Connectivity Structure of Systems

TL;DR

This paper investigates the extent to which the structure of linear systems can be determined solely from input/output data, revealing fundamental limitations and the need for additional assumptions or domain knowledge.

Contribution

It demonstrates that input/output data alone often cannot uniquely identify a system's structure, highlighting the importance of assumptions and domain knowledge in structural identification.

Findings

Linear transformations can alter the system's graph.

Input/output behavior does not determine graph components.

Additional assumptions are necessary for structure identification.

Abstract

In this paper, we consider to what degree the structure of a linear system is determined by the system's input/output behavior. The structure of a linear system is a directed graph where the vertices represent the variables in the system and an edge (x,y) exists if x directly influences y. In a number of studies, researchers have attempted to identify such structures using input/output data. Thus, our main aim is to consider to what degree the results of such studies are valid. We begin by showing that in many cases, applying a linear transformation to a system will change the system's graph. Furthermore, we show that even the graph's components and their interactions are not determined by input/output behavior. From these results, we conclude that without further assumptions, very few aspects, if any, of a system's structure are determined by its input/output relation. We consider a number of such assumptions. First, we show that for a number of parameterizations, we can characterize when two systems have the same structure. Second, in many applications, we can use domain knowledge to exclude certain interactions. In these cases, we can assume that a certain variable x does not influence another variable y. We show that these assumptions cannot be sufficient to identify a system's parameters using input/output data. We conclude that identifying a system's structure from input/output data may not be possible given only assumptions of the form x does not influence y.