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Problem 2 [40 marks] The control system of a DC motor is shown in Figure 3. In the diagram: is the angular position of the motor, 0, is the reference in position,d is a load torque disturbance. and e is the tracking error C) 100 +10 Figure 3 1. Let Cls=k, with k > O be a proportional controller. 1-) Show that the dosed-loop characteristic equation is: s +10+100k, a [4 Marks) 1-2) find the value of k, such that the poles of the closed loop system are real and coincident. [4 Marks 1-H) find an expression for the damping ratio of the closed-loop poles as a function of ks Marks Page 3 ENGD3113 Examination Paper 2021/2022 1-iv) find an expression for the steady-state tracking error (e.) caused by a unit step load disturbanced as a function of k 17 Marks] 1-v) use the expressions obtained in 1-1) and 1-tv) to illustrate the fact that the damping of oscillations and the rejection of load disturbances are conflicting design objectives. 16 Marks] 2. Let C(s)-ky+ks with k>0k > Obe an ideal proportional-derivative controllier. 24 Derive the closed-loop characteristic equation in this case. [4 Marks) 2-1) illustrate the fact that it is now possible to achieve a desired damping ratio ( and a desired steady-state tracking error (e) by proper selection of k, and ks [10 Marks