Observability and Controllability of Nonlinear Networks: The Role of Symmetry

Observing a system is a fundamental part of physics and control engineering. In control theory, observability and controllability are two important parameters; the former indicates how well scientists can reconstruct the full state of a system from incomplete measurements, and the latter measures how well the state of a system can be directed through control perturbations, that is, how much of the potential state space of the system can be reached through control. Observability and controllability were largely solved for linear systems by the 1970s. One of the implications from studies of linear systems is that symmetries compromise our ability to observe and control a system. In recent work to understand complex nonlinear networks, it has been assumed that linear theory should apply and that symmetries should be excluded.

We explore measures of observability and controllability in nonlinear networks with different topologies containing various degrees of explicit symmetries. We uncover a paradox in that symmetries do not always prevent observability and controllability. Using group representational theory, we are able to demonstrate that it is not symmetry per se that prevents observability and controllability of a nonlinear network but rather the type of group symmetry.

Our work changes a major underlying tenet of modern control theory and applies to all complex networks, from power grids to brains. Our findings open up a range of new possibilities using the symmetries of networks to further understand and control nonlinear networks.

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• Whalen AJ, Brennan SN, Sauer TD, Schiff SJ. Observability and Controllability of Nonlinear Networks: The Role of Symmetry. Physical Review X, 5, 011005.