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.
- Received 2 October 2014
DOI:https://doi.org/10.1103/PhysRevX.5.011005
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Published by the American Physical Society
Popular Summary
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.
