Minimax state estimation for linear continuous differential-algebraic equations

TL;DR

This paper develops a minimax state estimation method for linear differential-algebraic equations with uncertain parameters, introducing a generalized duality principle and a regularized filtering algorithm for improved estimation accuracy.

Contribution

It introduces a Generalized Kalman Duality principle and a Tikhonov regularization-based sub-optimal estimation algorithm for DAE systems with uncertainties.

Findings

The GKD principle links minimax estimates to dual control problems.

The regularized algorithm provides a practical filter for uncertain DAE systems.

Synthetic example demonstrates the effectiveness of the proposed approach.

Abstract

This paper describes a minimax state estimation approach for linear Differential-Algebraic Equations (DAE) with uncertain parameters. The approach addresses continuous-time DAE with non-stationary rectangular matrices and uncertain bounded deterministic input. An observation's noise is supposed to be random with zero mean and unknown bounded correlation function. Main results are a Generalized Kalman Duality (GKD) principle and sub-optimal minimax state estimation algorithm. GKD is derived by means of Young-Fenhel duality theorem. GKD proves that the minimax estimate coincides with a solution to a Dual Control Problem (DCP) with DAE constraints. The latter is ill-posed and, therefore, the DCP is solved by means of Tikhonov regularization approach resulting a sub-optimal state estimation algorithm in the form of filter. We illustrate the approach by an synthetic example and we discuss connections with impulse-observability.