Acoustic Boundary Output Feedback Stabilization of Dynamic n-τ Model Flame via Duct with Variable Cross Section

We propose an output feedback controller to stabilize thermoacoustic instabilities in a duct with variable cross section, by actuating and sensing the acoustic boundary opposite from the flame. A linear parametric state-space model, taking into account both time-delay effects from the classical <tex-math notation="LaTeX">n-τ</tex-math> model and low-pass filtering effects, models the flame subsystem. For the acoustics a distributed model taking into account variations in the cross-sectional area along the duct is developed and written as a <tex-math notation="LaTeX">2× 2</tex-math> linear hyperbolic PDE system. Next, a full-state feedback control law is proposed, designed by first mapping the considered plant into a simplified PDE-ODE cascade, for which control design is a solved problem. Subsequently the observer is designed by mapping the estimation error dynamics directly into a stable target PDE-ODE cascade. Combining the full-state feedback control law and observer we arrive at an output feedback controller. A simulation example to demonstrate the theory is presented, where the full-state feedback controller, observer and output feedback controller are shown to stabilize and estimate the pressure, velocity and heat release rate.