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-subsystems (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 system is stable if and only if every CS is stable. We then use the minimum-essential-set 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 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. We believe that decomposition techniques, coupled with other techniques such as model reduction, aggregation, singular, and nonsingular perturbations, will play key roles in large scale system design.