Adaptive observer design for uncertain hyperbolic PDEs coupled with uncertain LTV ODEs; Application to refrigeration systems

The problem of estimating the temperatures and the heat transfer coefficient of a concentric tube heat exchanger coupled with a heater is considered in this work. Measurements collected from the extremities of the exchanger tube are used to estimate the heat distribution over the length of the exchanger, which induces a boundary estimation problem. This system, which is part of any standard cooling plant, is particularly challenging due to the distributed nature of its variables. It is modeled by a system of (2 x 2) hyperbolic PDEs, coupled with an ODE at the boundary. To solve the estimation problem, we consider a general class of systems consisting of a (2 x 2) hyperbolic system coupled with a set of nX linear time-varying (LTV) ODEs at the boundary. Both the PDE and the ODEs have uncertain parameters to be estimated. The objective is to estimate the PDE states, the ODE states, and the parameters simultaneously with no assumption on the ODEs stability. We design a Luenberger state observer, and our method is mainly based on the decoupling of the PDE estimation error states from that of the ODEs via swapping design. We then derive the observer gains from the Lyapunov analysis of the decoupled system after proving the boundedness of the swapping filters. We give sufficient conditions of the exponential convergence of the adaptive observer through differential Lyapunov inequalities (DLIs). Finally, we apply the developed theory on the coupled heat exchanger-heater model to evaluate the performance of the observer in numerical simulations. & COPY; 2023 Elsevier Ltd. All rights reserved.

Automated summary

This work addresses the problem of estimating temperatures and heat transfer coefficients in a concentric tube heat exchanger coupled with a heater. The distributed nature of the system variables makes it challenging, as it requires estimating PDE and ODE states simultaneously. A Luenberger state observer is designed to decouple the estimation errors of the PDEs and ODEs, and sufficient conditions for the exponential convergence of the adaptive observer are derived. The performance of the observer is evaluated through numerical simulations on the heat exchanger-heater model.