Solution 8.8.5 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 -4 -3 -2 -1 0 1 2 3 4 Real Axis Imag Axis Figure 1: Root locus The overall forward loop transfer function is G c (s)G p (s)= 5K(s+3) s(s+3:2)(s+15) : The MATLAB program g=zpk([-3],[0 -3.2 -15],5) rlocus(g) print -deps rl885.eps plots and saves the root locus shown in Figure 1. Figure 2 shows howtocalculate the gain at a pointalong the negative real axis. That it, That is K =  jsjjs+3:2jjs+15j js+3j  : The MATLAB program 1 Re(s) Im(s) X X X -15 -3.2 s + 3.2 s s + 15 -3 s + 3 Figure 2: Calculating gain along negativerealaxis z=3 p1 =0 p2 = 3.2 p3 = 15 x=linspace(-10,-6,600);; K1 = (abs(x + p1).*abs(x+p2).*abs(x+p3))./abs(x + z) plot(x,K1) plots and saves the graph shown in Figure 3. As we mighthavesuspected, the break-out occurs in the vicinityofthe midpointbetween the poles at s = ;3:2 and the pole at s = ;15, at s = ;7:7563: The total gain of compensator and plantis K(s = ;7:7563) = 53:218: The MATLAB program z=3 p1 =0 p2 = 3.2 p3 = 15 x=linspace(-10,-6,600);; K1 = (abs(x + p1).*abs(x+p2).*abs(x+p3))./abs(x + z) plot(x,K1) 2 -10 -9.5 -9 -8.5 -8 -7.5 -7 -6.5 -6 48 49 50 51 52 53 54 Figure 3: Finding break-out point pause print -deps 885.eps s=x(337) K=(abs(s + p1)*abs(s+p2)*abs(s+p3))/abs(s+z) g=zpk([-3],[0 -3.2 -15],K) tc = feedback(g,1) T=linspace(0,2,200);; u=3.15*ones(1,200);; [Y,T] = lsim(tc,u,T);; plot(T,Y,'k-') print -deps sr885.eps plots and saves the step response shown in Figure 4. By comparing this response to Figure 8.22, weconlcude that wearepretty close to the break- out point. The gain of the plantis5,sothe gain required from the FAC and the power ampli er is 53:8218 5 =10:77 Since the rails of the power ampli er are 30 V, we should be able to get 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 Figure 4: Theoretical critically damped response to the break-out pointisthis case. With the zero of the PI compensator at s = ;1, the gain requirementtriples to about 30, whichisnear the rail of the power ampliferier. By moving the zero to the left wehaveimproved the performance. 4