2022 06 17 midterm answers

Pagina 1

Control Systems (Prof. Casella) Final Mid-Term Exam – June 17th, 2022 ANSWERS SHEET Question 1 Explain why the bandwidth of a linear feedback control system is limited in case the plant to be controlled has a non-minimum phase dynamic response and how to determine that limit. By appropriately designing the controller transfer function, the Bode plot of the loop transfer function can be made to cross the 0 dB axis at the required crossover frequency with a slope of -20 dB/dec for a sufficiently long frequency interval. However, the non-minimum phase part of the plant transfer function cannot be cancelled by the controller , because that would create an unstable hidden part in the case of positive zeros, or require a future-predicting controller in the case of pure time delay. Hence, the non-minimum phase factor Gnmp(s) of the plant transfer function is also found in the loop transfer function L(s). Therefore, the asymptotic phase margin when taking a very long streak of -20 dB/dec slope around the crossover frequency is approximately 90° - arg( Gnmp(jwc)). If one wants to have at least 40° phase margin, this entails the following limitation: arg( Gnmp(jwc)) < -50°. The maximum bandwitdh is thus approximately the value wcmax that brings the phase of Gnmp(jwcmax) = -50°. See also the lecture notes. Question 2 Consider the feedback control system represented in the figure, where G(s) is an asymptotically stable transfer function with a dominant pole of time constant T. Explain why step changes of the set point changes cause overshoots in the manipulated variable if the crossover frequency of the control system is larger than 1/T. The transfer function from the set point to the manipulated variable is Q(s) = C(s)/(1 + C(s)G(s)). The modulus diagram of Q(jw) can be…