Distributed <i>k</i>-Winner-Take-All Network Under Weight-Unbalanced Topology

Abstract

This brief investigates a distributed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -winner-take-all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(k$ </tex-math></inline-formula> WTA) network that effectively eliminates the lagging error problem. On this basis, a special case that the communication topology of robots is weight-unbalanced in the distributed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> WTA problem is considered for the first time. Then a distributed weight-unbalanced <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> WTA (DWU- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> WTA) model is proposed with the aid of a consensus estimator. Theoretical analyses substantiate the usability and exponential convergence of the proposed DWU- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> WTA model in encountering directed weight-unbalanced graphs. A simulation based on a multi-robot system further verifies the feasibility of the proposed model in dealing with dynamic task allocation problems.