On structure and stability of interconnected dynamical systems

In some recent works, graph-theoretic decomposition techniques were used in the stability analysis of large-scale interconnected dynamical systems. These results are applicable to systems with separable interconnecting structure. (By this term, we mean systems whose interconnecting structure enters additively into the system description and each term in the interconnecting structure connects at most two subsystems.) In this paper, we present results which are less conservative than the Lyapunov stability results developed previously and which are applicable to systems with nonseparable interconnecting structure. The graph-theoretic methods developed herein can be used to extend existing input-output stability results to be applicable to systems with nonseparable interconnecting structure as well. In arriving at the present results, generalizations of the usual concepts of a graph and a digraph (called a
G
-graph and an
M
-digraph, respectively) are introduced, and some of their properties are established. These graphtheoretic results, which we feel are of interest in their own right, make possible the systematic decomposition of large-scale systems into several useful and important equivalent forms. Indeed, these decomposition results constitute generalizations of earlier ones which use the usual notion of a digraph. To clarify concepts introduced and to demonstrate the usefulness of the results developed, several specific examples are included in the paper.