Stochastic gradient based extreme learning machines for stable online learning of advanced combustion engines

We propose and develop SG-ELM, a stable online learning algorithm based on stochastic gradients and Extreme Learning Machines (ELM). We propose SG-ELM particularly for systems that are required to be stable during learning; i.e., the estimated model parameters remain bounded during learning. We use a Lyapunov approach to prove both asymptotic stability of estimation error and boundedness in the model parameters suitable for identification of nonlinear dynamic systems. Using the Lyapunov approach, we determine an upper bound for the learning rate of SG-ELM. The SG-ELM algorithm not only guarantees a stable learning but also reduces the computational demand compared to the recursive least squares based OS-ELM algorithm (Liang et al., 2006). In order to demonstrate the working of SG-ELM on a realworld problem, an advanced combustion engine identification 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. The case studies demonstrate that the accuracy of the proposed SG-ELM is comparable to that of the OS-ELM approach but adds stability and a reduction in computational effort. & 2015 Elsevier B.V. All rights reserved.