Recursive Time Delay Estimation for Slowly Switching FIR Systems via Empirical Copula

This article develops a complete recursive time delay estimation framework for finite impulse response systems with slowly switching time delays, with rigorous and thorough theoretical proof provided to verify the consistency of the proposed scheme. The core of this framework is a filtered improved recursive algorithm that characterizes the dependence between system inputs and outputs via an online‐updated nonparametric dependence measure. This mechanism eliminates the mandatory requirements for off‐line batch computation and prior system model assumptions that plague traditional delay estimation methods. The designed filtered recursive algorithm introduces a decaying, time‐varying filter to weight incremental statistical terms in empirical copulas, enabling it to fully adapt to complex, diverse time delay variation patterns. To address the practical difficulty of unknown delay switching periods in industrial scenarios, a comprehensive stability index is further constructed. By sweeping a wide range of candidate filtering durations, this index can reliably extract the intrinsic slow switching period of time delays without any pre‐specified period hyperparameters. Extensive numerical simulations covering multiple test dimensions validate the effectiveness and strong robustness of the overall recursive estimation framework. Although the basic recursive algorithm can eventually converge to true delay values across all switching modes, it only yields relatively stable and smooth delay outputs under monotonically varying delay scenarios and suffers from long‐lasting transient fluctuations and slow convergence during non‐monotonic delay jumps. In sharp contrast, the filtered recursive variant is capable of accurately tracking all arbitrary time delay switching profiles, including monotonic rise, monotonic fall, rise‐then‐fall, and fall‐then‐rise. In addition, the constructed stability index can precisely identify the real switching period of time‐varying delays and suppress estimation lag caused by mismatched filtering durations.