Identification of averaged dynamics of a controlled combustion instability

Attenuation of thermoacoustic oscillations by active control has lately received considerable attention. In the present work, we identify the averaged dynamics or the dynamics of the amplitude of a controlled thermoacoustic instability. The identification is performed on a lean premixed combustor which has feedback control incorporated to attenuate the amplitude of the thermoacoustic oscillation. For the purpose of control, acoustic pressure in the combustor is sensed and fuel flow actuated by a phase-shifted version of the measured pressure signal. This control strategy, used by several authors in the literature, is motivated by the famous Rayleigh criterion for growth/attenuation of thermoacoustic oscillations. The averaged model we identify consists of a map of equilibrium amplitudes parametrized by the control phase shift, to which all the amplitudes settle exponentially fast from all initial conditions. An equilibrium map is obtained by fitting a curve through the equilibria found from steady state experiments, and decay rates at these equilibria are estimated from cumulative averaging of several step responses (the steps are in the controller parameter) to eliminate noise. The identified closed-loop model can be used to optimize feedback and for simulation studies.