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.