In many control and signal processing applications, state estimation must be performed using measurements from multiple sensors operating at different sampling rates. GPS updates at 1 Hz while inertial sensors run at 100 Hz — how do you optimally fuse these measurements? The standard Kalman filter assumes uniform sampling, but real-world multirate systems exhibit periodic time-varying dynamics in the measurement equations. This article explains a systematic LMI-based design framework for multirate steady-state Kalman filters that addresses a fundamental challenge: the cyclic measurement noise covariance becomes positive semidefinite (not positive definite) when sensors operate at different rates, preventing direct application of standard Riccati equation methods.

Author: Hiroshi Okajima, Associate Professor, Kumamoto University, Japan — 20 years of control engineering research

This article is based on the following paper.

Hiroshi Okajima, LMI Optimization-Based Multirate Steady-State Kalman Filter Design, IEEE Access (2025) (Open Access)

GitHub (MATLAB/Python Code)

MATLAB File Exchange: LMI-Based Multirate Kalman Filter Design

Contents

Why Multirate Kalman Filtering Matters
Problem Formulation: Multirate Sensing Systems
Cyclic Reformulation: From Periodic to Time-Invariant
The Semidefinite Challenge
LMI-Based Filter Design
Multi-Objective Extensions
Numerical Example: Automotive Navigation
Monte Carlo Validation
Multi-Objective Design Trade-offs
Design Guidelines
Connections to Related Research
MATLAB/Python Code
Related Articles and Videos
Paper Information

Why Multirate Kalman Filtering Matters

Multirate sensing environments are ubiquitous in modern engineering systems. In autonomous vehicles, GPS provides position measurements at 1–10 Hz while wheel speed sensors and IMUs operate at 50–100 Hz. Similar challenges arise in chemical process control (laboratory analysis vs. online sensors), power systems (PMUs vs. SCADA), robotics (visual and inertial sensor fusion for SLAM), and networked control systems with bandwidth-limited or event-triggered sampling.

The classical Kalman filter assumes all measurements arrive at a uniform rate. When sensors operate at different rates, naive approaches such as zero-order hold or measurement interpolation lead to suboptimal performance and fail to properly account for the uncertainty structure of the multirate measurements.

This paper presents an LMI-based framework that transforms the multirate Kalman filtering problem into a convex optimization problem that can be solved efficiently, even when standard Riccati equation methods fail.

Problem Formulation: Multirate Sensing Systems

Consider a discrete-time linear time-invariant system:

$\displaystyle x(k+1) = Ax(k) + Bu(k) + Q^{1/2} d_w(k)$

where $x(k) \in \mathbb{R}^{n}$ is the state, $u(k) \in \mathbb{R}^{p}$ is the control input, and $d_{w}(k)$ is a normalized disturbance with zero mean and unit covariance. The matrix $Q^{1/2}$ is obtained from the Cholesky factorization $Q = Q^{1/2}(Q^{1/2})^{T}$ .

The system is equipped with multiple sensors operating at different sampling rates. The measurement equation is:

$\displaystyle y(k) = S_k C x(k) + S_k R^{1/2} d_v(k)$

where $C \in \mathbb{R}^{q \times n}$ is the full measurement matrix and $S_{k} \in \mathbb{R}^{q \times q}$ is a periodic selection matrix representing measurement availability:

$\displaystyle S_k = \mathrm{diag}(s_{k,1}, s_{k,2}, \ldots, s_{k,q})$

with $s_{k,i} = 1$ if the $i$ -th measurement is available at time $k$ , and $s_{k,i} = 0$ otherwise. When sampling rates have rational ratios, there exists a period $N$ such that $S_{k+N} = S_{k}$ for all $k \geq 0$ .

The design objective is to find optimal periodic Kalman gains $L_{k}$ , $k = 0, 1, \ldots, N-1$ , for the predictor-form estimator:

$\displaystyle \hat{x}(k+1) = A\hat{x}(k) + Bu(k) + L_k(y(k) - S_k C \hat{x}(k))$

that minimize the estimation error $e(k) = x(k) - \hat{x}(k)$ in the mean square sense.

Cyclic Reformulation: From Periodic to Time-Invariant

The key technique for handling the periodic time-varying system is cyclic reformulation. For a signal $\xi(k)$ , its cycled representation $\check{\xi}(k)$ places the signal in the $(k \bmod N + 1)$ -th block position with zeros elsewhere.

