LMI-Based Multirate Kalman Filter Design

MATLAB/Python implementation of LMI-based multirate steady-state Kalman filter design using cyclic reformulation. This repository contains the code accompanying the paper:

H. Okajima, "LMI Optimization Based Multirate Steady-State Kalman Filter Design," IEEE Access (2026)

ArXiV ver.

Main Scripts

File	Description	Design Objective
MultirateKF_01.m	Basic optimal Kalman filter design	Minimize trace(P_e)
MultirateKF_02_eig.m	Multi-objective design with eigenvalue placement	Minimize trace(P_e) subject to |λ| < r̄
MultirateKF_03_l2.m	Multi-objective design with l2-induced norm	Minimize trace(P_e) subject to ||G||_{l2} < γ̄
MultirateKF_Simple.m	Basic optimal Kalman filter design for 1st order system	Minimize trace(P_e)
MultirateKF_Simple.ipynb	Basic optimal Kalman filter design for 1st order system (Python)	Minimize trace(P_e)
MultirateKF_Simple.py	Basic optimal Kalman filter design for 1st order system (Python)	Minimize trace(P_e)
MultirateKF_01.ipynb	Basic optimal Kalman filter design (Python)	Minimize trace(P_e)
MultirateKF_01.py	Basic optimal Kalman filter design (Python)	Minimize trace(P_e)

Required

MATLAB:

• Control System Toolbox
• Robust Control Toolbox

Python:

• numpy, scipy, matplotlib, cvxpy

Example System

Automotive navigation with GPS (1 Hz) and wheel speed sensor (10 Hz):

• State: [position; velocity; acceleration]
• Period: N = 10 steps
• Measurements:
  • k mod 10 = 0: GPS + wheel speed
  • k mod 10 ≠ 0: wheel speed only

Quick Start

git clone https://github.com/Hiroshi-Okajima/multirate-kalman-filter.git
cd multirate-kalman-filter

MATLAB:

MultirateKF_01                  % Basic optimal design
MultirateKF_02_eig              % With eigenvalue constraints
MultirateKF_03_l2               % With l2-induced norm constraints
MultirateKF_Simple              % Basic optimal Kalman filter design for 1st order system

Python:

pip install numpy scipy matplotlib cvxpy
python MultirateKF_Simple.py
python MultirateKF_01.py

Example Results

Automotive navigation (GPS 1Hz + Wheel speed 10Hz):

• Position RMSE: 0.600 m
• Velocity RMSE: 0.268 m/s
• Stable: max|λ| = 0.967