Solution 5.8.6.3 (a) The loop transfer function is GH(s)= K(s +2) s(s + 1)(s +40) The MATLAB program K=linspace(0,500,200);; gh = zpk([],[0 -4 -10],1);; [R,K] = rlocus(gh,K);; plot(real(R),imag(R),'kd') print -deps rl5863a.eps generates the root locus shown in Figure 1. If you don't likethediamomds -14 -12 -10 -8 -6 -4 -2 0 -8 -6 -4 -2 0 2 4 6 8 Figure 1: Root Locus using "diamonds" you can use the statement plot(real(R),imag(R),'k.') so that the plot looks likeFigure2. 1 -14 -12 -10 -8 -6 -4 -2 0 -8 -6 -4 -2 0 2 4 6 8 Figure 2: Root Locus using "dots" (b) The loop transfer function is GH(s)= K(s +5)(s+10) s(s +20) The MATLAB program K=linspace(0,500,200);; gh = zpk([-5 -10],[0 -20],1);; [R,K] = rlocus(gh,K);; plot(real(R),imag(R),'kd') print -deps rl5863b.eps generates the root locus shown in Figure 3. If you don't likethediamomds you can use the statement plot(real(R),imag(R),'k.') so that the plot looks likeFigure4. 2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 3: Root Locus using "diamonds" (c) The loop transfer function is GH(s)= K(s +3) s(s + 1)(s +40) The MATLAB program K=linspace(0,500,200);; gh = zpk([-3],[0 -1 -40],1);; [R,K] = rlocus(gh,K);; plot(real(R),imag(R),'kd') print -deps rl5863b.eps generates the root locus shown in Figure 5. If you don't likethediamomds you can use the statement plot(real(R),imag(R),'k.') so that the plot looks likeFigure6. (d) 3 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 4: Root Locus using "dots" The loop transfer function is GH(s)= K (s +5) 3 The MATLAB program K=linspace(0,500,200);; gh = zpk([],[-5 -5 -5],1);; [R,K] = rlocus(gh,K);; plot(real(R),imag(R),'kd') print -deps rl5863c.eps generates the root locus shown in Figure 7. If you don't likethediamomds you can use the statement plot(real(R),imag(R),'k.') so that the plot looks likeFigure8. Now consider GH(s)= K s +2 : 4 -40 -35 -30 -25 -20 -15 -10 -5 0 -8 -6 -4 -2 0 2 4 6 8 Figure 5: Root Locus using "diamonds" The characteristic equation is K s +2 =;1;; whichwe can write as K =;(s +2): Then @K @K =; @s @K ;; or @s @K =;1: Thus, wewould expect the points on the root locus to be evenly spaced. Indeed, the MATLAB program K=linspace(0,10,10);; gh = zpk([],[-2],1);; [R,K] = rlocus(gh,K);; plot(real(R),imag(R),'kd') print -deps rl5863e.eps 5 -40 -35 -30 -25 -20 -15 -10 -5 0 -8 -6 -4 -2 0 2 4 6 8 Figure 6: Root Locus using "dots" produces the plot shown in Figure 9 and weseethat the points are evenly spaced. Now consider GH(s)= K(s + a) s + b ;; from which, as before, weobtain from the characteristic equation Ks+ ka =;(s + a): Then @K @K s + K @s @K + a @K @K =; @s @K ;; or sK + K @s @K + a =; @s @K : This last expression can be rearranged as @s @K =; s + a 1+K : Thus, as K increases the value of s also gets closer to ;a,sowehave the denominator increasing as the numerator decreases. In the end, as K 6 -14 -12 -10 -8 -6 -4 -2 0 -8 -6 -4 -2 0 2 4 6 8 Figure 7: Root Locus using "diamonds" continues to increase, @s @K !0;; as it must, since the closed loop pole only \reaches" the zero at s = ;a when K =1.Weseethis behavior in Figure 10 for GH(s)= K(s +6) s +2 : Finally,consider GH(s)= K (s + c) 3 c>0: Then, using the characteristic equation, we can write K =;(s + c) 3 ;; leading to @K @K =;(s + c) 2 @s @K ;; or @s @K =; 1 3(s + c) 2 : 7 -14 -12 -10 -8 -6 -4 -2 0 -8 -6 -4 -2 0 2 4 6 8 Figure 8: Root Locus using "dots" -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 9: Root Locus for GH(s)= K s+2 8 -6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Figure 10: Root Locus for GH(s)= K(s+6) s+2 In this case we knowthat all the roots are complex, but even so, if we think in terms of absolute value, it should be clear that the farther s is from ;3 the smaller the rate of change of s with K. Wesee that in general, the rate of change of s with K decreases with increasing K.However, for the GH of part(c) wesee that the rate of change is rst fast then slowthenfastagain, then slow, then fast again, and nally slow. In particular, the rate of change speeds up near break-in and break-out points. To see whyconsider the simpler loop transfer function GH(s)= K (s +2)(s+4) ;; for which, proceeding as before, weobtain @s @K =; 1 2s +6 : Note that @s @K becomes in ntitely large in magnitude as weapproach the break-out point. This happens because the denominator of this last expression is the expres- 9 sion weset equal to zero to nd the critical points when weevaluate @K @s = @(s 2 +6s +8) @s =2s +6: 10