Adaptive Backstepping Stabilization of Thermoacoustic Instability in a Linearized ODE-PDE Rijke Tube Model

This paper proposes an adaptive scheme for the boundary stabilization of thermoacoustic instability in the Rijke tube system using the backstepping method. The mathematical model of the system is characterized by a
$2\times 2$
linear hyperbolic partial differential equation (PDE) coupled with a first-order ordinary differential equation (ODE) in a non-strict-feedback form. Recently, a full state feedback controller has been developed to stabilize the system, assuming that the parameters of the model are known. We take into account the most common uncertain parameters which result in the coefficients of the first-order ODE system being unknown parameters. The technique of adaptive identifier is then used along with the normalized gradient algorithm to achieve the parameter update laws. The adaptive control law is obtained by replacing the output of the identifier and estimated parameters in the non-adaptive state feedback control law. The adaptive control law is then manipulated such that it uses a few measurements of the PDE states. According to the stability analysis of the system, the proposed controller guarantees that all closed-loop system states are bounded, while the ODE-PDE system states are convergent to zero. Performance of the proposed scheme is evaluated by the simulation examples.