Solution: 10.8.6.6 Wehave GH(s)= K s(s+5) 2 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 Phase margin = 90 o Gain margin = 28 dB 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 slightly better overall indicator of system stability and 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 stabilityare useful. If we raise the gain byabout 20 dB we see that the phase margin is roughly 30  and the gain margin about 10 dB, so they both indicators are telling us that the system is less stable at higher gain. -7 -6 -5 -4 -3 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 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)