Solution 7.9.2.12 Σ GR C + c G p Figure 1: Nonunity feedbackand cascade compensation For Figure 1 wehave G p (s)= 5 (s + 2)(s +7) : Wewish to design G c so that the system has dominantclosed loop poles at s = ;4j4and zero steady state error to a step input, and then nd K v , and sketchthestep response. Weusethe MATLAB program s=-4+j*4 p1 = 0 p2 = 2 p3 = 7 theta1 = angle(s +p1) theta1d = theta1*180/pi theta2 = angle(s + p2) theta2d = theta2*180/pi theta3 = angle(s + p3) theta3d = theta3*180/pi alpha = theta1 + theta2 + theta3 -pi alphad = alpha*180/pi z1 = abs(real(s)) + (imag(s)/tan(alpha)) K=(abs(s+p1)*abs(s+p2)*abs(s +p3))/abs(s+z1) gcgp = zpk([-z1],[-p1 -p2 -p3],K) tc = feedback(gcgp,1) step(tc) print -deps sr79212.eps Kc = K/5 Kv = (K*z1)/(p2*p3) 1 Time (sec.) A mp li tu d e Step Response 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 0.4 0.6 0.8 1 1.4 1.2 Figure 2: Step resposne of compensated system We nd that K v = lim s!0 G c G p (s)2:29;; and the system will not trackaramp with anyaccuracy. 2