Lyapunov stability of interconnected systems: Decomposition into strongly connected subsystems

Abstract

New Lyapnnov results for uniform asymptotic stability, exponential stability, Instability, complete instability, and uniform ultimate boundedness of solutions for a class of interconnected systems described by nonlinear time-varying ordinary differential equations are established. The present results make use of graph theoretic decomposition techniques to transform large-scale systems into an interconnection of strongly connected components (subsystems), and they also make use of the properties of stability preserving mappings. The analysis of the overall interconnected system is then accomplished in terms of the qualitative properties of the subsystems (strongly connected components) and the stability preserving properties of the system interconnections. A typical result is of the following form. If every subsystem (strongly connected component) is uniformly asymptotically stable and if all system interconnections are stability preserving, then the overall interconnected system is uniformly asymptotically stable. To demonstrate the applicability of these results to physical systems, a specific example is considered. The principal advantages of the present method over existing Lyapunov results for interconnected systems are as follows. 1) The present results make it possible to address the "largeness" of complex systems, since the identification of the strongly connected components (subsystems) can be accomplished by means of efficient computer algorithms. 2) The present stability results are applicable to interconnected systems with unstable subsystems (i.e., the original nontransformed system description may involve unstable subsystems). The main disadvantages of the present method over many existing Lyapunov results are the following. 1) The analysis is not accomplished in terms of the original system structure. 2) When a system consists of only one strongly connected component, the present method cannot be used to advantage, while other methods may.