Non-parametric modeling of uncertain hyperbolic partial differential equations using pseudo-high order sliding mode observers

There are many examples in science and engineering that may be described by a set of partial differential equations (PDEs). The modeling process of such phenomenons is in general a complex task. Moreover, there exist some sources of uncertainties around that mathematical representation that sometimes are difficult to be included in the obtained model. Neural networks appear to be a plausible alternative to get a non parametric representation of the aforementioned systems. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set to an arbitrary accuracy. In this paper a strategy based on differential neural networks (DNNs) for the non parametric identification in a mathematical model described by hyperbolic partial differential equations is proposed. The identification problem is reduced, to finding an exact expression for the weights dynamics using the DNN properties. The adaptive laws for weights ensure the convergence of the DNN trajectories to the hyperbolic PDE states. To investigate the qualitative behavior of the suggested methodology, here the no-parametric modeling problem for the wave equation is solved successfully. Some three dimension graphic representations are used to demonstrate the identification abilities achieved by the DNN designed in this paper.