A differential-delay estimator for thermoacoustic oscillations in a Rijke tube using in-domain pressure measurements
Résumé
In this work, we study the observer design problem for estimating thermoacoustic instabilities in a Rijke tube, using an in-domain point pressure measurement. Writing the system model in Riemann coordinates and, after a "folding transformation", it takes the form of 4 × 4 linear hyperbolic partial differential equations (PDEs) coupled at the boundaries with a linear ordinary differential equation (ODE). This results in a PDE-ODE observer design problem, whose output is a combination of two infinite-dimensional states measured inside the domain. As a first step in our observer design, a second "folding" transformation is applied around the measurement point, resulting in a 6 × 6 PDE-ODE system with measurements at one boundary. Then, the observer is designed as a copy of these equations plus an output injection term, which is given by a linear operator in the right-hand side of the ODE part. This operator is then chosen so that convergence of estimates is guaranteed by using stability properties of differential-delay systems. The design extends a previous result, based on backstepping, that required two measurements (both pressure and velocity). Simulation results are presented to illustrate the effectiveness of the proposed observer design.
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