A comparison theorem for cooperative control of nonlinear systems

Asymptotic cooperative stability is studied in the paper, and explicit conditions are found for heterogeneous nonlinear systems to reach a consensus. Specifically, a new comparison theorem is proposed for concluding both cooperative stability and Lyapunov stability, and it is in terms of vector nonlinear differential inequalities (on Lyapunov function components). It is unique that the proposed result admits both heterogeneous dynamics of nonlinear systems and intermittent unpredictable changes in their associated sensing/communication network. Its proof is done using a combination of Lyapunov argument (in terms of the Lyapunov function components) and topology- dependent argument (in terms of structural properties of reducible matrices). Consequently, the proposed result does not impose any of the following assumptions required in the existing results: the knowledge of a successful Lyapunov function, system dynamics being convex, nonsmooth analysis, fixed or certain types of communication patterns, quasi- monotone property on differential inequalities. If the systems under consideration are all linear, the theorem reduces to the necessary and sufficient condition of cooperative controllability obtained using the matrix-theoretical approach, and the inequalities become equalities. For nonlinear systems, the proposed cooperative stability conditions are straightforward to verify. Several types of nonlinear systems are used as examples to illustrate application potentials of the comparison theorem in both cooperative stability analysis and cooperative control design.