Linearly Constrained Kalman Filter For Linear Discrete State-Space Models

TL;DR

This paper introduces the linearly constrained Kalman filter (LCKF) for linear discrete state-space models, extending the classical Kalman filter to incorporate linear constraints, thereby enabling robust filtering under uncertainties and misspecifications.

Contribution

The paper develops a general form of the LCKF that includes the linearly constrained minimum variance estimator, bridging deterministic and stochastic filtering frameworks.

Findings

LCKF encompasses existing constrained estimators.

Provides alternative robust filtering solutions.

Enhances filter performance under model uncertainties.

Abstract

For linear discrete state-space (LDSS) models, under certain conditions, the linear least mean squares filter estimate has a convenient recursive predictor/corrector format, aka the Kalman filter (KF). The aim of the paper is to introduce the general form of the linearly constrained KF (LCKF) for LDSS models, which encompasses the linearly constrained minimum variance estimator (LCMVE). Thus the LCKF opens access to the abundant litterature on LCMVE in the deterministic framework which can be transposed to the stochastic framework. Therefore, among other things, the LCKF may provide alternative solutions to $H_{\infty }$ filter and unbiased finite impulse response filter to robustify the KF, which performance are sensible to misspecified noise or uncertainties in the system matrices