Multiscale theory for linear dynamic processes: Part 1. Foundations

Multiscale models of linear dynamic systems offer a representation that links states at different time scales over distinct time periods. As such they allow a natural bridging of scales (ranges of frequencies) and time, and are more attractive than models defined in the time- or frequency-domain, alone. The term “multiscale model”, as used in this two-part series, is quite different from the notion of a “multiscale model”, as this is used in recent publications, where in essence such model bridges the states of a system at two distinct scales (an atomistic scale of fast elementary phenomena, and a macroscopic scale characterizing slow continuum-based physics), described by two distinctly different physical models. The multiscale dynamic models, introduced in this paper, link state dynamics over a number of distinct time scales, and are in essence “multiresolution” facets of a single model. The outgrowth of a series of developments, which came about with the advent of the wavelet decomposition for the analysis of discrete signals, multiscale models are defined on dyadic trees, which span the time-scale space. The nodes of such trees are used to index the values of states, inputs and outputs, modeling errors, and measurement errors of dynamic systems. 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 Part 1 of this series, which focuses on the foundations of a multiscale systems theory, we will describe the generation of multiscale models from their time-domain counterparts, and develop conditions governing their stability, controllability and observability. In addition, we will address the development of algorithms for the solution of basic problems, such as: optimal control, state estimation, and optimal fusion of measurements and control actions at various scales. It will be shown that multiscale models lend themselves nicely to construction of very effective parallelizable algorithms. In Part 2 of this series, we will apply all these developments to the formulation of a Multiscale Model Predictive Control (MS-MPC) approach and we will show that MS-MPC is a more attractive framework for the design and implementation of multivariable control systems.