Consider the system of Problem 8.2-2. It is desired that the steady-state error constant for a...

   

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Consider the system of Problem 8.2-2. It is desired that the steady-state error constant for a unit-ramp input, K, 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. 8.2-2. 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 G(2) 3²(5+1) (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 →→. 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. (f) Use MATLAB to find the rise time t,. The step response will show a 21 percent overshoot. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 T 20,0 30.0 40.0 TABLE P8.2-2 Frequency Response for PROBLEM 8.2-2 G(jo) G(jo) an ZG(jw) 0.100 9.95054 19.95 -96.28 0.200 4.90325 13.80 -102.45 0.299 3.19331 10.08 -108.41 0.399 2.32198 7.31 -114.08 0.499 1.78990 5.05 -119.40 0.599 1.43046 -124.36 0.698 1.17191 0.798 0.97794 0.897 0.82799 0.996 0.70945 1.974 0.22743 2,914 0.10973 3.805 0.06511 4.636 0.04372 5.404 0.03186 6.107 0.02459 6.747 0.01980 7.328 0.01646 7.854 0.01403 11.071 0.00558 12.490 0.00352 13.258 0.00259 0.01873z + 0.01752 (2-1)(z- 0.8187) D(z) 00 FIGURE P8.2-2 3.10 1.37 -0.19 -1.63 -2.98 -12.86 -19.19 s(x+1) -23.72 -27.18 -29.93 -32.18 -34.06 -35.67 -37.05 -45.07 -49.07 -51.73 -128.95 -133.17 -137.05 -140.61 -164.43 -177.74 -187.04 -194.33 -200.38 -205.55 -210.03 -213.95 -217.40 -236.77 -243.95 -246.94 C(s) Consider the system of Problem 8.2-2. It is desired that the steady-state error constant for a unit-ramp input, K, 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. 8.2-2. 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 G(2) 3²(5+1) (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 →→. 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. (f) Use MATLAB to find the rise time t,. The step response will show a 21 percent overshoot. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 T 20,0 30.0 40.0 TABLE P8.2-2 Frequency Response for PROBLEM 8.2-2 G(jo) G(jo) an ZG(jw) 0.100 9.95054 19.95 -96.28 0.200 4.90325 13.80 -102.45 0.299 3.19331 10.08 -108.41 0.399 2.32198 7.31 -114.08 0.499 1.78990 5.05 -119.40 0.599 1.43046 -124.36 0.698 1.17191 0.798 0.97794 0.897 0.82799 0.996 0.70945 1.974 0.22743 2,914 0.10973 3.805 0.06511 4.636 0.04372 5.404 0.03186 6.107 0.02459 6.747 0.01980 7.328 0.01646 7.854 0.01403 11.071 0.00558 12.490 0.00352 13.258 0.00259 0.01873z + 0.01752 (2-1)(z- 0.8187) D(z) 00 FIGURE P8.2-2 3.10 1.37 -0.19 -1.63 -2.98 -12.86 -19.19 s(x+1) -23.72 -27.18 -29.93 -32.18 -34.06 -35.67 -37.05 -45.07 -49.07 -51.73 -128.95 -133.17 -137.05 -140.61 -164.43 -177.74 -187.04 -194.33 -200.38 -205.55 -210.03 -213.95 -217.40 -236.77 -243.95 -246.94 C(s)