5.8.2.3 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 + 1)(s +2) The root locus is shown is shown in Figure 2. The pole zero excess(pze) is one, meaning one asymptote at  =180  .The gain to place a pole at s =0can be calcuated from the basic formula: jGH(s)j =1;; (1) whichinthe presentcaseyields K js; 1j js +1jjs +2j j s=0 =1;; or K = js +1jjs+2j js;1j j s=0 = 2 XX O Re(s) Im(s) 1 -1-2 Figure 2: Root locus 1 XX O Re(s) Im(s) 1 -1-2 s +2 s -1 s + 1 Figure 3: Gain calculation along real axis s -3 -4 -6 -5 K 0.5 1.2 2.857 2 Table 1: Gain values along negativereal axis The gain can also be found from the gure aboveby noting that js;1j = ` 1 ;; js +1j = ` 2 ;; js +2j = ` 3 ;; and K = ` 2 ` 3 ` 1 For this simple system wecould nd the second closed loop pole for a gain of 2 algebraically.However, for more complicated systems, it is more expedienttouseavector interpretation of equation 1. Figure 3 shows how the calculation is made. We simply pickpoints along the negativereal axis to the left of the pole of GH at s = ;2. Weknow that all these points are closed loop poles for some value of K. What wewantisthe particular value of s that corresponds toagainK =2.TheTable 1 summarizes the searchusing the equation K = js +1jjs +2j js; 1j Note that the value 0.5 for s = ;3tellsusto look further to the left, as we did to nd K =1:2 for s = ;4. The value K =2:857 for s = ;6tells us wehavegonetoofar, so wemovebacktothe rightto nd K =2for s = ;5. 2