Solution 5.8.6.5 The loop transfer function is GH(s)= K(s+3) s(s+ 1)(s+15) To nd the closed loop poles with real part of ;2wemust searchalong the horizontal line through s = ;j5until we nd where the root locus crosses this line. The point where the root locus crosses the vertical line is the point where the angle condition is satis ed. Referring to Figure 1 we see that we Im(s) Re(s)XX X -1 -3-15 s + 1 s s + 2 s + 15 α θ 1 θ 3 θ 2 5 -x Figure 1: Satisfying angle condition must nd s = ;x+j5suchthat ; 1 ; 2 ;  3 = ;180  : The MATLAB program z=3;; p1 = 0;; p2 =1;; p3 = 15;; s=-1.5 + j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-3+j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-4+j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-5+j*5 1 x 3 5 5.5 5.3 5.325 6 GH ;165:4  ;178:4  ;180:9  ;179:93  ;180:05  Table 1: Searchfor satisfaction of angle condition s ;13 ;10 ;5 ;5:5 ;5:47 ;5:46 ;5:465 ;5:467 K 31.2 64.29 100 94.05 94.34 94.4367 94.3877 94.3682 Table 2: Searchfor third closed loop pole ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-5.5 + j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-5.2 + j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-5.3 + j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-5.35 + j*5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-5.325 + j*5 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 = ;5:325 + j5. The gain that places twoofthe closed loop poles at s = ;5:325 j5is K = jsjjs+1jjs+15j js+3j j s=;5:325+j5 = 95:375 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+5:467)(s+5:325; j5)(s+5:325+ j5) Wecancheck this result with the MATLAB dialogue 2 EDU>g = zpk([-3],[0 -1 -15],94.3682) Zero/pole/gain: 94.3682 (s+3) -------------- s(s+1) (s+15) EDU>tc = feedback(g,1) Zero/pole/gain: 94.3682 (s+3) -------------------------------- (s+5.467) (s^2 + 10.53s + 51.78) EDU>p=[1 10.53 51.78];; EDU>roots(p) ans = -5.2650+ 4.9051i -5.2650- 4.9051i EDU> and weseethat weare reasonably close. 3