Consider the system of Problem 8.2-2. It is desired that the steady-state error constant for a unitramp input, \(K_{v}\) of (6-20), be 4, resulting in a steady-state error of 0.25.
(a) Repeat Problem 8.2-2(a), (b), (c), and (d) for this system.
(b) Use the phase margin \(\phi_{m}\) and Fig. 8-2 to estimate the characteristics \(\zeta, M_{r}\), and \(M_{p}\).
(c) Verify the value of \(M_{p}\) in part (b) using MATLAB. The step response will show that \(M_{p} \approx 1.62\).

Problem 8.2-2(a), (b), (c), and (d)

Fig. 8-2

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(a) Show that the pulse transfer function of the plant is given by 1 1 G(z) = 32 (s + 1)] = 0.01873z + 0.01752 (z - 1)(z-0.8187) (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.