2018 05 03 midterm

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Control Systems (Prof. Casella) Midterm Exam – May 3rd, 2018 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 Write down Lagrange’s formula for the motion of a LTI system and discuss what important properties about the motion of LTI systems can be drawn from it. Question 2 Assess the stability of the systems described by the following transfer functions: Asym. stable Simply stable Unstable s2−1 s2 −10 s+ 20    s+3 s2 +1    s s3 + s2    s ( s+1)3    s+1 s( s2 +1)    1 s2 −s    1 s( s+10)    1 ( s+1)(s2 +1)2    Question 3 Consider a cylindrical open tank of cross section A, containing a liquid of constant density r up to a level l, connected to a pump that feeds it with a prescribed mass flow rate wi and temperature Ti, and to a pump that discharges it with a prescribed flow rate wo. Assuming a constant specific heat capacity c and assuming that de = dh = cdT, where e and h are the specific energy and enthalpy of the liquid, the system is described as: 3.1 Write the system equations in standard state-space form, considering wi, wo, and…