Transformation of optimal centralized controllers into near-global static distributed controllers

This paper is concerned with the optimal decentralized control problem for linear discrete-time deterministic and stochastic systems. The objective is to design a stabilizing static distributed controller with a given structure, whose performance is close to that of the optimal centralized controller. To this end, we derive a necessary and sufficient condition under which there exists a distributed controller that generates the same input and state trajectories as the optimal centralized one. This condition is then translated into a convex optimization problem. Subsequently, a regularization term is incorporated into the objective of the proposed optimization problem to indirectly account for the stability of the distributed control system. The designed optimization has a closed-form solution (explicit formula), which depends on the optimal centralized controller as well as the prescribed controller structure. If the optimal objective value of the proposed optimization problem is small enough at the explicit solution, the resulting controller is stabilizing and has a high performance. The derived formula may help partially answer some open problems, such as finding the minimum number of free elements required in the distributed controller under design to achieve a performance close to the optimal centralized one. The proposed approach is tested on a power network and several random systems.