Distributed Actuator Selection: Achieving Optimality via a Primal-Dual Algorithm

Abstract

This letter addresses the actuator selection problem, i.e., given an interconnection of asymptotically stable linear dynamical systems on a network and m possible actuators choose ν among them to achieve a certain objective. In general, this is a combinatorial optimization problem which is hard to solve; convex relaxations do not usually yield an optimal solution for the original problem. In this letter we focus on a particular instance of the actuator selection problem, namely the formulation with the trace of the controllability Gramian matrix as the optimization metric, and show that such a choice gives rise to an integer linear program (LP). Using properties of integral polyhedra, we show through a sequence of reformulations that the optimal solution of this problem can be determined by means of an LP without introducing any relaxation gap. This allows us to obtain the optimal solution using a primal-dual distributed algorithm, thus providing a scalable approach to the problem of actuator placement which has been up to now performed in a centralized manner enumerating all possible placement alternatives. We illustrate the main features of our approach by means of a case study involving a simplified model of the European power grid.