A Concurrent Learning-Based Adaptive Observer for a Class of Uncertain Parabolic PDE Systems

This paper proposes a novel concurrent learning (CL)-based adaptive estimation law to solve the consensus observation problem of multi-observer for a class of parabolic partial differential equation (PDE) systems containing structured and unstructured uncertainties. We develop an observer network composed of multiple observers with only boundary information available, wherein information interaction is facilitated through the coupling of state estimation and parameter estimation. A novel adaptation law is proposed for the consensus estimate of uncertainties through the information transmission among observer agents in the network. To free the parameter identification from dependency on persistent excitation, the online data is stored in a historical information matrix. A relaxed and verifiable excitation condition is provided that the historical information matrix is built to full rank after sufficient but not persistent excitation (PE). For better observation performance, the unstructured uncertainty in the system is uniformly approximated by utilizing radial basis function neural networks (RBF NNs). By using recorded observations concurrently with online data, the theoretical stability analysis shows that the estimation error is exponentially convergent for structured uncertainties and ultimately bounded for unstructured uncertainties, both cases without requiring PE conditions. Simulation results illustrate the effectiveness of our proposed approaches via comparative simulations.