Adaptive Observer Design for Heat PDEs With Sensor Delay and Parameter Uncertainties

In this article, we consider the problem of observer design for heat diffusion-reaction partial differential equations (PDEs) that are monitored through delayed boundary sensor measurements. The novelty with respect to the existing observers lies in the fact that the PDE is subject to parameter uncertainties coming not only in the domain but also in the sensor. We develop an adaptive observer that provides online accurate estimates of both the state and parameter estimates. The observer design makes use of modal decomposition theory and the decoupling transformation method. The former is resorted to reformulate the observer design and analysis in a finite-dimensional space. The decoupling transformation makes the state estimation problem decoupled from the parameter estimation problem. Compared with the existing adaptive observers, the new observer features: 1) the injection of extra actions in the domain allowing a better compensation for the effects of parameter uncertainties and 2) a better consideration of all available information (at a given time) in the parameter estimator. The observer analysis is established using a Lyapunov-Krasovskii functional, under a persistent excitation assumption. The analysis highlights sufficient conditions, involving the maximal time delay, for the state and parameter estimation errors to be exponentially convergent to zero.