2023 06 16 midterm answers

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Control Systems (Prof. Casella) Final Mid-Term Exam – June 16th, 2023 ANSWERS SHEET Question 1 Consider the control system shown in the figure. Discuss why direct disturbance compensation can be much more effective than feedback control to reject fast changes of the disturbance d. The bandwidth of the feedback controller is limited to about 0.008 rad/s by the time delay in G(s). All disturbance components above that frequency are not rejected by the feedback control. The ideal disturbance compensator Cd(s) = -H(s)/G(s) = -0.25(1+s) is not strictly proper, but can be well approximated up to 1 rad/s by a static compensator C* d(s) = -0.25, which will reject disturbances over a bandwidth which is two orders of magnitude wider. So, there is a very strong advantage in using direct disturbance compensation. Question 2 Consider the feedback control system represented in the figure. Using Bode’s stability criterion, determine for which values of Kp the closed-loop system is asymptotically stable. The loop transfer function of this feedback system is The Bode plot of the modulus starts with a negative slope of -20 dB/dec at low frequency, and is then monotonously decreasing for all frequency values. Hence, whatever the value of Kp, there is always only one crossover frequency. Besides, all the poles of L(s) have negative or zero real part, so the preliminary assumptions of Bode’s criterion hold. The necessary and sufficient conditions for closed loop asymptotic stability are that the loop transfer function gain is positive and the phase margin is positive. The first condition is clearly met if Kp > 0. With that condition, the asymptotic phase plot is comprised between -90° and -180°, and so is the real phase plot. Hence, the phase margin is always positive. Summing up, the…