5.8.2.1 For the system of Figure 1 let G H C R + - Figure 1: Standard Closed Loop Con guration G(s)= K s +1 and H(s)= (s+2) (s;4) Webegin bylooking for the break-out pointonthepositive real axis. Table 1 summarizes the search. The root locus o the real axis is a circle centered s 0.44 0.45 0.455 0.46 K 2.10098 2.10102 2.1010 2.100975 Table 1: Searchforbreak-out point at the zero. Once we nd the break-out pointwe know the radius of the circle. From the data the break-out pointisvery close to s =0:45. Thus the radius of the circle is 2.45. Wecannow nd the point j! where the root locus crosses the imaginary axis. ! = p 2:45 2 ;2 2 = 1:415 Then the gain that places poles on the imaginary axis is K = p 1+1:415 2  p 4 2 +1:415 2 p 2 2 +1:415 2 = 3 Thus the system is stable for 3 <K<1. 1 We could also solve for the critical points of the equation dK ds = d ds ;(s 2 ;3s;4) s+2 = ;s 2 ;4s+2 (s+2) 2 : In whichcasewe nd the critical points are s = ;2 p 6: Thus, the break-out pointisats = p 6;2 and the break-in pointats = ;2; p 6. The root locus, shown in Figure 2, is generated bytheMATLAB program: K=linspace(0,20,1000) gh = zpk([-2],[-1 4],1) [R,K] = rlocus(gh,K);; plot(R,'k.') print -deps rl5821.eps 2 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Figure 2: Accurate root locus 3