A Sampling Complexity-aware Framework for Discrete-time Fractional-Order Dynamical System Identification

TL;DR

This paper introduces a sample complexity-aware identification algorithm for discrete-time fractional-order dynamical systems, addressing challenges of noisy data and avoiding complex polynomial solutions, with proven error bounds and validated simulations.

Contribution

It proposes a novel affine identification algorithm for fractional-order systems with detailed sample complexity analysis and error bounds, improving robustness in noisy environments.

Findings

Error decays at rate N = O(d/ε)

Algorithm avoids solving high-order polynomial equations

Simulation results confirm theoretical bounds

Abstract

A variety of complex biological, natural and man-made systems exhibit non-Markovian dynamics that can be modeled through fractional order differential equations, yet, we lack sample comlexity aware system identification strategies. Towards this end, we propose an affine discrete-time fractional order dynamical system (FoDS) identification algorithm and provide a detailed sample complexity analysis. The algorithm effectively addresses the challenges of FoDS identification in the presence of noisy data. The proposed algorithm consists of two key steps. Firstly, it avoids solving higher-order polynomial equations, which would otherwise result in multiple potential solutions for the fractional orders. Secondly, the identification problem is reformulated as a least squares estimation, allowing us to infer the system parameters. We derive the expectation and probabilistic bounds for the FoDS parameter estimation error, assuming prior knowledge of the functions \( f \) and \( g \) in the FoDS model. The error decays at a rate of \( N = O\left( \frac{d}{\epsilon} \right) \), where \( N \) is the number of samples, \( d \) is the dimension of the state variable, and \( \epsilon \) represents the desired estimation accuracy. Simulation results demonstrate that our theoretical bounds are tight, validating the accuracy and robustness of this algorithm.