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Unbiased Parameter Estimation via DREM with Annihilators

Anton Glushchenko, Konstantin Lastochkin

TL;DR

This paper addresses biased parameter estimation in perturbed regression problems by integrating Bias-Eliminated Least-Squares (BELS) with Dynamic Regressor Extension and Mixing (DREM). A continuous-time modification introduces a stable filter, perturbation decomposition, and annihilators to produce unbiased online estimates, with two cases handling different disturbance spectra. The main result is a pair of estimators (one derived without perturbation bias using Case 1, and a bias-eliminated Case 2 estimator using annihilators) that achieve convergence to an arbitrarily small neighborhood of the true parameters as the averaging window grows, under stated identifiability and excitation conditions. The approach is validated via numerical experiments showing improved robustness to perturbations and potential applications to adaptive observers and control systems.

Abstract

In adaptive control theory, the dynamic regressor extension and mixing (DREM) procedure has become widespread as it allows one to describe major of adaptive control problems in unified terms of the parameter estimation problem of a regression equation with a scalar regressor. However, when the system/parameterization is affected by perturbations, the estimation laws, which are designed on the basis of such equation, asymptotically provides only biased estimates. In this paper, based on the bias-eliminated least-squares (BELS) approach, a modification of DREM procedure is proposed to annihilate perturbations asymptotically and, consequently, asymptotically obtain unbiased estimates. The theoretical results are supported with mathematical modelling and can be used to design adaptive observers and control systems.

Unbiased Parameter Estimation via DREM with Annihilators

TL;DR

This paper addresses biased parameter estimation in perturbed regression problems by integrating Bias-Eliminated Least-Squares (BELS) with Dynamic Regressor Extension and Mixing (DREM). A continuous-time modification introduces a stable filter, perturbation decomposition, and annihilators to produce unbiased online estimates, with two cases handling different disturbance spectra. The main result is a pair of estimators (one derived without perturbation bias using Case 1, and a bias-eliminated Case 2 estimator using annihilators) that achieve convergence to an arbitrarily small neighborhood of the true parameters as the averaging window grows, under stated identifiability and excitation conditions. The approach is validated via numerical experiments showing improved robustness to perturbations and potential applications to adaptive observers and control systems.

Abstract

In adaptive control theory, the dynamic regressor extension and mixing (DREM) procedure has become widespread as it allows one to describe major of adaptive control problems in unified terms of the parameter estimation problem of a regression equation with a scalar regressor. However, when the system/parameterization is affected by perturbations, the estimation laws, which are designed on the basis of such equation, asymptotically provides only biased estimates. In this paper, based on the bias-eliminated least-squares (BELS) approach, a modification of DREM procedure is proposed to annihilate perturbations asymptotically and, consequently, asymptotically obtain unbiased estimates. The theoretical results are supported with mathematical modelling and can be used to design adaptive observers and control systems.
Paper Structure (6 sections, 55 equations, 2 figures)

This paper contains 6 sections, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Main result
  4. Numerical experiments
  5. Conclusion
  6. Proof of Theorem 1

Figures (2)

  • Figure 1: Behavior of $y\left( t \right)$ when $\delta \left( t \right){\rm{,\;}}v\left( t \right){\rm{:}} = 0$ and $\delta \left( t \right){\rm{,\;}}v\left( t \right){\rm{:}} = \ref{['eq25']}$.
  • Figure 2: Behavior of a) $\hat{\theta} \left( t \right)$ for \ref{['eq20']}, \ref{['eq28']}, \ref{['eq29']} and b) $\hat{\theta} \left( t \right)$ for \ref{['eq20']} using different values of $T$.