Solution 5.8.6.6 The loop transfer function is GH(s)= K(s+3) s(s+ 1)(s+15) To nd the closed loop poles that lie along the rayfrom the origin at angle 45  to the negativereal axis, wesearch along the rayuntil we nd where the root locus crosses this line. The pointwhere the root locus crosses the vertical line is the point where the angle condition is satis ed. Referring to Figure 1 weseethat wemust nd s = ;x+ jx suchthat Im(s) Re(s)XX X -1 -3-15 s + 1 s s + 15 α θ 1 θ 3 θ 2 -x x s + 3 Figure 1: Satisfying angle condition ; 1 ; 2 ;  3 = ;180  : The MATLAB program z=3;; p1 = 0;; p2 =1;; p3 = 15;; s=-3+j*3 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-2.9 + j*2.9 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3.3+j*3.3 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi 1 x 3 2.9 3.3 3.5 3.3675 6 GH ;182:7  ;183:6  ;180:43  ;179:3  ;180:03  Table 1: Searchfor satisfaction of angle condition s ;13 ;10 ;8 ;9 ;9:25 ;9:375 ;9:27 K 31.2 64.29 78.4 72 70.21 69.29 70.06 Table 2: Searchfor third closed loop pole s=-3.5 +j*3.5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3.4 + j*3.4 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3.35 + j*3.35 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3.36 + j*3.36 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3.365 + j*3.365 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3.3675 + j*3.3675 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi generates the Table 1. Thus the root locus crosses the horizontal line for s = ;3:3675+j3:3675. The gain that places twoofthe closed loop poles at s = ;3:3675j3:3675 is K = jsjjs+1jjs+15j js+3j s=;3:3675+j3:3675 = 70:08: To nd the third closed loop pole wesearchalong the real axis to the right of s = ;15 looking for this same value of gain. The calculation is exactly the same, weare just using di erenct values of s Table 2 summarizes this search. Thus the third pole is at s = ;5:467, and the characteristic polynomial is p(s)=(s+9:27)(s+3:3675; j3:3675)(s+3:3675+j3:3675) Wecancheck this result with the MATLAB dialogue 2 EDU>g = zpk([-3],[0 -1 -15],70.06) Zero/pole/gain: 70.06 (s+3) -------------- s(s+1) (s+15) EDU>tc = feedback(g,1) Zero/pole/gain: 70.06 (s+3) ------------------------------ (s+9.27) (s^2 + 6.73s + 22.67) EDU>p =[1 6.73 22.67] p= 1.0000 6.7300 22.6700 EDU>roots(p) ans = -3.3650+ 3.3685i -3.3650- 3.3685i EDU> and weseethat weare reasonably close. 3