A (Strongly) Connected Weighted Graph is Uniformly Detectable based on any Output Node

TL;DR

This paper proves that strongly connected weighted graphs ensure the uniform detectability of associated dynamical systems from any output node, facilitating observation of unstable states with limited measurements.

Contribution

It establishes that strongly connected graphs guarantee detectability from any node and extends this to parameter-varying systems maintaining the graph structure.

Findings

Fully connected graphs ensure detectability from any node.

Uniform detectability is preserved in parameter-varying systems with maintained graph structure.

Detectability can be achieved with limited discrete-time measurements.

Abstract

Many dynamical systems, including thermal, fluid, and multi-agent systems, can be represented as weighted graphs. In this paper we consider whether the unstable states of such systems can be observed from limited discrete-time measurement, that is, whether the discrete formulation of system is detectable. We establish that if the associated graph is fully connected, then a linear time invariant system is detectable by measuring any state. Further, we show that a parameter-varying or time-varying system remains uniformly detectable for a reasonable approximation of its state transition matrix, if the underlying graph structure is maintained.