2016 07 15 exam

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Control Systems (Prof. Casella) Written Exam – July 15th, 2016 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 With reference to the linear time-invariant dynamical system whose equations are shown here, define what is the transfer function between its input variable u(t) and its output variable y(t). Then, explain why it is a proper rational function. Question 2 Explain the phenomenon of wind-up in controllers with integral action, pointing out why it should be avoided. ˙x( t)= Ax( t)+ Bu(t ) y(t )=Cx( t)+ Du(t) Question 3 Consider the two-tank system shown in the figure. The system is described by the following equations, where l is the tank level, A the tank cross section, q is the volume flow rates, and k the valve flow coefficient. 3.1 Write down the state and output equations in standard state-space form, assuming q1 as input, A and k as constants, q2 and q3 as outputs. 3.2 Compute the equilibrium conditions for the system. 3.3 Write down the equations of the system linearized around a generic equilibrium. l1 l2 q1 q2 q3 2 A dl1 dt =q1−q2 A dl2 dt…