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Control Systems (Prof. Casella) Midterm Exam – May 7th, 2015 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 Define the transfer function of a linear time-invariant system described by following state-space equations and state its main properties in the case of single-input, single-output system. ˙x=Ax+ Bu y=Cx+ Du The transfer function G(s) is a matrix such that the relationship between the Laplace transforms of the system input U(s) and output Y(s) is Y(s) = G(s)U(s). G(s) = C(sI – A)–1B + D The transfer function is a rational function. If D = 0, the relative degree of the function is greater than zero, otherwise it is zero. The poles of G(s) are the eigenvalues of the matrix A, except in the case of pole/zero cancellations due to hidden system dynamics, in which case the set of the poles is strictly contained in the set of the eigenvalues. Question 2 Assess the stability of the systems described by the following transfer functions: Asym. stable Simply stable Unstable s−1 s2 −10 s+ 20   x s+3 s2 +1  x  s ( s+1)3 x   s ( s2 +1)3   x 5 s3 + s2  …