On the asymptotic output sensitivity problem for a discrete linear systems with an uncertain initial state

This paper studies a finite-dimensional discrete linear system whose initial state $x_0$ is unknown.  We assume that the system is augmented by two output equations, the first one $z_i$ being representing measurements made on the unknown state of the system and the other $y_i$ being representing the corresponding output.  The purpose of our work is to introduce two control laws, both in closed-loop of measurements $z_i$ and whose goal is to reduce asymptotically the effects of the unknown part of the initial state $x_0$.  The approach that we present consists of both theoretical and algorithmic characterization of the set of such controls.  To illustrate our theoretical results, we give a number of examples and numerical simulations.

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