Data-Driven Time-Varying Control Barrier Functions for Adaptive Safe-Set Learning with Online Decremental Support Vector Machines

Mission-critical intelligent systems often operate under time-varying constraints that reduce control authority and shrink the admissible safe operating envelope. A safety certificate learned under nominal conditions can quickly become invalid as system capabilities degrade. While control barrier functions (CBFs) are widely used to enforce safety in real time, standard methods typically rely on fixed, offline-designed barrier functions and predefined safe sets. This paper addresses the challenge of how to dynamically update a data-driven safe set online as system capabilities change, ensuring that safety filters remain compatible with a shrinking control authority without violating the smoothness required for rigorous theoretical guarantees.

The framework begins by learning a nominal safe operating envelope from operational data using a radial basis function (RBF) kernel support vector machine (SVM), whose decision function acts as the initial candidate control barrier function. To capture capability-induced safe-set contraction resulting from actuator degradation, the authors develop a continuous-time decremental SVM update law. Instead of retraining the classifier from scratch, selected support-vector coefficients are reduced according to a continuous degradation signal. To prevent discontinuous jumps in the learned decision boundary during active-set transitions, a homotopy-smoothed SVM-CBF is introduced, preserving the differentiability required for real-time safety enforcement.

The time-varying learned barrier is enforced through a quadratic-program-based (QP) safety filter under degraded polytopic input constraints. The authors rigorously establish the forward invariance of the learned time-varying safe set and the recursive feasibility of the safety filter. Simulation studies on a vertical takeoff and landing (VTOL) model demonstrate that the proposed framework successfully maintains safety under reduced control authority while eliminating abrupt barrier-switching artifacts during safe-set contraction.