Solution: 10.8.1.10 The nyquist plot is shown in Figure 1. I II Re(s) Im(s) Figure 1: Nyquist Plot The gain that makes 6 GH(j!)=;180  is shown in Figure 2. For K  1000 10 ;26=20 = 19;;952 The there are no encirclements and the system is stable. That is, z = N +P =0+0=0;; and since the contour in the s-plane encircles the righthalfofthes-plane, there are no closed loop poles in the righthalfofthes-plane. For K>19;;952 there are twoencirclements and the system is unstable. That is, z = N +P =2+0=2;; and since the contour in the s-plane encircles the righthalfofthes-plane, there are twoclosed loop poles in the righthalfofthes-plane, making the system unstable. Note there are also twoclosed loop poles in the left half plane for high gain. 1 -300 -250 -200 -150 -100 -50 0 Phase in Degrees -80 -60 -40 -20 0 20 40 Magnitude in Decibels 0.001 0.01 0.1 1 10 100 1000 Frequency in Radians/Sec. Phase in Degrees Magnitude in Decibels Figure 2: Bode Magnitude and Phase Plots of GH(s)= K(s+2) s(s+1)(s+10)(s+40) for K = 1000 2