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Necessary and sufficient conditions for distributed constrained optimal consensus under bounded input
Author(s) -
Qiu Zhirong,
Liu Shuai,
Xie Lihua
Publication year - 2018
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.4040
Subject(s) - subgradient method , bounded function , mathematical optimization , upper and lower bounds , convergence (economics) , consensus , mathematics , multi agent system , computer science , constraint (computer aided design) , network topology , convex function , topology (electrical circuits) , regular polygon , mathematical analysis , artificial intelligence , geometry , combinatorics , economics , economic growth , operating system
Summary In this paper, we study a distributed constrained optimal consensus problem for discrete‐time first‐order integrator systems under bounded input. Each agent is assigned with a local convex cost function, and all agents are required to achieve consensus at the minimum of the aggregate cost over a common convex constraint set, which is only accessible by part of the agents. A 2‐step control protocol is designed to solve the problem under bounded input. Firstly, at each time step, each agent moves a bounded step of the subgradient descent from an individual cost and another one along the projection direction if it can access the constraint. The second movement is then given by a bounded average of the relative positions to neighbors. Specifically, to coordinate the subgradient step within the network without using global information, we introduce an estimate of the upper bound of all agents' subgradients, which is updated by a unilateral consensus mechanism. Under the given control protocol, we obtain the necessary and sufficient conditions to achieve the constrained optimal consensus for a fixed topology. Under similar conditions, we also solve the problem for switching topologies and conduct a convergence rate analysis for strongly convex costs.