Solution 4.6.1.4 The characteristic equation is 1+ K(s +2) (s +4)(s +8) =0;; or s 2 +(12+K)s +(32+2K) (s +4)(s+8) =0;; or equivalently s 2 +(12+K)s +(32+2K)=0: The MATLAB program K=0.1 p=[1 12+K 32+2*K] roots(p) K=0.51 p=[1 12+K 32+2*K] roots(p) K=4 p=[1 12+K 32+2*K] roots(p) K=20 p=[1 12+K 32+2*K] roots(p) K=50 p=[1 12+K 32+2*K] roots(p) gh = zpk([-2],[-4 -8],1) rlocus(gh) print -deps rl4614.eps generates the following output EDU>sm4614 K= 0.1000 1 p= 1.0000 12.1000 32.2000 ans = -8.1482 -3.9518 K= 0.5100 p= 1.0000 12.5100 33.0200 ans = -8.7258 -3.7842 K= 4 p= 1 16 40 ans = 2 -12.8990 -3.1010 K= 20 p= 1 32 72 ans = -29.5647 -2.4353 K= 50 p= 1 62 132 ans = -59.7924 -2.2076 Zero/pole/gain: (s+2) ----------- (s+4) (s+8) 3 EDU> The root locus is shown in Figure ?? -10 -8 -6 -4 -2 0 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Real Axis I mag A x i s Figure 1: Root locus 4