Solution 5.8.5.4 For the system of Figure 1 let G H +R C Figure 1: Standard Closed Loop Con guration GH = K (s +3) 4 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 four. Thus four asymptotes are at  1 = 360  ` 4 ` =0;;1;;2;;3: Thus the asymptotes at  0 =0  ;;  1 =90   2 =180  ;; and  3 =270  : The root locus is shown in Figure 3. Figure 3. The MATLAB dialogue Im(s) Re(s) X 4 poles Figure 2: Root Locus on Real Axis 1 Im(s) Re(s) X 4 poles Figure 3: Probable root locus EDU>gh = zpk([-1],[0 -2],-1) Zero/pole/gain: -(s+1) ------- s(s+2) EDU>K = linspace(0,10,50);; EDU>rlocus(gh,K) EDU>print -deps rl5854f.eps EDU> draws the root locus shown in Figure 4. 2 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Real Axis Imag Axis Figure 4: MATLAB generated root locus 3