Data-driven identification of dynamical models using adaptive parameter sets

Abstract

This paper presents two data-driven model identification techniques for dynamical systems with fixed point attractors. Both strategies implement adaptive parameter update rules to limit truncation errors in the inferred dynamical models. The first strategy can be considered an extension of the dynamic mode decomposition with control (DMDc) algorithm. The second strategy uses a reduced order isostable coordinate basis that captures the behavior of the slowest decaying modes of the Koopman operator. The accuracy and robustness of both model identification algorithms is considered in a simple model with dynamics near a Hopf bifurcation. A more complicated model for nonlinear convective flow past an obstacle is also considered. In these examples, the proposed strategies outperform a collection of other commonly used data-driven model identification algorithms including Koopman model predictive control, Galerkin projection, and DMDc.

Data-Driven Inference of Low Order Representations of Observable Dynamics for an Airfoil Model

“…Previous work [27], [29] using the adaptive isostable reduction approach has been limited to prototype problems involving the Burgers' equation on simple 1 and 2-dimensional domains. Re-sults presented in this work provide proof of concept that this framework can be successfully used in realistic fluid flow models using geometries with practical relevance.…”

“…to keep ||Ψ|| small. General strategies for design of an appropriate update rule for p are discussed in [26] and [29]. Equations ( 16) with the parameter update rule (17) comprise the full form of the adaptive isostable reduction.…”

“…In situations where large magnitude inputs are necessary, references [26] and [27] suggest the inclusion of an adaptive parameter set, defining extended isostable coordinates that are functions of both the state and the adaptive parameters and appropriately updating the adaptive parameter set in order to limit truncation errors associated with neglected nonlinear terms. Preliminary evidence [28], [27], [29], suggests that this is a promising approach for reduced order modeling of fluid flows, although these techniques have thus far only been applied to relatively simple models based on simulations of the Burgers' equation.…”