A Convex Approximation of Optimal Distributed Controller in Frequency Domain

We study the optimal distributed control (ODC) problem and propose a design method based on the approximation of the H2 performance using the optimal centralized controller. The designed distributed controller obeys a prescribed sparsity pattern in the frequency domain. The proposed method enables a convex approximation of the distributed controller problem, which has a closed-form solution with sub-optimality guarantees. After introducing the notion of rank-preserving weights, we give sufficient conditions for the controller to be proper, and furthermore characterize the set of controllers that can be obtained by the proposed technique via fine-tuning the algorithm parameters. The results are applied to the linear-quadratic regulator problem and to the purely decentralized case with a diagonal sparsity pattern, where certain connections to the simple methods of thresholding and averaging are discovered. Numerical examples are provided to demonstrate the effectiveness of the developed method.