A symbolic network-based nonlinear theory for dynamical systems observability

• Published in: Scientific reports Vol. 8; no. 1; pp. 3785 - 15
• Main Authors: Letellier, Christophe, Sendiña-Nadal, Irene, Bianco-Martinez, Ezequiel, Baptista, Murilo S.
• Format: Journal Article
• Language: English
• Published: London Nature Publishing Group UK 28-02-2018; Nature Publishing Group
• Subjects: 639/766/530/2801; 639/766/530/2803; Algorithms; DNA Replication; Dynamical systems; Humanities and Social Sciences; Humans; Models, Theoretical; multidisciplinary; Neural Networks, Computer; Nonlinear Dynamics; Nonlinear Sciences; Observation; Science; Science (multidisciplinary); Systems Theory
• ISSN: 2045-2322; 2045-2322
• DOI: 10.1038/s41598-018-21967-w

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.