Abstract
This paper deals with state estimation on a pressurized water pipe modeled by nonlinear coupled distributed hyperbolic equations for non-conservative laws with three known boundary measures. Our objective is to estimate the fourth boundary variable, which will be useful for leakage detection. Two approaches are studied. Firstly, the distributed hyperbolic equations are discretized through a finite-difference scheme. By using the Lipschitz property of the nonlinear term and a Lyapunov function, the exponential stability of the estimation error is proven by solving Linear Matrix Inequalities (LMIs). Secondly, the distributed hyperbolic system is preserved for state estimation. After state transformations, a Luenberger-like PDE boundary observer based on backstepping mathematical tools is proposed. An exponential Lyapunov function is used to prove the stability of the resulted estimation error. The performance of the two observers are shown on a water pipe prototype simulated example.
Highlights
Supervision of physical transport plant has been an active research topic in recent years and water distribution network (WDN) monitoring is a major concern [1], [2], [3]
The indirect approach uses the discretized system; an exponential boundary nonlinear observer is designed by using the Lipschitz property of the nonlinear term and by solving Linear Matrix Inequalities (LMIs) to ensure the stability of estimation error
The direct approach, where some state transformations are used in order to design a Luenberger-like PDE boundary observer based on backstepping mathematical tools
Summary
Supervision of physical transport plant has been an active research topic in recent years and water distribution network (WDN) monitoring is a major concern [1], [2], [3]. 1) The indirect approach where Partial Differential Equations (PDE) operating in a functional infinite dimensional space are approximated by Ordinary Differential Equations (ODE) in a finite-dimensional space through a differentiation operator (finite differences schemes are commonly used [7], [8]) The advantage of such approach is to give access to many techniques well developed for observers design in finite dimension. 2) The direct approach that preserves all the system information for state estimation [9], [10] by designing directly PDE observers This approach should potentially lead to better estimation results since no model approximation is made, it is much more complex and needs mathematical background.
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