Backstepping stabilization of a linearized ODE–PDE Rijke tube model.

The problem of boundary stabilization of thermoacoustic oscillations in the Rijke tube is investigated using the backstepping method; as a first step, this work only considers the full-state design. This system consists of a vertical tube open at both ends and a heater placed in the lower half of the tube. To study this problem we consider that the mathematical model takes the form of 2 × 2 linear first-order hyperbolic partial differential equations (PDEs) with a point source term (induced by the Dirac delta distribution) on the right hand side, plus the coupling of an ordinary differential equation (ODE), and control input at one boundary condition. The presence of the Dirac delta distribution implies that the system solution has a discontinuity on a point of the domain, but is continuous everywhere else. We use a coordinate transformation to rewrite the equations into a system of four transport PDEs convecting in opposite directions and to translate the discontinuity to the boundary conditions. Then, a full state feedback backstepping controller is designed to exponentially stabilize the origin. However, the model is non-strict-feedback making unfeasible the use of standard backstepping designs. This issue is tackled by formulating a well-posed and invertible integral transformation with Volterra and Fredholm terms that maps the Rijke system into a target system with desirable stability properties. An exact piecewise-differentiable expression for the kernels of this transformation is found, allowing us in turn to derive an explicit feedback control law. Simulation results are presented to illustrate the effectiveness of the proposed control design. [ABSTRACT FROM AUTHOR]