2021 09 07 exam

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Control Systems (Prof. Casella) Written Exam – Sep 7th, 2021 Surname:.............................................. Name: ................................................... Reg. Number:............................... Signature:...................................... Notices:  This booklet is comprised of 7 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 various types of stability of equilibria in a dynamical system. Then, explain why all the equilibria in a given LTI dynamical system have the same stability property. Question 2 The step response of a LTI system approaches zero asymptotically as time approaches infinity. What can you say about its transfer function? Please motivate your answer clearly. Question 3 Consider a pipe carrying a liquid with constant density r, starting from a source at constant relative pressure Pi, ending with a valve with a variable flow coefficient Av discharging a mass flow rate w into the atmosphere. Assuming incompressible flow and neglecting friction, the equations of the system are given on the right, where L is the constant pipe length, A is the constant pipe cross-section, Po is the pipe outlet relative pressure. 3.1 Write down the state and output equations of the system in standard state-space form, considering the valve coefficient Av as input,…