On the Stability of Undesirable Equilibria in the Quadratic Program Framework for Safety-Critical Control

TL;DR

This paper analyzes the stability of undesirable equilibrium points in quadratic program-based safety control systems, proposing conditions and strategies to ensure safety and convergence in complex control scenarios.

Contribution

It introduces the concept of CLF-CBF compatibility and provides conditions for its satisfaction in LTI and drift-less systems, enhancing safety guarantees.

Findings

Undesirable equilibria are common in CLF-CBF-QP systems.

Compatibility conditions can prevent stable undesirable equilibria.

Proposed control strategy improves safety and convergence.

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. Numerical simulations illustrate the applicability of the proposed method.