The setup for controlling a radio telescope dish is shown in Figure Q2. The desired angular position is given by 0,(s), while the actual angular position is given by 0(s). The error between actual and desired position is amplified and used to actuate a motor. The dish is acted upon by the torque provided by the motor Um (s), and also by torque from external wind disturbances Ua(s). The actual position 0 (s) is measured and fed back by a sensor which has gain K2.

Derive the closed-loop transfer function of the system when there are no wind disturbances, theta (S)/thetar (s) Show that if there is a unit step change in 0,r, then the condition for zero steady-state error in position is K1 = K,K2 + 20 In addition, it is desired that a steady-state error of exactly 0.1 rad/s is observed-when the desired position of the dish is adjusted by a ramp input of 4 rad/s. By-substituting the condition derived in part (b) above into the closed-loop transfer-function, calculate suitable values for the gains K, and K2. Calculate the steady-state position 0ss resulting from a step change in wind disturbance torque Ua of 5Nm when the reference position 0, is zero.(Find an expression in terms of the gains K1 and K2, and then substitute the values from part (c))