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

In this contribution, the properties of sub-stochastic matrices and super-stochastic matrices are applied to solve 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 cooperation and competition among the agents, respectively. With the help of nonnegative matrix theory together with some key results related to the compositions of directed edge sets, we establish a systematic algebraic-graphical method of dealing with the product convergence of infinite sub-stochastic matrices and infinite super-stochastic matrices. By using this method, the stability of MASs over signed networks under different actual scenarios is analyzed, including asynchronous interactions, lossy links, random networks, matrix disturbance, external noise disturbance, and a leader of unmeasurable velocity and acceleration. Finally, the efficiency of the proposed method is verified by computer simulations.