Mechanisms for the stability and instability in interconnected systems

The purpose of this paper is to provide a brief informal exposition of the main ideas underlying the stability and instability criteria which have appeared in the literature in the context of feedback systems, and to show how these ideas generalize in such a way that they become applicable to much more general classes of interconnected systems. We hope to transmit at the same time an intuitive explanation of these results, an indication of the limitations of these methods, and a demonstration of their generality. Since these methods conclude stability or instability of interconnected systems by considering the individual subsystems and the interconnection laws separately, they are very well suited for the analysis of large scale systems. Another appealing aspect of these results is the fact that they make contact with various other areas of system theory as, for example, least squares optimal control, Lyapunov stability theory, and passive network synthesis. For the sake of exposition and notation, we will limit our attention to stationary continuous time systems. The ideas are easily generalized to time-varying systems and we will in fact give a time-varying example in which these results are used.