Stabilizability and Norm-Optimal Control Design subject to Sparsity Constraints

Abstract: Consider that a linear time-invariant (LTI) plant is given and that we wish to design a stabilizing controller for it. Admissible controllers are LTI and must comply with a pre-selected sparsity pattern. The sparsity pattern is assumed to be quadratically invariant (QI) with respect to the plant, which, from prior results, guarantees that there is a convex parametrization of all admissible stabilizing controllers provided that an initial admissible stable stabilizing controller is provided. This paper addresses the previously unsolved problem of determining necessary and sufficient conditions for the existence of an admissible stabilizing controller. The main idea is to cast the existence of such a controller as the feasibility of an exact model-matching problem with stability restrictions, which can be tackled using existing methods. Furthermore, we show that, when it exists, the solution of the model-matching problem can be used to compute an admissible stabilizing controller. This method also leads to a convex parametrization that may be viewed as an extension of Youla's classical approach so as to incorporate sparsity constraints. Applications of this parametrization on the design of norm-optimal controllers via convex methods are also explored. An illustrative example is provided, and a special case is discussed for which the exact model matching problem has a unique and easily computable solution.