2019 06 11 midterm answers

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Control Systems (Prof. Casella) Final Mid-Term Exam – June 11th, 2019 ANSWER SHEET Question 1 Consider the feedback control system shown in the figure. Define the sensitivity transfer function S(s) and discuss its role in determining the dynamic performance of the system, assuming that the phase margin jm > 40°. The sensitivity function is defined as S(s) = 1/(1+ C(s)G(s)). It is the transfer function between the forward path disturbance d and the controlled variable y, so it describes the disturbance rejection enacted by the feedback controller. If the loop transfer function satisfies the applicability conditions of Bode’s criterion, and the phase margin is above 40°C, the complementary sensitivity function is a high-pass filter, whose bandwidth is approximately equal to the cross-over frequency of the loop transfer function. Question 2 Consider the control system represented in the diagram of Question 1. Explain whether or not it is possible to increase the rejection of the effects of disturbance d on the controlled output y, without also increasing the effect of the sensor noise n on the controlled output y. The two involved transfer functions, which are the sensitivity function S(s) and minus the complementary sensitivity function T(s), have the same bandwidth, which is approximately equal to the cross-over frequency of the loop transfer function in both cases. Hence, increasing the disturbance rejection is only possible by increasing the cross-over frequency, which in turn widens the bandwidth of the complementary sensitivity function. This means that the controlled output will follow additional higher-frequency components of the sensor noise. So, it is not possible to increase the disturbance rejection without increasing the effect of the sensor noise on the…