A symbolic network-based nonlinear theory for dynamical systems observability

International audienceWhen 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. Variables spanning the state space of a dynamical system which is irreducible to a few smaller subsystems are always dependent on each other through linear and nonlinear interactions. Consequently, one may expect to be able to determine an adequate subset of variables together with their well-selected Lie derivatives to get a full observability of the underlying dynamics, that is, for distinguishing all possible states of the network 1,2. With the emergence of the Science of Complexity, complex networks are more and more often considered in various fields as well exemplified by power grids 3 , socio-economics networks 4-6 , or biological systems 7-10. To allow a reliable monitoring, dynamical analysis or control of these high-dimensional systems, suitable and systematic techniques are required to identify the subset of variables providing the best (if not the full) observability of their underlying dynamics. A related problem is how to unfold the whole dynamics by completing this subset of variables to reconstruct a space whose dimension is at least equal to the dimension of the original state space. Dealing with multivariate time series, specially those produced by high-dimensional dynamical networks, is not a trivial problem 11-13. Attempts to estimate network observability using symbolic techniques 14,15 were made to overcome the large computational times associated with the exact analytical calculations. In those approaches, a dimension reduction is performed in real time on a symbolic observability matrix until state estimation is possible from the selected measurements. However, linear and nonlinear interactions among variables are considered on an equal footing while it is strongly required to distinguish them for a reliable assessment of the observability of a system 2,13. In order to tackle such a challenging task, we propose a methodological approach that will be applied to reaction networks derived from dynamical systems with appropriately large dimension to corroborate our assessments with rigorous analytical calculations, and yet provide a framework making also possible the verification of observability in networked dynamical systems. The chosen reaction networks are models of interesting biological and physical systems: the circadian oscillation in the Drosophila period protein, the Rayleigh-Bénard convection, and the DNA replication. They also represent nonlinear systems with increasing nonlinear complexity , commonly observed in other natural and man-made systems. Therefore, they are an appropriate subset o