Exercise $-$MATLAB: Proportional control system using MATLAB

Suppose we have a simple mass, spring, and damper problem

The modeling equation of this system is

$Mx+b{x}^{}+kx=F$

Taking the Laplace transform of the modeling equation $(1)$

$M{s}^{2}x(s)+bsx(s)+kx(s)=F(s)$

The transfer function between the displacement $x(s)$ and the input $F(s)$ then becomes

$\frac{x((s))}{F((s))}=\frac{1}{M{s}^2+bs+k}$

Let

$M=1kg$

$b=10N.\frac{s}{m}$

$k=20\frac{N}{m}$

$F(s)=1$

Plug these values into the above transfer function

$\frac{x(s)}{F(s)}=\frac{1}{s^2+10s+20}$

Now consider the proportional controller $($ Kp $)$ reduces the rise time, increases the overshoot, and reduces the steady$-$state error. The closed$-$loop transfer function of the above system with a proportional controller is:$\frac{x(s)}{F(s)}=\frac{K_p}{s^2+10s+(20+K_p)}$

Now, for a step input and closed loop transfer function show the effect of Kp for increasing the stability of the system.

Consider values of $Kp=1,100,200,300$

$**$ Include the response figure in your lab report with the code.

$**$ Explain how the rise time, stability, steady$-$state error changes with the change of Kp$.$

Note: The Matlab function called cloop can be used to obtain a closed$-$loop transfer function directly from the open$-$loop transfer function