Interpolation and model reduction of nonlinear systems in the Loewner framework

Abstract

This thesis studies the problem of interpolation and model order reduction for dynamical systems, with the primary objective being the development of an enhancement of the Loewner framework for general families of nonlinear differential-algebraic systems. First, an interconnection-based interpretation of the Loewner framework for linear time-invariant systems is developed. This interpretation does not rely on frequency domain notions, yielding a natural approach for enhancement of the Loewner framework to more complex systems possessing nonlinear dynamics. Next, the interconnection-based interpretation is used to develop the framework, first for systems of nonlinear ordinary differential equations, then for systems of nonlinear differential-algebraic equations, and interpolants are constructed using the so-called tangential data mappings and Loewner functions. Following this, parameterized families of systems interpolating the tangential data mappings are given. The problem of constructing interpolants from tangential data mappings and Loewner functions is considered in the most general scenario, and a dynamic extension approach to interpolant construction is developed. As a result, all systems matching the tangential data mappings, and having dimension at least as large as that of the auxiliary interpolation systems, are parameterized under mild conditions. Hence, if an interpolant exists while possessing additional desired properties, then it is contained in the dynamically extended family of interpolants. Finally, the use of behaviourally equivalent representations of a system is investigated with the goal of selecting a representation having less stringent conditions guaranteeing the existence of solution to partial differential equations. This is accomplished for a class of semi-explicit nonlinear differential-algebraic systems by making use of the explicit algebraic constraints to simplify the model of the system.