Predictive Control for Transient Energy Growth of Thermoacoustic Systems in a Rijke Tube subject to Thermal Inertia

In combustion systems with stable eigenvalues, the presence of non-orthogonal eigenvalues can trigger instability due to transient growth caused by disturbances. This paper focuses on the impact of thermal inertia in Rijke tube unstable thermoacoustic systems with minimal transient energy growth. It aims to design predictive control strategy the Rijke tube system, ensuring the system's stability while minimizing the performance index including the acoustic energy, its variation rate, and control energy. Firstly, an ODE model for the thermoacoustic unstable system in a Rijke tube, incorporating nonlinear heat release and thermal inertia delay, was derived using the Galerkin method. Secondly, the optimization problem of the transient energy growth performance was proposed and the solvable condition was also presented by solving the convex optimization problem. Furthermore, an optimized predictive controller was designed. Finally, simulation results demonstrate that the proposed control and optimization method can asymptotically suppress the pressure oscillations in the Rijke tube achieve the optimized transient energy growth performance.