Convex relaxation for optimal distributed control problem

This paper is concerned with the optimal distributed control (ODC) problem. The objective is to design a fixed-order distributed controller with a pre-specified structure for a discrete-time system. It is shown that this NP-hard problem has a quadratic formulation, which can be relaxed to a semidefinite program (SDP). If the SDP relaxation has a rank-1 solution, a globally optimal distributed controller can be recovered from this solution. By utilizing the notion of treewidth, it is proved that the nonlinearity of the ODC problem appears in such a sparse way that its SDP relaxation has a matrix solution with rank at most 3. A near-optimal controller together with a bound on its optimality degree may be obtained by approximating the low-rank SDP solution with a rank-1 matrix. This convexification technique can be applied to both time-domain and Lyapunov-domain formulations of the ODC problem. The efficacy of this method is demonstrated in numerical examples.