2021 07 12 exam

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Control Systems (Prof. Casella) Written Exam – July 12th, 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 transfer function of a LTI dynamical system. Then, explain its relationship with the impulse response of the system. Question 2 Explain why a linear controller can perform in a predictable way when controlling a nonlinear system, and under which conditions this is actually the case. Question 3 Consider a power generation unit, comprising a turbine and a synchronous AC generator with N polar expansions, with total moment of inertia J and angular speed W, generating 3-phase AC power with angular frequency w. The generator feeds active power to a pool of loads, whose actual total power consumption Pe depends on their nominal consumption P°e, but also on the AC frequency. Assuming no electrical and mechanical losses for simplicity, the system is described by the equations on the right, where E is the total kinetic energy of the rotating machinery, Pm is the mechanical power input to the turbine, w0 is the nominal grid frequency, and a is a…