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.

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