Optimal state feedback controllers with strict row sparsity constraints

Abstract

This paper considers the problem of optimal row sparse state feedback controller design for LTI systems, where the controller is assumed to be static with pre-specified structural constraint. Incongruous to the existing literature on the sparsity promoting control synthesis, we do not employ convex relaxation of the sparsity representing terms, such as l0-norm of the controller gain, in our proposed framework. Borrowing the results from the theory of majorization, we develop an exact rank constrained reformulation of the s-sparse vector recovery from a convex set, and, then, utilized it to cast our row sparse control problem into a an optimization problem where all constraints are convex, except a single rank constraint. Furthermore, we propose a necessary and sufficient condition for the feasibility of a stabilizing row s-sparse controller, and exploited it to propose a bi-linear minimization problem, subject to convex constraints, which solve the derived equivalent rank constrained problem to deliver an optimal row sparse state feedback controller. The benefits of approach are demonstrated though several numerical simulations.