A Note on Structural Controllability and Observability Indices

TL;DR

This paper critically examines existing graph-theoretic characterizations of structural controllability and observability indices, revealing limitations, providing bounds, and proposing an efficiently computable lower bound validated by simulations.

Contribution

It identifies inaccuracies in current characterizations, extends bounds to non-controllable systems, and introduces a new tight lower bound for the indices.

Findings

Counter-examples show existing characterizations may not hold.

New bounds are applicable to non-structurally controllable systems.

A tight lower bound is proposed and validated through simulations.

Abstract

In this note, we investigate the structural controllability and observability indices of structured systems. We provide counter-examples showing that an existing graph-theoretic characterization for the structural controllability index (SCOI) may not hold, even for systems with self-loop at every state node. We further demonstrate that this characterization actually provides upper bounds, and extend them to new graph-theoretic characterizations applicable to systems that are not necessarily structurally controllable. Additionally, we reveal that an existing method may fail to obtain the exact SCOI. Consequently, complete graph-theoretic characterizations and polynomial-time computation of SCOI remain open. Given this, we present an efficiently computable tight lower bound, whose tightness is validated by numerical simulations. All these results apply to the structural observability index by the duality between controllability and observability.