A convex parameterization of all stabilizing controllers for non-strongly stabilizable plants under quadratically invariant sparsity constraints

This paper addresses the design of controllers, subject to sparsity constraints, for linear and time-invariant plants. Prior results have shown that a class of stabilizing controllers, satisfying a given sparsity constraint, admits a convex representation of the Youla-type, provided that the sparsity constraints imposed on the controller satisfy a certain condition (named quadratic invariance) with respect to the plant and that the plant is strongly stabilizable. Another important aspect of the aforementioned results is that the sparsity constraints on the controller can be recast as convex constraints on the Youla parameter, which makes this approach suitable for optimization using norm-based costs. In this paper, we extend these previous results to the general case of possibly non-strongly stabilizable plants. Our extension is conveyed in terms of a parametrization for the class of controllers that is very similar to the Youla parametrization. In our extension, under quadratically invariant constraints, the controller class also admits a representation where the free parameter is subject to only convex constraints. While the strong stabilizability assumption has been removed our result yields the same elegant simplicity from the strongly stabilizable case.