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Control Systems (Prof. Casella) Final Mid-Term Exam – June 25th, 2016 Answers sheet Question 1 Consider the feedback system shown in the figure. State Bode’s criterion for the asymptotic stability of the closed-loop system. Preliminary requirements: • L(s) must have poles with negative real part or in the origin • L(jw) must have a modulus equal to 1 for only one value wc of the frequency • Define j c = arg L(jw), j m = 180° - |j c |, m L gain of L(s) The closed-loop system is asymptotically stable if and only if j m > 0 and m L > 0 Question 2 Define what non-minimum phase linear dynamical systems are. Then, discuss how this property of of a process to be controlled influences its controller design and performance. In the class of LTI systems without poles with positive real part, a system is non-minimum phase if it has positive zeros and/or pure time delays. In the case of non-minimum phase systems, the system transfer function can be factored out in a minimum-phase part, which only contains the rational part without delay, with the positive poles replaced by their symmetric negative counterparts, and a non-minimum phase part, which contains the pure time delay and an all-pass filter (1 – sT)/(1 + sT) for each pole with positive real part. The non-minimum phase dynamics cannot be cancelled out by the controller’s dynamics, since the pure time delay cannot be cancelled by a pure time advance, while the positive poles cannot be cancelled by negative poles, because that would lead to hidden unstable dynamics in the system. The phase margin can be computed as j m = 180° + arg( Lmp(jwc)) + arg( Lmp(jwc)). By adding poles and zeros to the controller, one can always make sure that the modulus diagram crosses the 0 dB axis with a slope of -20 db/dec in a large enough…