Question

For the closed-loop system shown, and given:

G(s)=  20 * 2 / s2+  5.76s+  2  

For the closed-loop system shown, and given: G(s)-20 ,7; 576st 2- Part A - Controller Design Find the proportional gain (ie, C(s)-KP ) that would result in a rise time of tr-0.21 s vec Кр Previous Answers Request Answer Submit x Incorrect: Try Again

Answer 1

$F(s) = C(s)*G(s) = K_p* \frac{40}{s^2+5.76 s+2}$

Closed loop transfer function

$P = \frac{F}{1+F}$

$P(s) = \frac {K_p*\frac{40}{s^2+5.76 s+2}}{1+K_p*\frac{40}{s^2+5.76 s+2}}$

step input:

$C(s) = \frac {40K_p}{s(s^2+4.30s+2+40K_p)}$

Matlab Code:

clc;clear;
Kp = 0.01:0.01:10;
tr = 0.21;
err = zeros(1000,1);
for i = 1:1000
C = tf(40*Kp(i),[1 5.76 2+40*Kp(i)]);
[y,t] = step(C);
xss = y(end);
ye = y-xss;
trs = find(abs(ye)<0.05);
Tr = t(trs(1));
err(i) = Tr-tr;
end
[p a] = min(abs(err));
kp = Kp(a)

clc;clear; 1 0.01:0.01:10; tr 0.21; err = zeros (1000, 1) ; for i - 1:1000 5 C = tf ( 40 * Kp (1), [1 5.76 2+40*Kp (i)]); [y,t] - step (C) xss = y (end) ; ye = y-xss; trsfind (abs (ye) <0.05); Tr = t (trs (1)); err (1) = Tr-tr; 6 10 12 13- 14 15 en [p amin (abs (err)) kp= Kp(a) Command Window kp 2.2800