A combined B-spline-neural-network and ARX model for online identification of nonlinear dynamic actuation systems

This paper presents a block oriented nonlinear dynamic model suitable for online identification. The model has the well known Hammerstein architecture where as a novelty the nonlinear static part is represented by a B-spline neural network (BSNN), and the linear static one is formalized by an autoregressive exogenous model (ARX). The model is suitable as a feed-forward control module in combination with a classical feedback controller to regulate velocity and position of pneumatic and hydraulic actuation systems which present nonstationary nonlinear dynamics. The adaptation of both the linear and nonlinear parts is taking place simultaneously on a patter-by-patter basis by applying a combination of error-driven learning rules and the recursive least squares method. This allows to decrease the amount of computation needed to identify the model׳s parameters and therefore makes the technique suitable for real time applications. The model was tested with a silver box benchmark and results show that the parameters are converging to a stable value after 1500 samples, equivalent to 7.5 s of running time. The comparison with a pure ARX and BSNN model indicates a substantial improvement in terms of the RMS error, while the comparison with alternative nonlinear dynamic models like the NNOE and NNARX, having the same number of parameters but greater computational complexity, shows comparable performances.