A linear time algorithm to verify strong structural controllability

TL;DR

This paper introduces a new linear time algorithm for verifying strong structural controllability of pairs of matrices, significantly improving efficiency for large sparse systems.

Contribution

It presents the first linear time algorithm for strong structural controllability verification, utilizing advanced data structures and a novel update scheme.

Findings

Algorithm runs in linear time relative to non-zero entries

Efficiently handles large sparse matrices

Demonstrates practical performance on various system sizes

Abstract

We prove that strong structural controllability of a pair of structural matrices $(\mathcal{A},\mathcal{B})$ can be verified in time linear in $n + r + \nu$, where $\mathcal{A}$ is square, $n$ and $r$ denote the number of columns of $\mathcal{A}$ and $\mathcal{B}$, respectively, and $\nu$ is the number of non-zero entries in $(\mathcal{A},\mathcal{B})$. We also present an algorithm realizing this bound, which depends on a recent, high-level method to verify strong structural controllability and uses sparse matrix data structures. Linear time complexity is actually achieved by separately storing both the structural matrix $(\mathcal{A},\mathcal{B})$ and its transpose, linking the two data structures through a third one, and a novel, efficient scheme to update all the data during the computations. We illustrate the performance of our algorithm using systems of various sizes and sparsity.