On Submodularity and Controllability in Complex Dynamical Networks

TL;DR

This paper investigates the structural properties of controllability and observability metrics in complex networks, revealing that they are often submodular or modular, enabling efficient optimization methods.

Contribution

It demonstrates that key controllability and observability metrics are submodular or modular, facilitating efficient optimization in large complex networks.

Findings

Metrics based on controllability and observability Gramians are submodular or modular.

Greedy algorithms can efficiently approximate optimal sensor and actuator placements.

Results are validated on power grid models and random systems.

Abstract

Controllability and observability have long been recognized as fundamental structural properties of dynamical systems, but have recently seen renewed interest in the context of large, complex networks of dynamical systems. A basic problem is sensor and actuator placement: choose a subset from a finite set of possible placements to optimize some real-valued controllability and observability metrics of the network. Surprisingly little is known about the structure of such combinatorial optimization problems. In this paper, we show that several important classes of metrics based on the controllability and observability Gramians have a strong structural property that allows for either efficient global optimization or an approximation guarantee by using a simple greedy heuristic for their maximization. In particular, the mapping from possible placements to several scalar functions of the associated Gramian is either a modular or submodular set function. The results are illustrated on randomly generated systems and on a problem of power electronic actuator placement in a model of the European power grid.