Cooperative regulation based on virtual vector triangles asymptotically compressed in multidimensional space for time-varying nonlinear multi-agent systems

This study constructs virtual vector triangles in multidimensional space to address cooperative control issue in time-varying nonlinear multi-agent systems. The distributed adaptive virtual point and its dynamic equations are designed, with this virtual point, the leader, and the follower being respectively defined as the vertices of the virtual vector triangle. The virtual vector edges, decomposed by vectors into coordinate axis components, are organized to form a closed virtual vector triangle by connecting the three vertices with directed vector arrows that are oriented from the tail to the head. Specifically, these virtual vector edges are fictitious vector line segments connecting two vertices and used to compute the relative Euclidean distances between each vertex in multidimensional space. Based on the established virtual vector triangles, which are placed in multidimensional space, and the novel spatial coordinate transformation method, the cooperative regulation problem of the time-varying nonlinear multi-agent system is transformed into a mathematical problem of compressing the virtual vector triangles with exponential magnitude. The created distributed compression control protocol asymptotically shrinks the magnitude of the virtual vector triangles by exponential oscillatory decay towards the same dynamic point aligned with the motion trajectory of the leader or the leader, where the states of the time-varying nonlinear multi-agent systems achieve asymptotic convergence consensus. The reliable stability of the asymptotic compression convergence process of the virtual vector triangles was verified by establishing a Lyapunov function and relying on the Lyapunov stability theory. Finally, the example of time-varying nonlinear multi-agent systems are presented for simulation experiments to further validate the effectiveness and feasibility of the proposed control protocol in addressing the cooperative regulation issue.