On the Stability of Undesirable Equilibria in the Quadratic Program Framework for Safety-Critical Control (2402.08027v2)
Abstract: Control Lyapunov functions (CLFs) and Control Barrier Functions (CBFs) have been used to develop provably safe controllers by means of quadratic programs (QPs). This framework guarantees safety in the form of trajectory invariance with respect to a given set, but it can introduce undesirable equilibrium points to the closed loop system, which can be asymptotically stable. In this work, we present a detailed study of the formation and stability of equilibrium points with the CLF-CBF-QP framework with multiple CBFs. In particular, we prove that undesirable equilibrium points occur for most systems, and their stability is dependent on the CLF and CBF geometrical properties. We introduce the concept of CLF-CBF compatibility for a system, regarding a CLF-CBF pair inducing no stable equilibrium points other than the CLF global minimum on the corresponding closed-loop dynamics. Sufficient conditions for CLF-CBF compatibility for LTI and drift-less full-rank systems with quadratic CLF and CBFs are derived, and we propose a novel control strategy to induce smooth changes in the CLF geometry at certain regions of the state space in order to satisfy the CLF-CBF compatibility conditions, aiming to achieve safety with respect to multiple safety objectives and quasi-global convergence of the trajectories towards the CLF minimum. Numeric simulations illustrate the applicability of the proposed method.
- L. Lamport, “Proving the correctness of multiprocess programs,” IEEE transactions on software engineering, no. 2, pp. 125–143, 1977.
- B. Alpern and F. B. Schneider, “Defining liveness,” Information processing letters, vol. 21, no. 4, pp. 181–185, 1985.
- A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: Theory and applications,” arXiv preprint arXiv:1903.11199, 2019.
- A. D. Ames, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs with application to adaptive cruise control,” in 53rd IEEE Conference on Decision and Control. IEEE, 2014, pp. 6271–6278.
- M. F. Reis, A. P. Aguiar, and P. Tabuada, “Control barrier function-based quadratic programs introduce undesirable asymptotically stable equilibria,” IEEE Control Systems Letters, vol. 5, no. 2, pp. 731–736, 2021.
- G. Notomista and M. Saveriano, “Safety of dynamical systems with multiple non-convex unsafe sets using control barrier functions,” IEEE Control Systems Letters, vol. 6, pp. 1136–1141, 2022.
- P. Lancaster and Q. Ye, “Variational properties and rayleigh quotient algorithms for symmetric matrix pencils,” in The Gohberg Anniversary Collection. Springer, 1989, pp. 247–278.
- B. N. Parlett, “Symmetric matrix pencils,” Journal of Computational and Applied Mathematics, vol. 38, no. 1-3, pp. 373–385, 1991.
- A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861–3876, Aug 2017.
- R. P. Jain, A. P. Aguiar, and J. B. de Sousa, “Cooperative path following of robotic vehicles using an event-based control and communication strategy,” IEEE Robotics and Automation Letters, vol. 3, no. 3, pp. 1941–1948, July 2018.
- M. Reis, R. P. Jain, A. P. Aguiar, and J. Sousa, “Robust moving path following control for robotic vehicles: Theory and experiments,” IEEE Robotics and Automation Letters, pp. 1–1, 2019.