Consider the rotor shown below with inertia J and rotor angle (t). The rotor is subject to an applied torque (t) generated by a controller. A simple model of the systems dynamics is: JÖ = T The Laplace transform of the above equation, Js²(s) = T(s), can be rearranged to give the open-loop transfer function: G(s)= e(s) 1 T(s) Js2 angle Controller I Applied torque 3 Rotor 0 herba T We wish to design a controller such that it generates the appropriate torque 7 to achieve a particular desired rotor angle ea. The aim of the controller is to drive the error e(t) = a - e(t) to zero. This error is the difference between the desired and actual rotor angle. The torque controller chosen is of the form: T(f)-koe(f) + koe(f) + kie(r) where kp is a constant proportional control gain, kp is the constant derivative control gain, and k is the constant derivative control gain (forming a PID controller). The block diagram of the closed-loop system is shown below. Note that the H(s) = 1 simply means that the output rotor angle is measured directly and fed back into the control system. Recall that the integral block corresponds to the transfer function G(s) = 1, and the derivative block d/dt corresponds to the transfer function G(s) = s. KI Da(s) Ко EGO 8(5) Кр 0(5) H() 1 Question: Use block diagram reduction to simplify the above block diagram and arrive at a closed-loop transfer function of the form: GCL(S)= e(s) Od(s) For full credit express the transfer function in fully simplified polynomial form and show all intermediate block diagrams used during the reduction process.