Identification of minimal number of measurements allowing synchronization of a nodal observer for the wave equation

TL;DR

This paper investigates the minimal measurement requirements for ensuring the convergence of a nodal observer to the true state in a wave equation network, revealing conditions based on network topology.

Contribution

It identifies the minimal number of additional boundary measurements needed for observer synchronization in networked wave systems, considering both tree and cyclic structures.

Findings

Boundary measurements suffice for tree networks to ensure exponential synchronization.

Additional measurements are necessary in networks with cycles, at least one per cycle.

Results hold for systems with and without linear friction terms.

Abstract

We study a state estimation problem for a $2\times 2$ linear hyperbolic system on networks with eigenvalues with opposite signs. The system can be seen as a simplified model for gas flow through gas networks. For this system we construct an observer system based on nodal measurements and investigate the convergence of the state of the observer system towards the original system state. We assume that measurements are available at the boundary nodes of the network and identify the minimal number of additional measurements in the network that are needed to guarantee synchronization of the observer state towards the original system state. It turns out that for tree-shaped networks boundary measurements suffice to guarantee exponential synchronization, while for networks that contain cycles synchronization can be guaranteed if and only if at least one measurement point is added in each cycle. This is shown for a system without source term and for a system with linear friction term.