The paper addresses the problem of exact average-consensus reaching in a prescribed time. The communication topology is assumed to be defined by a weighted undirected graph and the agents are represented by integrators. A nonlinear control protocol, which ensures a finite-time convergence, is proposed. With the designed protocol, any prescribed convergence time can be guaranteed regardless of the initial conditions.
Abstract. We propose a randomized method for general convex optimization problems; namely, the minimization of a linear function over a convex body. The idea is to generate N random points inside the body, choose the best one and cut the part of the body defined by the linear constraint. We first analyze the convergence properties of the algorithm from a theoretical viewpoint, i.e., under a rather classical assumption that an algorithm for uniform generation of random points in the convex body is available. Under this assumption, the expected rate of convergence for such method is proved to be geometric. Moreover, explicit sample size results on convergence are derived. In particular, we compute the minimum number of random points that should be generated at each step in order to guarantee that, in a probabilistic sense, the method converges at least as fast as the deterministic center-of-gravity algorithm. From a practical viewpoint, the method can be implemented using Hit-and-Run versions of Markov-chain Monte Carlo algorithms, and we show how these convergence results can be extended to a Hit-and-Run implementation. A crucial notion for the Hit-and-Run implementation is that of Boundary Oracle, which is available for most optimization problems including LMIs and many other kinds of constraints. Preliminary numerical results for SDP problems are presented confirming that the randomized approach might be competitive to modern deterministic convex optimization methods.
A number of challenging problems in linear control theory are considered which admit simple formulation and yet lack efficient solution methods. These problems relate to the classical theory of linear systems as well as to the robust theory where the system description contains uncertainty. Various solution methods are discussed and the results of numerical simulations are given.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.