Efficient convex relaxation for stochastic optimal distributed control problem

This paper is concerned with the design of an efficient convex relaxation for the notorious problem of stochastic optimal distributed control (SODC). The objective is to find an optimal structured controller for a dynamical system subject to input disturbance and measurement noise. With no loss of generality, this paper focuses on the design of a static controller for a discrete-time system. First, it is shown that there is a semidefinite programming (SDP) relaxation for this problem with the property that its SDP matrix solution is guaranteed to have rank at most 3. This result is due to the extreme sparsity of the SODC problem. Since this SDP relaxation is computationally expensive, an efficient two-stage algorithm is proposed. A computationally-cheap SDP relaxation is solved in the first stage. The solution is then fed into a second SDP problem to recover a near-global controller with an enforced sparsity pattern. The proposed technique is always exact for the classical H₂ optimal control problem (i.e., in the centralized case). The efficacy of our technique is demonstrated on the IEEE 39-bus New England power network, a mass-spring system, and highly-unstable random systems, for which near-optimal stabilizing controllers with global optimality degrees above 90% are designed under a wide range of noise levels.