# A Unifying Framework for Strong Structural Controllability

## TL;DR

This paper introduces a comprehensive framework for analyzing strong structural controllability of linear systems with complex zero/nonzero/arbitrary patterns, providing algebraic and graph-theoretic conditions that generalize previous results.

## Contribution

It formalizes a new class of structured systems with zero/nonzero/arbitrary entries and establishes necessary and sufficient algebraic and graph-theoretic conditions for their strong controllability.

## Key findings

- Derived algebraic full rank conditions for pattern matrices.
- Developed a novel color change rule for graph analysis.
- Established a unified graph-theoretic controllability criterion.

## Abstract

This paper deals with strong structural controllability of linear systems. In contrast to existing work, the structured systems studied in this paper have a so-called zero/nonzero/arbitrary structure, which means that some of the entries are equal to zero, some of the entries are arbitrary but nonzero, and the remaining entries are arbitrary (zero or nonzero). We formalize this in terms of pattern matrices whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We establish necessary and sufficient algebraic conditions for strong structural controllability in terms of full rank tests of certain pattern matrices. We also give a necessary and sufficient graph theoretic condition for the full rank property of a given pattern matrix. This graph theoretic condition makes use of a new color change rule that is introduced in this paper. Based on these two results, we then establish a necessary and sufficient graph theoretic condition for strong structural controllability. Moreover, we relate our results to those that exists in the literature, and explain how our results generalize previous work.