Adaptive Neural Consensus Observer Networks Design for a Class of Semilinear Parabolic PDE Systems

This article concerns the investigation on the consensus problem for the joint state-uncertainty estimation of a class of parabolic partial differential equation (PDE) systems with parametric and nonparametric uncertainties. We propose a two-layer network consisting of informed and uninformed boundary observers where novel adaptation laws are developed for the identification of uncertainties. Particularly, all observer agents in the network transmit their information with each other across the entire network. The proposed adaptation laws include a penalty term of the mismatch between the parameter estimates generated by the other observer agents. Moreover, for the nonparametric uncertainties, radial basis function (RBF) neural networks are employed for the universal approximation of unknown nonlinear functions. Given the persistently exciting condition, it is shown that the proposed network of adaptive observers can achieve exponential joint state-uncertainty estimation in the presence of parametric uncertainties and ultimate bounded estimation in the presence of nonparametric uncertainties based on the Lyapunov stability theory. The effects of the proposed consensus method are demonstrated through a typical reaction-diffusion system example, which implies convincing numerical findings.