Applying this to all signals transforms the periodic system into a time-invariant cyclic system:

$\displaystyle \check{x}(k+1) = \check{A}\check{x}(k) + \check{B}\check{u}(k) + \check{Q}^{1/2}\check{d}_w(k)$

$\displaystyle \check{y}(k) = \check{C}\check{x}(k) + \check{R}^{1/2}\check{d}_v(k)$

where the cyclic system matrix $\check{A} \in \mathbb{R}^{Nn \times Nn}$ has the structure:

$\displaystyle \check{A} = \begin{pmatrix} 0 & 0 & \cdots & 0 & A \cr A & 0 & \cdots & 0 & 0 \cr 0 & A & \cdots & 0 & 0 \cr \vdots & \vdots & \ddots & \vdots & \vdots \cr 0 & 0 & \cdots & A & 0 \end{pmatrix}$

and the cycled measurement matrix and noise covariance are block-diagonal:

$\displaystyle \check{C} = \mathrm{diag}(S_0 C, S_1 C, \ldots, S_{N-1} C)$

$\displaystyle \check{R} = \mathrm{diag}(S_0 R S_0^T, S_1 R S_1^T, \ldots, S_{N-1} R S_{N-1}^T)$

This reformulation converts the periodic Kalman filtering problem into a time-invariant problem. A key property is that the stability of the cyclic closed-loop matrix $\check{A} - \check{L}\check{C}$ is equivalent to the stability of the monodromy matrix, with the spectral radius relationship $\rho(\check{A} - \check{L}\check{C}) = \rho(\Phi_{N})^{1/N}$ .

The Semidefinite Challenge

Here is the central finding of this paper: the cyclic measurement noise covariance $\check{R}$ is positive semidefinite ( $\check{R} \succeq 0$ ) but not positive definite. This occurs because at time instants when certain sensors do not provide measurements, the corresponding diagonal blocks $S_{k} R S_{k}^{T}$ have reduced rank.

For example, in the automotive navigation system considered in the paper:

When GPS is available ( $k \bmod 10 = 0$ ): $S_{0} = I_{q}$ , so $S_{0} R S_{0}^{T} = R \succ 0$ (full rank)
When only wheel speed is available ( $k \bmod 10 \neq 0$ ): $S_{k} = \mathrm{diag}(0, 1)$ , so $S_{k} R S_{k}^{T}$ has reduced rank

The standard DARE formulation requires $\check{R} \succ 0$ to compute the inverse $(\check{C}\check{P}\check{C}^{T} + \check{R})^{-1}$ . When $\check{R}$ is singular, standard DARE solvers may fail to converge or produce numerically unstable solutions. This fundamental limitation motivates the LMI-based approach.

LMI-Based Filter Design

The paper addresses the semidefinite challenge through a dual LQR formulation with LMI optimization. The estimation error dynamics is:

$\displaystyle \check{e}(k+1) = (\check{A} - \check{L}\check{C})\check{e}(k) + \check{Q}^{1/2}\check{d}_w(k) - \check{L}\check{R}^{1/2}\check{d}_v(k)$

For any stabilizing gain $\check{L}$ , the steady-state error covariance $\check{P}_{e}$ satisfies the discrete Lyapunov equation:

$\displaystyle \check{P}_e = (\check{A} - \check{L}\check{C})\check{P}_e(\check{A} - \check{L}\check{C})^T + \check{Q} + \check{L}\check{R}\check{L}^T$

Unlike the DARE, this equation does not involve the inverse of $\check{R}$ and is well-defined even when $\check{R}$ is only positive semidefinite.

Using the variable transformation $\check{Y} = -\check{X}\check{L}$ and introducing $\check{X} \succ 0$ as an upper bound variable satisfying $\check{P}_{e} \preceq \check{X}^{-1}$ , the design problem becomes the following SDP:

$\displaystyle \min_{\check{X}, \check{Y}, \check{W}} \mathrm{trace}(\check{W})$

subject to the LMI constraints:

$\displaystyle \begin{pmatrix} \check{X} & \check{X}\check{A} + \check{Y}\check{C} & \check{X}\check{Q}^{1/2} & \check{Y}\check{R}^{1/2} \cr (\check{X}\check{A} + \check{Y}\check{C})^T & \check{X} & 0 & 0 \cr (\check{Q}^{1/2})^T\check{X} & 0 & I_{Nn} & 0 \cr (\check{R}^{1/2})^T\check{Y}^T & 0 & 0 & I_{Nq} \end{pmatrix} \succeq 0$

$\displaystyle \check{X} \succeq \epsilon I_{Nn}, \quad \begin{pmatrix} \check{W} & I_{Nn} \cr I_{Nn} & \check{X} \end{pmatrix} \succeq 0$

After solving, the Kalman gain is recovered as $\check{L} = -\check{X}^{-1}\check{Y}$ , and the periodic gains $L_{k}$ are extracted using the cyclic indexing structure.

