Sub/super-stochastic matrix with applications to bipartite tracking control over signed networks

TL;DR

This paper explores the use of sub- and super-stochastic matrices to analyze bipartite tracking in multi-agent systems over signed networks, addressing various practical scenarios with theoretical and simulation validation.

Contribution

It introduces algebraic-graphical methods for analyzing product convergence of infinite sub- and super-stochastic matrices in complex multi-agent network scenarios.

Findings

Established systematic methods for convergence analysis of ISubSM and ISupSM.

Analyzed bipartite tracking under delays, switching topologies, and disturbances.

Validated methods through computer simulations.

Abstract

In this contribution, the properties of sub-stochastic matrix and super-stochastic matrix are applied to analyze the bipartite tracking issues of multi-agent systems (MASs) over signed networks, in which the edges with positive weight and negative weight are used to describe the cooperation and competition among the agents, respectively. For the sake of integrity of the study, the overall content is divided into two parts. In the first part, we examine the dynamics of bipartite tracking for first-order MASs, second-order MASs and general linear MASs in the presence of asynchronous interactions, respectively. Asynchronous interactions mean that each agent only interacts with its neighbors at the instants when it wants to update the state rather than keeping compulsory consistent with other agents. In the second part, we investigate the problems of bipartite tracing in different practical scenarios, such as time delays, switching topologies, random networks, lossy links, matrix disturbance, external noise disturbance, and a leader of unmeasurable velocity and acceleration. The bipartite tracking problems of MASs under these different scenario settings can be equivalently converted into the product convergence problems of infinite sub-stochastic matrices (ISubSM) or infinite super-stochastic matrices (ISupSM). With the help of nonnegative matrix theory together with some key results related to the compositions of directed edge sets, we establish systematic algebraic-graphical methods of dealing with the product convergence of ISubSM and ISupSM. Finally, the efficiency of the proposed methods is verified by computer simulations.