Sparse Control Node Scheduling in Networked Systems Based on Approximate Controllability Metrics

This article investigates a novel sparsity-constrained controllability maximization problem for continuous-time linear systems. For controllability metrics, we employ the minimum eigenvalue and the determinant of the controllability Gramian. Unlike the previous problem setting based on the trace of the Gramian, these metrics are not the linear functions of decision variables and are difficult to deal with. To circumvent this issue, we adopt a parallelepiped approximation of the metrics based on their geometric properties. Since these modified optimization problems are highly nonconvex, we introduce a convex relaxation problem for its computational tractability. After a reformulation of the problem into an optimal control problem to which Pontryagin’s maximum principle is applicable, we give a sufficient condition under which the relaxed problem gives a solution of the main problem.