Solution 10.8.9.6 The MATLAB program % % Note when matlab takes the transpose %ofarow matrix with complet elements %itCHANGES THE SIGN of the complex part % x=linspace(-0.001,-40,500);; s= j*x;; s=s-abs(x);; s=s';; K=10 z=2 p1 = 0 p2 = 1 p3 = 40 g=(K*(s + z))./((s + p1).*(s + p2).*(s+p3));; angg=angle(g)*180/pi;; realg = real(g);; imagg = imag(g);; figure(1) plot(realg,imagg) grid on axis([-1.0 0 -0.6 0.1]) print -deps 10896polara.eps figure(2) plot(realg,imagg) grid on axis([-0.5 0 -0.06 0.02]) print -deps 10896polarb.eps K=linspace(0 ,200,1000);; gcgp = zpk([-z],[-p1 -p2 -p3],10) [R,K] = rlocus(gcgp,K);; figure(3) plot(R,'k.') grid on axis([-40,1,-21,21]) print -deps 10896rl.eps Draws the polar plots shown in Figures 1 and 2, for the contour shown in Figure 3. The complete Nyquist plot is shown in Figure 4 For K =10 pointa = ;0:33 1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 Figure 1: Large scale polar plot -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 Figure 2: Small scale polar plot 2 Re(s) Im(s) Figure 3: Contour Im(G G ) cp Re( )G G cp 8 b c a Figure 4: Complete Nyquist plot 3 pointb = ;0:18 pointc = ;0:02 For K< 10 0:33 =30:303;; there are no encirclements. All the poles are inside the contour : Thus Z = N +P = 0+3 = 3;; For 30:303 <K< 10 0:18 =55:56 There is one clockwise encirclements and Z = N + P = ;2+3 = 1;; and there is one closed loop pole inside the contour. This means that the closed loop poles that started at s = ;1ands =0are now inside . The closed loop pole that originated at s = ;40 is still inside . However, the two dominantpoles are not far outside as can be seen from the root locus in Figure 5. For 55:56 <K<10=0:02= 500 there is one counterclockwise and one clockwise encirclementandso Z = N +P = ;1+1+3 = 3;; and the all three closed loop poles are now inside the contour again. Finaly, for 500 <K 4 weagain havetwp clockwise encirclements and Z = N + P = ;2+3 = 1;; and the pole that originated at s = ;40 is still inside the but the dominant poles that started at s =0and s = ;1nolonger are. Thus the dominantpoles are inside the triangular area for 0 <K<30;;303 and 55:56 <K<500: All this can be seen from the root locus in Figure 5, and, indeed, for all practical purposes, the dominantpoles are inside for 0 <K<500: 5 -40 -35 -30 -25 -20 -15 -10 -5 0 -20 -15 -10 -5 0 5 10 15 20 Figure 5: Root locus 6