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Q3. The DC motor equation is given by dl V-L-RI-E = 0 dt where V is the input voltage to the motor, L is the motor inductance, R is the motor internal resistance, I is the electrical current, and E is the back EMF. The motor is connected to a mobile robot with a wheel, and the dynamics is given by dw J==T-w dt where J is the wheel moment of inertia, w is the angular velocity, and T is the motor torque. Assume that the gear ratio between the motor and the wheel is 1:1. a) Obtain the transfer function between the angular velocity as the output and the motor input voltage as the input of the transfer function. Do not assume any steady-state. [10 Marks] b) Let L=0.5 [H], R=0.1 [22], the back EMF or the torque constant is equal to 0.01 [V/(rad/s)], and the wheel moment of inertia is 2 [kg m²]. The following input voltage is applied to the motor for a sufficiently long period of time: v (t) = sin(0.1 t) Calculate the wheel angular velocity in terms of a function of time. Show all calculation steps [10 Marks] c) Design a proportional controller to achieve a desired constant angular velocity, 10 [rad/s], for the DC motor transfer function obtained in b). Draw a block diagram showing connections between the motor, the controller, and the desired angular velocity. And, show that it is practically impossible to make the steady-state error equal to zero using the proportional controller. [10 Marks]