This paper deals with the analysis of a set of measurements collected on a lean premixed combustion process operating in a limit cycle. Due to the fact that the data are collected in closed-loop and the system has no external excitation, the identification task is particularly challenging. This work mainly focuses on the issue of the feasibility of the identification task. It will be shown that, despite the paucity of information available, a grey-box non-linear model can be estimated. The model provides an explanation both of the limit-cycle fundamental oscillation and of a non-harmonic high-frequency signal affecting the pressure of the combustion chamber.