Model reduction and minimality for uncertain systems

Abstract

The emphasis of this thesis is on the development of systematic methods for reducing the size and complexity of uncertain system models. Given a model for a large complex system, the objective of these methods is to find a simplified model which accurately describes the physical system, thus facilitating subsequent control design and analysis. Model reduction methods and realization theory are presented for uncertain systems represented by Linear Fractional Transformations (LFTs) on a block diagonal uncertainty structure. A complete generalization of balanced realizations, balanced Gramians and balanced truncation model reduction with guaranteed error bounds is given, which is based on computing solutions to a pair of Linear Matrix Inequalities (LMIs). A necessary and sufficient condition for exact reducibility of uncertain systems, the converse of minimality, is also presented. This condition further generalizes the role of controllability and observability Gramians, and is expressed in terms of singular solutions to the same LMIs. These reduction methods provide a systematic means for both uncertainty simplification and state order reduction in the case of uncertain systems, but also may be interpreted as state order reduction for multi-dimensional systems. LFTs also provide a convenient way of obtaining realizations for systems described by rational functions of several noncommuting indeterminates. Such functions arise naturally in robust control when studying systems with structured uncertainty, but also may be viewed as a particular type of description for a formal power series. This thesis establishes connections between minimal LFT realizations and minimal linear representations of formal power series, which have been studied extensively in a variety of disciplines, including nonlinear system realization theory. The result is a fairly complete development of minimal realization theory for LFT systems. General LMI problems and solutions are discussed with the aim of providing sufficient background and references for the construction of computational procedures to reduce uncertain systems. A simple algorithm for computing balanced reduced models of uncertain systems is presented, followed by a discussion of the application of this procedure to a pressurized water reactor for a nuclear power plant.