2020 07 23 exam answers

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Control Systems (Prof. Casella) Written Exam – July 23rd, 2020 ANSWERS SHEET Question 1 Consider the following control system. The controlled variable y is measured by a very fast sensor, that has negligible dynamics, but is affected by a lot of high- frequency noise n. The low-pass filter F(s) is thus introduced to reduce the effect of such sensor noise on the system performance. Discuss all the effects that the filter can have on the various aspects of the control system performance, particularly regarding the dynamic performance and the control sensitivity to the sensor noise. Assuming that you need to obtain a specified bandwidth and phase margin of the system, discuss possible design criteria for the filter F(s), highlighting potential trade-offs in the design process. The transfer function F(s) gets into the loop transfer function L(s) = C(s)G(s)F(s). Low-pass filters introduce a phase lag, so the first effect of introducing the filter is to reduce the phase margin, and possibly to reduce the crossover frequency if the bandwidth of the filter is lower than the original crossover frequency of the filter-less loop. These effects can be minimized by keeping the bandwidth of the filter sufficiently higher than the crossover frequency of the control loop, or by reducing the crossover frequency. The other effect is to mitigate the control sensitivity to noise. Without the filter, the transfer function from n to u would be -Q(s) = -C(s)/(1+L(s)), whose frequency response modulus is close to |C(jw)| for w > wc. When introducing the filter, the modulus of this transfer function is multiplied by | F(jw)|, which attenuates the influence of high frequency noise on the control variable u for frequencies above the bandwidth of the filter. So, the design criteria for the filter…