2018 06 26 exam

Page 1

Control Systems (Prof. Casella) Written Exam – June 26th, 2018 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 Give the most general possible definition of a dynamical system. Then, explain how systems described by state-space equations fit that definition. Question 2 Draw the diagrams of 2 degrees-of-freedom controllers and explain in which cases they are preferrable over standard 1 degree-of-freedom controllers. Question 3 Consider a system made by two water (density: r) reservoirs of surface A1 and A2, whose outlet flow rates w1 and w2 are determined by weirs. The first reservoir receives an incoming mass flow rate w0 which is an exogenous input; the second reservoir receives a mass flow rate w1 from the first reservoir. The system dynamics is described by the equations detailed here: 3.1 Write down the state and output equations in standard state-space form, assuming y1 and y2 as states, w0 as input, and w2 as output. 3.2 Compute the equilibrium conditions for the system. 3.3 Write down the equations of the system linearized around a generic…