Developing an IIR robust adaptive algorithm in the modified Filtered-x RLS form for active noise and vibration control systems

In this paper, a robust adaptive algorithm for active noise and vibration control applications is proposed and the robust stability of the algorithm is analyzed using a combination of the small gain theorem and Popov´s hyperstability theorem. The algorithm is developed based on the so-called Filtered-x RLS algorithm in the modified form. In design and analysis of the algorithm, it is assumed that the estimated model of the secondary path is associated with a set of uncertainties of additive structure; and sufficient conditions for stability of the algorithm are derived. In fact, by introducing a stabilizing filter, the aim is to design this filter in a way that the achieved sufficient conditions for robust stability are satisfied. The employed method is to transform the proposed control structure to an equivalent output error identification problem, and then formulate the governing adaptive algorithm in a way that is representable as a feedback control problem. In view of this approach, sufficient conditions for robust stability of the adaptive algorithm will be equivalent to find the conditions for the stability of the established feedback control system. The technique applied here to this end is established on the energy conservation relation that is valid for the general data models in adaptive filters.