Abstract-A system of two coupled Van der Pol equations is considered as a model for combustion instability. This system is analyzed using the Krylov-Bogoliubov method. The results of the analysis are compared with simulation results and with data obtained from a combustion system.

In this paper a new analytically tractable model for combustion instability is proposed. The model is based on two coupled resonators in a feedforward path, and a feedback path composed of a delay, generalized Van der Pol term and a low pass filter. The model is analyzed and approximated using a refined Krylov-Bogoliubov (K-B) method. The analysis shows that the model captures the phenomenon of two coexisting oscillating modes which has been noted in combustion instability in [1], [2]. Conditions for the occurrence of various operation regimes have been established. The importance of delay and low pass filtering is discussed in this article. A frequency domain comparison between K-B approximations and the true outputs of the model has been provided in the end of the paper.

The paper proposes and analyzes two prototype structures of coupled generalized van der Pol equations able to describe self-excitation of simultaneous oscillations with distinct frequencies. These structures are relevant for describing oscillations phenomena which may be encountered on systems subject to control. These structures are analyzed using the Krylov-Bogoliubov averaging method. This analysis allows to establish conditions for the occurrence of the various operation regimes. The usefulness of the results is illustrated by their application to the straightforward analysis of the properties of a combustion instability model.

This papers deals with active control of combustion instabilities through measurement and feedback of pressure oscillations. The measurement is used to construct a multiplicative feedback control. The aim of such a control is to quench the oscillations associated with the instability associated with lean pre-mixed combustion. This quenching is analyzed using the Krylov-Bogoliubov approach applied to a tractable gray-box model of the underlying process and fitted using experimental data. A linear and a nonlinear feedback law are considered and conditions for quenching the oscillations established. Both give successful results verified by the simulation tests.

PreludeIt is a great pleasure for us to contribute to this book dedicated to Alberto Isidori on the occasion of his sixty fifth birthday. It is also, for the first author, the occasion to acknowledge a very long period of useful and pleasant exchange which started in 1973 (bilinear systems) and has continued through the years on various specific subjects.The important contributions of Albereto Isidori in the control of nonlinear systems have had a tremendous impact in the control community. Feedback linearization was one of the important subjects developed by Alberto. While the oscillatory nonlinear system considered in this contribution requires specific techniques for its analysis, still a feedback linearization is used for quenching the oscillations.