Robust controllability assessment and optimal actuator placement in dynamic networks

Abstract

This paper presents a new framework for the optimal placement of actuators in uncertain dynamic networks. The objective is to select a subset of nodes from a set of potential actuator placements, such that the controllability of the network with norm-bounded perturbation is preserved over the entire uncertain region. Evaluation of robust controllability is known to be computationally intractable. The recent results of Babazadeh and Nobakhti (2016) are utilized to establish equivalent conditions for robust controllability of uncertain networks. It is shown that for a large class of dynamic systems, including undirected networks, the exact distance to uncontrollability is evaluated by solving a convex program. Properties of positive-definite polynomial matrices and duality theory facilitate formation of a non-convex optimization problem with linear matrix inequalities whose solution is the exact distance to uncontrollability for a general uncertain network. The underlying optimization averts the difficulty of gridding over the complex plane. A customized optimization algorithm is outlined to solve the equivalent non-convex problem whose solution is ensured to be a stationary point of the original problem. The method is further utilized to address the optimal placement of actuators by maximizing the network robust controllability index and simultaneously regularizing the number of actuators in the network graph. The proposed framework is implemented efficiently using convex optimization solvers and offers a fast and reliable tool for the assessment of robust controllability and optimal selection of control nodes. Analogous results can be established for robust observability and optimal sensor placement.