DNN-state identification of 2D distributed parameter systems

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. There are many examples in science and engineering which are reduced to a set of partial differential equations (PDEs) through a process of mathematical modelling. Nevertheless there exist many sources of uncertainties around the aforementioned mathematical representation. Moreover, to find exact solutions of those PDEs is not a trivial task especially if the PDE is described in two or more dimensions. 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 article, a strategy based on the differential neural network (DNN) for the non-parametric identification of a mathematical model described by a class of two-dimensional (2D) PDEs is proposed. The adaptive laws for weights ensure the 'practical stability' of the DNN-trajectories to the parabolic 2D-PDE states. To verify the qualitative behaviour of the suggested methodology, here a non-parametric modelling problem for a distributed parameter plant is analysed. 1. Introduction The dynamics description of natural phenomena is usually described by a set of differential equations (DEs). Those descriptions are obtained using well-known mathematical modelling rules. In particular, partial differential equations (PDEs) appear when the mathematical model considers time (t) as well as space (x) dependence of the state variables. For instance, linear second-order parabolic PDEs appear in time-dependent diffusion problems, such as the transient flow of heat conduction and others. These equations define a state (u :¼ u(x, t)) in both space and time. Nevertheless, to find exact solutions of those models is not an easy assignment (Banks 1994). On the other hand, some numerical methods have been developed to approximate the solution of such distributed parameter models. For example, the finite difference method (FDM; Smith 1978) and the finite …