Infinite horizon control and minimax observer design for linear DAEs

TL;DR

This paper develops an infinite horizon minimax state observer for linear DAEs with uncertainties, extending Kalman's duality principle and using geometric control theory to handle non-regular systems.

Contribution

It introduces a novel approach for minimax observer design for DAEs without regularity assumptions, utilizing a dual control problem and stable LTI systems.

Findings

Constructed a minimax state estimator as a stable LTI system.

Provided a solution to infinite-horizon LQ control for DAEs.

Extended Kalman's duality principle to non-regular DAEs.

Abstract

In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, in fact, the worst-case estimation error. Using geometric control theory, we construct an optimal control in the feed-back form and represent it as an output of a stable LTI system. The latter gives the minimax state estimator. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAEs.