On the Active Nodes of Network Systems

This paper studies interconnected systems (nodes) and exploits dissipation inequalities and the structure of the interconnection (the network) to derive analysis and design tools for stabilization. First, systems with quadratic supply rates in the dissipation inequalities are investigated. A stability condition based on the dissipation inequality associated to each node is given. This condition allows checking the sign of the dissipation inequality for the overall network system. By considering the underlying directed graph, the feasibility of controller design is discussed. The design problem is re-formulated into the problem of finding a solution to a system of linear inequalities. This allows the efficient search and computation of the design parameters of what we call active nodes. Then, systems with non-quadratic supply rates are considered. A vector of positive definite functions that is used as a basis of the non-quadratic supply rates is constructed: this requires augmenting the underlying directed graph. Similarly to the quadratic supply rates case, a stability condition for analysis and a graph-based criterion for checking the feasibility of controller design are discussed. Finally a design example to demonstrate how to exploit the so-called active nodes to design a controller without numerical computation is presented. The proposed method does not presume any stability property of the nodes and therefore can be applied to various scenarios occurring in the study of stability properties for network systems.