1.35 How to build and connect a closed loop control systems and show the response?

1.35.1 example 1, single input, single output closed loop

Given the following simple closed loop system, show the step response. Let mass $M=1\text{kg}$, damping coefficient $c=1\text{newton-seconds per meter}$ and let the stiffness coefficient be $k=20\text{newton per meter}$.

Using propertional controller $J(s)=k_p$ where $k_p=400$. First connect the system and then show $y(t)$ for 5 seconds when the reference $y_r(t)$ is a unit step function.

Mathematica

m = 1; c = 1; k = 20; kp = 400; 
plant      = TransferFunctionModel[1/(m s^2 + c s + k), s]; 
controller = TransferFunctionModel[kp, s]; 
sys        = SystemsModelSeriesConnect[plant, controller]; 
sys        = SystemsModelFeedbackConnect[sys]; 
o          = OutputResponse[sys, UnitStep[t], t]; 
Plot[o, {t, 0, 10}, Frame -> True, PlotRange -> All, 
  FrameLabel -> {{"y(t)", None}, 
    {"t (sec)", "response of closed loop"}}, 
    BaseStyle -> 14]
 

Matlab

close all; 
m = 1; c = 1; k = 20; 
s          = tf('s'); 
plant      = 1/(m*s^2+c*s+k); 
controller = 400; 
sys        = series(controller,plant); 
sys        = feedback(sys,1); 
[Y,T]      = step(sys,0:.01:10); 
plot(T,Y); 
xlabel('time (sec)'); 
ylabel('y(t)'); 
title('response of closed loop');
 

Another way to do the above is

m=1; 
c=1; 
k=20; 
s=tf('s'); 
sys3=1/(m*s^2+c*s+k); 
sys2=400; 
sys1=1; 
sys=append(sys1,sys2,sys3); 
Q=[2 1 -3;3 2 0]; 
input=[1]; 
output=[2 3]; 
sys=append(sys1,sys2,sys3); 
sys=connect(sys,Q,input,output); 
T=0:.01:10; 
[Y,T,X]=step(sys,T);