Abstract

We are considering the problem of state observation for a class of infinite dimensional systems modeled by parabolic type PDEs. The model is subject to parametric uncertainty entering in both the domain equation and the boundary condition. An adaptive boundary observer, providing online estimates of the system state and parameters, is designed using finite- and infinite-dimensional backstepping-like transformations. The observer is exponentially convergent under an ad hoc persistent excitation condition.

Highlights

  • The problem of observer design for infinite dimensional systems (IDSs) is given an increasing interest

  • Output-feedback adaptive controllers have been proposed for some classes of IDSs including specific parabolic PDEs (Hong & Bentsman, 1994; Smyshlyaev & Krstic, 2007b) and wave PDEs subject to a boundary harmonic disturbance linearly parameterized along a known set of functions (Smyshlyaev & Krstic, 2006)

  • A quite complete description of the literature on adaptive controllers of IDSs systems described by parabolic equations, where both sensing and actuation are performed at the boundary and the unknown parameters are allowed to be spatially varying, can be found in Smyshlyaev and Krstic (2010)

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Summary

Introduction

The problem of observer design for infinite dimensional systems (IDSs) is given an increasing interest. A quite complete description of the literature on adaptive controllers of IDSs systems described by parabolic equations, where both sensing and actuation are performed at the boundary and the unknown parameters are allowed to be spatially varying, can be found in Smyshlyaev and Krstic (2010). This study is focused on the problem of designing adaptive observers featuring exponential convergence of the state estimate and the unknown parameter vector estimate. The problem has recently been addressed in Ahmed-Ali, Giri, Krstic, Burlion, and Lamnabhi-Lagarrigue (2015a); Ahmed-Ali, Giri, Krstic, LamnabhiLagarrigue, and Burlion (2015b) for a class of semilinear parabolic PDEs. In Ahmed-Ali et al (2015a), an exponentially convergent adaptive observer has been proposed for parabolic PDEs containing a single unknown parameter in the boundary condition.

Observation problem statement
Observer structure
Adaptive observer analysis
Conclusion
Simulation

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