Solution 5.8.2.6 For the system of Figure 1 let G H +R C Figure 1: Standard Closed Loop Con guration G(s)H(s)= K(s+1) s 2 (s + 2)(s +4) The root locus is shown above. The pole zero/excess(pze) is three, mean- ing three asymptotes at  1 =60  ;;  2 =180  ;;  3 =300  : The asymptotes intersect at  i = P pole locations (of GH) ; P zero locations (of GH ) pole/zero excess = (0;0;2;4);(;1) 3 = ;1:67 The pole locations on the j!-axis can be found bysatisfying the angle condition, namely 6 GH(j!)=;180  (1) for some !. We nd ! bysearching along the j!-axis until wesatisfy 1. The calculation can be set up using Figure 2 That is 6 GH(j!)=tan ;1 (!=1);90  ;tan ;1 (!=2);tan ;1 (!=4) (2) Table 1 summarizes the search for the! that satis es this equation. From the table we see that the root locus crosses at approximately s = j1:415. Students resist this approachbecause it does not lead to 1 XXX ω α θ 1 θ 2 θ 3 2 Im(s) Re(s) Figure 2: Finding crossing pointonimaginary axis s j1 j1:5 j1:4 j1:42 1:415 6 GH(s) ;175:6  ;181:12  ;179:8  ;180:1  ;180:01  Table 1: Searchforcrossing of imaginary axis 2 s -1.5 -1.3 -1.2 -1.25 -1.27 -1.26 -1.268 K 5.63 10.64 16.1 12.9 11.9 12.3 11.99 Table 2: Search for closed loop pole to the rightofs = ;2 s -5 -4.5 -4.7 -4.8 -4.75 -4.73 4.732 K 18.75 7.23 11.28 13.58 12.4 11.95 12.0 Table 3: Searchforclosed loop pole to left of s = ;4 an analytical solution. However, as can be seen, we nd the crossing points relatively quickly.Inaddition, the process is self correcting. If wemakea mistakeinanyparticular calculation, it will generally correct itself if wedo the next one right. The gain to place poles at s = j1:415 can be calcuated from the basic formula: jGH(s)j=1;; (3) whichinthe presentcaseyields K = (jsj 2 js +2jjs+4j) js +1j s=j1:415 = 1:415 2 j2+j1:415jj4+j1:415j j1+j1:415j = 1:415 2  p 2 2 +1:415 2  p 4 2 +1:415 2 p 1+1:415 2 = 12:01 For this system we can nd the other twoclosed loop poles bysimply picking points along the negativereal axis to the rightofthe pole of GH at s = ;2andto the left of the pole at s = ;4and compute the gain. We know that all these points are closed loop poles for some value of K. What wewantisthe particular values of s that correspond to a gain K =12:01. Table 2 summarizes the searchfortheclosed loop pole to the rightofthe pole of GH at s = ;2. Wesee there is a closed loop pole very close to s = ;1:268 Table 3 summarizes the search for the closed loop pole to the left of s = ;4. 3 -6 -5 -4 -3 -2 -1 0 1 2 -4 -3 -2 -1 0 1 2 3 4 Real Axis I mag A x i s Figure 3: Accurate root locus Thus we know that for K =12 C(s)= 12 (s) s;j1:415)(s+ j:1415)(s+1:268)(s+4:732) (4) and the system is stable for 0 <K<12. The root locus is shown in Figure 3. 4