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# Steady state error using proportional controller

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#### LostInTran

##### Newbie level 2

A system is to be controlled using a proportional controller. If the transfer function G(s) = 3/s^2 + 4s + 2;
with a unit negative feedbacj,determine the steady state error to a unit step change in set-point. Also show that if a PI controller is used in the same system the steady state offset will reduce to 0.

I will be very grateful

For a feedback system with the forward path as A(s) and feedback path is B(s), the closed loop system transfer function is given by :

T(s) = A(s) / ( 1 + A(s)B(s) )

The output to a step response would be the inverse Laplace transform of T(s) / s.

Replace A(s) with your G(s) and put B(s) = 1 since you have unity gain feedback.

This will give the transient step response for the system with proportional controller.

To get just the steady state error, you can use the final value theorem to get the final value. 1 minus that would give the steady state error.

A PI controller would have the transfer function as C(s) = Kp + (Ki / s)
Where Kp is the proportional gain and Ki is the integral gain.

For the second part of your question, replace A(s) with C(s) * G(s) and B(s) = 1
Set Kp and Ki as 1 for now and solve like before to get the transient response and the steady state error.

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