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Bias-VarianceTrade-off in Kalman Filter-Based Disturbance Observers

Shilei Li, Dawei Shi, Xiaoxu Lyu, Jiawei Tang, Ling Shi

TL;DR

This work investigates how incomplete disturbance models affect KF-DOB performance, revealing a fundamental bias-variance trade-off. It proves that SISE corresponds to KF-DOB in the limit of infinite disturbance covariance and characterizes the impact of mismatch via a series of theorems. To mitigate the trade-off, two remedies are proposed: MKCKF-DOB, which uses multi-kernel correntropy, and IMMKF-DOB, which blends multiple disturbance covariances through an IMM framework. Simulations across vehicle tracking and simple mechanical systems show both remedies reduce bias at disturbance jumps while preserving smoothness, offering practical improvements over conventional KF-DOB. The results provide a principled pathway to balancing responsiveness and estimation reliability in disturbance-observer design, with future work focusing on adaptive tuning of kernel bandwidths and transition probabilities.

Abstract

The performance of disturbance observers is strongly influenced by the level of prior knowledge about the disturbance model. The simultaneous input and state estimation (SISE) algorithm is widely recognized for providing unbiased minimum-variance estimates under arbitrary disturbance models. In contrast, the Kalman filter-based disturbance observer (KF-DOB) achieves minimum mean-square error estimation when the disturbance model is fully specified. However, practical scenarios often fall between these extremes, where only partial knowledge of the disturbance model is available. This paper investigates the inherent bias-variance trade-off in KF-DOB when the disturbance model is incomplete. We further show that SISE can be interpreted as a special case of KF-DOB, where the disturbance noise covariance tends to infinity. To address this trade-off, we propose two novel estimators: the multi-kernel correntropy Kalman filter-based disturbance observer (MKCKF-DOB) and the interacting multiple models Kalman filter-based disturbance observer (IMMKF-DOB). Simulations verify the effectiveness of the proposed methods.

Bias-VarianceTrade-off in Kalman Filter-Based Disturbance Observers

TL;DR

This work investigates how incomplete disturbance models affect KF-DOB performance, revealing a fundamental bias-variance trade-off. It proves that SISE corresponds to KF-DOB in the limit of infinite disturbance covariance and characterizes the impact of mismatch via a series of theorems. To mitigate the trade-off, two remedies are proposed: MKCKF-DOB, which uses multi-kernel correntropy, and IMMKF-DOB, which blends multiple disturbance covariances through an IMM framework. Simulations across vehicle tracking and simple mechanical systems show both remedies reduce bias at disturbance jumps while preserving smoothness, offering practical improvements over conventional KF-DOB. The results provide a principled pathway to balancing responsiveness and estimation reliability in disturbance-observer design, with future work focusing on adaptive tuning of kernel bandwidths and transition probabilities.

Abstract

The performance of disturbance observers is strongly influenced by the level of prior knowledge about the disturbance model. The simultaneous input and state estimation (SISE) algorithm is widely recognized for providing unbiased minimum-variance estimates under arbitrary disturbance models. In contrast, the Kalman filter-based disturbance observer (KF-DOB) achieves minimum mean-square error estimation when the disturbance model is fully specified. However, practical scenarios often fall between these extremes, where only partial knowledge of the disturbance model is available. This paper investigates the inherent bias-variance trade-off in KF-DOB when the disturbance model is incomplete. We further show that SISE can be interpreted as a special case of KF-DOB, where the disturbance noise covariance tends to infinity. To address this trade-off, we propose two novel estimators: the multi-kernel correntropy Kalman filter-based disturbance observer (MKCKF-DOB) and the interacting multiple models Kalman filter-based disturbance observer (IMMKF-DOB). Simulations verify the effectiveness of the proposed methods.
Paper Structure (37 sections, 2 theorems, 77 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 37 sections, 2 theorems, 77 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SISE Algorithm
  4. Kalman Filter-Based Disturbance Observer
  5. Multi-kernel Correntropy Kalman Filter
  6. Interacting Multiple Models Kalman Filter
  7. Problem Description
  8. Bias-Variance Trade-off in KF-DOB
  9. Bias-Variance Effects with Incorrect Initialization in KF
  10. Effects of $\Delta \mathrm{Q}$ on Convergence Speed
  11. Effects of $\Delta \mathrm{Q}$ on the Error Covariance
  12. Bias-Variance Trade-off in KF-DOB
  13. Remedies
  14. Native Kalman Filter-based Disturbance Observer
  15. Kalman filter-based Disturbance Observer
  16. ...and 22 more sections

Key Result

Corollary 1

If KF is stable, one has where $\bar{\Phi}_k=(\mathrm{I}-\mathrm{K}_k \mathrm{H}_k)\Phi_k$ and $\bar{\mathrm{x}}_0^{\bar{s}}=\bar{\mathrm{x}}_0$.

Figures (7)

  • Figure 1: The disturbance estimation performances of NKF-DOB and KF-DOB with different $\eta$.
  • Figure 2: The visualization of bias-variance trade-off in KF-DOB with different $\eta$.
  • Figure 3: Identity property of SISE, NKF-DOB, and KF-DOB. (a) The disturbance estimate of different estimators. (d) Error covariance of different estimators.
  • Figure 4: The disturbance estimate of SISE, KF-DOB, MKCKF-DOB, and IMMKF-DOB.
  • Figure 5: (a) The bias-variance visualization of KF-DOB, MKCKF-DOB, and IMMKF-DOB with 100 Monte Carlo runs. (b) Performance loss of different estimators with different $\eta$. Note that in MKCKF-DOB, the kernel bandwidth for the disturbance channel is set as $\varsigma_d=3+\eta$. In IMMKF-DOB, we use $D_1=D^{*}$ and $D_2=\exp(\eta)D^{*}$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Corollary 1
  • Corollary 2