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Convex Reformulation of Information Constrained Linear State Estimation with Mixed-Binary Variables for Outlier Accommodation

Wang Hu, Zeyi Jiang, Hamed Mohsenian-Rad, Jay A. Farrell

TL;DR

This work tackles outlier accommodation in information-constrained linear state estimation with abundant measurements by reframing Risk-Averse Performance-Specified (RAPS) measurement selection as a convex optimization problem. It introduces a convex reformulation that converts mixed-binary decision variables into linear constraints, making the problem amenable to standard convex solvers. The study analyzes Full-RAPS (full information-matrix constraint) and Diag-RAPS (diagonal constraint) and finds that Diag-RAPS is about $100\times$ faster while achieving the required performance and minimizing risk. Across simulations against a Kalman filter and threshold-based decisions, Diag-RAPS consistently yields the lowest risk under the performance bound, demonstrating practical viability for real-time, outlier-robust state estimation in signal-rich settings, with potential applicability to GNSS and vision-based navigation.

Abstract

This article considers the challenge of accommodating outlier measurements in state estimation. The Risk-Averse Performance-Specified (RAPS) state estimation approach addresses outliers as a measurement selection Bayesian risk minimization problem subject to an information accuracy constraint, which is a non-convex optimization problem. Prior explorations into RAPS rely on exhaustive search, which becomes computationally infeasible as the number of measurements increases. This paper derives a convex formulation for the RAPS optimization problems via transforming the mixed-binary variables into linear constraints. The convex reformulation herein can be solved by convex programming toolboxes, significantly enhancing computational efficiency. We explore two specifications: Full-RAPS, utilizing the full information matrix, and Diag-RAPS, focusing on diagonal elements only. The simulation comparison demonstrates that Diag-RAPS is faster and more efficient than Full-RAPS. In comparison with Kalman Filter (KF) and Threshold Decisions (TD), Diag-RAPS consistently achieves the lowest risk, while achieving the performance specification when it is feasible.

Convex Reformulation of Information Constrained Linear State Estimation with Mixed-Binary Variables for Outlier Accommodation

TL;DR

This work tackles outlier accommodation in information-constrained linear state estimation with abundant measurements by reframing Risk-Averse Performance-Specified (RAPS) measurement selection as a convex optimization problem. It introduces a convex reformulation that converts mixed-binary decision variables into linear constraints, making the problem amenable to standard convex solvers. The study analyzes Full-RAPS (full information-matrix constraint) and Diag-RAPS (diagonal constraint) and finds that Diag-RAPS is about faster while achieving the required performance and minimizing risk. Across simulations against a Kalman filter and threshold-based decisions, Diag-RAPS consistently yields the lowest risk under the performance bound, demonstrating practical viability for real-time, outlier-robust state estimation in signal-rich settings, with potential applicability to GNSS and vision-based navigation.

Abstract

This article considers the challenge of accommodating outlier measurements in state estimation. The Risk-Averse Performance-Specified (RAPS) state estimation approach addresses outliers as a measurement selection Bayesian risk minimization problem subject to an information accuracy constraint, which is a non-convex optimization problem. Prior explorations into RAPS rely on exhaustive search, which becomes computationally infeasible as the number of measurements increases. This paper derives a convex formulation for the RAPS optimization problems via transforming the mixed-binary variables into linear constraints. The convex reformulation herein can be solved by convex programming toolboxes, significantly enhancing computational efficiency. We explore two specifications: Full-RAPS, utilizing the full information matrix, and Diag-RAPS, focusing on diagonal elements only. The simulation comparison demonstrates that Diag-RAPS is faster and more efficient than Full-RAPS. In comparison with Kalman Filter (KF) and Threshold Decisions (TD), Diag-RAPS consistently achieves the lowest risk, while achieving the performance specification when it is feasible.
Paper Structure (18 sections, 24 equations, 3 figures)

This paper contains 18 sections, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. State Estimation Using Selected Measurements
  4. Threshold Decisions (TD)
  5. Risk-Averse Optimization
  6. Full-RAPS
  7. Diag-RAPS
  8. Comparison of Full and Diagonal RAPS
  9. Convex Reformulation
  10. Evaluation in Simulation
  11. System Model
  12. Measurements and Outliers
  13. Performance Specification
  14. Runtime comparison between Full-RAPS and Diag-RAPS
  15. Estimator Implementations
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Runtime comparison between Full-RAPS (blue) vs Diag-RAPS (orange). Each point represents the mean of the per epoch runtime for an experiment with the number of measurements specified along the horizontal axis. The values above the points represent the STD.
  • Figure 2: Simulation results comparing the square root of the posterior information and of the risk for Kalman filter (KF) in blue, Kalman filter with threshold detection (TD) in red, and Diag-RAPS in green. The solid black line shows the corresponding square root of the diagonal element of $J_d$.
  • Figure 3: Diag-RAPS computation time probability histogram.