Somayeh Sojoudi, Salar Fattahi and Javad Lavaei, Convexification of Generalized Network Flow Problem, to appear in Mathematical Programming, pp. 1-39, 2017.
Abstract: This paper is concerned with the optimal static 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, we fist consider deterministic systems, where the initial state is either given or belongs to a known bounded region. Given an arbitrary centralized controller, we derive a condition under which there exists a distributed controller that generates input and state trajectories close to their counterparts in the centralized closedloop system. This condition for the design of a distributed controller is translated into an optimization problem, where the optimal objective value of this problem quantifies the closeness of the designed distributed and given centralized control systems. The results are then extended to stochastic systems that are subject to input disturbance and measurement noise. The proposed optimization problem has a closed-form solution (explicit formula) and can be efficiently solved for large-scale systems. The mathematical framework developed in this paper is utilized to design a near-globally optimal distributed controller based on the optimal centralized controller, and strong theoretical lower bounds on the global optimality guarantee of the obtained distributed controller are derived. 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. To illustrate the effectiveness of the proposed method, case studies on aircraft formation and frequency control of power systems are offered.