A chordal decomposition approach to scalable design of structured feedback gains over directed graphs

This paper considers the problem of designing static feedback gains subject to a priori structural constraints, which is in general a non-convex problem. By exploiting the sparsity properties of the problem, and using chordal decomposition, a scalable algorithm is proposed to compute structured stabilizing feedback gains for large-scale systems over directed graphs. Specifically, we first present a chordal decomposition theorem for block-semidefinite matrices. A relaxation is then used to recast the design of structured feedback gains into a convex problem. Combining the decomposition with the relaxation, we propose a sequential design algorithm to obtain structured feedback gains clique-by-clique over a clique tree of the underlying chordal graph. Numerical simulations demonstrate the efficiency of the proposed method.