Robust SOC estimation for lithium-ion batteries: Combination of GRU and FOMIAUKF approach with an improved state transition matrix

The fractional-order model (FOM) exhibits superior capability in capturing electrochemical dynamics for accurate state of charge (SOC) estimation, particularly within the low-frequency impedance spectrum. Nevertheless, two enduring limitations restrict its practical deployment: (1) uncertainty in initial state estimation, and (2) reliance on ampere-hour integration, which leads to cumulative errors. To overcome these challenges, this study proposes a novel dual-process estimation algorithm that addresses the inherent shortcomings of conventional FOM-based state transition models: (a) A Gated Recurrent Unit (GRU) network is employed to learn temporal dependencies and provide data-driven initial state estimates, thereby eliminating the need for manual initialization. (b) The outputs of the GRU are refined in real time through an unscented transform resampling mechanism incorporated within the Fractional-Order Multi-Innovations Adaptive Unscented Kalman Filter (FOMIAUKF), enabling dynamic correction of state deviations. (c) A multi-innovation weighting strategy is integrated to improve the accuracy of posterior state estimation, enhancing the ability of the algorithm to suppress stochastic disturbances and accelerate convergence. The proposed framework establishes a tightly coupled system that synergistically combines data-driven learning with model-based filtering, thereby improving robustness across varying operational conditions. Experimental validation across diverse temperature scenarios demonstrates that the proposed approach consistently maintains MAE and RMSE below 2%, with deviations remaining within 2.5% under significant thermal fluctuations. Furthermore, it consistently outperforms standalone FOMIAUKF (1.92%) and GRU (2.05%) implementations, highlighting its robustness and practical applicability for real-world battery management systems.