Question

A uncompensated (un-controlled) feedback system with and plant transfer function are shown below. Design a PI controller that you could add that will drive the steady-state error to zero for a unity step reference, and operate with a damping ratio of 0.5. Provide the resulting %OS, and 2% settling time. You must show the analytical process and all steps you took to design your controller.

Use MATLAB/Simulink to simulate the system and your feed-back controller for a unity step input R(s). Provide the time domain plot of Y(s) from 0 s ? t ? 10s

73 G(s) R(s) Your Controller Y( s)

Answer 1

G(s) = 73/(s + 1) (s + 3) (s + 10) The damping ratio of the system is, zeta = 0.5 The percentage overshoot is Given by, %OS = e^- zeta pi/squareroot 1 - zeta^2 times 100 = e^- 0.5 pi/squareroot 1 - (0.5)^2 times 100 %OS = 16.30% clc clear all Gs = zpk ([], [- 1 - 3 - 10], 73): rlocus (Gs): sgrid (0.5, 0): axis ([- 12 0 - 6 6]) From the root locus plot of the Given uncompensated system, the dominant pole of the system corresponding to the zeta = 0.5 is s_1 2 = - 1.54 plusminus 2.66j The settling time of the system is, T_s = 4/zeta omega_n = 4/1.54 T_s = 2.597 s For the uncompensated system, The static error constant is, k_ = lim_ x rightarrow 0 G(s) = lim_x rightarrow 0 (73/(s + 1) (s + 3) (s + 10)) = 73/(1) (3) (10) k_p = 2.43 The steady state error for step response is Given