Solution: 10.8.6.7 Wehave GH(s)= K(s+1) (s+2) 2 (s+6)(s+20) The Bode phase and magnitude plots are shown in Figure 1, The root locus 10 -2 10 -1 10 0 10 1 10 2 10 3 -80 -60 -40 -20 0 20 10 -2 10 -1 10 0 10 1 10 2 10 3 -250 -200 -150 -100 -50 0 Gain margin = 18 dB Phase margin = 65 o Figure 1: Bode phase and magnitude plots of GH in Figure 2, and the polar plot of GH(I)inFigure 3. Weseefrom the root locus that we can achievealarge damping ratio over a wide range of gains. The polar plot shows that jGH(j!)j will decline rapidly once a magnitude of one is reached. The Bode magnitude plot is drawn for K =10,and wesee that wegetsimultaneously a large phase margin and a large gain margin, so both of these indicators of stabilityareuseful. Athghergains wecould still haveareasonable gain margin but very little phase margin. Thus phase margin is the better overall indicator of system stability and performance. -25 -20 -15 -10 -5 0 5 -15 -10 -5 0 5 10 15 Real Axis Imag Axis Figure 2: Root locus -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Circle of radius 1 Figure 3: Polar plot of GH(I)