Solution 4.6.1.12 The characteristic equation is 1+ K(s +3) s(s + 2)(s +10) =0;; or s 3 +12s 2 +(20+K)s+3K s(s +2)(s +10) =0;; or equivalently s 3 +12s 2 +(20+K)s+3K =0: The MATLAB program K=0.5 p=[1 12 20+K 3*K] roots(p) K=1 p=[1 12 20+K 3*K] roots(p) K=5 p=[1 12 20+K 3*K] roots(p) K=10 p=[1 12 20+K 3*K] roots(p) K=20 p=[1 12 20+K 3*K] roots(p) K=50 p=[1 12 20+K 3*K] roots(p) gh = zpk([-3],[0 -2 -10],1) [R,K] = rlocus(gh,K) plot(R,'kd') print -deps rl46112.eps generates the following output EDU>sm46112 1 K= 0.5000 p= 1.0000 12.0000 20.5000 1.5000 ans = -9.9561 -1.9673 -0.0766 K= 1 p= 1 12 21 3 ans = -9.9119 -1.9314 -0.1567 K= 5 p= 2 1 12 25 15 ans = -9.5456 -1.2272+ 0.2557i -1.2272- 0.2557i K= 10 p= 1 12 30 30 ans = -9.0519 -1.4740+ 1.0684i -1.4740- 1.0684i K= 20 p= 1 12 40 60 ans = 3 -7.8968 -2.0516+ 1.8409i -2.0516- 1.8409i K= 50 p= 1 12 70 150 ans = -4.0454+ 4.6911i -4.0454- 4.6911i -3.9091 K= 0.5000 1.0000 5.0000 10.0000 20.0000 50.0000 Zero/pole/gain: (s+3) -------------- s(s+2) (s+10) R= Columns 1 through 4 -9.9561 -9.9119 -9.5456 -9.0519 -1.9673 -1.9314 -1.2272+ 0.2557i -1.4740+ 1.0684i -0.0766 -0.1567 -1.2272- 0.2557i -1.4740- 1.0684i 4 Columns 5 through 6 -7.8968 -4.0454+ 4.6911i -2.0516+ 1.8409i -3.9091 -2.0516- 1.8409i -4.0454- 4.6911i K= 0.5000 1.0000 5.0000 10.0000 20.0000 50.0000 EDU> The plot of the points is shown in Figure 1 5 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Figure 1: Plot of solutions 6