Boundary proportional control of thermoacoustic instability in a Rijke tube

Abstract

Thermoacoustic instability has always been a critical concern of low pollution combustion chambers and combustion components of ground gas turbines. It is also a severe challenge for dynamic system design. In order to reveal the thermoacoustic instability, a typical Rijke tube based on partial differential equations (PDEs) cascaded with the Heckl heat release is studied. A boundary proportional feedback control to stabilize the thermoacoustic instability in the Rijke tube is proposed. First of all, one-dimensional gas dynamic equations coupled with the Heckl heat release are presented to describe the thermoacoustic unstable dynamic system in the Rijke tube. Then by using the small deviation linearization and characteristic coordinate transformation, the thermoacoustic coupling inside the domain is transferred into a group of 4 × 4 PDEs with the boundary coupling. Next, a boundary proportional feedback control, characterized by its practical implementation feasibility, is proposed to ensure the exponential stabilization of the thermoacoustic unstable system. A solvability condition is also provided to theoretically design the boundary proportional controller. Furthermore, the optimal control parameters are given by solving the nonlinear programming problem to achieve the fastest convergence rate of the system. Finally, the effectiveness of the boundary proportional feedback control is verified by simulations, and its robustness to the uncertain time delay and the nonlinearity of the heat release is also discussed.