Stability analysis of interconnected dynamical systems: Hybrid systems involving operators and difference equations

We address the stability analysis of interconnected feedback systems of the type depicted in Fig. 1, which consists of a linear interconnection of subsystems. Each subsystem is a feedback system in its own right, consisting of a local plant (which is described by an operator ) and of a digital controller (which is described by a system of difference equations and which includes A/D and D/A converters). We establish conditions for the attractivity, asymptotic stability, asymptotic stability in the large, and boundedness of solutions for such systems. The hypotheses of our results are phrased in terms of the I/O properties of the operators and of the entire interconnected system, and in terms of the Lyapunov stability properties of digital controllers described by the indicated difference equations. In all cases, our results allow a stability analysis of complex interconnected systems in terms of the qualitative properties of the simpler free subsystems and in terms of the properties of the system interconnecting structure. The applicability of our results is demonstrated by means of a specific example (Fig. 2).