Distributed Observer Design for Discrete-Time LTI Systems via Jordan Canonical Form

TL;DR

This paper develops two distributed state estimation schemes for discrete-time LTI systems using Jordan canonical form, improving flexibility and reducing conditions compared to prior methods.

Contribution

It introduces two novel distributed observer designs leveraging Jordan form, with less restrictive conditions and enhanced flexibility over existing approaches.

Findings

Both schemes guarantee asymptotic convergence of estimates to the true state.

The first scheme matches the original state dimension but requires many inequalities.

The second scheme is less restrictive but results in higher-order observers.

Abstract

This paper addresses the problem of distributed state estimation for discrete-time linear time-invariant systems. Building on the framework proposed in Gao & Yang (2025), we exploit the Jordan canonical form of the system matrix to develop two distributed estimation schemes that ensure asymptotic convergence of local estimates to the true system state. In both approaches, each node reconstructs the components of the state that are locally detectable for it via a Luenberger observer, while employing a consensus-based mechanism to estimate the components that are not directly detectable. The first scheme relies on local observers whose dimension matches that of the original state vector; however, its applicability requires the satisfaction of a large set of inequalities. The second scheme, in contrast, can be implemented under less restrictive conditions, but results in observers of increased (augmented) order. For both methods, we derive necessary and sufficient conditions - expressed in terms of the eigenvalues of the system matrix and certain submatrices of the communication network Laplacian - that guarantee the existence of a distributed observer achieving asymptotically accurate estimation. Compared to Gao & Yang (2025), the proposed approaches offer greater flexibility in the selection of coupling gains and impose less stringent solvability conditions.