1 Solution 9.10.4.2 Σ Σc (s)G G p (s) R(s) C(s) + _ + + D(s) Figure 1: UnityFeedbackwith Disturbance at the Output For the system of Figure 1, G p (s)= 2 s(s +2) and wewant T c (s)= ! 2 n s 2 +2! n s + ! 2 n ;; With  =0:8 and ! n =10. Then T c (s)= 100 (s +8;j6)(s+8+j6) To ndG c wesetD =0andnotethat T c = G c G p 1+G c G p : Solving rst for G c G p ,wehave G c G p = T c 1;T c : Then G c = 1 G p T c 1;T c : In the presentcase T c 1;T c = 100 (s +8;j6)(s+8+j6) 1; 100 (s +8;j6)(s+8+j6) 2 = 100 s 2 +16s +100;100 = 100 s(s +16) : Then G c (s) = 100 s(s +16) s(s +2) 2 = 50(s+2) (s +16) : Thus, G c simply cancels the pole at s = ;2 and replaces it byanother pole at s = ;16. The closed loop transfer function between D and C with R =0,canbe found as follows. With R =0, C d = D;C d G;; whichcan be rearranged as C d = D 1+G ;; or T d = C d D = 1 1+G = 1 1+G c G p = 1 1+ T c 1;T c = 1;T c : Wethus havetwoexpressions for T d : T d = 1 1+G c G p =1;T c : Wenow compute T d bybothmethods. T d (s) = 1 1+G c G p 3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 -100 -80 -60 -40 -20 0 20 40 Figure 2: Frequency Response of T d (j!) . = 1 1+ 50(s+ 2)(2) s(s +2)(s+16) = s(s +16) s 2 +16s +100 = 1;T c = 1; 100 (s +8;j6)(s+8+j6) = s 2 +16s+100;100 (s +8;j6)(s+8+j6) = s(s +16) (s +8;j6)(s+8+j6) b. The frequency response of T d (j!)isshown in Figure 2 The disturbance rejection at 10 r./s. is ver poor, about 0 db. To see howtoimproveitwe 4 generalize the analysis somewhat. Wehavechosen T c (s)= ! 2 n s 2 +2! n s + ! 2 n : Then T c 1;T c = ! 2 n s 2 +2! n s + ! 2 n 1; ! 2 n s 2 +2! n s + ! 2 n = ! 2 n s 2 +2! n s : Then G c (s) = T c 1;T c 1 G p = ! 2 n s 2 +2! n s s(s +2) 2 = (! 2 n =2)(s+2) s +2! n T d (s) = 1;T c (s) = 1; ! 2 n s 2 +2! n s + ! 2 n = s(s +2! n ) s 2 +2! n s + ! 2 n The situation should nowbeclear. Wecanget anynoise attentuation we wantat10rad/sbymerelyincreasing ! n .Theprice we will payisincreased gain in our compensator, since the gain of the compensator is ! n =2. As long as gain is not a problem, noise suppression is not a problem. If, how- ever, there is a limitation on gain, then there will be a limitation on noise suppression.