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Recursive Estimation for Dynamical Systems with Measurement Bias, Outliers and Constraints

Krishan Mohan Nagpal

TL;DR

This work tackles recursive state estimation for linear dynamical systems when measurements may be biased or contaminated by outliers. It introduces two convex losses, $f_\epsilon$ (epsilon-insensitive quadratic) and $f_{Huber}$ (epsilon-insensitive Huber), to suppress small residuals while providing linear penalties for large errors, enhancing robustness. The authors derive one-step-ahead recursive filters that resemble Kalman-Bucy updates, but compute the innovation term via solving a small quadratic program with linear constraints, and extend the framework to cases with linear inequality constraints on states and exogenous signals. The resulting Kalman-like, constraint-satisfying estimators offer robustness to noise, bias, and outliers and are practical to implement since the per-step QP size remains fixed and does not grow with the number of observations.

Abstract

This paper describes recursive algorithms for state estimation of linear dynamical systems when measurements are noisy with unknown bias and/or outliers. For situations with noisy and biased measurements, algorithms are proposed that minimize $ε$ insensitive loss function. In this approach which is often used in Support Vector Machines, small errors are ignored making the algorithm less sensitive to measurement bias. Apart from $ε$ insensitive quadratic loss function, estimation algorithms are also presented for $ε$ insensitive Huber M loss function which provides good performance in presence of both small noises as well as outliers. The advantage of Huber cost function based estimator in presence of outliers is due to the fact the error penalty function switches from quadratic to linear for errors beyond a certain threshold. For both objective functions, estimation algorithms are extended to cases when there are additional constraints on states and exogenous signals such as known range of some states or exogenous signals or measurement noises. Interestingly, the filtering algorithms are recursive and structurally similar to Kalman filter with the main difference being that the updates based on the new measurement ("innovation term") are based on solution of a quadratic optimization problem with linear constraints.

Recursive Estimation for Dynamical Systems with Measurement Bias, Outliers and Constraints

TL;DR

This work tackles recursive state estimation for linear dynamical systems when measurements may be biased or contaminated by outliers. It introduces two convex losses, (epsilon-insensitive quadratic) and (epsilon-insensitive Huber), to suppress small residuals while providing linear penalties for large errors, enhancing robustness. The authors derive one-step-ahead recursive filters that resemble Kalman-Bucy updates, but compute the innovation term via solving a small quadratic program with linear constraints, and extend the framework to cases with linear inequality constraints on states and exogenous signals. The resulting Kalman-like, constraint-satisfying estimators offer robustness to noise, bias, and outliers and are practical to implement since the per-step QP size remains fixed and does not grow with the number of observations.

Abstract

This paper describes recursive algorithms for state estimation of linear dynamical systems when measurements are noisy with unknown bias and/or outliers. For situations with noisy and biased measurements, algorithms are proposed that minimize insensitive loss function. In this approach which is often used in Support Vector Machines, small errors are ignored making the algorithm less sensitive to measurement bias. Apart from insensitive quadratic loss function, estimation algorithms are also presented for insensitive Huber M loss function which provides good performance in presence of both small noises as well as outliers. The advantage of Huber cost function based estimator in presence of outliers is due to the fact the error penalty function switches from quadratic to linear for errors beyond a certain threshold. For both objective functions, estimation algorithms are extended to cases when there are additional constraints on states and exogenous signals such as known range of some states or exogenous signals or measurement noises. Interestingly, the filtering algorithms are recursive and structurally similar to Kalman filter with the main difference being that the updates based on the new measurement ("innovation term") are based on solution of a quadratic optimization problem with linear constraints.

Paper Structure

This paper contains 4 sections, 8 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation and Background Results
  3. Main Results
  4. Summary

Key Result

Theorem 2.1

(Problem $1$) : Let $\hat{\Theta} , \hat{\zeta} \in \mathbb{R}^{Nm}$ be the solution of the following quadratic optimization problem with linear constraints: Let $\lambda_k$ be obtained from optimal $\hat{\Theta}$ as follows: With $\lambda_k$ obtained as above, the optimal values for $\hat{x}_k$, $\hat{w}_k$ that minimize (problem1) subject to constraints (constrdynamics) and (constr2b) are obta

Figures (1)

  • Figure 1: Comparison of the two loss functions considered in this paper with $\epsilon=1$. Both loss functions are the same for $-4 \leq$Error$\leq4$ and both are zero for $-1 \leq$Error$\leq1$. For larger errors (when |Error|$>4$), Huber loss function (right plot) is linear and has smaller value than the quadratic loss function (left plot)

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4