Discrete-time Control Barrier Function High-order Case And Adaptive Case

This paper proposes the novel concepts of high order discrete-time control barrier function and adaptive discrete-time control barrier function. The high order discrete-time control barrier function is used to guarantee forward invariance of a safe set for discrete-time systems of high relative degree. An optimization problem is then established unifying high order discrete-time control barrier functions with discrete-time control Lyapunov functions to yield a safe controller. To improve the feasibility of such optimization problems, the adaptive discrete-time control barrier function is designed, which can relax constraints on system control input through time-varying penalty functions. The effectiveness of the proposed methods in dealing with high relative degree constraints and improving feasibility is verified on the discrete-time system of a three-link manipulator.