Solution 5.8.3.5 For the system of Figure 1 let G H +R C Figure 1: Standard Closed Loop Con guration GH = K(s+4) 2 s(s +1)(s+30) 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 two nite zeros. Twoofthe limbs of the root locus will end at the nite zeros, and the third limb will migrate towards s = ;1.Theroot locus is shown in Figure 3. The angle of arrival can be found using Figure 4. 2 1 ;2 1 ; 2 = ;180  : Im(s) Re(s) X X -30 X 2 zeros Figure 2: Root Locus on Real Axis 1 Im(s) Re(s) X X -30 X 2 zeros Figure 3: Probable root locus As the circle shrinks in radius all the angles except  1 can be computed. That is, 1 =  1 + 2 +  3 ;180  2 = 180  +180  +0  ;180  2 = 90  : Since the angles are modulo 360  ,wealsohave  1 = 540  2 =270  : The MATLAB dialogue: EDU>gh = zpk([-4 -4],[0 -1 -30,1) (s + 4)^2 -------------------------- s(s+1)(s+30) EDU>rlocus(gh) EDU>print -deps rl5835f.eps EDU> draws and saves the root locus shown in Figure 5. The MATLAB root locus program "rlocus' is not very good so the angle of departure appears to be positiveeven though we knowitisnegative. 2 Im(s) Re(s) θ 2 θ 3 α 1 θ 1 Figure 4: Calculation of angle of arrival -35 -30 -25 -20 -15 -10 -5 0 5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Real Axis I mag A x i s Figure 5: MATLAB generated root locus 3