LMI Optimization Based Multirate Steady-State Kalman Filter Design

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.