Learning Fractional-Order Dynamics from a Single Trajectory

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 *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 stability and regularity assumptions, 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 $\mathcal{O}(t^{-1/4})$. Through experiments, we show that *FO-GS* outperforms existing baselines in recovering both the fractional order and the underlying system dynamics.