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This paper investigates the safety analysis and verification of nonlinear systems subject to high-relative-degree constraints and unknown disturbance. The closed-form solution of the high-order control barrier functions (HOCBF) optimization problem with and without a nominal controller is first provided, making it unnecessary to solve the quadratic program problem online and facilitating the analysis. Further, the tunable input-to-state safety (ISSf) framework is investigated to accommodate systems with high relative degrees, and a new tunable function in conjunction with HOCBF is provided. When combined with the existing ISSf theorem, produces controllers for constrained nonlinear systems with external disturbances. The theoretical results are proven and supported by numerical simulations.
Yan et al. (Mon,) studied this question.