Figure P11.26a is the circuit diagram of a speed-control system in which the dc motor voltage v a is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel...

Figure P11.26a is the circuit diagram of a speed-control system in which the dc motor voltage v_a
is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not so sensitive to speed and thus can be used to drive the locomotive at various speeds. The motor voltage v_a
is varied by changing the generator input voltage v_f. The voltage v_a
is related to the generator field current i_f
by va = K_f
i_f.

Figure P11.26b is a diagram of a feedback system for controlling the speed by measuring it with a tachometer and varying the voltage v_f. Use the following values in SI units.

a. Develop a state variable model of the plant that includes the generator, the motor, and the load. Include the load torque T_L
as a disturbance.

b. Develop a proportional controller assuming all the state variables can be measured. Analyze its steady-state error for a step command input and for a step disturbance.