Answer Happy

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, Co is the reference in position, d is a load torque disturbance, and e is the tracking error. + 0 1 C(s) 100 $+10 S 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 +100kp = 0. [4 Marks] 1-ii) find the value of k, such that the poles of the closed-loop system are real and coincident. [4 Marks] 1-iii) 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 (eco) 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 + kas with kp > 0, ka > 0 be an ideal proportional-derivative con- troller. = 2-i) Derive the closed-loop char teristic equation 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 (eo) by proper selection of kp and kd. [10 Marks]