Asymptotic stability and instability of large-scale systems

The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov´s theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations. With minor technical adjustments, the theory is broadened to include considerations of unstable subsystems as well as instability of composite systems.