Observability of Dynamical Networks from Graphic and Symbolic Approaches
Abstract
A dynamical network, a graph whose nodes are dynamical systems, is usually characterized by a large dimensional space which is not always accessible due to the impossibility of measuring all the variables spanning the state space. Therefore, it is of the utmost importance to determine a reduced set of variables providing all the required information to non-ambiguously distinguish its different states. Inherited from control theory, one possible approach is based on the use of the observability matrix defined as the Jacobian matrix of the change of coordinates between the original state space and the space reconstructed from the measured variables. The observability of a given system can be accurately assessed by symbolically computing the complexity of the determinant of the observability matrix and quantified by symbolic observability coefficients. In this work, we extend the symbolic observability, previously developed for dynamical systems, to networks made of coupled d-dimensional node dynamics (d > 1). From the observability of the node dynamics, the coupling function between the nodes, and the adjacency matrix, it is indeed possible to construct the observability of a large network with an arbitrary topology.
Keywords
Dynamical network ObservabilityNotes
Acknowledgements
ISN acknowledges partial support from the Ministerio de Economía y Competitividad of Spain under project FIS2017-84151-P and from the Group of Research Excelence URJC-Banco de Santander.
References
- 1.Bianchin, G., Frasca, P., Gasparri, A., Pasqualetti, F.: The observability radius of networks. IEEE Trans. Autom. Control 62(6), 3006–3013 (2017).MathSciNetCrossRefGoogle Scholar
- 2.Chan, B.Y., Shachter, R.D.: Structural controllability and observability in influence diagrams. In: Proceedings of the Eighth International Conference on Uncertainty in Artificial Intelligence (UAI’92), pp. 25–32. Morgan Kaufmann Publishers, San Francisco (1992)Google Scholar
- 3.Hasegawa, T., Takaguchi, T., Masuda, N.: Observability transitions in correlated networks. Phys. Rev. E 88, 042809 (2013)ADSCrossRefGoogle Scholar
- 4.Hermann, R., Krener, A.: Nonlinear controllability and observability. IEEE Trans. Autom. Control 22(5), 728–740 (1977)MathSciNetCrossRefGoogle Scholar
- 5.Letellier, C., Aguirre, L.A.: Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables. Chaos 12(3), 549–558 (2002).ADSMathSciNetCrossRefGoogle Scholar
- 6.Letellier, C., Aguirre, L.A., Maquet, J.: Relation between observability and differential embeddings for nonlinear dynamics. Phys. Rev. E 71(6), 066213 (2005)ADSMathSciNetCrossRefGoogle Scholar
- 7.Letellier, C., Sendiña-Nadal, I., Aguirre, L.A.: A nonlinear graph-based theory for dynamical network observability. Phys. Rev. E 98, 020303(R) (2018)Google Scholar
- 8.Letellier, C., Sendiña-Nadal, I., Bianco-Martinez, E., Baptista, M.S.: A symbolic network-based nonlinear theory for dynamical systems observability. Sci. Rep. 8, 3785 (2018)ADSCrossRefGoogle Scholar
- 9.Lin, C.T.: Structural controllability. IEEE Trans. Autom. Control 19(3), 201–208 (1974)ADSMathSciNetCrossRefGoogle Scholar
- 10.Liu, Y.Y., Slotine, J.J., Barabási, A.L.: Observability of complex systems. Proc. Natl. Acad. Sci. 110(7), 2460–2465 (2013)ADSMathSciNetCrossRefGoogle Scholar
- 11.Newman, M.E.: Networks: an introduction. Oxford University Press, Oxford (2010)CrossRefGoogle Scholar
- 12.Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57(5), 397–398 (1976)ADSCrossRefGoogle Scholar
- 13.Sendiña Nadal, I., Boccaletti, S., Letellier, C.: Observability coefficients for predicting the class of synchronizability from the algebraic structure of the local oscillators. Phys. Rev. E 94(4), 042205 (2016)ADSCrossRefGoogle Scholar
- 14.Sevilla-Escoboza, R., Buldú, J.Mss.: Synchronization of networks of chaotic oscillators: structural and dynamical datasets. Data in Brief 7, 1185–1189 (2016)CrossRefGoogle Scholar
- 15.Van Mieghem, P., Wang, H.: The observable part of a network. IEEE/ACM Trans. Netw. 17(1), 93–105 (2009)CrossRefGoogle Scholar