Multi-Scale Aspects in Linear and Nonlinear Estimation and Control

Abstract

Multi-scale models of processing systems offer an attractive alternative to models defined in the time- or frequency-domain. They are the outgrowth of a series of developments which came about with the advent of the wavelet decomposition for the analysis of discrete signals. Multi-scale models are defined on dyadic or higher-order trees. The nodes of such trees are used to index the values states, inputs and outputs, modeling errors, and measurement errors. These values are localized in both time and scale (range of frequencies), and thus they offer a hybrid domain that is particularly conducive for estimation and control problems. In this paper we introduce a formal framework for the formulation of multi-scale models on trees, which are consistent with their time-domain counterparts. Such models lead to a multi-scale control theory and the definition of the corresponding transfer functions, stability, controllability, and observability concepts for systems on trees. Fusion of control actions and measurements at different rates as well as their implications on the controllability and observability of dynamic systems are also examined. Of particular significance is the issue of closure between the models in the time- and the time-scale domains, which constrains the type of physico-chemical processes that can be modeled on a tree.