Active Nodes of Network Systems With Sum-Type Dissipation Inequalities

This article studies a small gain-like analysis and control synthesis tool for large-scale interconnected dynamical systems (networks). By exploiting the structure of the interconnection one can derive (or enforce it via control synthesis) an algebraic condition (the small gain-like condition) to allow the construction of a network storage function with a network dissipation inequality in a desired form (the small gain-like property), which can be further used for establishing convergence or stability properties. Small gain-like conditions, for systems with quadratic, general-form nonlinear, and parametrized supply rates, respectively, are derived and interpreted using the underlying graph. This allows enforcing such a condition via the design of a class of controlled nodes, called active nodes, provided their locations satisfy a graph-based condition. The article then proceeds to discuss control synthesis methods using the notion of active nodes, including the placement, the parameter computation, and the adaptation of the active nodes. Finally, an example of public-health-related control for interconnected settlements, to demonstrate the implementation of the active-node-based scheme, is presented.