Observability and Controllability of Nonlinear Networks: The Role of Symmetry

TL;DR

This paper investigates how symmetries in nonlinear networks influence their observability and controllability, revealing that certain symmetries can hinder control and observation, with implications for network design.

Contribution

It introduces a numerical and group-theoretic framework to analyze the impact of symmetries on nonlinear network controllability and observability, extending beyond linear models.

Findings

Symmetries can reduce network controllability and observability.

Networks with only rotational symmetries remain controllable and observable.

Symmetry considerations alter traditional views on network controllability.

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.