Convex relaxations and Gramian rank constraints for sensor and actuator selection in networks

Abstract

We propose a convex relaxation heuristic for sensor and actuator selection problems in dynamical networks using Gramian metrics. We also propose heuristic algorithms to enforce a rank constraint on the Gramian that can be used in conjunction with combinatorial greedy algorithms and the convex relaxation. This allows selection of sensor or actuator sets that optimize an objective function while preserving a certain amount of observability or controllability throughout the state space, combining previous methods that focus exclusively on either rank or Gramian metrics. We illustrate and compare the greedy and convex relaxation heuristics in several numerical examples involving random and regular networks.