THE SIMPLE PENDULUM AND THE PERIODIC LQG CONTROL PROBLEM

Some recent developments in linear periodic control theory are illustrated in terms of the optimal control of a simple pendulum. This problem, with a long history in classical mechanics, is examined here under a periodic control theoretic light, and is taken as a prototype of a large class of possible applications of the theory. The pendulum control problem under partial observation is carefully posed, critically reviewing the various assumptions underlying such a formulation. A cyclostationary model of the modelling error is adopted, thereby taking periodicity into account. The problem thus posed is then solved along the lines of optimal periodic control theory, by resorting to the so-called Linear Quadratic and Gaussian (LQG) approach. This requires the determination of the periodic solutions of two periodic Riccati equations. The existence of such solutions, as well as the numerical algorithm for their determination, is then discussed. Finally, the main features of PENDULUM are presented. PENDULUM is a user-friendly computational environment devised in order to test the applicability and limitations of linear periodic control theory.