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Volume 12 Issue 4
Apr.  2025

IEEE/CAA Journal of Automatica Sinica

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Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2025  >  12(4): 694-704
J. Qu, Z. Ji, J. Wang, and  Y. Liu,  “Necessary and sufficient conditions for controllability and essential controllability of directed circle and tree graphs,” IEEE/CAA J. Autom. Sinica, vol. 12, no. 4, pp. 694–704, Apr. 2025. doi: 10.1109/JAS.2024.124866
Citation: J. Qu, Z. Ji, J. Wang, and  Y. Liu,  “Necessary and sufficient conditions for controllability and essential controllability of directed circle and tree graphs,” IEEE/CAA J. Autom. Sinica, vol. 12, no. 4, pp. 694–704, Apr. 2025. doi: 10.1109/JAS.2024.124866
shu

Necessary and Sufficient Conditions for Controllability and Essential Controllability of Directed Circle and Tree Graphs

doi: 10.1109/JAS.2024.124866
Funds:  This work was supported by the National Natural Science Foundation of China (62373205, 62033007), Taishan Scholars Climbing Program of Shandong Province of China, and Taishan Scholars Project of Shandong Province of China (tstp20230624, ts20190930)
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    Abstract
  • The multi-agent controllability is intrinsically affected by the network topology and the selection of leaders. A focus of exploring this problem is to uncover the relationship between the eigenspace of Laplacian matrix and network topology. For strongly connected directed circle graphs, we elaborate how the zero entries in the left eigenvectors of Laplacian matrix L arise. The topologies arising from left eigenvectors with zero entries are filtered to construct essentially controllable directed circle graphs regardless of the choice of leaders. We propose two methods for constructing a substantial quantity of essentially controllable graphs, with a focus on utilizing essentially controllable circle graphs as the foundation. For a special directed graph-OT tree, the controllability is shown to be related with its substructure-paths. This promotes the establishment of a sufficient and necessary condition for controllability. Finally, a method is presented to check the controllable subspace by identifying the left eigenvectors and generalized left eigenvectors.

     

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    Highlights

    • For strongly connected directed circle graphs, we elaborate how the zero entries in the left eigenvectors of Laplacian matrix L arise. The topologies arising from left eigenvectors with zero entries are filtered to construct essentially controllable directed circle graphs regardless of the choice of leaders
    • We propose two methods for constructing a substantial quantity of essentially controllable graphs, with a focus on utilizing essentially controllable circle graphs as the foundation
    • We find that the controllable subspace is intimately tied to the left eigenvectors and generalized left eigenvectors of L

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