Synchronization of unknown nonlinear networked systems via cooperative adaptive control

Traditional research on cooperative networked systems has focused on analysis of given protocols for simple linear systems, mainly the integrator or double integrator dynamics. Pinning control has been developed for nonlinear systems, generally using either a Jacobian linearization method or a Lipschitz assumption. Work has also progressed on nonlinear coupling protocols. In this paper we present a design method for adaptive controllers for distributed systems having non-identical unknown nonlinear dynamics, and for a target dynamics to be tracked that is also nonlinear and unknown. The development is for general digraph communication structures. A Lyapunov technique is presented for designing a robust synchronization control protocol. The proper selection of the Lyapunov function is the key to ensuring that the resulting control laws thus found are implementable in a distributed fashion. Lyapunov functions are defined in terms of a local neighborhood tracking synchronization error and the Frobenius norm. The resulting protocol consists of a linear protocol and a nonlinear control term with adaptive update law at each node. Connections are made between the convergence rate/convergence residual set and graph structural properties such as the sum of the in-degrees and out-degrees, and the difference between the in-degrees and out-degrees. Singular value analysis is used. It is shown that the singular values of certain key matrices are intimately related to structural properties of the graph.