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arxiv: 2602.01537 · v4 · submitted 2026-02-02 · 📡 eess.SY · cs.SY

LMI Optimization Based Multirate Steady-State Kalman Filter Design

Hiroshi Okajima This is my paper

Pith reviewed 2026-05-16 08:40 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords multirate Kalman filterLMI optimizationsteady-state designsensor fusionperiodic systemsautomotive navigationdual LQR
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The pith

The LMI optimization method designs steady-state Kalman filters for multirate systems by solving a dual LQR problem that handles semidefinite noise covariances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to create a design method for Kalman filters in systems with sensors at different sampling rates, like combining slow GPS with fast wheel speed sensors in vehicles. Standard approaches fail after reformulating the periodic system into a time-invariant one because the noise covariance matrix becomes only semidefinite. By recasting the problem as a dual linear quadratic regulator and solving it with linear matrix inequalities, the method finds optimal steady-state gains while allowing extra constraints for performance. This matters because it enables reliable state estimation from mixed-rate sensors without needing to solve time-varying Riccati equations at each step. Validation in an automotive navigation setup confirms lower position errors than using GPS alone.

Core claim

The central claim is that the steady-state Kalman filter for the multirate system can be obtained by cyclic reformulation of the periodic system followed by solving an LMI optimization problem derived from the dual LQR formulation, which accommodates the resulting semidefinite measurement noise covariance and supports additional design specifications such as pole placement and l2-induced norm bounds.

What carries the argument

The dual LQR formulation solved via LMI optimization, applied to the cyclically reformulated periodic multirate system to handle semidefinite covariances.

If this is right

  • The designed gains are periodic and repeat every frame period for the multirate system.
  • The LMI solution guarantees valid upper bounds on the estimation error covariance.
  • Multi-objective performance can be achieved by incorporating pole placement for convergence speed and induced-norm constraints.
  • Effective fusion of multirate sensors yields position estimation RMSE below the GPS noise level in the navigation example.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may allow similar LMI techniques for designing multirate controllers or observers in other engineering domains.
  • The method could reduce computational load in real-time applications by providing constant periodic gains instead of recalculating at each time step.
  • Connections to other periodic system problems suggest the framework might apply when noise models lead to singular covariances.

Load-bearing premise

The multirate system with differing sensor rates can be modeled as a periodic time-varying system whose Kalman gains converge to periodic steady-state values.

What would settle it

If the computed estimation error covariance upper bound from the LMI is violated in Monte Carlo simulations of the automotive system, or if the gains do not converge to periodic values.

Figures

Figures reproduced from arXiv: 2602.01537 by Hiroshi Okajima.

Figure 1
Figure 1. Figure 1: Position estimation results. The green circles indicate GPS m [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Velocity estimation results. Wheel speed measurements a [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Acceleration estimation results. Acceleration is not direct [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Kalman filter with pole placement: trace( [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Kalman filter with l2-induced norm constraint: trace(Wˇ ) and maxi￾mum eigenvalue magnitude vs. ¯γ/γopt [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
read the original 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. I 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 $l_2$-induced norm constraints for balancing average and worst-case performance. Numerical validation using an automotive navigation system with GPS and wheel speed sensors, including Monte Carlo simulation with 500 independent noise realizations, demonstrates that the proposed filter achieves a position RMSE well below the GPS noise level through effective multirate sensor fusion, and that the LMI solution provides valid upper bounds on the estimation error covariance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents an LMI-based design framework for multirate steady-state Kalman filters by reformulating the system as periodic time-varying, applying cyclic reformulation (which yields a semidefinite measurement noise covariance), and using a dual LQR formulation solved via LMI optimization to obtain periodic gains. It incorporates multi-objective constraints such as pole placement for convergence rate and l2-induced norm bounds, and validates the method on an automotive navigation example with GPS and wheel-speed sensors via 500-run Monte Carlo simulations claiming position RMSE below GPS noise level and valid upper bounds on error covariance.

