Input-output stability of interconnected systems using decompositions: An improved formulation

We study the input-output stability of an arbitrary interconnection of multi-input, multi-output subsystems which may be either continuous-time or discrete-time. We consider throughout three types of dynamics: nonlinear time-varying, linear time-invariant distributed and linear time-invariant lumped. First, we use the strongly connected component decomposition to aggregate the subsystems into strongly-connected sub-systems (SCS´s) and interconnection-subsystems (IS´s). These SCS´s and IS´s are then aggregated into column subsystems (CS´s) so that the overall system becomes a hierarchy of CS´s. The basic structural result states that the overall systems is stable if and only if every CS is stable. We then use the minimum-essentialset decomposition on each SCS so that it can be viewed as a feedback interconnection of aggregated subsystems where one of them is itself a hierarchy of subsystems. Based on this decomposition, we present the results which lead to sufficient conditions for the stability of an SCS. For linear time-invariant (transfer function) dynamics, we obtain a characteristic function which gives the necessary and sufficient condition for the overall system stability. We point out the computational saving due to the decompositions in calculating this characteristic function.