On the Stability of Undesirable Equilibria in the Quadratic Program Framework for Safety-Critical Control
Matheus F. Reis, A. Pedro Aguiar
TL;DR
This work examines the coexistence of stability and safety in the CLF-CBF-QP framework, showing that multiple CBFs can generate undesirable equilibria beyond the CLF minimum. It introduces the notion of CLF compatibility and provides necessary and sufficient conditions for quadratic CLFs in LTI and driftless full-rank systems, using a matrix-pencil and a boundary-equilibrium analysis to identify when boundary equilibria are stable. A compatibilization approach is proposed to compute compatible CLFs and a CLF-shape controller that adaptively changes the CLF geometry in regions where boundary equilibria arise, aiming to remove stable non-minimum equilibria while preserving safety and quasi-global convergence to the CLF minimum. Numerical simulations on a two-dimensional, low-order system demonstrate that the proposed method can eliminate stable boundary equilibria and drive trajectories toward the CLF minimum, validating the approach for safety-critical control with multiple objectives. Overall, the paper provides a practical framework for achieving safety with multiple CBFs without incurring deadlocks or unsafe attractors, by shaping the CLF geometry in a region-dependent manner.
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.