The Collective Behavior Of Multi-Agent Cooperative Dynamic Systems

This article study the multi-agent dynamic system under time-invariant directed network. The main theoretical results of previous research will be systematically re-viewed, and will be advanced in the followings:(1) The issue of the status consistency under some specific network will be ad-vanced to the issue of collective behavior under any directed network.(2) The state of agents (individuals) will be substantially defined in the m-dimensional space(m=1,2,3).(3) The linear dynamic model will be advanced to affine dynamic model.In addition,(4) Some properties of simple directed graph and graph properties of matrix will be targeted collected, to make it an effective tool for structural analysis of the multi-agent dynamic system.(5) A background of elementary mathematics for the multi-agent Dynamic, el-ementary diagonally dominant matrix theory which based on graph theory and linear algebra, will be provided.The structure of a multi-agent system is determined by action relations (behav-ior relations, communication relations) between the agents in the system. The action relations of a system may be described by a directed graph G which be called action network(behavior network, communication network) of the system. A system may be decomposed into some basic subsystems by the structure of its action network. Each basic subsystem correspond with a strong component of G(a maximum strongly con-nected subgraph of G).Some of basic subsystems are independent from the others and some are non-independent.In this study, we consider the initial value problem of multi-agent cooperative dynamic system which is a basic type of multi-agent dynamic systems.If the agents are with dynamical order one, then ones have the followings:(1) Each basic independent subsystems will reach their own consensus state;(2) The agents in the non-independent subsystems will asymptotically go inside of a convex set which is formed by the consensus states of basic independent subsystems.(3) The states of all agents of system reach consensus if and only if there is only one independent subsystem in the system.The second order linear dynamic system also have the same collective behavior be described above as it achieves the inertia states.