A geometric approach to nonlinear dissipative balanced reduction

Abstract

In almost every field of applied science and engineering, dynamical models are widely used as a professional tool to describe in a compact format the scientific knowledge we have about the phenomena we are analyzing. Such models are widely used for the estimation, optimization, detection of failures and even for real-time control. The relevance of models is such, that huge budgets are invested yearly in super-computers to comply with the ever growing need of expensive simulation software to simulate complex dynamical models. But larger models are harder to analyze for humans, harder to simulate on computers and harder to control. In contrast, experience has shown that the dimension of the models used for simulation and control can be significantly reduced if techniques of model order reduction (MOR) are used. Most of the tools for MOR use a technique for linear operators called principal component analysis (PCA). Given a full-order dynamical model (FOM), the problem of MOR consists in finding a reduced-order model(ROM) which keeps structural properties like stability, reachability / controllability and observability and other desirable characteristic properties of the FOM into the ROM. For these and other reasons, MOR is a topic of growing concern for every scientific and engineering field, especially when such models are nonlinear since the few reduction methods that exist are of local validity and empirical formulation. This dissertation contributes to the understanding of this problem with a theory for structure-preserving MOR for nonlinear dissipative control systems. In particular, this thesis provides a theoretical framework for nonlinear balanced reduction for a rather large class of dynamical models with the property of being dissipative, a property that includes electrical, mechanical, hydrodynamic and thermodynamic models, among others. In this work is asserted that the nonlinear balanced reduction problem for dissipative systems is isomorphic to the problem of an isometric transformation between two Hilbert manifolds, where the Gramians are components of two Riemannian metrics and duality suffices for a balanced realization. Furthermore the decomposition of isometric operators provided here is a nonlinear generalization to principal component analysis (NL-PCA).