Solution: 7.9.3.6 Σ G C G P + R C Figure 1: Cascade Compensation with UnityFeedback For the system of Figure 1 wehave G p = 10 s(s+1) : and G c = K(s+3) s+18 : The following MATLAB program g1 = zpk([-3],[0 -1 -18],62) h=1 tc1 = feedback(g1,h) [c1,t1] = step(tc1,5);; g2 = zpk([-3],[0 -1 -18],78) tc2 = feedback(g2,h) [c2,t2] = step(tc2,5);; g3 = zpk([-3],[0 -1 -18],94) tc3 = feedback(g3,h) [c3,t3] = step(tc3,5);; plot(t1,c1,'r-',t2,c2,'b:',t3,c3,'g-.') print -deps lead6step.eps generates: EDU>lead6 Zero/pole/gain: 62 (s+3) -------------- s (s+1) (s+18) h= 1 1 Zero/pole/gain: 62 (s+3) -------------------------------- (s+14.32) (s^2 + 4.679s + 12.99) Zero/pole/gain: 78 (s+3) -------------- s (s+1) (s+18) Zero/pole/gain: 78 (s+3) ---------------------- (s+13) (s^2 + 6s + 18) Zero/pole/gain: 94 (s+3) -------------- s (s+1) (s+18) Zero/pole/gain: 94 (s+3) -------------------------------- (s+11.29) (s^2 + 7.706s + 24.97) EDU> The three step responses are shown in Figure 2 As would be expected the system with K =94isthe fastest, although not bymuch. The advantage of this system is that the percentovershoot stays exactly the same as the gain varies by 10%. 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 2: Time Responses of Three Systems 3