The stabilization of digraphs of variable parameter systems

Directed acyclic graphs of dynamical systems are considered. Such graphs possess a natural and unique layering of their nodes. Each node is taken to be associated with a dynamical system in such a way that each parameter of a given system is some nonlinear function of the states of systems connected to it and lying in higher layers of the network. Such an arrangement we call a digraph of variable parameter systems. It is shown that digraphs of a large class of variable parameter systems may be stabilized by associating a compensator (of observer-linear feedback type) with each variable parameter system. Each compensator need only observe the output and input of the variable parameter system with which it is associated; however, an argument is presented to show that the performance of the controlled digraph is enhanced if the compensators signal to each other in a manner consistent with the flow of the parametric disturbances through the network. In this sense, it is shown that a network of variable parameter systems is efficiently stabilized by a multilevel, or hierarchical, control system.