Abstract
This chapter examines strong structural controllability of linear-time-invariant networked systems. We provide necessary and sufficient conditions for strong structural controllability involving constrained matchings over the bipartite graph representation of the network. An \(\mathcal{O}(n^{2})\) algorithm to validate if a set of inputs leads to a strongly structurally controllable network and to find such an input set is proposed. The problem of finding such a set with minimal cardinality is shown to be NP-complete. Minimal cardinality results for strong and weak structural controllability are compared.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A matrix is unreduced upper-Hessenberg if all entries on the first superdiagonal nonzero and all entries above this diagonal are zero.
- 2.
The bandwidth of a graph is the minimum \(\max \left \{\left \vert i - j\right \vert \vert \left \{i,j\right \} \in E\right \}\) over all labeling of the nodes.
- 3.
Directed Erdős–Rényi random graphs are randomly generated graphs with an edge \(\left (i,j\right ) \in E\) independently existing with probability p [20]. The mean degree is defined as \(\left \langle k\right \rangle = 2np\).
- 4.
A directed graph is connected if its underlying undirected graph is connected.
References
M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks. Princeton: Princeton University Press, 2010.
M. Nabi-Abdolyousefi and M. Mesbahi, “On the controllability properties of circulant networks,” IEEE Transactions on Automatic Control, vol. 58, no. 12, pp. 3179–3184, 2013.
G. Parlangeli and G. Notarstefano, “Observability and reachability of grid graphs via reduction and symmetries,” in Proc. 50th IEEE Conference on Decision and Control, 2011, pp. 5923–5928.
S. Zhang, M. Camlibel, and M. Cao, “Controllability of diffusively-coupled multi-agent systems with general and distance regular coupling topologies,” in Proc. 50th IEEE Conference on Decision and Control, 2011, pp. 759–764.
Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabási, “Controllability of complex networks,” Nature, vol. 473, no. 7346, pp. 167–73, May 2011.
A. Rahmani and M. Mesbahi, “Pulling the strings on agreement: anchoring, controllability, and graph automorphisms,” in Proc. American Control Conference, 2007, pp. 2738–2743.
C.-T. Lin, “Structural Controllability,” IEEE Transactions on Automatic Control, vol. AC-19, no. 3, pp. 201–208, Aug. 1974.
H. Mayeda and T. Yamada, “Strong structural controllability,” SIAM Journal on Control and Optimization, vol. 17, no. 1, pp. 123–138, 1979.
Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabási, “Controllability of complex networks (Supplementary),” Nature, vol. 473, no. 7346, pp. 167–73, May 2011.
K. J. Reinschke, F. Svaricek, and H.-D. Wend, “On strong structural controllability of linear systems,” in Proc. 31st IEEE Conference on Decision and Control, no. 1, 1992, pp. 203–206.
J. C. Jarczyk, F. Svaricek, and B. Alt, “Strong structural controllability of linear systems revisited,” in Proc. 50th IEEE Conference on Decision and Control, Dec. 2011, pp. 1213–1218.
C.-T. Chen, Linear Systems Theory and Design. New York: Oxford University Press, 1999.
C. Paige, “Properties of numerical algorithms related to computing controllability,” IEEE Transactions on Automatic Control, vol. 26, no. 1, pp. 130–138, Feb. 1981.
R. Shields and J. Pearson, “Structural controllability of multiinput linear systems,” IEEE Transactions on Automatic Control, vol. 21, no. 2, pp. 203–212, Apr. 1976.
R. Cantó, M. Boix, and B. Ricarte, “On the minimal rank completion problem for pattern matrices,” WSEAS Transactions on Mathematics, vol. 3, no. 3, pp. 711–716, 2004.
D. Hershkowitz and H. Schneider, “Ranks of zero patterns and sign patterns,” Linear and Multilinear Algebra, vol. 34, no. 90, pp. 3–19, 1993.
M. Golumbic, T. Hirst, and M. Lewenstein, “Uniquely restricted matchings,” Algorithmica, vol. 31, no. 2, pp. 139–154, 2001.
S. Micali and V. V. Vazirani, “An \(O(\sqrt{\vert V \vert }\vert E\vert )\) algorithm for finding maximum matching in general graphs,” in 21st Annual Symposium on Foundations of Computer Science, Oct. 1980, pp. 17–27.
D. D. Olesky, M. Tsatsomeros, and P. van den Driessche, “Qualitative controllability and uncontrollability by a single entry,” Linear Algebra and its Applications, vol. 187, pp. 183–194, 1993.
B. Bollobás, Random Graphs. New York: Cambridge University Press, 2001.
S. Mishra, “On the maximum uniquely restricted matching for bipartite graphs,” Electronic Notes in Discrete Mathematics, vol. 37, pp. 345–350, Aug. 2011.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Chapman, A. (2015). Strong Structural Controllability of Networked Dynamics. In: Semi-Autonomous Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-15010-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-15010-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15009-3
Online ISBN: 978-3-319-15010-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)