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In this article, a stochastic gradient based online learning algorithm forExtreme Learning Machines (ELM) is developed (SG-ELM). A stability criterionbased on Lyapunov approach is used to prove both asymptotic stability ofestimation error and stability in the estimated parameters suitable foridentification of nonlinear dynamic systems. The developed algorithm not onlyguarantees stability, but also reduces the computational demand compared to theOS-ELM approach based on recursive least squares. In order to demonstrate theeffectiveness of the algorithm on a real-world scenario, an advanced combustionengine identification problem is considered. The algorithm is applied to twocase studies: An online regression learning for system identification of aHomogeneous Charge Compression Ignition (HCCI) Engine and an onlineclassification learning (with class imbalance) for identifying the dynamicoperating envelope of the HCCI Engine. The results indicate that the accuracyof the proposed SG-ELM is comparable to that of the state-of-the-art but addsstability and a reduction in computational effort.

Stochastic Gradient Based Extreme Learning Machines For Online Learning of Advanced Combustion Engines