Optimal Decentralized Control Problem as A Rank-Constrained Optimization

This paper is concerned with the long-standing optimal decentralized control (ODC) problem. The objective is to design a fixed-order decentralized controller for a discretetime system to minimize a given finite-time cost function subject to norm constraints on the input and output of the system. We cast this NP-hard problem as a quadratically-constrained quadratic program, and then reformulate it as a rank-constrained optimization. The reformulated problem is a semidefinite program (SDP) after removing its rank-1 constraint. Whenever the SDP relaxation has a rank-1 solution, a globally optimal decentralized controller can be recovered from this solution. This paper studies the rank of the minimum-rank solution of the SDP relaxation since this number may provide rich information about the level of the approximation needed to make the ODC problem tractable. Using our recently developed notion of "nonlinear optimization over graph", we propose a methodology to compute the rank of the minimum-rank solution of the SDP relaxation. In particular, we show that in the case where the unknown decentralized controller being sought needs to be static with a diagonal matrix gain, this rank is upper bounded by 4. Since the upper bound is close to 1 and does not depend on the order of the system, the ODC problem may not be as hard as it is thought to be. This paper also proposes a penalized SDP relaxation to heuristically enforce the few unwanted nonzero eigenvalues of the solution to diminish.