Convex Reformulation of LMI-Based Distributed Controller Design with a Class of Non-Block-Diagonal Lyapunov Functions

This study addresses a distributed state feedback controller design problem for continuous-time linear time-invariant systems by means of linear matrix inequalities (LMIs). As structural constraints on a control gain result in non-convexity in general, the block-diagonal relaxation of Lyapunov functions has been prevalent despite its conservatism. In this work, we target a class of non-block-diagonal Lyapunov functions with the same sparsity pattern as distributed controllers. By leveraging a block-diagonal factorization of sparse matrices and Finsler's lemma, we first present a nonlinear matrix inequality for stabilizing distributed controllers with such Lyapunov functions, which boils down to a necessary and sufficient condition for such controllers if the sparsity pattern is chordal. As its relaxation, we derive novel LMIs, one of which strictly covers the conventional relaxation, and then provide analogous results for $H_\infty$ control. Lastly, numerical examples underscore the efficacy of our results.