In this note we construct minimax observers for linear stationary DAEs with bounded uncertain inputs, given noisy measurements. We prove a new duality principle and show that a finite (infinite) horizon minimax observer exists if and only if the DAE is @math -impulse observable ( @math -detectable) . Remarkably, the regularity of the DAE is not required.
Abstract This paper presents a generalization of the minimax state estimation approach for singular linear Differential-Algebraic Equations (DAE) with uncertain but bounded input and observation's noise. We apply generalized Kalman Duality principle to DAE in order to represent the minimax estimate as a solution of a dual control problem for adjoint DAE. The latter is then solved converting the adjoint DAE into ODE by means of a projection algorithm. Finally, we represent the minimax estimate in the form of a linear recursive filter. 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. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAE.  In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence ( u _ ) for the dual problem applying Tikhonov method. Finally we represent ( u _ ) in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.