Exponential Convergence of Piecewise-Constant Parameters Identification under Finite Excitation Condition

TL;DR

This paper introduces an improved identification method for piecewise-constant parameters in linear regression, achieving exponential convergence under finite excitation conditions and robustness to regressor amplitude variations.

Contribution

It develops an enhanced I-DREM procedure with finite interval filtering, ensuring exponential error decay during excitation and convergence to zero outside it, with proven stability.

Findings

Exponential convergence of parameter error during excitation interval

Robustness to regressor amplitude variations

Numerical validation including noise scenarios

Abstract

A problem of identification of piecewise-constant unknown parameters of a linear regression equation (LRE) is considered. Such parameters change their values over the interval of the regressor finite (rather than persistent) excitation. To solve it, the previously proposed I-DREM procedure and the integral-based filtering method with the exponential forgetting and resetting are improved: the integral of the filter equations is taken over the finite time intervals, which belong to the finite excitation time range. This allows one to obtain an exponentially bounded identification error over the excitation interval, and, when the LRE parameters are constant outside such interval, to provide exponential convergence of the identification error to zero. In addition, the applied method of the regressor excitation normalization makes it possible to obtain the same rate of convergence of the parameter error for the regressors of various amplitudes. The stability and all the above-mentioned properties are proved for the derived identification method. The results of numerical experiments (including the case of the noise-contaminated regressor and output measurements) fully support the obtained theoretical results.