Solution 4.6.1.1 The characteristic equation is 1+ K s(s +4) =0;; or s 2 +4s + K s(s +4) =0;; or equivalently s 2 +4s + K =0: The MATLAB program K=0.1 p=[1 4 K] roots(p) K=0.51 p=[1 4 K] roots(p) K=4 p=[1 4 K] roots(p) K=20 p=[1 4 K] roots(p) K=50 p=[1 4 K] roots(p) K=[0.1 0.51 4 20 50] gh = zpk([],[0 -4],1) [R,K] = rlocus(gh,K);; plot(R,'kd') print -deps rl4611.eps generates the following output K= 0.1000 1 p= 1.0000 4.0000 0.1000 ans = -3.9748 -0.0252 K= 0.5100 p= 1.0000 4.0000 0.5100 ans = -3.8682 -0.1318 K= 4 p= 1 4 4 ans = 2 -2 -2 K= 20 p= 1 4 20 ans = -2.0000+ 4.0000i -2.0000- 4.0000i K= 50 p= 1 4 50 ans = -2.0000+ 6.7823i -2.0000- 6.7823i K= 0.1000 0.5100 4.0000 20.0000 50.0000 3 Zero/pole/gain: 1 ------- s(s+4) EDU> The plot of the points is shown in Figure 1 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 -8 -6 -4 -2 0 2 4 6 8 Figure 1: Plot of solutions 4