Figure 1 shown a DC motor including free body diagrams of its electric circuit of the amature and the rotor. The imput is the amature voltage V in Volts (driven by a voltage source). Measured variables are the angular velocity of the shaft w in radians per second, and the shaft angle e in radians. R L. T. V, = Ko Figure 1 The rotor and the shaft are assumed to be rigid, and the values for the physical parameters are listed in Table 1 below. Table 1 moment of inertia of the rotor damping (friction) of the mechanical system (back-Jelectromotive force constant J =0.01 kg m b =0.1 Nms K =0.01 Nm/A R =1N L = 0.5 H electric resistance electric inductance (a) Develop the mathematical model of the system intem of ODE, Transfer Function, and State-Space Forms. (15 marks) (b) Simulate and plot the responses of the system based on intial, step and impulse inputs using Matlab, respectively, where tspan=[0 5] and all initial conditions zero. (20 marks) (c) By referring to the response results on (b), give your comments relating to stability and performance parameters (rise time, over shoot, settling time, etc). (15 marks)