Consider the feedback control system as shown in Figure 10.93.

a. Assuming \(C(s)=k_{\mathrm{p}}\), determine the value of the proportional gain that makes the closed-loop system marginally stable. Find the frequency of the sustained oscillation.

b. Using the gain and the frequency obtained in Part (a), apply the ultimate sensitivity method of Ziegler-Nichols tuning rules to design a PID controller.

c. Plot the unit-step response of the resulting closed-loop system. Find the values of the rise time \(t_{\mathrm{r}}\), overshoot \(M_{\mathrm{p}}\), peak time \(t_{\mathrm{p}}\), and \(1 \%\) settling time \(t_{\mathrm{s}}\).

Transcribed Image Text:

FIGURE 10.93 Problem 3. $+5 Y(s) R(s) C(s) (s+s)(s+4s+13)