An adaptive algorithm with information-dependent data forgetting

An adaptive algorithm is derived, based on a weighted least-squares criterion incorporating a penalty on parameter variations. The result is a recursive algorithm whose covariance matrix update is identical to existing regularized or stabilized least-squares algorithms. In particular, the covariance matrix and its inverse are bounded under weak conditions. The parameter update equation is different from conventional algorithms and exhibits a second-order response, as opposed to the first-order response of the standard weighted least-squares algorithm. An averaging analysis shows that a linear time-invariant approximation of the parameter response gives useful information about the system. The approximate system has two sets of poles, with one set being the same as the one obtained for the weighted least-squares algorithm. The other set of poles is close to the origin in the z-plane when there is significant excitation, and close to the number one when the excitation vanishes. The results allow one to select design parameters so that the algorithm reacts with a fast response under normal conditions, but slows down when the excitation is insufficient to yield adequate noise rejection.