On the Stability of Undesirable Equilibria in the Quadratic Program Framework for Safety-Critical Control
This addresses stability issues in safety-critical control systems, particularly for robotics and autonomous systems, but is incremental as it builds on existing CLF-CBF-QP methods.
The paper tackles the problem of undesirable stable equilibrium points in the CLF-CBF-QP framework for safety-critical control, proving their prevalence and stability dependence on geometric properties, and proposes a control strategy to achieve safety with multiple objectives and quasi-global convergence, as demonstrated in simulations.
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. Numerical simulations illustrate the applicability of the proposed method.