Minimax state estimation for linear descriptor systems

TL;DR

This paper develops a generalized minimax state estimation theory for linear differential-algebraic equations, extending Kalman duality to unbounded operators and providing new algorithms for robust estimation and model validation.

Contribution

It introduces a generalized Kalman duality principle for DAEs with unbounded operators and proposes new minimax estimation algorithms and validation tools.

Findings

Generalized duality principle for unbounded operators.

Existence theorems for minimax estimates.

New fast estimation algorithms for DAEs.

Abstract

Author's Summary of the dissertation for the degree of the Candidate of Science (physics and mathematics). The aim of the dissertation is to develop a generalized Kalman Duality concept applicable for linear unbounded non-invertible operators and introduce the minimax state estimation theory and algorithms for linear differential-algebraic equations. In particular, the dissertation pursues the following goals: - develop generalized duality concept for the minimax state estimation theory for DAEs with unknown but bounded model error and random observation noise with unknown but bounded correlation operator; - derive the minimax state estimation theory for linear DAEs with unknown but bounded model error and random observation noise with unknown but bounded correlation operator; - describe how the DAE model propagates uncertain parameters; - estimate the worst-case error; - construct fast estimation algorithms in the form of filters; - develop a tool for model validation, that is to assess how good the model describes observed phenomena. The dissertation contains the following new results: - generalized version of the Kalman duality principle is proposed allowing to handle unbounded linear model operators with non-trivial null-space; - new definitions of the minimax estimates for DAEs based on the generalized Kalman duality principle are proposed; - theorems of existence for minimax estimates are proved; - new minimax state estimation algorithms (in the form of filter and in the variational form) for DAE are proposed.