A procedure for real-time mode decomposition, observation, and prediction for active control of combustion instabilities

This paper presents and examines the behavior of a nonlinear observer. The observer is a system whose time evolution is described by a set of nonlinear differential equations. The input to the observer is a time dependent signal, assumed equal to a sum of N sinusoids, each described by a frequency, amplitude, and phase that do not change with time. An N-mode observer is designed for a nominal input composed of N modes. The observer´s task is to track these three characteristics for each component mode of the input. When the actual input to the observer is indeed composed of N modes, then the frequencies, amplitudes, and phases of these modes correspond to an equilibrium point of the observer. Numerical computer simulations, supported by analytical arguments, strongly suggest that this equilibrium point is asymptotically stable and has a rather large domain of attraction. Such simulations also suggest that the observer is robust, namely, it is able to sufficiently identify the characteristics of N dominant modes when forced with an M-mode input, when M is greater than N. This is the case most likely to be encountered in practice. From the point of view of combustion instabilities, the usefulness of the observer rests on the premise that the unstable pressure oscillations in a combustion chamber be quasi-steady and have a discrete spectral content. In such a case, an N-mode observer with a large enough N could, in principle, track the frequencies, amplitudes, and phases of the individual unstable modes in real-time. The information obtained during this process can be used to cancel any time delays in the actuators, and form, in accordance with Rayleigh´s criterion, the secondary feedback fuel flow for optimal damping of the instability. A preliminary version of this observer has performed this task extremely well in numerous experiments with an unstable combustor.