On the Feasibility and Continuity of Feedback Controllers Defined by Multiple Control Barrier Functions

• Published in: IEEE transactions on automatic control Vol. 69; no. 11; pp. 7326 - 7339
• Main Authors: Isaly, Axton, Ghanbarpour, Masoumeh, Sanfelice, Ricardo G., Dixon, Warren E.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.11.2024; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive control; Constraints; Continuity (mathematics); Control systems design; Design optimization; Dynamical systems; Feasibility; Feedback control; Iterative methods; Lyapunov methods; nonlinear control systems; Optimization; Polynomials; Process control; Safety; Weapons
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2024.3383069

Control barrier functions (CBFs) are a popular method for encoding safety specifications for dynamical systems. In this article, a notion of CBF is defined that permits vector-valued barrier functions and flow constraints involving both the state and the control input. CBFs induce constraints on the control input that, when satisfied, guarantee the forward invariance of a safe set of states. The constraints are enforced using a pointwise-optimal feedback controller. Sufficient conditions for the continuity of the controller are given. The existence of a CBF is defined to be equivalent to the feasibility of the optimal feedback controller. Polynomial optimization problems based on sums of squares are formulated that can be used to certify that a given function is a CBF. An example of the CBF design procedure is presented illustrating the process of formulation, synthesis, and verification.