8.2-4. Consider the system of Problem 8.2-2. The plant frequency response \(G\left(j \omega_{w}ight)\) is given in Table P8.2-2.
(a) Let the digital controller realize a gain \(K\), that is, \(D(z)=K\). Find \(K\) such that the system phase margin is \(45^{\circ}\).
(b) Obtain the system step response of part (a) using MATLAB, and find the rise time \(t_{r}\) and the percent overshoot.

Problem 8.2-2

Table P8.2-2

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Consider the system of Fig. P8.2-2 with T = 0.2. (a) Show that the pulse transfer function of the plant is given by 1 G(z) T = 1 [36+D] [s (s + D(z) = (b) The frequency response for G(z) is given in Table P8.2-2. From this frequency response, sketch the Bode and the Nyquist diagrams for the uncompensated system, indicating the gain and phase margins. (c) Calculate the value of G(jo) as w. It is not necessary to find G(w) to calculate this value. (d) Use the results of part (b) to estimate the overshoot in the unit-step response. (e) Based on the results in part (d), are the zeros of the uncompensated system characteristic equation real or complex? Why? (f) Use MATLAB to find the rise time t,. The step response will show a 21 percent overshoot. 0.01873z + 0.01752 (z - 1)(z- 0.8187) 1--Ts S m(t) 1 s(s+1) C(s)