Koopman Lifted Finite Memory Identification via Truncated Grunwald Letnikov Kernels

TL;DR

This paper introduces a novel data-driven linear modeling approach for nonlinear hereditary systems by combining Koopman lifting with truncated Grunwald-Letnikov kernels, enabling effective memory incorporation and improved prediction accuracy.

Contribution

It develops a new Koopman-based framework that models history dependence directly in lifted coordinates using fractional-difference weights, extending standard methods beyond Markovian assumptions.

Findings

Enhanced multi-step prediction accuracy over baseline models

Effective modeling of hereditary nonlinear systems with finite memory

Demonstrated improvements on a benchmark with non-Markovian dynamics

Abstract

We propose a data-driven linear modeling framework for controlled nonlinear hereditary systems that combines Koopman lifting with a truncated Grunwald-Letnikov memory term. The key idea is to model nonlinear state dependence through a lifted observable representation while imposing history dependence directly in the lifted coordinates through fixed fractional-difference weights. This preserves linearity in the lifted state-transition and input matrices, yielding a memory-compensated regression that can be identified from input-state data by least squares and extending standard Koopman-based identification beyond the Markovian setting. We further derive an equivalent augmented Markovian realization by stacking a finite window of lifted states, thereby rewriting the finite-memory recursion as a standard discrete-time linear state-space model. Numerical experiments on a nonlinear hereditary benchmark with a non-Grunwald-Letnikov Prony-series ground-truth kernel demonstrate improved multi-step open-loop prediction accuracy relative to memoryless Koopman and non-lifted state-space baselines.