Boundary adaptive observer design for heat PDEs with spatially varying parameters

• Published in: IFAC-PapersOnLine Vol. 56; no. 2; pp. 9930 - 9935
• Main Authors: Bouklata, A., Giri, F., Krstic, M., Brouri, A., Chaoui, F.Z.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.01.2023
• Subjects: Adaptive observer design; parameters estimation; PDEs; states estimation; Adaptive observer design; parameters estimation; states estimation; PDEs
• ISSN: 2405-8963; 2405-8963
• DOI: 10.1016/j.ifacol.2023.10.689

We are considering the problem of observer design for parabolic partial differential equations (PDEs) that are subject to parameter uncertainty. The novelty lies in the fact that the unknown parameters are allowed to be spatially varying and the associated regressor is subject to a mild assumption. Then, the initial observer design problem, involving a scalar PDE with spatially-varying unknown parameters, is shown to be transformable into a new problem involving coupled PDEs with spatially-invariant parameters and a bounded modelling error. Invoking the high-gain principle and the so-called decoupling transformation technique, an adaptive observer is designed that provides online estimates of the state and parameter vectors. The original spatially varying parameters are recovered using least-squares estimators. All state and parameter estimation errors are shown, under appropriate persistent-excitation (PE) conditions, to be exponentially convergent to balls cantered on the origin with radiuses depending on the adaptive observer gain: the higher the observer gain the better the accuracy of estimates.