Problem 2: The open-loop transfer function of a DC-motor is a second order transfer function, P(s) ė(s) V(8) K (Js+b)(Ls + R) + K2 rad/sec V where the rotational speed è is considered the output, the armature voltage V is considered the input, the physical parameters for the considered DC motor are: (J) moment of inertia of the rotor (b) motor viscous friction constant (K) static DC gain (R) electric resistance 0.01 kg.m² 0.1 N.m.s 0.01 V/rad/sec 1 Ohm 0.5 H (L) electric inductance An automatic controller is used to control the DC-motor rotational speed, as shown the block diagram. controller C($) The transfer function for a PID controller is: M plant P(s) K₁ C(s) = Kp + + Kas = Kas²+Kps+K 8 For a 1-rad/sec step reference, the design criteria are the following. Settling time less than 2 seconds Overshoot less than 5% Steady-state error less than 1% Use MATLAB to create a Simulink block diagram representing the closed-loop system shown in the block diagram. Obtain the response (t) to the unit step input for the following cases: Use proportional controller with K, = 100. • Use a PID controller with Kp = 75, K-1 and Ka= 1. • Use a PID controller with Kp = 100, K, • Use a PID controller with Kp = 100, K, = 200 and Ka = 1. 200 and Ka = 10. Then discuss the effect of the controller on the system response, taking into consideration the rise time, the overshot, the settling time and the steady state error.