Solution 5.8.6.4 The loop transfer function is GH(s)= K(s +2) s(s + 1)(s +30) To nd the closed loop poles with real part of ;2wemust searchalong the vertical line through s = ;2until 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 -2-30 s + 1 s s + 2s + 30 α θ 1 θ 3 θ 2 Figure 1: Satisfying angle condition must s = ;2+j! such that ;  1 ; theta 2 ; 3 = ;180  : The MATLAB program z=2;; p1 = 0;; p2 =1;; p3 = 30;; s=-2+j*1 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-2+j*1.5 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-2+j*1.49 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-2+j*1.495 1 ! 1 1.5 1.49 1.495 1.496 1.4965 6 GH ;200:5  ;179:9  ;180:23  ;180:057  ;180:023  180:006  Table 1: Searchfor satisfaction of angle condition s ;28 ;26 ;27 K 58.15 108.33 84.24 Table 2: Searchfor third closed loop pole ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-2+j*1.496 ang = (angle(s + z) -angle(s +p1)-angle(s+p2)-angle(s +p3) )*180/pi s=-2+j*1.4965 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 vertical line for s = ;2+j1:4965. The gain that places twoofthe closed loop poles at s = ;2 j1:4965 is K = jsjjs +1jjs+30j js +2j j s=;2+j1:4965 = 84:24 To nd the third closed loop pole wesearchalong the real axis to the right of s = ;30 looking for this same value of gain. The calculation is exactly the same, wearejust using di erenct values of s.Table 2 summarizes this search. Thus the third pole is at s = ;27, and the characteristic polynomial is p(s)=(s + 27)(s+2;j1:4965)(s+2+j1:4965) Wecancheck this result with the MATLAB dialogue EDU>g = zpk([-2],[0 -1 -30],84.24) Zero/pole/gain: 84.24 (s+2) -------------- 2 s(s+1) (s+30) EDU>tc = feedback(g,1) Zero/pole/gain: 84.24 (s+2) ------------------------ (s+27) (s^2 + 4s + 6.24) EDU>p = [1 4 6.24];; EDU>roots(p) ans = -2.0000+ 1.4967i -2.0000- 1.4967i EDU> and weseethat wearevery close. 3