A key result (Theorem 4.1 in the paper) guarantees that if the LMI optimization yields a feasible solution with $\check{X} \succ 0$ , then the resulting filter is Schur stable regardless of whether $\check{R}$ is positive definite or only positive semidefinite.

When $\check{R} \succ 0$ , the LMI solution coincides with the classical DARE-based Kalman filter gain, as confirmed both theoretically through the Kalman–LQR duality and numerically (the discrepancy is below $10^{-6}$ in Frobenius norm).

Multi-Objective Extensions

A key advantage of the LMI framework is its ability to incorporate multiple design objectives simultaneously.

Pole Placement Constraints

To guarantee a minimum convergence rate, the eigenvalues of $\check{A} - \check{L}\check{C}$ are constrained to lie within the disk $\mathcal{D}_{\bar{r}} = \lbrace z \in \mathbb{C} : |z| \lt \bar{r} \rbrace$ by adding the LMI:

$\displaystyle \begin{pmatrix} \bar{r}^2 \check{X} & \check{X}\check{A} + \check{Y}\check{C} \cr (\check{X}\check{A} + \check{Y}\check{C})^T & \check{X} \end{pmatrix} \succ 0$

Smaller $\bar{r}$ implies faster convergence but may require larger gains and increased estimation error covariance.

Kalman Filter with l2-induced Norm Constraint

For balancing average and worst-case performance, the $l_{2}$ -induced norm from the combined disturbance to the estimation error is constrained by $\gamma$ :

$\displaystyle \begin{pmatrix} \check{X} & \check{X}\check{A} + \check{Y}\check{C} & \check{X}\check{Q}^{1/2} & \check{Y}\check{R}^{1/2} \cr (\check{X}\check{A} + \check{Y}\check{C})^T & \check{X} - \check{C}_z^T \check{C}_z & 0 & 0 \cr (\check{Q}^{1/2})^T\check{X} & 0 & \gamma^2 I_{Nn} & 0 \cr (\check{R}^{1/2})^T\check{Y}^T & 0 & 0 & \gamma^2 I_{Nq} \end{pmatrix} \succ 0$

All extensions can be combined into a single convex optimization problem.

The paper demonstrates the proposed method using an automotive navigation system with the following configuration:

Sampling time: $\Delta t = 0.1$ s
Period: $N = 10$ (GPS at 1 Hz, wheel speed at 10 Hz)
State: $x(k) = \lbrack p(k), v(k), a(k) \rbrack^{T}$ (position, velocity, acceleration)
Measurements: GPS position + wheel speed velocity

The system matrices are:

$\displaystyle A = \begin{pmatrix} 1 & 0.1 & 0.005 \cr 0 & 1 & 0.1 \cr 0 & 0 & 0.8 \end{pmatrix}, \quad B = \begin{pmatrix} 0 \cr 0 \cr 1 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 0 & 0 \cr 0 & 1 & 0 \end{pmatrix}$

with noise covariances $Q = \mathrm{diag}(0.01, 0.1, 0.5)$ and $R = \mathrm{diag}(1.0, 0.1)$ .

The cyclic covariance $\check{R} \in \mathbb{R}^{20 \times 20}$ has rank 11, confirming the semidefinite structure. The observability matrix has full rank $\mathrm{rank}(\mathcal{O}) = 30 = Nn$ with condition number $1.11 \times 10^{1}$ .

The LMI solver converges with $\mathrm{trace}(\check{W}) = 18.07$ and maximum closed-loop eigenvalue magnitude 0.9673, confirming stability.

Periodic Kalman Gains

The extracted periodic gains show the expected structure:

k mod 10	Sensors Active	$L_{k}$
0	GPS + Wheel	$\lbrack 0.2827, 0.1017; 0.0042, 0.6979; 0.0062, 0.3755 \rbrack$
1	Wheel only	$\lbrack 0, 0.1094; 0, 0.6980; 0, 0.3757 \rbrack$
5	Wheel only	$\lbrack 0, 0.1148; 0, 0.6981; 0, 0.3758 \rbrack$

When only wheel speed measurements are available, the first column of $L_{k}$ is zero, reflecting the absence of GPS. The gains exhibit periodic variation, with larger values at GPS update times.

Monte Carlo Validation

Monte Carlo simulations with 500 independent noise realizations were conducted to validate the filter performance. The RMSE was computed over the steady-state interval (time steps 51–200).

