2022 04 11 midterm answers

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Control Systems (Prof. Casella) Midterm Exam – Apr 11th, 2022 Answers sheet Question 1 Define the open-loop and closed-loop control strategies. Then, comment on how and to which extent they can manage uncertainty (unknown disturbances and uncertainty on dynamic behaviour) In open-loop control strategies, the controller does not have access to a measurement of the controlled variable, while in closed-loop ones it does. See also the lecture notes for more details. Closed-loop strategies in general can handle uncertainty better than open loop ones: they can reject unmeasured disturbances, which open-loop strategies can’t do, and are relatively tolerant to mismatches between the assumed behaviour of the system and the actual one. However, severe mismatch between the assumed behaviour of the system and the actual one can lead to unstable behaviour in closed-loop systems, which is something that cannot happen in open-loop ones. Question 2 Consider a number of SISO nonlinear systems, each linearized around a certain equilibrium. Based on the transfer functions between the input and the output of the linearized systems, classify the stability of the corresponding equilibria of the nonlinear system. Asym. stable Simply stable Unstable Can’t say s+1 s3 −10 s2 + s   X  s+3 s2 +10    X s+3 s2 −10   X  s+4 ( s+10)2 (s+3)2 X    s+1 s( s+1)    X 1 s2 ( s2 +1)    X 1+10 s− s2 ( s+10)3 ( s+2)2 ( s+1) X    1 s4 +2 s2 +1    X 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) 2 s+1 s+2 (2) 2 s+2 s+1 (3) 2 s+2 (4) (5) 50 s2 +3 s+2 (6) 9 s2 +s+9 (7) 9 s2 +2 s+9 (8) (9) s+3 s2 +3 s+2 (10) s+1 s2 (11) s+1 s (12) s+1 s+2 10 s2 +3 s+2 3 1+3 s s2 +3 s+2 (3) (9) (7)…