Dynamic Factor and Multi-Innovation-Based Output–Input Feedback Elman Network Modeling From Measurements

Accurate parameter estimation is vital in instrumentation and measurement, especially for modeling dynamic systems based on measurement data. Recurrent networks have been widely adopted for this purpose, with the output–input feedback (OIF) Elman network offering enhanced modeling through its dual-feedback structure. However, most existing parameter estimation algorithms for such a network underuse historical measurements and exhibit limited adaptability to its dual-feedback mechanism. This article proposes a dynamic factor and multi-innovation-based gradient (DFMIG) parameter estimation algorithm for the OIF Elman network. The algorithm is derived from a multi-innovation cost function augmented with a weight decay factor, enabling dynamic balancing between current and past measurements to improve estimation accuracy. A dynamic factor adjustment strategy is introduced to adaptively tune both the weight decay factor and the self-feedback factors of the network based on the evolving weight parameter estimates. Unlike batch-based methods, the proposed algorithm supports recursive updates, enabling real-time parameter estimation from sequential measurement data. Convergence and computational complexity analyses confirm the theoretical properties of the proposed algorithm, while numerical simulations and a real-world electric load forecasting case study demonstrate its superior modeling performance compared with existing methods.