Solution: 10.8.6.8 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 = 10 dB Phase margin = 90 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. Thus phase margin is the better overall indicator of system stabilityand performance. The Bode magnitude plot is drawn for K =10, and weseethat wegetsimultaneously a large phase margin and a large gain margin, so both of these indicators of stabilityareuseful. However, looking at the Bode plots it appears that gain margin declines somewhat faster than phase margin at higher gains and hence gain margin is a slightly better indicator of stabilityathighgain. -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 -10 -8 -6 -4 -2 0 2 4 6 8 10 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)