  In this article, a stochastic gradient based online learning algorithm for
Extreme Learning Machines (ELM) is developed (SG-ELM). A stability criterion
based on Lyapunov approach is used to prove both asymptotic stability of
estimation error and stability in the estimated parameters suitable for
identification of nonlinear dynamic systems. The developed algorithm not only
guarantees stability, but also reduces the computational demand compared to the
OS-ELM approach based on recursive least squares. In order to demonstrate the
effectiveness of the algorithm on a real-world scenario, an advanced combustion
engine identification problem is considered. The algorithm is applied to two
case studies: An online regression learning for system identification of a
Homogeneous Charge Compression Ignition (HCCI) Engine and an online
classification learning (with class imbalance) for identifying the dynamic
operating envelope of the HCCI Engine. The results indicate that the accuracy
of the proposed SG-ELM is comparable to that of the state-of-the-art but adds
stability and a reduction in computational effort.
