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

In this report we address the linear state estimation problem: to estimate a linear transformation $\mathrm{\ell }(\phi )$ of the state $\phi$ through an algorithm $\widehat{\mathrm{\ell }(\phi )}$ 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 $\widehat{\widehat{\mathrm{\ell }(\phi )}}$ that minimizes the worst case error ${\sup }_{\phi ,\eta }d(\mathrm{\ell }(\phi ),\widehat{\mathrm{\ell }(\phi )})$. 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 $\mathrm{\ell }$ 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.