# Proportional-Rate Feedback Control

Consider the block diagram in the attached figure Q3.1, describing the process under Proportional-Rate Feedback control.

1) Find a closed loop system transfer function in terms of controller gains, Kp, Kd.

2) Find the value gains Kp,Kd such that the resulting Percent Overshoot of the closed loop step response will be equal to 20% and the settling time, T[settle 2%], will be equal to 1 second. Write the closed loop transfer function expression for the computed values of the controller gains. For the computed values of the controller gains, find the closed loop system DC gain, Kdc, and the closed loop steady state error for the step response, e[SS%].

3) The attached figure Q3.2 shows the system response of a certain system under two control schemes - the first one a Proportional-Rate Feedback control as show in figure Q3.1, and the second one a Proportional-Derivative control, with a unit feedback loop but with the controller Kp in the forward loop replaced by Kp(KdS+1) term. Which of the two responses correspond to which controller scheme? Label them on the graph and briefly justify your answer.

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#### Solution Preview

1. We can see that the part in the feedback can be shown as (Kds+ 1). Now considering the forward part of Kp/(s^2+ 3s+ 1), we have:

Y(s)/R(s)= [Kp/(s^2+ 3s+ 1)]/[1+ Kp(Kds+ 1)/(s^2+ 3s+ 1)]

Y(s)/R(s)= Kp/(s^2+ (KpKd+ 3)s+ 1+ Kp)

2. We have that:

s^2+ (KpKd+ 3)s+ 1+ Kp and that means: (w_n)^2= 1+Kp ...

#### Solution Summary

Calculations, answers and explanation. 205 words.