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.
Highlights
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
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 possible the verification of observability in networked dynamical systems
Since full observability warrants that every distinct point of the original state space x ∈ d can be univocally identified, there is a great interest to target the minimal set of variables to measure for accomplishing such a full observability condition
Summary
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. 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. 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 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 represent nonlinear systems with increasing nonlinear complexity, commonly observed in other natural and man-made systems. Our results are in full agreement with the analytical prediction of getting a no null determinant of the observability matrix and the technique drastically reduces the search for candidate variables, providing a key step to observe and model natural and man-made complex systems of large dimension
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