Solution 4.6.1.9 The characteristic equation is 1+ K(s +1) s 2 (s +4)(s +60) =0;; or s 4 +64s 3 +240s 2 + Ks+ K s 2 (s +4)(s +60) =0;; or equivalently s 4 +64s 3 +240s 2 + Ks+ K =0: The MATLAB program K=1 p=[1 64 240 K K] roots(p) K=5 p=[1 64 240 K K] roots(p) K=10 p=[1 64 240 K K] roots(p) K=100 p=[1 64 240 K K] roots(p) K=800 p=[1 64 240 K K] roots(p) K=900 p=[1 64 240 K K] roots(p) K=2000 p=[1 64 240 K K] roots(p) K=[1 5 10 100 800 900 2000] gh = zpk([-1],[0 0 -4 -60],1) [R,K] = rlocus(gh,K) plot(R,'kd') print -deps rl4619.eps 1 generates the following output EDU>sm4619 K= 1 p= 1 64 240 1 1 ans = -60.0003 -3.9967 -0.0015+ 0.0646i -0.0015- 0.0646i K= 5 p= 1 64 240 5 5 ans = -60.0015 -3.9832 -0.0077+ 0.1444i -0.0077- 0.1444i 2 K= 10 p= 1 64 240 10 10 ans = -60.0029 -3.9663 -0.0154+ 0.2044i -0.0154- 0.2044i K= 100 p= 1 64 240 100 100 ans = -60.0292 -3.6468 -0.1620+ 0.6562i -0.1620- 0.6562i K= 800 3 p= 1 64 240 800 800 ans = -60.2323 -1.2056+ 2.8874i -1.2056- 2.8874i -1.3566 K= 900 p= 1 64 240 900 900 ans = -60.2611 -1.2213+ 3.1669i -1.2213- 3.1669i -1.2964 K= 2000 p= 1 64 240 2000 2000 4 ans = -60.5740 -1.1610+ 5.3442i -1.1610- 5.3442i -1.1039 K= Columns 1 through 6 1 5 10 100 800 900 Column 7 2000 Zero/pole/gain: (s+1) ---------------- s^2 (s+4) (s+60) R= Columns 1 through 4 -60.0003 -60.0015 -60.0029 -60.0292 -3.9967 -3.9832 -3.9663 -3.6468 -0.0015+ 0.0646i -0.0077+ 0.1444i -0.0154+ 0.2044i -0.1620+ 0.6562i -0.0015- 0.0646i -0.0077- 0.1444i -0.0154- 0.2044i -0.1620- 0.6562i Columns 5 through 7 -60.2323 -60.2611 -60.5740 -1.2056+ 2.8874i -1.2213+ 3.1669i -1.1610+ 5.3442i 5 -1.3566 -1.2964 -1.1039 -1.2056- 2.8874i -1.2213- 3.1669i -1.1610- 5.3442i K= 1 5 10 100 800 900 2000 EDU> The plot of the points is shown in Figure ?? -70 -60 -50 -40 -30 -20 -10 0 -6 -4 -2 0 2 4 6 Figure 1: Plot of solutions 6