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Performance of the Kalman Filter and Smoother for Benchmark Studies

Batin Kurt, Umut Orguner

TL;DR

This work addresses predicting mean square error performance of the Kalman filter and smoother when the true dynamics are unknown but a fixed trajectory is available for benchmarking. It develops an exact, linear-time recursive framework based on a modified two-filter smoothing approach, decomposing MSE into covariance and bias components and computing final MSEs from recursive statistics. The method markedly reduces computational cost while maintaining accuracy, demonstrated by close alignment with Monte Carlo benchmarks and prior analytic methods in long trajectories. The results enable efficient, reliable performance prediction in benchmark studies and suggest avenues for extending the framework to nonlinear smoothing problems.

Abstract

We propose analytical mean square error (MSE) expressions for the Kalman filter (KF) and the Kalman smoother (KS) for benchmark studies, where the true system dynamics are unknown or unavailable to the estimator. In such cases, as in benchmark evaluations for target tracking, the analysis relies on deterministic state trajectories. This setting introduces a model mismatch between the estimator and the system, causing the covariance estimates to no longer reflect the actual estimation errors. To enable accurate performance prediction for fixed state trajectories without relying on computationally intensive Monte Carlo simulations, we derive recursive MSE expressions with linear time complexity. The proposed framework also accounts for measurement model mismatch and provides an efficient tool for performance evaluation in benchmark studies with long trajectories. Simulation results confirm the accuracy and computational efficiency of the proposed method.

Performance of the Kalman Filter and Smoother for Benchmark Studies

TL;DR

This work addresses predicting mean square error performance of the Kalman filter and smoother when the true dynamics are unknown but a fixed trajectory is available for benchmarking. It develops an exact, linear-time recursive framework based on a modified two-filter smoothing approach, decomposing MSE into covariance and bias components and computing final MSEs from recursive statistics. The method markedly reduces computational cost while maintaining accuracy, demonstrated by close alignment with Monte Carlo benchmarks and prior analytic methods in long trajectories. The results enable efficient, reliable performance prediction in benchmark studies and suggest avenues for extending the framework to nonlinear smoothing problems.

Abstract

We propose analytical mean square error (MSE) expressions for the Kalman filter (KF) and the Kalman smoother (KS) for benchmark studies, where the true system dynamics are unknown or unavailable to the estimator. In such cases, as in benchmark evaluations for target tracking, the analysis relies on deterministic state trajectories. This setting introduces a model mismatch between the estimator and the system, causing the covariance estimates to no longer reflect the actual estimation errors. To enable accurate performance prediction for fixed state trajectories without relying on computationally intensive Monte Carlo simulations, we derive recursive MSE expressions with linear time complexity. The proposed framework also accounts for measurement model mismatch and provides an efficient tool for performance evaluation in benchmark studies with long trajectories. Simulation results confirm the accuracy and computational efficiency of the proposed method.

Paper Structure

This paper contains 11 sections, 47 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Modified Two-Filter Formulation of the Kalman Smoother
  4. MSE Recursions with Linear Time Complexity
  5. Recursions for $\mathbf{\hat{x}}{}_{k|0:k}^*$ and $\mathbf{\hat{x}}{}_{k|0:K}^*$
  6. Recursions for $\mathbf{e}{}_{k|0:k}$ and $\mathbf{e}{}_{k|0:K}$
  7. Recursions for $\mathbf{b}{}_{k|0:k}$ and $\mathbf{b}{}_{k|0:K}$
  8. Recursions for $\mathbf{C}{}_{k|0:k}$ and $\mathbf{C}{}_{k|0:K}$
  9. Expressions for $\operatorname{MSE}_{k|k}$ and $\operatorname{MSE}_{k|K}$
  10. Simulation Results
  11. Conclusions and Future Work

Figures (4)

  • Figure 1: Fixed true target trajectory. The red dot indicates the start of the trajectory.
  • Figure 2: True target position and velocities.
  • Figure 3: Comparison of analytical (horizon-recursive and proposed) and numerical RMS position and velocity errors for the Kalman filter.
  • Figure 4: Comparison of analytical (batch, horizon-recursive, and proposed) and numerical RMS position and velocity errors for the Kalman smoother.