Indistinguishability in Localization and Control with Coarse Information

We study localization and control problems in which agent dynamics are described by difference or differential equations, while output measurements are collected at discrete times and given by finite-valued maps depending on possibly unknown landmark locations. Guided by the goal of understanding fundamental limitations imposed by such coarse measurements, we focus on characterizing indistinguishable states, i.e., agent-landmark pairs that produce identical observations under all control inputs. We show that indis-tinguishability relations can be checked automatically under mild assumptions and, being a special type of bisimulation, we develop an iterative algorithm for approximately computing them. We then introduce an analytical approach, rooted in observability theory of linear control systems, which iteratively computes a sequence of subspaces converging in finitely many steps to the indistinguishable subspace; a differential-geometric extension to nonlinear systems is also outlined.