# Proportional-Rate Feedback Control System

Consider the block diagram in Figure Q3.1, describing the process from Question 2, under Proportional-Rate Feedback control (see attached).

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

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

3. Figure Q3.2 shows the system response of a certain system under two control schemes - the first one a Proportional-Rate Feedback control as shown in Figure Q3.1 and the second one a Proportional-Derivative control, with a unit feedback loop but with the controller in the forward loop replaced by a 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) The closed loop transfer function in terms of Kp and Kd will be

Kp/(s^2+3s+1)+Kp(Kds+1)

Or Kp/s^2+(3+KpKd)s+(1+Kp)

wn^2 = Kp

2*lambda*wn=(3+KpKd)

2) If settling time 2% = 1 second

Then 3.91/ lambda*wn= 1

KpKd= 4.82

Closed loop ...

#### Solution Summary

Solution includes a nice explaination with necessary plots in an attached Word file.