Stochastic Averaging for Identification of Feedback Nonlinearities in Thermoacoustic Systems

We present algorithms based on stochastic averaging for estimating nonlinear feedback parameters obtained from time series data with application to noise-driven nonlinear vibration systems, with particular emphasis on limit-cycling thermo-acoustic systems. The harmonic and Gaussian components of relevant signals are estimated from the probability density function (pdf) of an output signal from a single experiment. The respective feedback gains, along with a phase-shifting element are fit to a nominal (given) linear oscillator model from which the parameters of a nonlinearity are fit. When input-output data are available from multiple experiments, the feedback nonlinearity can be estimated point-wise via an iterative algorithm, applicable when the appropriate input signals have a constant (Gaussian) variance. The estimation procedures are demonstrated on a benchmark thermo-acoustic model and applied to time-series data obtained from a limit-cycling combustor rig experiment. In the latter case, relations between the feedback parameters and the fuel to air ratio are briefly discussed.