j Im(s) Re(s) -3-5 Figure 1: Root Locus on Real Axis Solution 5.8.1.19 The rst step is to plot the poles and zeros of GH.Thepoles of GH are not the closed loop pole locations, but they can be used to nd the closed loop poles. The closed loop zeros can be found immediately: they are the zeros of G and the poles of H. The poles and zeros of GH serves as landmarks that help in nding the poles of the closed loop system. The portion of the root locus on the real axis is the shaded regions shown in Figure 1. These regions are determined byinvoking the rule that states that root locus on the real axis is found to the left of an odd countofpoles and zeros of GH. The root locus has four poles and one nite zero. One of the limbs of the root locus will end at the nite zero for K = 1.Theother three limbs will end at so-called `zeros at in nity.' These zeros are at the `end' of each of the three asymptotes. The asymptotes are at  =60  ;;180  ;; and 300  . The asymptotes intersect at  i = [0+ (;2)+ (;2)+ (;3)+ (;5)];[0] 4;1 = ;4 The calculation of the angle of departure from the pole at s =;2+j is shown in g 2. Tosatisfy the angle condition wemust have ;( 1 +  2 +  3 +  4 )=;180  ;; where  4 is the desired angle of departure. Solving for  4 yields  4 = +180  ;( 1 +  2 +  3 ) = [180  ;tan ;1 (1=2)]+ 180  ;90  ;tan ;1 (1=1);tan ;1 (1=3) = 153:43  +90  ;45  ;18:43  = ;180  The root locus is shown in Fig. 3. 1 j Im(s) Re(s) -3-5 θ 1 θ 2 θ 3 α Figure 2: Calculation of Angle Departure from s = ;2+j1 j Im(s) Re(s) -3 -5 Figure 3: Final Root Locus 2