A Lyapunov Approach to Cascade Interconnection of Integral Input-to-State Stable Systems

This technical note deals with the stability of cascade interconnection of integral input-to-state (iISS) time-varying systems. A new technique to construct smooth Lyapunov functions of cascaded systems is proposed. From the construction, sufficient conditions for internal stability and stability with respect to external signals are derived. One of the derived conditions is a trade-off between slower convergence of the driving system and steeper input growth of the driven system. The trade-off is no more necessary if the speed of convergence of the driven system is not radially vanishing. The results are related to trajectory-based approaches and small-gain techniques for feedback interconnection. The difference between the feedback case and the cascade case is viewed from the requirement on convergence speed of autonomous parts.