Imitation Learning Control for Thermoacoustic Stabilization of a Rijke Tube

This work studies the use of Neural Network (NN) boundary controllers to stabilize thermoacoustic instabilities in a Rijke tube. The dynamics of this phenomenon are governed by a system of
$4\times 4$
hyperbolic linear partial differential equations (PDEs) for the acoustic wave propagation, plus a linear ordinary differential equation (ODE) for the heat release. The control action is applied in one of the left boundary conditions, characterizing this system as underactuated. Previous results in the literature showed that this control problem can be solved by the backstepping methodology with stability guarantees in the
$\mathcal{L}_{2}$
sense. However, the stabilization and closed-loop system performance are usually affected by uncertainties. To tackle this issue, we rewrite this PDE-ODE boundary control problem as an imitation learning problem for stabilizing the system by observing the state values of a numerical simulator of the Rijke tube system under different operating conditions. Additionally, we present a Lyapunov-based method with local sector quadratic constraints to analyze the stability of the closed-loop system with the NN controllers. We demonstrate by simulations that the NN controller is able to stabilize the system under uncertain conditions, with the potential to overcome the performance of the backstepping.