KKL set-valued observers for non-observable systems

KKL observer design consists in transforming the system dynamics into a filter of the output, which admits a trivial observer, and left-inverting the transformation to recover an estimate of the state in the system coordinates. This left-inversion is typically guaranteed under a backward-distinguishability condition. In this paper, instead, we demonstrate how this KKL approach may also be applied without any such observability assumption. We show that there exist appropriate choices of the filter such that any filter solution asymptotically contains the full information about the state indistinguishable class, namely the set of points generating the same output. Then, we investigate the existence of a set-valued left-inverse allowing to estimate asymptotically this indistinguishable class, in the Hausdorff sense. We prove that the estimate tends to be included asymptotically in the indistinguishable classes of the limit points of the system solution. Finally, we provide a numerical example illustrating this convergence. Copyright (c) 2023 The Authors. This is an open access article under the CC BY-NC-ND license (<THESTERM>https://creativecommons.org/licenses/by-ne-nd/4.0/</THESTERM>)