Lyapunov-Like Characterization of Stipulated-Time Stability: Controller and Observer Design

There is a lack of rigorous stability concept which can stipulate the actual settling time of a dynamic system in existing studies. In this article, a stipulated-time stability for nonautonomous dynamic systems is proposed, which is then extended to stipulated-time boundedness for uncertain systems. By using a class of bounded time-varying functions, Lyapunov-like conditions to ensure a dynamic system to exhibit stipulated-time stability/boundedness are developed. It is interesting that previous Lyapunov-like theorems for predefined-time (PDT) stability can be unified into our framework to achieve stipulated-time stability. To validate the framework, a stipulated-time controller is first designed for a general affine system, which requires a smaller initial control signal than that of PDT control. Furthermore, a generalized design of stipulated-time distributed observer for leader-following multiagent systems is proposed.