Probabilistic Safety Bounds for Discrete-Time High Relative Degree Systems with Unknown Dynamics

This paper develops a theoretical framework for the construction and analysis of high-order discrete-time control barrier functions (CBFs). Two main results are presented. First, we show that the difference between the true and nominal auxiliary functions associated with a high-order discrete-time CBF is linear if and only if the class–$\mathcal{K}$ functions $\alpha$ satisfy $\alpha(x) = \gamma x$ for some $\gamma \in (0,1)$. Second, we prove that if a corresponding optimization problem admits a solution at each time step, then the resulting control law renders a prescribed safe set forward invariant, thereby guaranteeing system safety. The theoretical results are numerically demonstrated on a discrete-time nonlinear system representing a type-1 diabetic patient, showing that the proposed control framework ensures safety in the regulation of blood glucose levels while respecting safety constraints.