Convex Relaxation for Optimal Distributed Control Problems

This paper is concerned with the optimal distributed control (ODC) problem for linear discrete-time deterministic and stochastic systems. The objective is to design a static distributed controller with a prespecified structure that is globally optimal with respect to a quadratic cost functional. 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 an SDP relaxation of this problem has a matrix solution with rank at most 3. Since the proposed SDP relaxation is computationally expensive for a large-scale system, a computationally cheap SDP relaxation is also developed with the property that its objective function indirectly penalizes the rank of the SDP solution. Various techniques are proposed to approximate a low-rank SDP solution with a rank-1 matrix, leading to near globally optimal controllers together with a bound on the optimality degree of each controller. The above results are developed for both finite-horizon and infinite-horizon ODC problems. The SDP relaxations developed in this work are exact for the design of a centralized controller, hence serving as an alternative for solving Riccati equations. The efficacy of the proposed SDP relaxations is elucidated through a case study on the distributed frequency control of power systems.