Type 1 Systems Examples

Let's say that we have a system that looks like this:

where G(s) is:

1
-----------------
s (s + 2)(s + 3)
Let's look at the closed-loop response for this system when we use different inputs:

## Step Input

num = 1;
den = conv([1 2],[1 3]);
den = conv(den,[1 0]);
sys = tf(num,den);
sys_cl = feedback(sys,1);
[y,t] = step(sys_cl);
u = ones(size(t));
plot(t,y,'y',t,u,'g')

## Ramp Input

num = 1;
den = conv([1 2],[1 3]);
den = conv(den,[1 0]);
sys = tf(num,den);
sys_cl = feedback(sys,1);
t = 0:0.1:100;
u = t;
[y,t,x] = lsim(sys_cl,u,t);
plot(t,y,'b',t,u,'m')
xlabel('Time(secs)')
ylabel('Amplitude')
title('Input-purple, Output-blue')

Our steady-state error is a constant.

## Parabolic Input

num = 1;
den = conv([1 2],[1 3]);
den = conv(den,[1 0]);
sys = tf(num,den);
sys_cl = feedback(sys,1);
t = 0:0.1:150;
u = 0.5*t.*t;
[y,t,x] = lsim(sys_cl,u,t);
plot(t,y,'b',t,u,'m')
xlabel('Time(secs)')
ylabel('Amplitude')
title('Input-purple, Output-blue')

Our steady-state error is infinity (the error continually increases as time goes to infinity).