Tracking Distributed Parameters System Dynamics with Recursive Dynamic Mode Decomposition with Control

Abstract

The goal of this work is to introduce a new online data driven modeling solution for tracking the dynamics of systems, the behavior of which is governed by (nonlinear) partial differential equations (PDEs). In order to reach this goal, this paper introduces a recursive algorithm for black box time varying model identification along with an online algorithm for reducing the order of the identified model. The resulting system model is a linear time varying low order state space representation usable, e.g., for controller design or prediction. The model reduction step relies on the recursive determination of the orthonormal basis of a specific data based proper orthogonal decomposition. The system identification procedure is initialized with the dynamic mode decomposition with control and updated recursively via the introduction of a specific recursive least squares algorithm. In both steps, a forgetting factor is used to focus on the most recent variations of the system. As case studies, a linear heat diffusion equation with a time varying diffusion parameter is first considered. The Burgers equation, a nonlinear hyperbolic PDE, is then utilized to assess the introduced methods applicability with nonlinear systems. Identification results based on simulation show the high accuracy of this new approach.