2023 04 20 midterm

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Control Systems (Prof. Casella) Midterm Exam – Apr 20th, 2023 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 Consider two asymptotically stable LTI dynamical systems. Discuss the stability of their feedback connection. Question 2 Give the most general definition of state variable for a generic dynamical system. Then, define what are the state variables for a continuous-time state-space model of a continuous-time dynamical system, explaining why it fits the general definition. Question 3 For each of the following four diagrams indicate what is the transfer function, among those listed below, of which they are the step response. (1) 10 s2 (2) 4 (1+s)4 (3) 4 1+0.5 s (1+ s)2 (4) (5) 10 1+0.8 s (1+ s)2 (6) 4 s2 + s+ 25 (7) 4 s2 + 4 s+ 25 (8) (9) s+3 s2 +3 s+2 (10) 4 s2 + 20 s+25 (11) 2 1+0.9 s s2 +4 s+3 (12) 10 s+1 s2 10 (1+s)4 2 1+ 2 s s2 +4 s+3 Question 4 Consider the nonlinear RLC circuit shown in the diagram, where L is a constant inductance, C is a constant capacitance, v0 and i0 are constant parameters describing the nonlinear resistor behaviour. The…