Input-output stability theory of interconnected systems using decomposition techniques

We study the input-output stability of arbitrary interconnections of nonlinear, time-varying, multivariable subsystems which may be either continuous-time or discrete-time. All subsystems are specified by their input-output description. We use an algorithmic decomposition of the overall system into a hierarchy of strongly connected subsystems (SCS) interacting through inter-connection subsystems (IS). Theorem I establishes that the overall system is stable once the SCS\´s and the IS\´s are stable. Theorem II shows that, under very reasonable assumptions, these sufficient conditions are actually necessary. Using the concept of minimum essential set, we partition each SCS into two parts: one part corresponds to the "forward subsystem" whose only feedbacks are self-loops; the other part corresponds to the vertices of the minimum essential set; together they form the overall feedback system. Simplified stability conditions for the SCS are obtained by exploiting this structural decomposition. Theorem III gives sufficient condition for stability of nonlinear time-varying SCS\´s. The other three theorems consider linear time-invariant continuous-time, lumped as well as distributed SCS\´s. We provide translation rules for reformulating these three theorems for the discrete-time case. We also show how to reduce the amount of calculation involved in finding the SCS characteristic polynomial.