Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability

In this paper, we study decomposition techniques for nonlinear large-scale systems, which have the feature that the interactions between the various subsystems are nonadditive. Using the technique of decomposing a graph into its strongly connected components, we first rewrite the system differential equations into a hierarchical form, by renumbering and aggregating the original state variables, if necessary. In this hierarchical form, each subsystem interacts only with "lower" subsystems but not with "higher" subsystems. Once the system equations have been rearranged in this hierarchical form, we show that the overall system is uniformly asymptotically stable (respectively exponentially stable, globally exponentially stable) if and only if each of the subsystems is uniformly asymtotically stable (respectively exponentially stable, globally exponentially stable). The main technique used to do this is the converse Lyapunov theory. We then turn to problems of stabilizability, and show that, once the system equations have been arranged in hierarchical form, the overall system can be stabilized by a decentralized control law if and only if each of the subsystems can be stabilized. Several examples are presented to illustrate the various theorems.