Solution 10.8.9.4 The MATLAB program % %Note when matlab takes the transpose %ofarowmatrix with complet elements %itCHANGES THE SIGN of the complex part % x=linspace(-0.001,-4,100);; s= j*x;; s=s-abs(x);; s=s' y=linspace(4,0,100);; s1 = -j*y;; s1 = s1 -4;; s1= s1' s=[s;; s1] K=10 z=3 p1 = 0 p2 = 1 p3 = 18 g=(K*(s + 3))./((s + p1).*(s + p2).*(s+p3)) realg = real(g) imagg = imag(g) figure(1) plot(realg,imagg) grid on axis([-4 0 -0.5 0.5]) print -deps 10894polara.eps figure(2) plot(realg,imagg) grid on axis([-0.5 0 -0.1 0.1]) print -deps 10894polarb.eps K=linspace(0 ,50,1000);; gcgp = zpk([-z],[-p1 -p2 -p3],10) [R,K] = rlocus(gcgp,K);; figure(3) 1 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 1: Large scale polar plot plot(R,'k.') grid on axis([-9,1,-5,5]) print -deps 10894rl.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 pointd = ;2:3 pointc = ;0:13 pointd = ;0:105 pointd = ;0:06 For K< 10 2:3 =4:3478;; there are no encirclements The poles at s =0and s = ;1areinside the contour : Thus Z = N + P =0+2=2;; 2 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 Figure 2: Small scale polar plot Re(s) Im(s) Figure 3: Contour 3 123 45 Im(G G ) cp Re( )G G cp a b c d Figure 4: Complete Nyquist plot and the closed loop poles that originate from s =0and s = ;1are inside the contour. For 4:3478<K< 10 0:13 =76:9231 There are twoclockwise encirclements and Z = N +P = ;2+2=0;; and there are no closed loop poles inside the contour. Actually 76.9 is an approximate value. The exact value is 78, as wefound in Chapter 7. For 78 <K<10=0:105 = 95:2381 there is one counterclockwise and one clockwise encirclementandso Z = N +P =0+2;; and the two dominantpolesarenowinside the contour again. For 95:2381 <K<10=0:06 = 166:7;; 4 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 Figure 5: Root locus weagain havetwocounterclockwise encirclementand Z = N +P = ;2+2=0: Finally for K>166:7 there is one counterclockwise encirclementand Z = N +P = ;1+2=1: This corresponds to the the closed loop pole that originated at s = ;18 reaching and passing to the rightofs = ;4. Thus the dominantpoles are inside the triangular area for 0 <K<4:3478 and 78 <K<95:2381: All this can be seen from the root locus in Figure 5. 5