Answer Happy

Problem 2 [40 marks] The control system of a DC motor is shown in Figure 3. In the diagram: 0 is the angular position of the motor, Go is the reference in position, d is a load torque disturbance, and e is the tracking error. C(s) 100 $+10 $ Figure 3 1. Let C(s) = kp with kp > 0 be a proportional controller, 1-i) Show that the closed-loop characteristic equation is: 52 + 10s +100k, = 0. [4 Marks) 1-ii) find the value of kp such that the poles of the closed-loop system are real and coincident. [4 Marks) 1-ii) find an expression for the damping ratio (5) of the closed-loop poles as a function of kp. [5 Marks) 1-iv) find an expression for the steady-state tracking error (Co) caused by a unit step load disturbance d as a function of kp. [7 Marks) 1-v) use the expressions obtained in 1-iii) and 1-iv) to illustrate the fact that the damping of oscillations and the rejection of load disturbances are conflict- ing design objectives. [6 Marks) 2. Let C(s) = kp +kds with kp > 0, kd > 0 be an ideal proportional-derivative con- troller. 2-1) Derive the closed-loop characteristic equation in this case. [4 Marks] 2-ii) illustrate the fact that it is now possible to achieve a desired damping ratio (5) and a desired steady-state tracking error (coo) by proper selection of kp and kd. [10 Marks)