5.8.2.5 For the system of Figure 1 let G H C R + - Figure 1: Standard Closed Loop Con guration G(s)H(s)= K(s+1) s(s; 2)(s+40) : The root locus is shown is shown in Figure 2. The pole zero excess(pze) is two, meaning two asymptotes at 90  . The gain to place a poles the imaginary axis can be found in a number of ways. The easiest wayisto search along the imaginary axis until we nd where the angles contributions of the pole and zeros of GH sum to ;180  . If wewantto nd the break-in and break-out points then we use jGH(s)j =1;; (1) whichinthe presentcaseyields Kjs+1j jsjjs; 2jjs+40j =1;; or K = jsjjs; 2jjs+40j js+1j : (2) The vector interpretation of equation 1 is shown in Figure 3. Wesimply pickpoints along the positivereal axis to the left of the pole of GH at s =2. We knowthat all these points are closed loop poles for some value of K. Table 1 summarizes the searchusing Eq. 2. The break-out pointisnear s =0:741. To nd where the root locus crosses the imaginary axis wesolve the angle equation 6 (GH)=;180  ;; 1 XXXO Re(s) Im(s) -1 2-40 Figure 2: Root locus X X O Re(s) Im(s) 2-1 s + 1 s s - 2 -40 s + 40 Figure 3: Gain calculation along real axis s 0.72 0.73 0.74 0.735 0.741 0.739 K 21.818 21.827 21.8310 21.8296 21.8311 21.8308 Table 1: Gain values along positiverealaxis 2 X X O Re(s) Im(s) 2-1-40 s + 40 s + 1 s - 2 ω Figure 4: Finding crossing pointonimaginary axis ! 1.5 1.51 1.49 1.47 6 (GH(j!) ;178:97  ;178:62  ;179:3  ;180:02  Table 2: Gain values along positiverealaxis as shown in Figure 4. Wethus have 6 (s+1); 6 (s+40); 6 (s;2) = ;180  : Table 2 shows the search. Wesee from the table that the root locus crosses the imaginary axis at ! =1:47 rad: The gain to place poles on the imaginary axis is then K = jsjjs;2jjs+40j js+1j s=j1:47 = =82:15: Thus, the system is stable for K>82:15. 3 The MATLAB program z=1 p1 =0 p2= -2 p3 = 40 gh=zpk([-1],[0 2 -40],1) K=linspace(1,1000);; R=rlocus(gh,K);; figure(1) plot(R,'k.') print -deps 5825rl.eps s=linspace(-25,-1.5);; K1 = (abs(s +p1).*abs(s+p2).*abs(s+p3))./abs(s +z);; figure(2) plot(s,K1) print -deps 5825K1.eps generates the rootlocus shown in Figure 5, and the graph ofK versuss along the the negativereal axis, shown in Figure 6. Clearly there is a break-in point near s = ;3, and a break-out pointnears = ;17. 4 -40 -35 -30 -25 -20 -15 -10 -5 0 5 -25 -20 -15 -10 -5 0 5 10 15 20 25 Figure 5: MATLAB generated root locus 5 -25 -20 -15 -10 -5 0 250 300 350 400 450 500 Figure 6: MATLAB generated root locus 6