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Generalized Forgetting Recursive Least Squares: Stability and Robustness Guarantees

Brian Lai, Dennis S. Bernstein

TL;DR

This paper introduces GF-RLS, a discrete-time generalization of RLS derived from a composite least-squares cost that unifies diverse forgetting-based extensions. It provides rigorous stability results (Lyapunov, uniform Lyapunov, GAS, and GUES) under clearly stated matrix and regressor conditions, and robustness guarantees in the presence of time-varying parameters and bounded noises, including an errors-in-variables specialization. The framework demonstrates that many RLS extensions are special cases of GF-RLS, enabling a cohesive analysis and a set of practical design guidelines (A1–A8) linking algorithm structure to data properties. The work thus offers a principled approach to designing and analyzing RLS-based estimators with guaranteed stability and bounded-error performance in challenging identification scenarios.

Abstract

This work presents generalized forgetting recursive least squares (GF-RLS), a generalization of recursive least squares (RLS) that encompasses many extensions of RLS as special cases. First, sufficient conditions are presented for the 1) Lyapunov stability, 2) uniform Lyapunov stability, 3) global asymptotic stability, and 4) global uniform exponential stability of parameter estimation error in GF-RLS when estimating fixed parameters without noise. Second, robustness guarantees are derived for the estimation of time-varying parameters in the presence of measurement noise and regressor noise. These robustness guarantees are presented in terms of global uniform ultimate boundedness of the parameter estimation error. A specialization of this result gives a bound to the asymptotic bias of least squares estimators in the errors-in-variables problem. Lastly, a survey is presented to show how GF-RLS can be used to analyze various extensions of RLS from the literature.

Generalized Forgetting Recursive Least Squares: Stability and Robustness Guarantees

TL;DR

This paper introduces GF-RLS, a discrete-time generalization of RLS derived from a composite least-squares cost that unifies diverse forgetting-based extensions. It provides rigorous stability results (Lyapunov, uniform Lyapunov, GAS, and GUES) under clearly stated matrix and regressor conditions, and robustness guarantees in the presence of time-varying parameters and bounded noises, including an errors-in-variables specialization. The framework demonstrates that many RLS extensions are special cases of GF-RLS, enabling a cohesive analysis and a set of practical design guidelines (A1–A8) linking algorithm structure to data properties. The work thus offers a principled approach to designing and analyzing RLS-based estimators with guaranteed stability and bounded-error performance in challenging identification scenarios.

Abstract

This work presents generalized forgetting recursive least squares (GF-RLS), a generalization of recursive least squares (RLS) that encompasses many extensions of RLS as special cases. First, sufficient conditions are presented for the 1) Lyapunov stability, 2) uniform Lyapunov stability, 3) global asymptotic stability, and 4) global uniform exponential stability of parameter estimation error in GF-RLS when estimating fixed parameters without noise. Second, robustness guarantees are derived for the estimation of time-varying parameters in the presence of measurement noise and regressor noise. These robustness guarantees are presented in terms of global uniform ultimate boundedness of the parameter estimation error. A specialization of this result gives a bound to the asymptotic bias of least squares estimators in the errors-in-variables problem. Lastly, a survey is presented to show how GF-RLS can be used to analyze various extensions of RLS from the literature.
Paper Structure (30 sections, 19 theorems, 155 equations, 2 figures)

This paper contains 30 sections, 19 theorems, 155 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cost Function
  3. Stability
  4. Robustness
  5. Contributions
  6. Notation and Terminology
  7. Generalized Forgetting Recursive Least Squares (GF-RLS)
  8. Stability of Fixed Parameter Estimation
  9. Discussion of Conditions \ref{['item: cond F PSD']} through \ref{['item: cond weighted regressor PE and bounded']}
  10. Condition \ref{['item: cond F PSD']}
  11. Conditions \ref{['item: cond inv(Pkinv - F) upper bound']} and \ref{['item: cond Pk lower bound']}
  12. Condition \ref{['item: cond weighted regressor PE and bounded']}
  13. Robustness to Time-Varying Parameters, Measurement Noise, and Regressor Noise
  14. Discussion of Conditions \ref{['item: cond delta theta bound']} through \ref{['item: cond theta true bound']}
  15. Condition \ref{['item: cond delta theta bound']}
  16. ...and 15 more sections

Key Result

Theorem 1

For all $k \ge 0$, let $\Gamma_k \in {\mathbb R}^{p \times p}$ be positive definite, let $\phi_k \in {\mathbb R}^{p \times n}$, and let $y_k \in {\mathbb R}^{p}$. Furthermore, let $P_0 \in {\mathbb R}^{n \times n}$ be positive definite, and let $\theta_0 \in {\mathbb R}^{n}$. For all $k \ge 0$, let For all $k \ge 0$, define $J_k \colon {\mathbb R}^n \rightarrow {\mathbb R}$ by where Then, $J_k$

Figures (2)

  • Figure 1: This flowchart summarizes how different extensions of RLS (blue) can be derived as special cases of GF-RLS (red). Furthermore, this chart summarizes how certain RLS extensions are special cases of other RLS extensions (black).
  • Figure 2: Proof "roadmap" of Theorem \ref{['theo: GFRLS Lyapunov stability v2']} and Theorem \ref{['theo: GFRLS UUB']}.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Definition 3
  • Theorem 2
  • proof
  • Corollary 2
  • ...and 31 more