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Control Systems (Prof. Casella) Midterm Exam – May 5th, 2016 Answers Sheet Name:.............................................. Surname:................................................... Reg. Number:............................... Signature:...................................... Notices:  This booklet is comprised of 6 sheets – Check that it is complete and fill in the cover.  Write your answers in the blank spaces with short arguments, including only the main steps in the derivation of the results.  You are not allowed to leave the classroom unless you hand in the exam paper or withdraw from the exam.  You are not allowed to consult books or lecture notes of any kind.  Please hand in only this booklet at the end of the exam – no loose sheets.  The clarity and order of your answers will influence how your exam is graded. Question 1 Consider two asymptotically stable systems described by their transfer functions A(s) and B(s). Discuss the stability of the systems obtained by connecting the above-mentioned systems in cascaded, parallel, and feedback configurations. Let A(s) = nA(s) / dA(s) and B(s) = nB(s) / dB(s), where nA and dA are coprime, as well as nB and dB. By hypothesis, the poles of both transfer functions have strictly negative real part, i.e., the real part of all the roots of the polynomials dA(s) and dB(s) is strictly negative. The transfer function of the cascaded connection is nA(s)nB(s) / dA(s)dB(s), so its poles are the roots (or a subset of the roots in case of cancellations) of dA and dB, which are strictly negative. Hence, the system is asymptotically stable. The transfer function of the parallel connection is [ nA(s)dB(s) + nB(s)dA(s)] / dA(s)dB(s), so its poles are the roots (or a subset of the roots in case of cancellations) of dA and dB,…