A Backstepping-based observer for estimation of thermoacoustic oscillations in a Rijke tube with in-domain measurements

Abstract

This paper presents an observer design for estimation of thermoacoustic instabilities in a Rijke tube. To study this problem, we consider that the acoustic dynamics is represented by the wave equation with a point source term representing the heat release. In turn, the heat release dynamics is given by a first-order ordinary differential equation (ODEs). The observer, whose design is based on the backstepping methodology, relies on measurements of acoustic pressure and velocity at an arbitrary point of the domain. The design employs a folding transformation (with two folds) around the measurements and the heat release point, allowing to write the system into a form with more states but boundary measurements and ODE couplings. Then, we formulate a well-posed and invertible integral transformation with both triangular and full terms that maps the observer error dynamics into an exponentially stable target system. The theoretical results were tested through numerical simulations in order to show the effectiveness of the design.