Properties of pattern matrices with applications to structured systems

TL;DR

This paper investigates the algebraic properties of pattern matrices with symbolic entries indicating zero, nonzero, or arbitrary parameters, and applies these properties to analyze controllability and observability in structured linear systems.

Contribution

It introduces formal definitions and operations for pattern matrices and applies them to derive conditions for controllability and observability in structured systems.

Findings

Pattern matrices can be added and multiplied with well-defined rules.

Structural controllability and observability can be characterized using pattern matrix algebra.

The results facilitate analysis of systems with uncertain or incomplete parameter information.

Abstract

The exact parameter values of mathematical models are often uncertain or even unknown. Nevertheless, we may have access to crude information about the parameters, e.g., that some of them are nonzero. Such information can be captured by so-called pattern matrices, whose symbolic entries are used to represent the available information about the corresponding parameters. In this paper, we focus on pattern matrices with three types of symbolic entries: those that represent zero, nonzero, and arbitrary parameters. We formally define and study addition and multiplication of such pattern matrices. The results are then used in the study of three strong structural properties, namely, controllability of linear descriptor systems, and input-state observability and output controllability of linear systems.