Identification of instantaneous flame transfer functions

Most existing system identification methods are designed for time-invariant systems. However, for many practical applications, data collection over a wide range of parameters under stationary conditions is either infeasible or costly. To address this limitation, we propose a time-domain, nonparametric methodology for linear, time-varying (LTV) systems, extending the classical paradigm of impulse response function estimation from broadband data using least-squares regression. We introduce the time-varying impulse response function (TV-IRF), which uniquely characterizes the dynamic behavior of LTV systems, and represent it as a series expansion over an orthonormal basis. The collected nonstationary data is projected onto each basis function, and the TV-IRF is estimated using least-squares regression. To validate and analyze this methodology, we first apply it to data generated from measurements of a swirled, hydrogen-enriched flame. Subsequently, we apply it to identify the TV-IRF and time-varying flame transfer functions (TV-FTF) of a canonical slit flame. Using both stationary and nonstationary direct numerical simulations across a wide range of mean flow velocities in the burner, we demonstrate that the instantaneous flame transfer functions derived from the TV-FTF closely match those identified in a stationary setting. Notably, this accuracy is maintained even when the length of nonstationary time series is equivalent to that used for stationary identification at a single velocity. This methodology promises substantial reductions in computational and experimental costs, paving the way for efficient exploration and identification of dynamical systems across large parameter spaces.