Minimax state estimation for linear discrete-time differential-algebraic equations

TL;DR

This paper develops a minimax state estimation method for uncertain linear discrete-time differential-algebraic equations, extending Kalman filtering to non-causal DAEs with a new observable subspace concept.

Contribution

It introduces a novel minimax observable subspace and derives recursive estimators for discrete-time DAEs, generalizing Kalman filter results for regular DAEs.

Findings

The estimator's equation matches the Kalman filter for regular DAEs.

A new notion of minimax observable subspace is proposed.

Numerical example demonstrates the estimator's effectiveness.

Abstract

This paper presents a state estimation approach for an uncertain linear equation with a non-invertible operator in Hilbert space. The approach addresses linear equations with uncertain deterministic input and noise in the measurements, which belong to a given convex closed bounded set. A new notion of a minimax observable subspace is introduced. By means of the presented approach, new equations describing the dynamics of a minimax recursive estimator for discrete-time non-causal differential-algebraic equations (DAEs) are presented. For the case of regular DAEs it is proved that the estimator's equation coincides with the equation describing the seminal Kalman filter. The properties of the estimator are illustrated by a numerical example.