On the Local Stability Properties of Adaptive Parameter Estimators with Composite Errors and Split Algorithms

Abstract

We examine the stabllity characteristics of a generalized error system structure for adaptive parameter estimation systems. This error system form encompasses the structure of a number of particular applications of adaptive parameter estimation theory which fall outside the framework of more familiar error system models. In the basic model, one recursively updates the parameter estimate vector with a term which is the.product of a small step size, a filtered version of the regressor vector, and the system prediction error. The prediction error is a filtered inner product of the regressor and parameter error vectors. By contrast, the prediction error form entering our generalized error system is a sum of differently filtered products of corresponding entries in the regressor and parameter error vectors, termed a composite error. Similarly, the algorithm form, which we call a split algorithm, updates each parameter estimate individually by a term composed as the product of the step size, a filtered regressor element, and the composite error. In the update of different parameter estimates, the filtering operation applied to the appropriate regressor elements may be different, thereby “splitting” the algorithm. This paper analyzes the consequences for error system stabllity which derive from the added generality. For a particular choice of split algorithm, the generalized error system has stability properties similar to those for the basic error system. However, for other split algorithm selections, fundamental differences between the two error system forms arise. We demonstrate that, when using averaging theory techniques to analyze the error system, one must augment stability conditions for the basic error system in order to ensure stability for the generalized error system. We rigorously establish the inadequacy of regressor spectral restrictions and persistent spanning conditions to guarantee local error system stability of the generalized structure, but we demonstrate alternative conditions which will yield such local stability. To illustrate the concepts involved, we examine the recursive identification of parameters in a parallel-form realization of a linear system. Show more