State	RMSE	95% Pctl	Theory UB	Ratio
Position	0.561 m	[0.317, 0.919]	0.587 m	1.046
Velocity	0.431 m/s	[0.372, 0.490]	0.432 m/s	1.002
Acceleration	1.125 m/s²	[0.889, 1.380]	1.130 m/s²	1.005

The theoretical upper bound (Theory UB) is computed by period-averaging the diagonal entries of the error covariance upper bound $\check{X}^{-1}$ . The ratios exceed unity for all three states (1.002–1.046), confirming that the LMI solution provides valid and tight upper bounds with low conservatism.

The position RMSE (0.561 m) is well below the GPS noise standard deviation (1.0 m), confirming that multirate sensor fusion effectively reduces position estimation error. The velocity RMSE (0.431 m/s) exceeds the wheel speed noise standard deviation (0.316 m/s); this is expected because the dominant error source for velocity is not measurement noise but the propagation of acceleration process noise.

Multi-Objective Design Trade-offs

Kalman Filter with Pole Placement

The trade-off between estimation performance and convergence rate is highly nonlinear:

$\bar{r}$	$\mathrm{trace}(\check{W})$	$\max \lvert \lambda \rvert$
0.975	19.64	0.953
0.950	24.91	0.910
0.900	41.19	0.826
0.850	76.45	0.746
0.800	165.9	0.707
0.750	422.1	0.684

Compared to the optimal Kalman filter ( $\mathrm{trace}(\check{W}) = 18.07$ ), even a mild constraint $\bar{r} = 0.975$ increases the cost to 19.64. Moderate constraints ( $\bar{r} \approx 0.90$ ) roughly double the cost, while aggressive constraints ( $\bar{r} \lt 0.80$ ) increase it by more than an order of magnitude.

Kalman Filter with l2-induced Norm Constraint

The minimum achievable $l_{2}$ -induced norm is $\gamma_{\mathrm{opt}} = 1.0214$ .

$\bar{\gamma}/\gamma_{\mathrm{opt}}$	$\bar{\gamma}$	$\mathrm{trace}(\check{W})$	$\max \lvert \lambda \rvert$
10.0	10.21	18.71	0.961
5.0	5.11	19.12	0.957
2.0	2.04	21.40	0.936
1.5	1.53	23.08	0.921
1.1	1.12	26.10	0.897
1.01	1.03	34.65	0.815

As $\bar{\gamma}$ approaches $\gamma_{\mathrm{opt}}$ , $\mathrm{trace}(\check{W})$ increases from 18.71 to 34.65, while the maximum eigenvalue magnitude decreases from 0.961 to 0.815, indicating improved stability margins at the cost of estimation performance.

Comparison of Optimal Gains

The periodic Kalman gain $L_{0}$ (at GPS + wheel speed update) differs significantly between the optimal Kalman filter and the optimal $l_{2}$ -induced norm design:

Design	$L_{0}$	$\max \lvert \lambda \rvert$
Optimal Kalman filter	$\lbrack 0.283, 0.102; 0.004, 0.698; 0.006, 0.376 \rbrack$	0.967
Optimal $l_{2}$ -induced norm	$\lbrack 1.000, 0.086; 0.002, 0.787; 0.006, 0.558 \rbrack$	0.653

The $l_{2}$ -optimal design yields a position gain of 1.000 (compared to 0.283 for the Kalman filter), indicating that worst-case robustness optimization prioritizes GPS measurements more aggressively.

Design Guidelines

Based on the numerical results, the paper recommends the following design strategy:

Standard applications: The optimal Kalman filter design provides the best average estimation performance under stochastic disturbances.
Fast convergence required: Add a pole placement constraint with $\bar{r} = 0.7\text{--}0.8$ for a guaranteed minimum decay rate.
Worst-case performance critical: Use the $l_{2}$ -induced norm constraint with $\bar{\gamma} = 1.5\text{--}2.0 \times \gamma_{\mathrm{opt}}$ to bound worst-case gain from disturbances.
Multiple requirements: The combined formulation provides systematic balancing within a single convex optimization.

This work builds upon and connects to several of the author's previous publications on multirate systems.

Multirate State Observer Design (IEEE Access, 2023): The earlier paper H. Okajima, Y. Hosoe, and T. Hagiwara, "State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm," IEEE Access, vol. 11, pp. 20079–20087, 2023 established the multirate observer framework using the $l_{2}$ -induced norm. The present paper extends this to the stochastic setting by incorporating the Kalman filter criterion (trace minimization) and addressing the semidefinite noise covariance issue.

