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LMI Optimization Based Multirate Steady-State Kalman Filter Design

Hiroshi Okajima

TL;DR

The paper addresses multirate sensor fusion for state estimation by formulating a cyclic, periodic Kalman filtering problem where the measured covariance becomes semidefinite due to intermittent sensor availability. It introduces an LMI-based dual LQR framework that handles semidefinite covariances and yields periodic steady-state Kalman gains via SDP, enabling offline design with stability guarantees. The approach supports multi-objective extensions including pole placement for guaranteed convergence and $l_2$-induced norm constraints for robustness, demonstrated on a multirate automotive navigation scenario with GPS and wheel-speed sensors. The results show estimation errors well below raw measurement noise and illustrate practical guidelines for balancing convergence rate, accuracy, and robustness in real-world multirate sensing applications.

Abstract

This paper presents an LMI-based design framework for multirate steady-state Kalman filters in systems with sensors operating at different sampling rates. The multirate system is formulated as a periodic time-varying system, where the Kalman gains converge to periodic steady-state values that repeat every frame period. Cyclic reformulation transforms this into a time-invariant problem; however, the resulting measurement noise covariance becomes semidefinite rather than positive definite, preventing direct application of standard Riccati equation methods. We address this through a dual LQR formulation with LMI optimization that naturally handles semidefinite covariances. The framework enables multi-objective design, supporting pole placement for guaranteed convergence rates and mixed H_2/l_2-induced norm design for balancing average and worst-case performance. Numerical validation using an automotive navigation system with GPS and wheel speed sensors demonstrates that the proposed filter achieves estimation errors well below raw measurement noise levels.

LMI Optimization Based Multirate Steady-State Kalman Filter Design

TL;DR

The paper addresses multirate sensor fusion for state estimation by formulating a cyclic, periodic Kalman filtering problem where the measured covariance becomes semidefinite due to intermittent sensor availability. It introduces an LMI-based dual LQR framework that handles semidefinite covariances and yields periodic steady-state Kalman gains via SDP, enabling offline design with stability guarantees. The approach supports multi-objective extensions including pole placement for guaranteed convergence and -induced norm constraints for robustness, demonstrated on a multirate automotive navigation scenario with GPS and wheel-speed sensors. The results show estimation errors well below raw measurement noise and illustrate practical guidelines for balancing convergence rate, accuracy, and robustness in real-world multirate sensing applications.

Abstract

This paper presents an LMI-based design framework for multirate steady-state Kalman filters in systems with sensors operating at different sampling rates. The multirate system is formulated as a periodic time-varying system, where the Kalman gains converge to periodic steady-state values that repeat every frame period. Cyclic reformulation transforms this into a time-invariant problem; however, the resulting measurement noise covariance becomes semidefinite rather than positive definite, preventing direct application of standard Riccati equation methods. We address this through a dual LQR formulation with LMI optimization that naturally handles semidefinite covariances. The framework enables multi-objective design, supporting pole placement for guaranteed convergence rates and mixed H_2/l_2-induced norm design for balancing average and worst-case performance. Numerical validation using an automotive navigation system with GPS and wheel speed sensors demonstrates that the proposed filter achieves estimation errors well below raw measurement noise levels.
Paper Structure (33 sections, 6 theorems, 58 equations, 5 figures, 4 tables)

This paper contains 33 sections, 6 theorems, 58 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Standard Kalman Filter
  4. Periodic Time-Varying Systems
  5. Multirate Kalman Filter Expression
  6. Problem Formulation
  7. Cyclic Reformulation
  8. Observability Analysis
  9. Challenges with Standard DARE Approach
  10. Stability Analysis
  11. LMI-based Filter Design for Multirate Systems
  12. LMI-based Filter Design
  13. Dual LQR Formulation
  14. LMI Formulation
  15. Optimization Problem
  16. ...and 18 more sections

Key Result

Proposition 1

The cyclic reformulation $(\check{A}, \check{B}, \check{C},$$\check{Q}^{1/2}, \check{R}^{1/2})$ is equivalent to the original periodic system $(A, B, S_k C)$ with noise covariances $(Q, S_k R S_k^T)$ in the sense that if $\check{x}(0) = [x(0)^T, 0, \ldots, 0]^T$, then the non-zero block of $\check{x

Figures (5)

  • Figure 1: Position estimation results. The green circles indicate GPS measurements available at 1 Hz.
  • Figure 2: Velocity estimation results. Wheel speed measurements are available at 10 Hz.
  • Figure 3: Acceleration estimation results. Acceleration is not directly observed but estimated from the dynamics.
  • Figure 4: Kalman filter with pole placement: trace$(\check{W})$ versus pole constraint $\bar{r}$.
  • Figure 5: Kalman filter with $l_2$-induced norm constraint: trace$(\check{W})$ and maximum eigenvalue magnitude vs. $\bar{\gamma}/\gamma_{\text{opt}}$.

Theorems & Definitions (13)

  • Proposition 1: bittanti2000invariant
  • Remark 3.1: Critical Observation
  • Proposition 2: bittanti2000invariantokajima2024design
  • Remark 3.2
  • Definition 3.1: Monodromy Matrix
  • Theorem 3.1: Equivalence of Stability Conditions bittanti2000invariantbittanti1986analysis
  • Proof 1
  • Corollary 3.1
  • Theorem 5.1: Pole Placement via LMI boyd1994lmi
  • Remark 5.1
  • ...and 3 more