Solution 4.6.1.5 The characteristic equation is 1+ K(s +1) s(s + 4)(s+8) =0;; or s 3 +12s +(32+K)+K (s +4)(s+8) =0;; or equivalently s 3 +12s +(32+K)+K =0: The MATLAB program K=0.1 p=[1 12 32+K K] roots(p) K=0.51 p=[1 12 32+K K] roots(p) K=1.5 p=[1 12 32+K K] roots(p) K=1.8 p=[1 12 32+K K] roots(p) K=2 p=[1 12 32+K K] roots(p) K=3.4 p=[1 12 32+K K] roots(p) K=20 p=[1 12 32+K K] roots(p) K=50 p=[1 12 32+K K] roots(p) K=[0.1 0.51 1.5 1.8 2 3.4 20 50] gh = zpk([-1],[0 -4 -8],1) [R,K] = rlocus(gh,K) 1 plot(R,'kd') print -deps rl4615.eps generates the following output EDU>sm4615 K= 0.1000 p= 1.0000 12.0000 32.1000 0.1000 ans = -7.9780 -4.0189 -0.0031 K= 0.5100 p= 1.0000 12.0000 32.5100 0.5100 ans = -7.8854 -4.0988 -0.0158 2 K= 1.5000 p= 1.0000 12.0000 33.5000 1.5000 ans = -7.6420 -4.3124 -0.0455 K= 1.8000 p= 1.0000 12.0000 33.8000 1.8000 ans = -7.5614 -4.3843 -0.0543 K= 2 p= 3 1 12 34 2 ans = -7.5055 -4.4344 -0.0601 K= 3.4000 p= 1.0000 12.0000 35.4000 3.4000 ans = -7.0406 -4.8600 -0.0994 K= 20 p= 1 12 52 20 ans = 4 -5.7876+ 3.6860i -5.7876- 3.6860i -0.4248 K= 50 p= 1 12 82 50 ans = -5.6639+ 6.5043i -5.6639- 6.5043i -0.6722 K= Columns 1 through 7 0.1000 0.5100 1.5000 1.8000 2.0000 3.4000 20.0000 Column 8 50.0000 Zero/pole/gain: (s+1) ------------- s(s+4) (s+8) R= 5 Columns 1 through 4 -7.9780 -7.8854 -7.6420 -7.5614 -4.0189 -4.0988 -4.3124 -4.3843 -0.0031 -0.0158 -0.0455 -0.0543 Columns 5 through 8 -7.5055 -7.0406 -5.7876- 3.6860i -5.6639- 6.5043i -4.4344 -4.8600 -5.7876+ 3.6860i -5.6639+ 6.5043i -0.0601 -0.0994 -0.4248 -0.6722 K= 0.1000 0.5100 1.5000 1.8000 2.0000 3.4000 20.0000 50.0000 EDU> The plot of the points is shown in Figure 1 6 -8 -7 -6 -5 -4 -3 -2 -1 0 -8 -6 -4 -2 0 2 4 6 8 Figure 1: Plot of solutions 7