On the stability of adaptation process in active noise control systems

The stability analysis of the adaptation process, performed by the filtered-x least mean square algorithm on weights of active noise controllers, has not been fully investigated. The main contribution of this paper is conducting a theoretical stability analysis for this process without utilizing commonly used simplifying assumptions regarding the secondary electro-acoustic channel. The core of this analysis is based on the root locus theory. The general rules for constructing the root locus plot of the adaptation process are derived by obtaining root locus parameters, including start points, end points, asymptote lines, and breakaway points. The conducted analysis leads to the derivation of a general upper-bound for the adaptation step-size beyond which the mean weight vector of the active noise controller becomes unstable. Also, this analysis yields the optimum step-size for which the adaptive active noise controller has its fastest dynamic performance. The proposed upper-bound and optimum values apply to general secondary electro-acoustic channels, unlike the commonly used ones which apply to only pure delay channels. The results are found to agree very well with those obtained from numerical analyses and computer simulation experiments.