A Lyapunov approach to integral input-to-state stability of cascaded systems with external signals

This paper deals with the stability of cascade interconnection of integral input-to-state (iISS) time-varying systems. A new technique is introduced for the purpose of constructing smooth Lyapunov functions of cascaded systems explicity. From the construction, sufficient conditions for internal stability and stability with respect to external signals are derived. One of the proposed conditions is a generalized version of trade-off between slower convergence of the driving system and steeper input growth of the driven system. It is demonstrated that 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 in the literature and small-gain techniques for feedback interconnection. The difference between the feedback case and the cascade case is also viewed from the requirement on convergence speed of autonomous parts.