Question

A transfer function is given by

G(s) H (s) = s(s+1 ) (s + 8 (a) Design a Lead Compensator or PD controller such that the closed loop has the following specifications: Percent Overshoot (PO) 16 % Rise time 0.4 sec-2.16 ? + 0.6 (b) Determine the velocity error constant (Kv) of the uncompensated and compensated systems.

Answer 1

G(s) H(s) = k/s (s + 1) (s + 8) percent overshoot (p_0) = 16% t_r = 0.4 sec e^-zeta pi/squareroot 1 - zeta^z = 0.16 - zeta pi/squareroot 1 - zeta^2 = -1.83 zeta^2 pi^2 = 3.3489 - 3.3489 zeta^2 zeta^2 [pi62 - 3.3489] = 3.3489 zeta = 0.716 T_r = pi - theta/wd, theta = tan^-1 squareroot 1 - zeta^2/zeta = 44.27 6 degree = 0.75 radians. Wd = w_n squareroot 1 - zeta^2 w_n = wd/squareroot 1 - zeta^2 = 5.97/squareroot 1 - (0.716)^2 = 8.55 ras/s rightarrow p = 3, z = 0, N = p = 3, p = z = 3 starting points: 0, -1, - infinity terminating points: infinity. Infinity, infinity the angle of asymptotes: 60 degree, 180 degree, 300 degree centroid: 0 - 1 - 8/3 = -3 breakaway point: 1 + G(s) H(s) = 0 1 + k/s (s + 1) (s + 8) = 0, (s^2 + s) (s + 8) + k = 0,