Consider the rotational mass–spring–damper system. A PD controller, τ = −kpθ − kD, is designed to adjust the input torque τ so that the rotational disk can quickly return to the equilibrium position regardless of disturbances applied to the system. The performance requirements of the closed-loop system are overshoot Mp <>5% and rise time tr <>0.004 s.

a. Design a PD controller to meet the performance requirements.

b. Build a block diagram of the feedback control system, in which the plant is constructed using Simscape blocks and the controller is constructed using Simulink blocks. Find the closed-loop response if the disk is initially 0.1 rad away from the equilibrium position.