5.8.1.37 For the system of Figure 1 let For the system of Figure 1 G H C R + - Figure 1: Standard Closed Loop Con guration G(s)H(s)= K(s +1) s 2 (s +20)(s +40) : From the Table 1 weseethat there is a break-in pointnear s = ;2:2and abreak-out pointnears = ;8 s -2.1 -2.2 -2.3 -2.4 -9 -8 -6 K 2720 2714 2715 2723 3453 3511 3427 Table 1: Locating break-in and break-out points 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  : The asymptotes intersect at  i = P poles of GH ; P zeros of GH pze = (;20;40);(;1) 3 = ;19:67 The root locus, shown in Figure 2, is generated bytheMATLAB dialogue: 1 EDU>K=linspace(0,1000,1000);; EDU>gh=zpk([-1],[0 0 -20 -40],10) Zero/pole/gain: 10 (s+1) ----------------- s^2 (s+20) (s+40) EDU>rlocus(gh,K) EDU>print -deps rl58137.eps EDU> -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -15 -10 -5 0 5 10 15 Real Axis I mag A x i s Figure 2: Accurate root locus 2