A Note on Structural Controllability and Observability Indices
Yuan Zhang, Ranbo Cheng, Ziyuan Luo, Yuanqing Xia
TL;DR
This work shows that a prominent graph-theoretic characterization of the Structural Controllability Index (SCOI) can fail to yield the exact index and may only provide upper bounds, even when self-loops are present. It introduces a new cactus-structure-based upper-bound framework applicable to structurally uncontrollable systems and demonstrates limitations of existing methods, including a counterexample to a well-known lemma and a critique of the Sueur approach. The authors derive a tight, efficiently computable lower bound mu_low via a single minimum-cost maximum-flow construction in a dynamically extended graph, with strong empirical support from random ER graphs. By exploiting the dynamic-graph gammoid structure, the results offer practical tools for bounding SCOI and extend to the Structural Observability Index by duality, while leaving a complete graph-theoretic, polynomial-time SCOI characterization as an open problem.
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.