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Adaptive Kalman Filtering Developed from Recursive Least Squares Forgetting Algorithms

Brian Lai, Dennis S. Bernstein

TL;DR

The paper derives a Kalman-filter based least-squares cost (KFLS) whose recursive minimizer reproduces the Kalman filter, revealing that generalized forgetting RLS (GF-RLS) and many RLS extensions are special cases of the Kalman filter under appropriate process-noise modeling. It then introduces an adaptive Kalman filtering framework that injects forgetting from RLS extensions into the prior covariance update, bridging RLS forgetting strategies with Kalman updates. A robust variable forgetting factor example in a mass-spring-damper with impulsive disturbances demonstrates improved tracking and faster adaptation after disturbances. These results suggest that a broad family of forgetting strategies from the RLS literature can enhance Kalman-based state estimation in non-classical disturbance environments, with future work exploring additional forgetting mechanisms.

Abstract

Recursive least squares (RLS) is derived as the recursive minimizer of the least-squares cost function. Moreover, it is well known that RLS is a special case of the Kalman filter. This work presents the Kalman filter least squares (KFLS) cost function, whose recursive minimizer gives the Kalman filter. KFLS is an extension of generalized forgetting recursive least squares (GF-RLS), a general framework which contains various extensions of RLS from the literature as special cases. This then implies that extensions of RLS are also special cases of the Kalman filter. Motivated by this connection, we propose an algorithm that combines extensions of RLS with the Kalman filter, resulting in a new class of adaptive Kalman filters. A numerical example shows that one such adaptive Kalman filter provides improved state estimation for a mass-spring-damper with intermittent, unmodeled collisions. This example suggests that such adaptive Kalman filtering may provide potential benefits for systems with non-classical disturbances.

Adaptive Kalman Filtering Developed from Recursive Least Squares Forgetting Algorithms

TL;DR

The paper derives a Kalman-filter based least-squares cost (KFLS) whose recursive minimizer reproduces the Kalman filter, revealing that generalized forgetting RLS (GF-RLS) and many RLS extensions are special cases of the Kalman filter under appropriate process-noise modeling. It then introduces an adaptive Kalman filtering framework that injects forgetting from RLS extensions into the prior covariance update, bridging RLS forgetting strategies with Kalman updates. A robust variable forgetting factor example in a mass-spring-damper with impulsive disturbances demonstrates improved tracking and faster adaptation after disturbances. These results suggest that a broad family of forgetting strategies from the RLS literature can enhance Kalman-based state estimation in non-classical disturbance environments, with future work exploring additional forgetting mechanisms.

Abstract

Recursive least squares (RLS) is derived as the recursive minimizer of the least-squares cost function. Moreover, it is well known that RLS is a special case of the Kalman filter. This work presents the Kalman filter least squares (KFLS) cost function, whose recursive minimizer gives the Kalman filter. KFLS is an extension of generalized forgetting recursive least squares (GF-RLS), a general framework which contains various extensions of RLS from the literature as special cases. This then implies that extensions of RLS are also special cases of the Kalman filter. Motivated by this connection, we propose an algorithm that combines extensions of RLS with the Kalman filter, resulting in a new class of adaptive Kalman filters. A numerical example shows that one such adaptive Kalman filter provides improved state estimation for a mass-spring-damper with intermittent, unmodeled collisions. This example suggests that such adaptive Kalman filtering may provide potential benefits for systems with non-classical disturbances.
Paper Structure (10 sections, 6 theorems, 39 equations, 4 figures, 1 table)

This paper contains 10 sections, 6 theorems, 39 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Background Material
  4. The one-step Kalman Filter
  5. Discrete-Time LTV State Transition Function
  6. A Least Squares Cost Function which Derives the Kalman Filter
  7. RLS Extensions as Special Cases of the Kalman Filter
  8. Adaptive Kalman Filtering Developed from RLS Forgetting Algorithms
  9. Kalman Filter with Robust Variable Forgetting Factor
  10. Conclusion

Key Result

Theorem 1

For all $k \ge 0$, let $A_k \in {\mathbb R}^{n \times n}$ be nonsingular, $B_k \in {\mathbb R}^{n \times m}$, $C_k \in {\mathbb R}^{p \times n}$, $\Gamma_k \in {\mathbb R}^{p \times p}$ be positive definite, $u_k \in {\mathbb R}^{m}$, and $y_k \in {\mathbb R}^{p}$. Furthermore, let $P_0 \in {\mathbb For all $k \ge 0$, define $J_k \colon {\mathbb R}^n \rightarrow {\mathbb R}$ as where Then, there

Figures (4)

  • Figure 1: Mass-spring-damper system diagram. The mass can collide with the wall at $z=2$, reversing direction and keeping the same speed.
  • Figure 2: Vertical displacement ($z$) and velocity ($\dot{z}$) estimation using Kalman filter (KF) and adaptive Kalman filter (KF*). $\lambda$ shows the forgetting factor used in KF*.
  • Figure 3: Estimation error of vertical displacement (top) and velocity (bottom) using Kalman filter (KF) and adaptive Kalman filter (KF*).
  • Figure 4: Marginal variance of vertical displacement (top) and velocity (bottom) using Kalman filter (KF) and adaptive Kalman filter (KF*).

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 2
  • ...and 1 more