 1

-180 -4900 118 1000

David Clark MAE 488 Homework #8

Transfer function: 34 s + 160 -----------------2 s^2 + 36 s + 160

David Clark MAE 488 Homework #8

Transfer function: 58 s + 400 -----------------2 s^2 + 60 s + 400

David Clark MAE 488 Homework #8

Transfer function: 118 s + 1000 -------------------2 s^2 + 120 s + 1000

10.42) Input s1a=(-0.5) s2a=(-5)-5i s3a=(-5)+5i A=[(s1a)^2, (s1a), 1, (-10*(s1a)^3-2*(s1a)^2); (s2a)^2, (s2a), 1, (-10*(s2a)^3-2*(s2a)^2); (s3a)^2, (s3a), 1, (-10*(s3a)^3-2*(s3a)^2)] Arref=rref(A) Kd1=Arref(1,4) Kp1=Arref(2,4) Ki1=Arref(3,4) s1b=(-0.5) s2b=-1 s3b=-2 B=[(s1b)^2, (s1b), 1, (-10*(s1b)^3-2*(s1b)^2);

David Clark MAE 488 Homework #8

(s2b)^2, (s2b), 1, (-10*(s2b)^3-2*(s2b)^2); (s3b)^2, (s3b), 1, (-10*(s3b)^3-2*(s3b)^2)] Brref=rref(B) Kd2=Brref(1,4) Kp2=Brref(2,4) Ki2=Brref(3,4) sys1=tf([Kd1, Kp1, Ki1],[10, (2+Kd1), Kp1, Ki1]) step(sys1) sys2=tf([Kd2, Kp2, Ki2],[10, (2+Kd2), Kp2, Ki2]) step(sys2) bode(sys1) bode(sys2) Output s1a = -0.5000 s2a = -5.0000 - 5.0000i s3a = -5.0000 + 5.0000i A= 1.0e+003 * 0.0003 -0.0005 0.0010 0 + 0.0500i -0.0050 - 0.0050i 0.0010 0 - 0.0500i -0.0050 + 0.0050i 0.0010 Arref = 1 0 0 103 0 1 0 550 0 0 1 250 Kd1 = 103 Kp1 = 550 Ki1 = 250 s1b = -0.5000 s2b = -1 s3b = -2 B= 0.2500 -0.5000 1.0000 0.7500 1.0000 -1.0000 1.0000 8.0000 4.0000 -2.0000 1.0000 72.0000 Brref = 1 0 0 33

0.0008 -2.5000 + 2.4000i -2.5000 - 2.4000i

David Clark MAE 488 Homework #8

0 0

1 0

0 1

35 10

Kd2 = 33 Kp2 = 35 Ki2 = 10 Transfer function: 103 s^2 + 550 s + 250 -----------------------------10 s^3 + 105 s^2 + 550 s + 250 Transfer function: 33 s^2 + 35 s + 10 --------------------------10 s^3 + 35 s^2 + 35 s + 10

David Clark MAE 488 Homework #8

David Clark MAE 488 Homework #8

10.53) Input Kp1=18; Kl1=40; Kp2=18; Kl2=20; Kp3=108; Kl3=200; sys1=tf( [Kp1, Kl1-20*Kp1, -20*Kl1],[-100, -40-20*Kp1, -20*Kl1] ) sys2=tf( [Kp2, Kl2-20*Kp2, -20*Kl2],[-100, -40-20*Kp2, -20*Kl2] ) sys3=tf( [Kp3, Kl3-20*Kp3, -20*Kl3],[-100, -40-20*Kp3, -20*Kl3] ) step(sys1, sys2, sys3) sys1T=tf( [5*Kp1, (5*Kl1-100*Kp1+2*Kp1), (-100*Kl1+2*Kl1-40*Kp1), (-40*Kl1)], [-100, -40, -40*Kp1-20*Kl1]) step(sys1T)

David Clark MAE 488 Homework #8

Output

10.55) Input Kd=58; Kp=55; sys=tf([10,(2+Kd),Kp,0],[1]) rlocus(sys)

David Clark MAE 488 Homework #8

Output