C. Letellier, I. Sendiņa-Nadal, E. Bianco-Martinez & M. S. Baptista
A symbolic network-based nonlinear theory for dynamical systems observability
Scientific Reports, 8, 3785, 2018. Online

- Abstract
When the state of the whole reaction network can be inferred by just measuring the dynamics of a limited set of nodes the system is said to be fully observable. However, as the number of all possible combinations of measured variables and time derivatives spanning the reconstructed state of the system exponentially increases with its dimension, the observability becomes a computationally prohibitive task. Our approach consists in computing the observability coefficients from a symbolic Jacobian matrix whose elements encode the linear, nonlinear polynomial or rational nature of the interaction among the variables. The novelty we introduce in this paper, required for treating large-dimensional systems, is to identify from the symbolic Jacobian matrix the minimal set of variables (together with their time derivatives) candidate to be measured for completing the state space reconstruction. Then symbolic observability coefficients are computed from the symbolic observability matrix. Our results are in agreement with the analytical computations, evidencing the correctness of our approach. Its application to efficiently exploring the dynamics of real world complex systems such as power grids, socioeconomic networks or biological networks is quite promising.

Electronic supplementary material

C. Letellier, I. Sendina-Nadal & L. A. Aguirre
A nonlinear graph-based theory for dynamical network observability
eprint arXiv:1803.00851 Online

- Abstract
A faithful description of the state of a complex dynamical network would require, in principle, the measurement of all its d variables, an unfeasible task for systems with practical limited access and composed of many nodes with high dimensional dynamics. However, even if the network dynamics is observable from a reduced set of measured variables, how to reliably identifying such a minimum set of variables providing full observability remains an unsolved problem. From the Jacobian matrix of the governing equations of nonlinear systems, we construct a pruned fluence graph in which the nodes are the state variables and the links represent only the linear dynamical interdependences encoded in the Jacobian matrix after ignoring nonlinear relationships. From this graph, we identify the largest connected sub-graphs where there is a path from every node to every other node and there are not outcoming links. In each one of those sub-graphs, at least one node must be measured to correctly monitor the state of the system in a d-dimensional reconstructed space. Our procedure is here validated by investigating large-dimensional reaction networks for which the determinant of the observability matrix can be rigorously computed

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