In Figure, the block is pulled up the incline by the tension force f in the inextensible cable. The motor torque T is controlled to regulate the speed v of the block to obtain some desired speed vr. The precise value of the friction coefficient vf is unknown, as is the slope angle α, so we model them as a disturbance. Neglect all masses and inertias in the system except for the block mass m. Also neglect the field time constant of the field-controlled motor. Feedback of the block speed v is provided by a sensor that measures the pulley rotational speed ω, which is directly related to v by v = Rω.

a. Obtain the equation of motion of the block speed v, with the voltage vm, friction force F = μmg cosα, and the weight component Wx = mg sinα as the inputs.
b. Draw a block diagram representing the control system, with the command input iv, the output v, and the disturbance D = F + Wx. Model the speed sensor as directly sensing the speed ω. Show the necessary transfer functions for each block in the diagram. c. Obtain the output, error, and torque transfer functions from the block diagram.

Pulley Controller Pulley Speed Sensor IN Motor Block Amplifier