Identification of noisy dynamical systems with parameter estimation based on Hopfield neural networks

An algorithm for estimating time-varying parameters of dynamical systems is proposed, within the large family of prediction error methods. The algorithm is based on the ability of Hopfield neural networks to solve optimisation problems, since its formulation can be summarized as minimisation of the prediction error by means of a continuous Hopfield network. In previous work, it was proved, under mild assumptions, that the estimates converge towards the actual values of parameters and the estimation error remains asymptotically bounded in the presence of measurement noise. The novelty of this work is the advance in the robustness analysis, by considering deterministic disturbances, which do not fulfil the usual statistical hypothesis such as normality and uncorrelatedness. A model of HIV epidemics in Cuba is used as suitable benchmark, which is confirmed by the computation of the sensitivity matrix. The results show a promising performance, in comparison to the conventional Least Squares Estimator. Indeed, the estimation error is almost always lower in the proposed method that in least squares, and it is never significantly higher. Further, from a qualitative point of view, the estimate provided by the Hopfield estimator is smoother, with no overshoot that could eventually destabilize a closed control loop. A significant finding is the fact that the form of the perturbation affects critically the dynamical behaviour and magnitude of the estimation, since the estimation error asymptotically vanishes when the disturbances are additive, but not when they are multiplicative. To summarize, we can conclude that the proposed estimator is an efficient and robust method to estimate time-varying parameters of dynamical systems.