Solution 5.8.3.4 For the system of Figure 1 let G H +R C Figure 1: Standard Closed Loop Con guration GH = K(s+2;j6)(s+2+j6)(s+10) s(s+3;j8)(s+3+j8) The rst step in drawing the root locus 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.Thepoles and zeros of GH serves as landmarks that help in nding the poles of the closed loop system. The portion of the root locus on the real axis is the shaded regions shown in Figure 2. These regions are determined byinvoking the rule that states that root locus on the real axis is found to the left of an odd countofpoles and zeros of GH. The root locus has three poles and three nite zeros. All of the limbs of the root locus will end at the nite zeros, and there are no asymptotes. The most probable root locus is shown in Figure 3. Break-in and break-out points seem unlikely since all the poles and zeros are close together. Im(s) Re(s) X X X -10 Figure 2: Root Locus on Real Axis 1 Im(s) Re(s) -10 X X X Figure 3: Probable root locus The angle of departure can be found using Figure 4. 1 + 2 + 3 ; 1 ; 2 ; 3 = ;180  : As the circle shrinks in radius all the angles except  1 can be computed. That is,  1 = 1 + 2 + 3 +180  ; 2 ; 3 = 48:81  +94:09  +116:57  ;180  ;110:57  ;90  = ;121:1  The angle of arrival can be found in the same wayusing Figure 5. 1 =  1 + 2 +  3 ; 2 ; 3 ;180  = 63:43  +108:43  +85:91  ;180  ;36:87  ;90  = ;175:96  The MATLAB dialogue EDU>gh = zpk([-4+j*4 -4-j*4 -10],[0 -2+j*2 -2-j*2],100) 2 Im(s) Re(s) -10 θ 1 θ 2 α 1 θ 3 α 3 α 2 Figure 4: Calculation of angle of Departure Im(s) Re(s) -10 θ 1 θ 2 α 1 θ 3α 3 α 2 Figure 5: Calculation of angle of arrival 3 -10 -8 -6 -4 -2 0 -8 -6 -4 -2 0 2 4 6 8 Real Axis Imag Axis Figure 6: MATLAB generated root locus Zero/pole/gain: 100 (s+10) (s^2 + 8s + 32) -------------------------- s(s^2+4s+8) EDU>rlocus(gh) EDU>print -deps rl5834f.eps EDU> draws the root locus shown in Figure 6. The MATLAB root locus program "rlocus' is not very good so the angle of departure appears to be positive even though we knowitisnegative. Figures 7 and 8, generated by the MATLAB program z1 = 2-j*6 z2 = 2+j*6 z3 = 10 p1 = 0 p2 = 3-j*8 4 p3 = 3+j*8 gh = zpk([-z1 -z2 -z3],[p1 -p2 -p3],1) K=linspace(0,0.1,1600);; [R,K] = rlocus(gh,K);; figure(1) plot(R,'k.') grid on axis([-3.3,-2.8,7.8,8.2]) print -deps 5834rl.eps K=linspace(10,100,1600);; [R,K] = rlocus(gh,K);; figure(2) plot(R,'k.') grid on axis([-2.05,-2,5.95,6.05]) print -deps 5834rla.eps s=-3+j*8 alpha1 = angle(s + z1);; alpha1d = alpha1*180/pi alpha2 = angle(s + z2);; alpha2d = alpha2*180/pi alpha3 = angle(s + z3);; alpha3d = alpha3*180/pi theta2 = angle(s + p3);; theta2d = theta2*180/pi theta3 = angle(s + p1);; theta3d = theta3*180/pi theta1d = alpha1d + alpha2d +alpha3d -theta2d -theta3d -180 showmore clearly the angles of arrival and departure 5 -3.25 -3.2 -3.15 -3.1 -3.05 -3 -2.95 -2.9 -2.85 -2.8 7.8 7.85 7.9 7.95 8 8.05 8.1 8.15 Figure 7: Angle of departure 6 -2.045 -2.04 -2.035 -2.03 -2.025 -2.02 -2.015 -2.01 -2.005 -2 5.95 5.96 5.97 5.98 5.99 6 6.01 6.02 6.03 6.04 6.05 Figure 8: Angle of arrival 7