j Im(s) Re(s) -40 -1 -4 Figure 1: Solution 5.8.1.17 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 on the real axis. 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 countanoddnumber of poles and zeros. The twopoles o the root locus havenoe ect, because at anypointalongreal axis their angle contributions sum to zero. The next step is to compute the asymptotes. p ex =4;1=3:  0 =  1+20 p ex  180  =  1 3  180  =60   1 =  1+21 p ex  180  =  3 3  180  = 180   1 =  1+21 p ex  180  =  5 3  180  = 300   i = Sum of poles of GH ;Sum of zeros of GH p ex = [;10;6];[;1] 3 = ;15 One nite zero, four poles so three poles migrate to `zeros' in nitely far away at ends of asymptotes (called zeros at in nity). Calculate angle of departure of root locus branch from pole at s = j4, as shown in Fig. 3. ; 1 ; 2 ; 3 ; 4 = ;180  ;; Where  1 is the angle of departure from pole at s = j4. Thus  1 = +180  ; 2 ; 3 ; 4 1 j Im(s) Re(s) -40 -1 -4 j4 -j4 θ 2 θ 3 θ 4 α Figure 2: Angle of Departure Calculation = tan ;1 (4=1);90  tan ;1 (4=6);tan ;1 (4=10) = 75:96  +180  ;90  ;33:69  ;21:8  = 110:5  The root locus is shown in Figure 4 2 j Im(s) Re(s) -10 -6 j 4 -j 4 -1 Figure 3: Angle of Departure Calculation 3