Fully Distributed Algorithms for Constrained Nonsmooth Optimization Problems of General Linear Multiagent Systems and Their Application

This article investigates the constrained nonsmooth distributed optimization problems (DOPs) of general linear multiagent systems. Our problem involves the general linear dynamics of agents, and the cost functions are nondifferentiable. Moreover, the decisions of agents are constrained by both local and coupled inequalities. Without considering the general linear dynamics, the nonsmooth cost functions, and/or the inequality constraints, existing distributed algorithms are ineffective for our problem. Besides, due to the general linear dynamics, the decisions of agents cannot be determined directly by their control inputs, while the optimal solution must satisfy the constraints. Therefore, it is not easy to design algorithms for our problem. In addition, noteworthily, the global Lipschitz continuity of gradients is an indispensable condition for existing results of DOPs with physical systems. However, the subgradients of nonsmooth cost functions are not Lipschitz continuous, which consequently puts up barriers to algorithm analysis. Based on state/output feedback, subgradient descents, and primal-dual methods, we develop two distributed algorithms. Compared with existing algorithms for DOPs with physical systems, our algorithms are fully distributed. By set-valued Lasalle invariance principle and convex analysis, we analyze the two algorithms rigorously. The first algorithm can solve strictly convex DOPs, in contrast to most results of DOPs with physical systems. The second algorithm only depends on the output information, different from many related algorithms. Finally, the proposed methods are applied to smart grids. With the methods, doubly-fed induction generators can achieve the optimal economic dispatch autonomously.