Question

S2+ S*

Derive separate formulas for $k_{p}$ and $k_{i}$ using the equation above then calculate $k_{p}$ and $k_{i}$ to have (Peak Time) $t_{p}$ = 0.2 seconds and (Overshoot) OS = 15%.

Answer 1

Given: - s^2 + s [1 + k_pk/tau] + k_ik/tau = 0 Characteristic equation: - s^2 + 2 zeta w_n s + w^2_s = 0 Also: peak time, t_p = 0.2 sec M_p(overshoot) = 0.15 M_p = e^-zeta pi squareroot 1 -zeta^2 0.15 = e^-zeta pi squareroot 1 - zeta^2 implies zeta = 0.517 t_p = pi/w_n squareroot 1- zeta^2 implies 0.2 = pi/w_n squareroot 1 - (0.517)^2 w_n = 18.35 rad/sec On comparing the characteristic equation with given equation: 1 + k_p k/tau = 2 zeta w_n 1 + k_p k/tau = 18.975 k_p = 18.975 tau - 1/k k_i k/tau = w^2_n k_i = 336.75 tau/k

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