Solution 5.8.1.14 G H + R CΣ Figure 1: Standard Closed Loop Con guration For the system of Figure 1 wehave GH(s)= K(s + 1)(s +3) s(s +2) 2 (s +10) The rst step 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.Thezeros of GH also 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. Im(s) Re(s) -1-2-3-10 XXX Figure 2: Root Locus on Real Axis The root locus has four poles and two nite zeros. Twoofthe limbs of the root locus will end at the nite zeros for K = 1.The other two limbs 1 s ;9 ;8 ;7 ;6 ;5 ;4 ;3:5 K 9.19 16.46 21.88 25.6 28.13 32 40.95 Table 1: Gain Values for Selected Points in [;10;;;3] will end at so-called `zeros at in nity.' These zeros are at the `end' of each of the twoasymptotes. The asymptotes are at  0 =90  and  1 =270  . The asymptotes intersect at  i = [0+ (;2)+ (;2)+ (;10)];[;1+(;3)] 4;2 = ; 10 2 = ;5 There are twopotential root loci as shown in Figure 3 (a) and (b). To decide whichlocus is the correct one, weneedtolookforpotential break-in and break-out points along the real axis between s = ;10 and s = ;3. We do this byplotting the gain along this segmentofthe real axis. If there are no break-in or break out points the plot of K versus position along the real axis will look likeFigure 4 (a). If there a break-in and break-out pointdo exist the plot will look likeFigure 4 (b). The gain is plotted using Figure 5. A representativepointonthe line segmentisshown in the Figure. For this choice of s weknow that jGH(s)j = Kjs +1jjs +3j jsjjs +2j 2 js +10j =1 Thus K = jsjjs +2j 2 js +10j js +1jjs +3j For instance, for s = ;9, K = (9)(7) 2 (1) (8)(6) = 9:1875 Table 1 summarizes the calculation of the gain on the interval [;10;;;3]. Since the gain increases constantly across the interval the correct root locus is that shown in Figure 3 (a). 2 Im(s) Re(s) -1 -2-3 -10 (a) Im(s) Re(s) -1 -2 -3 -10 (b) -5 -5 x x xxx x 2 2 Figure 3: TwoPotential Root Loci 3 Im(s) Re(s) -1-2-3-10 -3-10 Re(s) K Im(s) Re(s) -1 -2 -3-10 -3-10 Re(s) K break-out break-in (b) (a) 2 2 Figure 4: TwoPotential Gain Patterns 4 Im(s) Re(s) -1 -2 -3-10 s s + 10 s + 3 s + 2 s + 1 2 Figure 5: Computation of Gain Along Real Axis 5