Distributed unknown input observers for linear time-invariant systems in discrete time

This paper introduces a distributed unknown input observer (D-UIO) for discrete-time LTI plants monitored by a sensor network with limited communication. The design builds on a node-wise decomposition of the state space into locally detectable and undetectable components, which separates the estimation task driven by local measurements from the one enabled by inter-node cooperation. We present two complementary synthesis routes: (i) a split design that independently stabilizes the detectable component (via local output-injection gains) and the undetectable component (via diffusive consensus gains), and (ii) a unified design that computes all gains at once. Both routes rely on linear matrix inequalities (LMIs), enabling computationally efficient synthesis as well as secondary objectives such as minimizing H∞ error bounds or enforcing a prescribed consensus convergence rate. Under mild rank conditions, the resulting estimators guarantee ultimately bounded errors in the presence of measurement noise and unknown inputs. Two simulation studies, using alternative node-wise decompositions, illustrate the effectiveness and trade-offs of the proposed D-UIO.