Abstract
Observability and controllability are essential concepts to the design of predictive observer models and feedback controllers of networked systems. For example, noncontrollable mathematical models of real systems have subspaces that influence model behavior, but cannot be controlled by an input. Such subspaces can be difficult to determine in complex nonlinear networks. Since almost all of the present theory was developed for linear networks without symmetries, here we present a numerical and group representational framework, to quantify the observability and controllability of nonlinear networks with explicit symmetries that shows the connection between symmetries and nonlinear measures of observability and controllability. We numerically observe and theoretically predict that not all symmetries have the same effect on network observation and control. Our analysis shows that the presence of symmetry in a network may decrease observability and controllability, although networks containing only rotational symmetries remain controllable and observable. These results alter our view of the nature of observability and controllability in complex networks, change our understanding of structural controllability, and affect the design of mathematical models to observe and control such networks.
Highlights
An observer model of a natural system has many useful applications in science and engineering, including understanding and predicting weather or controlling dynamics from robotics to neuronal systems [1]
The presence of symmetries in the system’s differential equations makes observation difficult from variables around which the invariance of the symmetry is manifested [31,32]. We extend this analysis to networks of ordinary differential equations and investigate the effects of symmetries on observability and controllability of such networks as a function of connection topology, measurement function, and connection strength
In motifs 1 and 3 the effect of the symmetry is partially broken by introducing a variation in the coupling terms, and the results show nonzero observability indices in the plots for such heterogeneous coupling [plots (a) and (b) in Figs. 2 and 3] with a dependence on the coupling strength
Summary
An observer model of a natural system has many useful applications in science and engineering, including understanding and predicting weather or controlling dynamics from robotics to neuronal systems [1]. [11], the requirements of structural observability incorporated explicit use of transitive components of directed graphs—fully connected subgraphs where paths lead from any node to any other node—to identify the minimal number of sites required to observe from a network. All of these prior works depend critically on the dynamics being linear and generic, in the sense that network connections are essentially random. Our present work is motivated by the following question: What role do the symmetries and network coupling strengths play when reconstructing or controlling network dynamics? Our findings apply to any complex network, including power grids, the internet, genomic and metabolic networks, food webs, electronic circuits, social organization, and brains [8,11,18,21]
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