On a general structure for adaptation/learning algorithms. — Stability and performance issues

Abstract

The paper introduces a general structure for parameter adaptation/learning algorithms (PALA). This structure is characterized by the presence of an embedded ARMA (poles-zeros) filter in the PALA. The key question is how to select the coefficients of this filter in order, on the one hand, to guarantee the stability of the parameter estimator for any (positive) value of the adaptation gain/learning rate and for any initial conditions and on the other hand to accelerate the adaptation transient. In order to achieve this, it is shown that on one hand the embedded ARMA filter should be characterized by a positive real transfer function and on the other hand the filter acting on the correcting term (the dynamic adaptation gain) should be characterized by a strictly positive real transfer function. Specific conditions for the design of a second order ARMA embedded filter (ARIMA2 algorithm) are provided.It is shown in the paper that many parameter adaptation/learning algorithms (PALA) used in adaptive control, system identification and neural networks (Nesterov, Conjugate gradients, Momentum back propagation, Averaged gradient, Integral+proportional+derivative, …) are particular cases of the PALA structure introduced in this paper and specific conditions for the stable operation of these algorithms are given.Performance of the ARIMA2 algorithm as well as of the other algorithms reviewed in the paper will be comparatively evaluated by simulations and experimental results obtained on an active noise control system.