5.8.1.44 G H + R CΣ Figure 1: Standard Closed Loop Con guration For the system of Figure 1 G(s)H(s)= K(s +1) (s;1;j)(s;1+j)(s+10)(s +20) : Tables 1, 2, and 3 provide information on possible break-in and break-out points and crossings of the imaginary axis. s -9.9 -9.5 -9 -8.5 -8 -7 -6 -5 -4 -3 -2 K 13.6 68.7 138.9 209.9 281.1 422.5 560 693.8 832 1012 1440 Table 1: Searchfor break-in/break-out points ! 1.1 2 2.5 2.6 3 4 6 6 GH(j!) ;251:5  ;197  ;182:6  ;180:5  ;174:3  ;166:9  ;166:6  Table 2: Searchforcrossing points on imaginary axis From the tables we see that there are no break-in or break-out points and the root locus crosses the imaginary axis near ! =2:6and! =10:4. The pole/zero excess (pze) is 4 ;1=3, and there is one nite zero at s = ;1. This means that one of the limbs of the root locus will terminate at the zero at s = ;1, and the other three limbs will terminate at zeros at in nity,located at the ends of the asymptotes at  =60  ;;=180  ;;and 300  : 1 ! 8 10 10.4 12 14 6 GH(j!) ;172:1  ;178:8  ;180:2  ;185:5  ;191:7  Table 3: Searchfor crossing points of imaginary axis The asymptotes intersect at  i = P polesofGH ; P zeros of GH pze = (;25;50);(;2) 3 = ;24:33: The root locus, shown in Figure 2, is generated bytheMATLAB program: K=linspace(0,1000,1000);; gh = zpk([-1],[1+j*1 1-j*1 -10 -20],10) [R,K] = rlocus(gh,K);; plot(R,'k.') print -deps rl58144.eps 2 -35 -30 -25 -20 -15 -10 -5 0 5 -20 -15 -10 -5 0 5 10 15 20 Figure 2: Accurate root locus 3