Abstract
We study the strong structural controllability (SSC) of networks, where the external control inputs are injected to only some nodes, namely the leaders. For such systems, one measure of controllability is the dimension of strong structurally controllable subspace (SSCS), which is equal to the smallest possible rank of controllability matrix under admissible coupling weights among the nodes In this paper, we compare two tight lower bounds on the dimension of SSCS: one based on the distances of followers to leaders, and the other based on the graph coloring process known as zero forcing. We first show that each of these two bounds can be arbitrarily better than the other in some special cases. We then show that the distance-based lower bound is usually better than the zero-forcing-based bound when the value of the latter is less than the dimensionality of the overall network state, n. On the other hand, we also show that any set of leaders that makes the distance-based bound equal to n necessarily makes the zero-forcing-based bound equal to n (the converse is not true). These results indicate that while the zero-forcing-based approach may be preferable when the focus is only on verifying complete SSC (dimension of SSCS is equal to n), the distance-based approach usually yields a closer bound on the dimension of SSCS when the bounds are both smaller than n. Furthermore, we also present a novel bound based on combining these two approaches, which is always at least as good as, and in some cases strictly greater than, the maximum of the two original bounds. Finally, we support our analysis with numerical results on various graphs.
Original language | English (US) |
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Article number | 110562 |
Journal | Automatica |
Volume | 146 |
DOIs | |
State | Published - Dec 2022 |
Bibliographical note
Funding Information:Xenofon Koutsoukos is a Professor and the Chair of the Department of Computer Science and a Senior Research Scientist with the Institute for Software Integrated Systems (ISIS), Vanderbilt University, Nashville, TN, USA. He was a Member of Research Staff at the Xerox Palo Alto Research Center (PARC) (2000, 2002). His research work is in the area of cyber–physical systems with emphasis on learning-enabled systems, formal methods, distributed algorithms, security and resilience, diagnosis and fault tolerance, and adaptive resource management. He has published more than 300 journal and conference papers and he is co-inventor of four US patents. Prof. Koutsoukos was the recipient of the NSF Career Award in 2004, the Excellence in Teaching Award in 2009 from the Vanderbilt University School of Engineering, and the 2011 NASA Aeronautics Research Mission Directorate (ARMD) Associate Administrator (AA) Award in Technology and Innovation. He was named a Fellow of the IEEE for his contributions to the design of resilient cyber–physical systems.
Publisher Copyright:
© 2022 Elsevier Ltd
Keywords
- Control of networks
- Controllability
- Graph theory