Multirate Feedback Controller Design (IEEE Access, 2023): The paper H. Okajima, K. Arinaga, and A. Hayashida, "Design of Observer-Based Feedback Controller for Multi-Rate Systems with Various Sampling Periods Using Cyclic Reformulation," IEEE Access, vol. 11, pp. 121956–121965, 2023 develops the cyclic reformulation framework for observer-based feedback control in multirate systems, providing the foundational system representation used in the present work.

Multirate System Identification (J. Robot. Mechatron., 2025): The paper H. Okajima, R. Furukawa, and N. Matsunaga, "System Identification Under Multirate Sensing Environments," J. Robot. Mechatron., vol. 37, no. 5, pp. 1102–1112, 2025 addresses the upstream problem of obtaining the plant model required for Kalman filter design. Together with the present paper, this completes the multirate pipeline: identification → estimation → control.

Periodically Time-Varying Observer (Trans. SICE, 2019): The Japanese-language paper [H. Okajima, Y. Hosoe, and T. Hagiwara, "Periodically Time-Varying State Observer Design for Multi-Rate Systems Using L2-Induced Norm," Trans. SICE, vol. 55, no. 12, pp. 792–799, 2019] provides the theoretical foundation for the $l_{2}$ -induced norm analysis and synthesis via cyclic reformulation.

MATLAB/Python Code

The MATLAB and Python implementation of the proposed LMI-based multirate Kalman filter design is available at:

GitHub: multirate-kalman-filter

MATLAB File Exchange: LMI-Based Multirate Kalman Filter Design

Blog Articles (blog.control-theory.com)

State Observer and State Estimation: A Comprehensive Guide — Hub article covering all state estimation methods
Kalman Filter: From Basic Algorithm to Multi-Rate Extensions — Standard Kalman filter, steady-state design, and multi-rate extensions
H-infinity Filter: Robust State Estimation Using LMI Optimization — LMI-based robust filter design
State Observer Under Multi-Rate Sensing Environment and Its Design Using l2-Induced Norm — Multirate observer with l2-induced norm optimization
System Identification: Obtaining Dynamical Model — Hub article for system identification methods

Research Web Pages (www.control-theory.com)

Multi-rate System — Overview of multirate control and estimation research
State Estimation — State estimation research overview
Linear Matrix Inequality — LMI-based control and estimation design

Video

Video Abstract: LMI Optimization Based Multirate Steady-State Kalman Filter Design — Video abstract of this paper

Paper Information

Title: LMI Optimization Based Multirate Steady-State Kalman Filter Design
Author: Hiroshi Okajima (Kumamoto University)
Journal: IEEE Access Open Access
arXiv Preprint: arXiv:2602.01537
GitHub: multirate-kalman-filter
MATLAB File Exchange: LMI-Based Multirate Kalman Filter Design

Related Japanese Paper:

1. Okajima, Y. Hosoe, and T. Hagiwara, "Periodically Time-Varying State Observer Design for Multi-Rate Systems Using L2-Induced Norm" (in Japanese), Trans. SICE, vol. 55, no. 12, pp. 792–799, 2019.

Self-Introduction

Hiroshi Okajima received his M.E. and Ph.D. degrees from Osaka University in 2004 and 2007, respectively. He is currently an Associate Professor at Kumamoto University, Japan. His research interests include model error compensator design, multirate control and estimation, quantization for networked control systems, and state estimation. He has been actively publishing in IEEE Access and other international journals, with a focus on making control engineering research accessible through open-access publications, MATLAB code sharing, and educational content.

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MultiRateKalmanFilter #KalmanFilter #LMI #SensorFusion #StateEstimation #CyclicReformulation #SemidefiniteProgramming #PeriodicSystems #RobustEstimation #ControlEngineering #IEEEAccess #MATLAB

Why Multirate Kalman Filtering Matters

Problem Formulation: Multirate Sensing Systems

Cyclic Reformulation: From Periodic to Time-Invariant

The Semidefinite Challenge

LMI-Based Filter Design

Multi-Objective Extensions

Pole Placement Constraints

Kalman Filter with l2-induced Norm Constraint

Numerical Example: Automotive Navigation

Periodic Kalman Gains

Monte Carlo Validation

Multi-Objective Design Trade-offs

Kalman Filter with Pole Placement

Kalman Filter with l2-induced Norm Constraint

Comparison of Optimal Gains

Design Guidelines

Connections to Related Research

MATLAB/Python Code

Related Articles and Videos

Blog Articles (blog.control-theory.com)

Research Web Pages (www.control-theory.com)

Video

Paper Information

MultiRateKalmanFilter #KalmanFilter #LMI #SensorFusion #StateEstimation #CyclicReformulation #SemidefiniteProgramming #PeriodicSystems #RobustEstimation #ControlEngineering #IEEEAccess #MATLAB