A data-based stability-preserving model order reduction method for hyperbolic partial differential equations

This paper proposes a data-based approach for model order reduction that preserves incremental stability properties. Existing data-based approaches do typically not preserve such incremental system properties, especially for nonlinear systems. As a result, instability of the constructed model commonly occurs for inputs outside the training set, which seriously limits the usefulness of such models. Therefore, we propose to construct incrementally stable or incrementally ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document}-gain stable reduced-order nonlinear models to ensure robustness for a broad class of (bounded) input signals. Hereto, nonlinear discrete-time state-space equations are fitted to input-state-output data, obtained by simulations with the original model. We conjecture that certain classes of hyperbolic partial differential equations enjoy such incremental stability properties. Given the fact that complexity reduction in such PDE models is desirable, we employ the developed data-based reduction method to the discretized version of the hyperbolic equations thereby preserving the incremental stability features of the original system. In particular, this method is applied to a linear advection equation, for which stability properties are proved analytically. Finally, simulation results show the successful application of the method to the nonlinear Burgers’ equation.