Minimax State Estimation for a Dynamic System Described by a Differential-Algebraic Equation

Abstract

In this report we address the linear state estimation problem: to estimate a linear transformation $\ell (\phi )$ of the state $\phi$ through an algorithm $\widehat{\ell (\phi )}$ operating on measurements $y$, where $L\phi =f,y=H\phi +\eta$. We study the estimation problem in terms of the minimax estimation framework: to find a linear algorithm $\widehat{\hat{\ell (\phi )}}$ that minimizes the worst case error ${\sup }_{\phi ,\eta }d(\ell (\phi ),\widehat{\ell (\phi )})$. A key feature of the presented estimation approach is to fix a class of linear operators $L$, $H$; given any pair $L,H$ from that class we describe a class $\mathcal{L}$ of all solution operators $\ell$ such that the worst case error is finite. We formulate a duality theorem (like Kalman duality principle) that is the estimation problem is equal to the optimal control problem if $G$ is convex bounded subset of the corresponding Hilbert space, $L$ is a closed linear mapping. We obtain optimal estimations as solutions of the linear operator equations if $G$ is an ellipsoid. Then we apply this to the state estimation for the linear differential-algebraic equations (DAE). The minimax observer for DAE is represented in the form of the minimax filter. For discrete time DAEs we present the online minimax estimator.

This report was presented at the International Conference "Differential Equations and Topology", Moscow, 2008