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Koopman Lifted Finite Memory Identification via Truncated Grunwald Letnikov Kernels

Navid Mojahed, Mahdis Rabbani, Shima Nazari

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.

Koopman Lifted Finite Memory Identification via Truncated Grunwald Letnikov Kernels

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.
Paper Structure (8 sections, 5 theorems, 46 equations, 1 figure, 1 table)

This paper contains 8 sections, 5 theorems, 46 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement & Background
  3. Koopman--GL Finite-Memory Identification
  4. Memory-Compensated Regression
  5. Finite-Memory Approximation Error
  6. Equivalent Augmented Markovian Realization
  7. Numerical Experiments
  8. Conclusion

Key Result

Theorem 1

Suppose $\Omega$ has full row rank and satisfies $\Omega\Omega^\top \succeq \mu I_{p+m}$ for some $\mu>0$. If the stacked regression satisfies eq:stacked, then the least-squares estimator $\hat{\Theta}=\mathbf{Y}\Omega^\dagger$ obeys and therefore

Figures (1)

  • Figure 1: Performance of the proposed Koopman--GL framework. (a) Rollout NRMSE as a function of $N$ and $\alpha$, with the best performance attained at $(N,\alpha)=(100,0.2)$. (b) Distribution of NRMSE across the unseen test set for the selected best configuration, compared with the baselines. (c) Mean relative error over the rollout horizon, with the bound from Theorem \ref{['prop:kernel_mismatch']}.

Theorems & Definitions (10)

  • Theorem 1: Identification error bound
  • proof
  • Theorem 2: Finite-memory modeling error bound
  • proof
  • Lemma 1: Decay and tail mass of GL coefficients
  • proof
  • Corollary 1: Pure truncation bound for exact GL kernel
  • proof
  • Theorem 3: Exact augmented Markovian realization
  • proof