A Graphical Characterization of Structurally Controllable Linear Systems with Dependent Parameters

TL;DR

This paper extends the concept of structural controllability to include parameter dependencies across multiple matrix entries, providing a graph-theoretic characterization under the binary assumption.

Contribution

It introduces a broadened definition of structural controllability allowing shared parameters and derives an explicit graph-based characterization under specific conditions.

Findings

Provides a new graph-theoretic criterion for dependent-parameter controllability

Extends existing controllability theory to more complex parameterizations

Offers insights for designing controllable systems with shared parameters

Abstract

One version of the concept of structural controllability defined for single-input systems by Lin and subsequently generalized to multi-input systems by others, states that a parameterized matrix pair $(A, B)$ whose nonzero entries are distinct parameters, is structurally controllable if values can be assigned to the parameters which cause the resulting matrix pair to be controllable. In this paper the concept of structural controllability is broadened to allow for the possibility that a parameter may appear in more than one location in the pair $(A, B)$. Subject to a certain condition on the parameterization called the "binary assumption", an explicit graph-theoretic characterization of such matrix pairs is derived.