On the Reachability of Networked Systems

TL;DR

This paper investigates the reachability of networked linear systems controlled via a base-station communicating with some subsystems called leaders, establishing algebraic conditions for reachability and demonstrating generic parameter cases.

Contribution

It introduces the concepts of leader-reachability and base-reachability, providing algebraic conditions and showing that generic parameters ensure both are satisfied.

Findings

Base-reachability always holds for minimal state space representations.

Leader-reachability depends on specific algebraic conditions.

Generic system parameters lead to both leader- and base-reachability.

Abstract

In this paper, we study networks of discrete-time linear time-invariant subsystems. Our focus is on situations where subsystems are connected to each other through a time-invariant topology and where there exists a base-station whose aim is to control the subsystems into any desired destinations. However, the base-station can only communicate with some of the subsystems that we refer to as leaders. There are no direct links between the base-station and the rest of subsystems, known as followers, as they are only able to liaise among themselves and with some of the leaders. The current paper formulates this framework as the well-known reachability problem for linear systems. Then to address this problem, we introduce notions of leader-reachability and base-reachability. We present algebraic conditions under which these notions hold. It turns out that if subsystems are represented by minimal state space representations, then base-reachability always holds. Hence, we focus on leader-reachability and investigate the corresponding conditions in detail. We further demonstrate that when the networked system parameters i.e. subsystems' parameters and interconnection matrices, assume generic values then the whole network is both leader-reachable and base-reachable.