  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.
