Transformation of Optimal Centralized Controllers Into Near-Globally Optimal Static Distributed Controllers
نویسندگان
چکیده
This paper is concerned with the optimal distributed control problem for linear discrete-time deterministic and stochastic systems. The objective is to design a stabilizing static distributed controller whose performance is close to that of the optimal centralized controller. To this end, a necessary and sufficient condition is derived to guarantee the existence of a distributed controller that generates the same input and state trajectories as the optimal centralized controller for a given deterministic system. 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 problem has a closedform solution (explicit formula) and can be efficiently solved for large-scale systems that require low computational efforts. Furthermore, strong theoretical lower bounds are derived on the optimality guarantee of the designed distributed controller in the case where the proposed conditions do not hold. We show that if the optimal objective value of the proposed convex program is sufficiently small, the designed controller is stabilizing and nearly globally optimal. The results are then extended to stochastic systems that are subject to input disturbance and measurement noise. By building upon the developed methodology, we partially address some long-standing 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. Numerical results on a power network and several random systems are reported to demonstrate the efficacy of the proposed method.
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