Solution 5.8.1.8 For the system of Figure 1 let G H +R C Figure 1: Standard Closed Loop Con guration GH = 10K(s +6) s(s + 4)(s +40) The rst step is to plot the poles and zeros of GH.Thepoles of GH are not the closed loop pole locations, but they can be used to nd the closed loop poles. The closed loop zeros can be found immediately: they are the zeros of G and the poles of H.Thezeros of GH also help in nding the poles of the closed loop system. The root locus is shown in Figure 2. The root locus on the real axis occurs only between the poles at s =0ands = ;4andbetween the pole at s = ;40 and the zero at s = ;6. There is a break-out pointandabreak-in pointbetween the pole at s = ;40 and the zero at s = ;6. This can be seen bycalculating the gain along the real axis in this region. The gain at selected points in this region is shown in Table 1. There is a break-out near s -22 -20 -18 -16 -14 -12 -10 K 44.55 45.7 46.2 46.1 45.5 -44.8 45.0 Table 1: Selected Gains Along Real Axis ;20 <s<;10 s = ;16 and a break-in nears = ;12. The pole zero excess is p =3;1=2: The numberofasymptotes is equal to p and hence there will be twoasymp- totes at  ` =  (1+2`) p  180  ` =0;;1 = 90  ;; 270  1 Re(s) Im(s) -4 -40 -6 Figure 2: Root Locus The twoasymptotes intersect the real axis at:  i = P polesofGH ; P zeros of GH p = ;40;4;0;(;6) 2 = ;19 The break-out pointbetween s =0and s = ;4can be found either by computing some gains along this stretchofthe real axis, or by nding the critical points of dK=ds. The break-out will be roughly in the middle of the interval [0;;;4]. 2