Question

The transfer function of a position control system, with load angular position as an output and motor armature voltage, is given as 1. G(s) s(s +10) For this system design the following controllers 1. Proportional controller to obtain 0.7 2. PD controller to obtain 0.7 and 2% steady-state error due to a ramp input. 3. PI controller to have a dominant pair of poles with ? = 0.7 , ??-4 rad/sec and zero steady-state error due to a ramp input 4. PID controller to obtain with ?= 0.7 and ??-4 rad/sec Compare the controllers based on their time -domain response. State your conclusions on basis of the results and graphs. Submit a report which will contain a. The mathematical work used to design the controllers in each case. b. Simulation plots showing the closed-loop response in each case. c. Simulation plots showing clearly the rise time, percent overshoot, settling time and steady state error for all states in each case.

Answer 1

Given that the transfer function G(0) = 1/s(s + 10) therefore Let G(s) for proportional controller = kp/s^2 + 103+k_p z = 07 2z wn = 10 [0.7]wn = 10_5 k_p = 51.02 G(s) = 51.02 Propotinal derivative controller G(s) = kos + k_p CLiF = k_p + k_p/s(s + 10)+k_p + k_p s^2+(10 + kD)5 + k_p = 0 ramp input k_a = lt G(s) G(s) k_a = kp/100 Ess = A/ka = 10/k_p = 10/51.02 ESS = 0.02 k_p = 500 PI controller G(s) = 21.304 + 5005 = k_ps + kI/s s^2 + 7.925 + 16.47 + kp = 0 - 16.47 + k_p = 16 k_p = 5.69 35.265 KI + 2.08 k_p = 0 K_I = - 0.45 G(s) = 5.69 - 0.45/5 PID controller G(s) = k_p + k_D3 + K_I/5 CLIF = kps + k_Ds + KI/s(s + 10)+(Kps + K_Ds^2 + KD) (10 + k_D)s^2 (10 + k_D)s^2 + KI = 0 s = squareroot - KI/10 + k_D W_d = KI/10 + k_0 28.56 + K_D 28.56 + K_I = 0 => s = - z W_n + w_n squareroot 1 - z^2 s = - 2.08