Question

Problem 2 Consider a unity feedback system with open-loop transfer function: K(s+10) s(s+5) (s+6) (s+8) a) Determine the segments of root locus on real axis. b) Sketch the center and the angle of the asymptotes c) Determine the value of K corresponding to point s =-4.

Answer 1

Given: - unity feedback s/m with open loop T.F G_1(s) G(s) = k(s + 10)/s(s + 5)(s + 6)(s + 8) poles: 0, -5, -6, -8 zeros: -10 it should be drawn in root locus diagram as below centroid: sigma = xi(real part of poles - xi (real part of zeros)/p - z sigma = (5 - 6 - 8) - (-10)/4 - 1 = -9/3 = -3 Angle of Assymptotes = (2q + 1) 180 degree/p - z = > theta = (2q + 1)180 degree/p - z theta = (2q + 1)/3 times 180 degree theta_1 = 60 degree rightarrow q = 0 theta_2 = 180 degree rightarrow q = 1 theta_3 = 300 degree rightarrow q = 2 Break away point: - d G(s)/d(s) = d/d(s) (s + 10)/s^4 + 19s^3 + 118s^2 + 240s = 0 1(s^4 + 19s^3 + 118s^2 + 240s) - (s + 10)(4s^3 + 57s^2 + 236s + 24) = 0 4s^4 + 57s^3 + 236s^2 + 240s + 40s^3 + 570s^2 + 2360s +