A DC motor speed control is described by the differential equation y + 60y = 600va - 1500w, where y( t) is the motor speed, Va ( t) is the armature voltage, and w ( t) is the load torque. Assume the armature voltage is computed using the PI control law Kp [r(t)- y(t)] + K 1 lt [r(t)- y(t)] dt,

where r(t) is the reference motor speed.

(a) Deri'ue a block diagram representation of the system.

(b) (i) Compute the transfer function from r(t) to y(t) when w(t) = 0.
(ii) What is the steady state error due to a ramp reference motor speed, r( t) = t?
(iii) Deduce the system type with respect to the reference input r(t).

(c) (i) Compute the transfer fu,nction from w(t) to y(l.) when r(t) = 0.
(ii) What is the steady state error due to a distw·bance input of the form w(t) = t ?
(iii) Deduce the system type with respect to the disturbance w(t).

(d) For the transfer function in

(b) (i), compute the values for Kp and Kr so that the characte'ristic equation of the closed-loop system will have roots at -60 ± 60j.

(e) Using the computed values of Kp and Kr, der'ive the system natuml response d·ue to the transfer function from w(t) and y(t).

(f) Verify the solut'ions to parts b (i'i), c (ii), and

(e) using MATLAB.