Solution 5.8.1.9 G H C R + - Figure 1: For the system shown above GH(s)= K (s + 2)(s+4))(s +10) 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 countanodd number of poles and zeros. The next step is to compute the asymptotes. p ex =3;0=3:  0 =  1+20 p ex  180  =  1 3  180  =60   1 =  1+21 p ex  180  =  3 3  180  = 180  Im(s) Re(s) -10 -2 -4 Figure 2: 1 Im(s) Re(s) -10 -4 -2 Figure 3: Completed Root Locus  1 =  1+21 p ex  180  =  5 3  180  = 300   i = Sum of poles of GH ;Sum of zeros of GH p ex = [;2;4;10];[0] 3 = ;5:33 No nite zeros, three poles so three poles migrate to `zeros' in nitely far awayatends of asymptotes (called zeros at in nity). The root locus is shown in Figure 3 2