2022 07 11 exam answers

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Control Systems (Prof. Casella) Written Exam – July 11th, 2022 ANSWERS SHEET Question 1 Consider a simple frictionless pendulum, where the input variable is a horizontal force applied to the point mass and the output is the angle formed by the pendulum with a vertical axis. The pendulum is in the lower equilibrium conditions with a zero applied force and a zero angle. Based on the general definition of external stability, explain whether the equilibrium is asymptotically stable, simply stable or unstable. Rewriting the conditions with the specific variables of the prolem, the equilibrium is stable if, for each A > 0, there exist a > 0 such that F(t) = ed(t) with |e| < a implies |q(t)| < A, where F(t) is the applied force and q(t) is the pendulum angle. This is actually the case because of the conservation of energy: the initial impulse transfers a certain amount of energy E to the pendulum, which allows it to move away from the equilibrium until it reaches a height h such that Mgh = E, then it swings back. In that position, L(1-cos(qmax)) = hmax, so By taking a sufficiently small a, the energy E can be chosen to be as small as possible, hence the angle qmax will be as close as one wants to zero, accomodating any value of A one may choose. So, the equilibrium is stable. However, since there are no dissipative phenomena, the motion goes on forever, hence angle q(t) does not asymptotically return to zero as time approaches infinity. Hence, the equilibrium is not asymptotically stable, only simply stable. Question 2 Consider the control system shown on the right. Assume that the system is subject to significant process disturbances d and measurement noise n. Discuss the trade-offs in system performance when choosing a low or a high cross-over frequency for the controller…