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Design of Unitless Normalized Measure of Nonlinearity for State Estimation

Ondřej Straka, Jindřich Havlík

TL;DR

The paper addresses quantifying nonlinearity in stochastic state estimation and demonstrates that conventional MSE-based MoNs are sensitive to unit choices. It introduces a unitless, normalized MoN built on a weighted norm $\|\cdot\|_W$, with a systematic weight-design framework ensuring affine invariance and $[0,1]$-normalization; the approach yields closed-form deterministic and stochastic components, $J_{A,b}$ and $J_n$, and explicit forms for additive and multiplicative noise. A normalization bound $M \le \sqrt{\mathrm{tr}(\mathbf W\boldsymbol\Sigma_{\mathbf g\mathbf g})}$ is derived, with weight choices able to preserve unitlessness (e.g., $\mathbf W^{\mathrm{diag}}$ and $\mathbf W^{\mathrm{full}}$). Numerical experiments on BOT, GMTI, and RDCOS tracking problems illustrate reduced unit-sensitivity and the ability to distinguish varying degrees of nonlinearity across measurement models.

Abstract

The paper deals with measures of nonlinearity. In state estimation, they are utilized i) to select a suitable state estimation algorithm by assessing the nonlinearity of a system model, ii) to adapt the estimation algorithm structure or parameters, or iii) to indicate the possible effect of strong nonlinearity that leads to estimate credibility loss. This paper summarizes the state of the art of nonlinearity measures, focusing on the mean-square-error-based measure of nonlinearity. Its weak point related to unit selection is illustrated, and based on this, requirements for a new measure of nonlinearity are formulated. A new nonlinearity measure that is both unitless and normalized is designed. Its properties are demonstrated using numerical tracking experiments.

Design of Unitless Normalized Measure of Nonlinearity for State Estimation

TL;DR

The paper addresses quantifying nonlinearity in stochastic state estimation and demonstrates that conventional MSE-based MoNs are sensitive to unit choices. It introduces a unitless, normalized MoN built on a weighted norm , with a systematic weight-design framework ensuring affine invariance and -normalization; the approach yields closed-form deterministic and stochastic components, and , and explicit forms for additive and multiplicative noise. A normalization bound is derived, with weight choices able to preserve unitlessness (e.g., and ). Numerical experiments on BOT, GMTI, and RDCOS tracking problems illustrate reduced unit-sensitivity and the ability to distinguish varying degrees of nonlinearity across measurement models.

Abstract

The paper deals with measures of nonlinearity. In state estimation, they are utilized i) to select a suitable state estimation algorithm by assessing the nonlinearity of a system model, ii) to adapt the estimation algorithm structure or parameters, or iii) to indicate the possible effect of strong nonlinearity that leads to estimate credibility loss. This paper summarizes the state of the art of nonlinearity measures, focusing on the mean-square-error-based measure of nonlinearity. Its weak point related to unit selection is illustrated, and based on this, requirements for a new measure of nonlinearity are formulated. A new nonlinearity measure that is both unitless and normalized is designed. Its properties are demonstrated using numerical tracking experiments.

Paper Structure

This paper contains 14 sections, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Statement and MSE-based MoN
  3. MSE-based MoN
  4. Motivation Example
  5. Requirements for MoN
  6. Unitless MSE-based MoN
  7. Minimization
  8. Normalization of MoN
  9. Weighting Matrix Choice
  10. Numerical Illustration
  11. Concluding Remarks
  12. Acknowledgement
  13. Minimization of $J_\mathbf n$
  14. Set of weights

Figures (2)

  • Figure 1: Effect of various bearing units on the MSE-based MoN.
  • Figure 2: Comparison of MoNs for BOT, GMTI, and RDCOS problems: unitless MoN with $\mathbf W^\text{full}$ (green), unitless MoN with $\mathbf W^\text{diag}$ (red), original MSE-based MoN (blue and dashed blue).