Solution 5.8.1.30 G H C R + - Figure 1: For the system shown abovewehave GH(s)= K(s+20) s(s +1)(s +40) : The rst step is to plot the poles and zeros of GH in the s-plane and then nd the root locus on the real axis. The shaded regions of the real axis in Figure 2 showwhere the root locus occurs. The rule is that root locus occurs on the real axis to the left of an odd countofpoles and zeros. That is, if you stand on the real axis and look to your right you must countan odd numberofpoles and zeros. The next step is to compute the asymptotes. The numberofasymptotes is p ex =Number of poles of GH ;Numberof nite zeros of GH: Im(s) Re(s) -40 -1 -20 Figure 2: 1 The asymptotes occur at the angles  ` =  1+2` p ex  180  ` =0;;1;;:::p ex ;1: In the presentcase there are three poles and one nite zero so p ex =2. Thus, there are two asymptotes at  0 =  1+(2)(0) 2  180  =90   1 =  1+(2)(1) 2  180  =270  The asymptotes intersect the real axis at  i = Sum of poles of GH ;Sum of zeros of GH p ex = [(;1)+ (;40))];[(;20)] 2 = ;10:5 Since there is one nite zero and three poles, twoclosed loop poles will migrate to zeros at in nity,onesuchzeroatthe`end' of eachasymptote. Twopossible root loci are shown in Figure 3 The following MATLAB dialogue generartes the root locus shown in Figure 4 EDU>K = linspace(0,10000,1000);; EDU>gcgp = zpk([-20],[0 -1 -40],10) Zero/pole/gain: 10 (s+20) -------------- s(s+1) (s+40) EDU>rlocus(gcgp,K) EDU>print -deps rl58130.eps EDU> 2 Im(s) Re(s) -40 -1 -20 -10.5 Figure 3: Completed Root Locus 3 -40 -35 -30 -25 -20 -15 -10 -5 0 -400 -300 -200 -100 0 100 200 300 400 Real Axis Imag Axis Figure 4: 4