On-line learning of dynamical systems in the presence of model mismatch and disturbances

This paper is concerned with the online learning of unknown dynamical systems using a recurrent neural network. The unknown dynamic systems to be learned are subject to disturbances and possibly unstable. The neural-network model used has a simple architecture with one layer of adaptive connection weights. Four learning rules are proposed for the cases where the system state is measurable in continuous or discrete time. Some of these learning rules extend the σ-modification of the standard gradient learning rule. Convergence properties are given to show that the weight parameters of the recurrent neural network are bounded and the state estimation error converges exponentially to a bounded set, which depends on the modeling error and the disturbance bound. The effectiveness of the proposed learning rules for the recurrent neural network is demonstrated using an illustrative example of tracking a Brownian motion.