. Consider a system consisting of a motor driving two masses that are connected by a torsional spring, as shown in the diagram below. Note that w₁ = 1 and w₂ = 2. Φι Motor 001 J₁ 92 02 √2 Figure 1: The schematic of a motor drive. This system can represent a motor with a flexible shaft that drives a load. As- suming that the motor delivers a torque that is proportional to the current, the dynamics of the system can be described by the equations J₁² + c (dip-dyz) + k (41 - 42) = kI d² с dt J222 + c (dip-do) + k (42 − 1) = Ta. dt - (1) Derive a state space model for the system by introducing the normalized state vari- ables x1 = 1, x2 = 42, x3 = w1/wo, and x4 = w2/wo, where wo = √√k (J1J2)/(J1J2) is the undamped natural frequency of the system when the control signal is zero. (Such a state space model was derived in homework 2. You may refer to that model.) Using the following normalized parameters, J₁ = 10/9, J2 = 10, c = 0.1, k = 1, k₁ = 1, verify that the eigenvalues of the open loop system are 0, 0, -0.05 ± i. Design a state feedback controller in the form u = -k1x1 - k2x2 - k3x3 - k4x4 + krxr that gives a closed-loop system with eigenvalues -2, -land - 1±i. This choice implies that the oscillatory eigenvalues will be well damped and that the eigenval- ues at the origin are replaced by eigenvalues on the negative real axis. Simulate the responses of the closed loop system to step changes in the command signal for 02 and a step change in a disturbance torque on the second rotor.