Enhanced safe control for arbitrary relative degree: A generalized discrete-time CBF approach

Control Barrier Functions (CBFs) are widely used for ensuring safety in control systems and can be implemented on real-world systems via Approximate Sampled-Data Systems (ASDSs). However, the relative degree may be altered during the time-discretization process when generating corresponding ASDSs. Additionally, since the relative degree is a local property, it may vary across the state space. These two facts introduce significant challenges in designing safe controllers. In this paper, we propose a novel approach termed the Generalized Discrete-Time Control Barrier Function (GD-CBF), which provides safety guarantees for ASDSs under certain conditions, regardless of whether the relative degree is constant or variable. A key feature of the GD-CBF framework is the introduction of a safety predictive horizon, which enables improved safety performance and endows the method with predictive capabilities. In addition, it is proved that the safety of the original continuous-time system can be guaranteed in the sense that its state remains within a bounded deviation from the safe set, provided the sampling period is less than a threshold. We also explicitly characterize the dependence of this bound on the sampling period and the properties of the continuous-time system.