Learning Fractional-Order Dynamics from a Single Trajectory

2026-09-16

AI Summary

研究通过单轨迹识别分数阶系统，提出FO-GS方法解耦历史依赖性问题，提高估计精度。

Abstract

Many real-world processes exhibit long-range dependence, where the current state depends on a slowly decaying trace of past states rather than on the most recent state alone. This paper studies system identification for discrete-time fractional-order linear time-invariant systems from a single observed trajectory of length $t$, a setting that captures such non-Markovian dynamics through the Grünwald--Letnikov difference operator. Unlike Markovian systems, fractional-order systems couple estimation across the entire history, making both statistical analysis and practical identification more challenging. We propose \emph{Fractional-Order Ordinary-Least-Squares Grid-Search (FO-GS)}, a simple two-stage estimator that exploits the diagonal structure of the fractional-difference operator to decouple the identification problem row-wise. Under the stability assumption, we establish high-probability, non-asymptotic error bounds for estimating both the fractional order and the system matrix in the heterogeneous setting, with both estimation errors scaling as \(\mathcal{O}(t^{-1/2})\). Through experiments, we show that \emph{FO-GS} outperforms existing baselines in recovering both the fractional order and the underlying system dynamics.

Problem

• Fractional-Order Systems
• System Identification
• Non-Markovian Dynamics
• Single Trajectory

Innovation

• Fractional-Order Systems
• Non-Markovian Dynamics
• Grünwald–Letnikov Difference Operator
• System Identification