2019 04 15 midterm answers

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Control Systems (Prof. Casella) Midterm Exam – Apr 15th, 2019 ANSWER SHEET Question 1 Consider a room heating system, in which the temperature of the water sent to the radiators is a function of the outside temperature, while the flow rate is fixed. Discuss whether the control strategy of the system is open-loop or closed-loop. Then, discuss if this changes with the installation of thermostatic valves on the radiators. The controlled variable in this case is not directly used by the controller, which only acts based on the value of a measured disturbance. Hence, the control strategy is open-loop. Thermostatic valves modulate the flow rate in the radiators in response to the sensed room temperature, so their installation changes the strategy to closed-loop, generally improving the control system performance, and thus both the comfort and the fuel economy. Question 2 Assess the stability of the systems described by the following transfer functions: Asym. stable Simply stable Unstable s+1 s3 +10 s2 +s    s+3 s2 +10    s+3 s2 −10    s+4 ( s+10)2 (s+3)2    s+1 s( s2 +1)    1 s2 ( s2 +1)    1 s(s2 +1)2    1 s4 +2 s2 +1    Question 3 Consider the systems shown in the figures 1.A and 1.B, in which a ball that can roll without any friction is sitting in equilibrium condition. The input of the system is a force applied to the ball in the horizontal direction, while the output is the horizontal displacement of the ball center. Based on physical intuition and on the general definition of external stability, discuss the stability of the two equilibria; motivate your answers by making explicit reference to the general definition. Figure 1.A Figure 1.B In case A, it is not true that if we select an arbitrarily small interval around the equilibrium position,…