On the performance and complexity of a class of hybrid controller switching policies

Abstract

Based on observations of the past inputs and outputs of an unknown system Σ, a countable set of predictors O p , p e P, is used to predict the system output sequence. Using performance measures derived from the resultant prediction errors a decision rule is to be designed to select a peP at each time k. We study the structure and memory requirements of decision rules that converge to some qeP such that the qth prediction error sequence has desirable properties, e.g., is suitably bounded or converges to zero. In a very general setting we give a positive result that there exist stationary decision rules with countable memory that converge (in finite time) to a ‘good’ predictor. These decision rules are robust in a sense made precise in the paper. In addition, we demonstrate that there does not exist a decision rule with finite memory that has this property. This type of problem arises in a variety of contexts but one of particular interest is the following. Based on the decision rule's selection at time k a controller for the system Σ is chosen from a family Π p , p e P, of predesigned control systems. We show t for certain mimo linear systems the resultant closed loop controlled system is stable and can asymptotically track an exogenous reference input.