Quantitative Stability Analysis Using Real-Valued Frequency Response Data

Models for the analysis of thermoacoustic instabilities are conveniently formulated in the frequency domain. In this case one often faces the difficulty that the response behavior of some elements of the system is only known at real-valued frequencies, although the transfer behavior at complex-valued frequencies is required for the quantification of the growth rates of instabilities. The present paper discusses various methods for extrapolation of frequency response data at real-valued frequencies into the complex plane. Some methods have been used previously in thermoacoustic stability analysis; others are newly proposed. First the pertinent mathematical background is reviewed, then the sensitivity of predicted growth rates on the extrapolation scheme is explored. This is done by applying different methods to a simple thermoacoustic system, i.e., a ducted premixed flame, for which an analytical solution is known. A short analysis determining the region of confidence of the extrapolated transfer function is carried out to link the present study to practical applications. The present study can be seen as a practical guideline for using frequency response data collected for a set of real-valued frequencies in quantitative linear stability analysis.