Solution 5.8.5.10 For the system of Figure 1 let G H +R C Figure 1: Standard Closed Loop Con guration GH = K (s +1)(s+4)(s+20) The portion of the root locus on the real axis is the shaded regions shown in Figure 2. These regions are determined byinvoking the rule that states that root locus on the real axis is found to the left of an even countofpoles and zeros of GH. The root locus has a pole zero excess of three. Thus there are asymptotes at  1 = 360  ` 3 ` =0;;1;;2: Thus the asymptotes at  0 =0  ;;  1 =120  and 240  : The asymptotes intersect at  i = ;2;4;20 3 = ;8:67: Im(s) Re(s)XXX -4-20 -2 Figure 2: Root Locus on Real Axis 1 Im(s) Re(s)XXX -4 -20 Figure 3: Probable root locus The root locus is shown in Figure 3. Figure 3. The MATLAB dialogue EDU>gh = zpk([],[-2 -4 -20],-1) Zero/pole/gain: -0.1 ------------------ (s+2) (s+4) (s+10) EDU>K = linspace(0,1000,1000);; EDU>rlocus(gh,K) EDU>print -deps rl58510f.eps EDU> draws the root locus shown in Figure 4. 2 -20 -15 -10 -5 0 5 -20 -15 -10 -5 0 5 10 15 20 Real Axis I mag A x i s Figure 4: MATLAB generated root locus 3