Input-to-State Safety for Discrete-Time Nonlinear Systems: Sufficiency and Converse Results

Input-to-state safety (ISSf) extends comparison-function-based robustness analysis from stability to safety by quantifying degradation of safety margin under exogenous inputs. For continuous-time systems, sufficiency conditions for the existence of ISSf comparison function have been established, while its converse result remains an open problem. Also, the extension of the ISSf notion and its characterization to the discrete-time (DT) setting are largely unavailable. This letter develops ISSf notions for general discrete-time nonlinear systems. We introduce a global practical ISSf notion and two tube-local notions namely local ISSf and strong local ISSf properties. We show that an ISSf barrier certificate on a compact tube implies local ISSf property under an explicit state-input compatibility condition. Conversely, strong local ISSf property under bounded inputs implies the existence of an ISSf barrier certificate.