Adaptive Multi-Timescale Particle Filter for Nonlinear State Estimation in Wastewater Treatment: A Bayesian Fusion Approach with Entropy-Driven Feature Extraction

We propose an adaptive multi-timescale particle filter (AMTS-PF) for nonlinear state estimation in wastewater treatment plants (WWTPs) to address multi-scale temporal dynamics. The AMTS-PF decouples the problem into minute-level state updates and hour-level parameter refinements, integrating adaptive noise tuning, multi-scale entropy-driven feature extraction, and dual-timescale particle weighting. It dynamically adjusts noise covariances via Bayesian fusion and uses wavelet-based entropy analysis for adaptive resampling. The method interfaces seamlessly with existing WWTP control systems, providing real-time state estimates and refined parameters. Implemented on a heterogeneous computing architecture, it combines edge-level parallelism and cloud-based inference. Experimental validation shows superior performance over extended Kalman filters and single-timescale particle filters in handling nonlinearities and time-varying dynamics. The proposed AMTS-PF significantly enhances the accuracy of state estimation in WWTPs compared to traditional methods. Specifically, during the 14-day evaluation period using the Benchmark Simulation Model No. 1 (BSM1), the AMTS-PF achieved a root mean square error (RMSE) of 54.3 mg/L for heterotroph biomass (XH) estimation, which is a 37% reduction compared to the standard particle filter (PF) with an RMSE of 68.9 mg/L. For readily biodegradable substrate (Ss) and particulate products (Xp), the AMTS-PF also demonstrated superior performance with RMSE values of 7.2 mg/L and 9.8 mg/L, respectively, representing improvements of 24% and 21% over the PF. In terms of slow parameters, the AMTS-PF showed a 37% reduction in RMSE for the maximum heterotrophic growth rate (μH) estimation compared to the PF. These results highlight the effectiveness of the AMTS-PF in handling the multi-scale temporal dynamics and nonlinearities inherent in WWTPs. This work advances the state-of-the-art in WWTP monitoring by unifying multi-scale temporal modeling with adaptive Bayesian estimation, offering a practical solution for improving operational efficiency and process reliability.