Significance. If the LMI-derived P is shown to provably upper-bound the true periodic steady-state covariance under semidefinite R, the framework would provide a practical, multi-objective tool for multirate sensor fusion in navigation and control applications where standard Riccati solvers fail due to rank-deficient noise covariances.

major comments (2)
  1. [dual LQR/LMI formulation] The central claim that the LMI optimum P provides a valid upper bound on the periodic error covariance when the effective R is only semidefinite (after cyclic reformulation) is load-bearing for the validation results, yet the manuscript provides no explicit theorem, derivation, or reference establishing that the dual LQR/LMI solution preserves this bounding property (standard Riccati theory does not apply directly).
  2. [numerical validation / Monte Carlo section] In the numerical validation, the 500-run Monte Carlo results report low position RMSE and state that the LMI solution provides valid upper bounds, but contain no direct comparison of the computed P matrix against the empirical sample covariance matrix obtained from the simulations.
minor comments (1)
  1. [abstract and introduction] The abstract and introduction could more explicitly state the precise conditions (e.g., observability assumptions after lifting) under which the periodic steady-state gains are guaranteed to exist.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the theoretical foundations and strengthen the validation of our LMI-based multirate Kalman filter framework. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [dual LQR/LMI formulation] The central claim that the LMI optimum P provides a valid upper bound on the periodic error covariance when the effective R is only semidefinite (after cyclic reformulation) is load-bearing for the validation results, yet the manuscript provides no explicit theorem, derivation, or reference establishing that the dual LQR/LMI solution preserves this bounding property (standard Riccati theory does not apply directly).

    Authors: We agree that an explicit derivation is required to rigorously support the bounding property under semidefinite R. In the revised manuscript, we will add a new theorem (with proof) in Section III showing that the dual LQR/LMI solution yields a valid upper bound on the periodic steady-state error covariance. The proof relies on the equivalence of the LMI to a discrete Lyapunov inequality for the lifted periodic system, combined with a regularization argument: the semidefinite case is obtained as the limit of positive-definite perturbations, for which the standard Riccati bound holds, and the LMI optimum remains feasible and bounding in the limit. revision: yes

  2. Referee: [numerical validation / Monte Carlo section] In the numerical validation, the 500-run Monte Carlo results report low position RMSE and state that the LMI solution provides valid upper bounds, but contain no direct comparison of the computed P matrix against the empirical sample covariance matrix obtained from the simulations.

    Authors: We acknowledge the lack of direct comparison. In the revised manuscript, we will augment the Monte Carlo section with a table and/or plot comparing the diagonal elements (and trace) of the LMI-derived P matrix against the sample covariance matrix estimated from the 500 independent runs. This will quantitatively confirm that the computed P upper-bounds the empirical covariance for the automotive navigation example. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on standard periodic-system and LMI techniques

full rationale

The paper's chain formulates the multirate system as periodic time-varying, applies cyclic reformulation to obtain a time-invariant problem, and substitutes a dual LQR/LMI optimization for the Riccati equation to accommodate semidefinite measurement noise. These steps invoke established results from periodic Kalman filtering and LMI-based covariance bounding without any reduction of the central upper-bound claim to a fitted parameter, self-defined quantity, or load-bearing self-citation. Monte Carlo validation supplies independent empirical checks rather than tautological confirmation, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The approach rests on standard periodic-system theory and LMI feasibility from control literature; no new free parameters or invented entities are introduced in the abstract.

axioms (3)
  • domain assumption Multirate system can be represented as a periodic time-varying system with known frame period
    Invoked to enable cyclic reformulation into time-invariant problem
  • domain assumption Kalman gains converge to periodic steady-state values
    Required for steady-state filter design
  • domain assumption Dual LQR formulation with LMI optimization handles semidefinite measurement noise covariances
    Central to bypassing the positive-definiteness requirement of standard Riccati methods

pith-pipeline@v0.9.0 · 5478 in / 1399 out tokens · 46284 ms · 2026-05-16T08:40:28.091129+00:00 · methodology

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