Sampled-Data Safe Stabilization of Nonlinear Time-Delay Systems

In this paper, the sampled-data stabilization problem under safety constraints is studied for nonlinear time-delay systems. In particular, a methodology for the design of sampled-data safe stabilizers is provided for nonlinear systems affected by state delays. As a first step, the new notion of Safe Steepest Descent Feedback (SSDF), continuous or not, is introduced by suitably combining the well-known notions of Steepest Descent Feedbacks and Barrier functions considered to cope with safety requirements. Then, the stabilization in the sample-and-hold sense theory is used as a tool to show the existence of a suitably fast sampling such that: the digital implementation of SSDFs, continuous or not, ensures the semi–global practical safe stability property of the related closed-loop system with arbitrarily small final target ball of the origin. In the theory developed here, time-varying sampling periods are allowed. The proposed theoretical results are validated through an application concerning the plasma glucose regulation problem in Type-2 diabetic patients via artificial pancreas.