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# Step response of control system with PI controller

In this post, we will see the step response of control system with and without PI controller.

Step response without PI controller

A unity feedback closed loop system has a open loop transfer function  $G(s)= \frac {2}{s^2+s+2}$  as shown below

We can see that the steady state error for a unit step input  is 0.5 for the given system.

It is also seen that the response has settled at 0.5, so steady state error is 1-0.5=0.5 (Refer the response below)

Step response with PI controller

We saw in the video lecture that PI controller is used to decrease the steady state error without affecting the stability.

Let us see, how the  PI controller denoted by $G_c(s)$ will improve the steady state error of the system G(s) described above. So after adding PI controller in the forward path , overall closed loop looks like the diagram below, where (R(s) is the input i.e unit step function, since we are interested in step response)

Let $G_c(s)$ be $1+ \frac{1}{s}$

i.e $G_c(s) = \frac {s+1}{s}$

So the combined open loop transfer function (OLTF)  is  $G(s) G_c(s)$ = $\frac{2s+2}{s^3+s^2+2s}$

After unit step input is applied as shown in the block diagram, the unit step response looks like below.

So , we can see the difference in step response before and after adding Proportional Integral controller(PI). PI controller is used to improve steady state response in cases like these. Always remember that increasing the type of the system decreases the steady state error.

In this case, introducing PI controller in the forward path has helped us achieve  zero steady state error for step input instead of  0.5.  If you have any doubts regarding this topic, please use comments section.