Robust Passivity and Control for Directed and Multiweighted Coupled Dynamical Networks

In this article, the passivity and control issues for directed uncertain coupled dynamical networks are solved. The presented model is directly coupled with multiple coupling matrices and parametric uncertainty, while previous literatures of multiweighted networks usually suppose that outer coupling matrices (OMs) are connected, undirected, and certain. The viewpoint of inner coupling matrices (IMs) in this article is added and OMs can be directed and not connected, which is a great improvement on the existing results. First, for all diagonal IMs, considering each dimension separately, we can derive if the weighted combination of multiple OMs for each dimension is strongly connected, then passivity and pinning control rules can be established. In addition, we also discuss the situation that IMs are positive definite but not diagonal. By means of the weighted combination of normalized left eigenvectors (NLEVec) corresponding to zero eigenvalue for multiple coupling matrices, we prove if the Chebyshev distance (Cheb-Dist) among these NLEVec is less than a tolerant deviation interval, then passivity, synchronization, and pinning control criteria are acquired. Moreover, a matter of adaptive coupling strengths is also settled. Examples are provided to verify the validity of established results.