Distributed Bipartite State Consensus Tracking for Discrete-Time Singular Multi-agent Systems

This paper studies the distributed bipartite state consensus tracking problem of discrete-time linear leader-following singular multi-agent systems (MASs). Compared with the traditional continuous-time linear dynamics, it is difficult to achieve the desired cooperative control of discrete-time counterpart since their stability regions are limited to the unit circle. In addition, in some practical scenarios, there are both cooperation and competition coexisting among followers. First, a new distributed state feedback control protocol is proposed based on the Gerschgorin circle theorem and the discrete algebraic Riccati equation. At the same time, by introducing the constant coupling gain associated with the system topology matrix, the global tracking error system can locate in the stable region of the unit circle. Under the structurally balanced condition of directed signed graph, it can be proved that all the agents of two opposite subgroups can achieve bipartite state tracking by Lyapunov stability theory and separation principle. Then, by using the cooperative–competitive interaction information of neighbours, a new distributed state observer is introduced to estimate the state of leader. Furthermore, when the states of followers are not known, an observer-based output feedback bipartite control protocol is proposed, which can also be applied to more general consensus control scenarios. Meanwhile, a bipartite state formation controller is given to achieve the formation control by making the followers to keep the desired relative positions with the leader. Finally, two simulation results are given to verify the feasibility and effectiveness of proposed algorithms.