% Example 8.5
% Revised February 6, 2004
% Transfer function G(s)
num=[1 6]
d1=[1 10]; d2=[1 2 2]; den=conv(d1,d2)
K=10
% Steady state unit step error
Kp=K*num(2)/den(4)
ess=1/(1+Kp)
% Closed-loop system with unity feedback (1, 1 indicate unity feedback)
[cnum, cden]=feedback(K*num,den,1,1,-1)
% Step response of the original system
t=0:0.1:5;
y=step(cnum,cden,t);
% PI controller design
nc=[1 0.1]
dc=[1 0]
numc=conv(nc,num)
denc=conv(dc,den)
[cnumc,cdenc]=feedback(K*numc,denc,1,1,-1)
yc=step(cnumc,cdenc,t)
figure (1)
plot(t,y,t,yc,'--')
xlabel('Time [s]'); ylabel('Step responses'); grid
% Step response in the different time scale
t=0:0.1:50;
yc=step(cnumc,cdenc,t)
figure (2)
plot(t,yc)
xlabel('Time [s]'); ylabel('Step response of the controlled system'); grid
% Closed loop eigenvalues
lam=roots(cden)
lamc=roots(cdenc)
% Root loci of the original and controlled systems
figure (3)
rlocus(num,den)
figure (4)
rlocus(numc,denc)