Model and controller selection policies based on output prediction errors

Based on observations of the past inputs and outputs of an unknown system Σ, a countable set of predictors, O_p, p ε 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 p ε P at each time k. We study the structure and memory requirements of decision rules that converge to some q ε P 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 Γ_p, p ε P of predesigned control systems. We show that for certain multi-input/multi-output linear systems the resultant closed-loop controlled system is stable and can asymptotically track an exogenous reference input.