Newton-Like Extremum-Seeking for the Control of Thermoacoustic Instability

In practice, the convergence rate and stability of perturbation based extremum-seeking schemes can be very sensitive to the curvature of the plant map. An example of this can be seen in the use of extremum-seeking to reduce the amplitude of thermoacoustic oscillations in premixed, gas-turbine combustors. This sensitivity to the plant map curvature arises from the use of a gradient descent adaptation algorithm. Such extremum-seeking schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions, resulting in slower optimization than could be achieved for a fixed operating condition. This can severely reduce the effectiveness of perturbation based extremum-seeking schemes in some applications. In this paper, a sinusoidally perturbed extremum-seeking scheme using a Newton-like step is developed. Non-local stability results for the scheme are formulated using a Lyapunov analysis. A local analysis of the scheme is given to investigate the influence of plant dynamics and to show that the local rate of convergence is independent of the plant map curvature. The benefit of this plant map curvature independence is then experimentally demonstrated in minimizing the thermoacoustic oscillations in a model premixed combustor.