Optimal Input Placement in Lattice Graphs

TL;DR

This paper analyzes how the placement of control inputs affects the controllability of lattice networks, deriving explicit formulas and showing exponential scaling of control energy with node distance.

Contribution

It provides a closed-form expression for the controllability Gramian of infinite lattice systems and analyzes the impact of node placement on control energy.

Findings

Control energy scales exponentially with maximum distance between driver and target nodes.

Derived explicit formulas for the controllability Gramian in lattice systems.

Insights into optimal input placement for efficient control of networked systems.

Abstract

The control of dynamical, networked systems continues to receive much attention across the engineering and scientific research fields. Of particular interest is the proper way to determine which nodes of the network should receive external control inputs in order to effectively and efficiently control portions of the network. Published methods to accomplish this task either find a minimal set of driver nodes to guarantee controllability or a larger set of driver nodes which optimizes some control metric. Here, we investigate the control of lattice systems which provides analytical insight into the relationship between network structure and controllability. First we derive a closed form expression for the individual elements of the controllability Gramian of infinite lattice systems. Second, we focus on nearest neighbor lattices for which the distance between nodes appears in the expression for the controllability Gramian. We show that common control energy metrics scale exponentially with respect to the maximum distance between a driver node and a target node.