Safe and Stable Filter Design Using a Relaxed Compatibitlity Control Barrier -- Lyapunov Condition

In this paper, we propose a quadratic programming-based filter for safe and stable controller design, via a Control Barrier Function (CBF) and a Control Lyapunov Function (CLF). Our method guarantees safety and local asymptotic stability without the need for an asymptotically stabilizing control law. Feasibility of the proposed program is ensured under a mild regularity condition, termed relaxed compatibility between the CLF and CBF. The resulting optimal control law is guaranteed to be locally Lipschitz continuous. We also analyze the closed-loop behaviour by characterizing the equilibrium points, and verifying that there are no equilibrium points in the interior of the control invariant set except at the origin. For a polynomial system and a semi-algebraic safe set, we provide a sum-of-squares program to design a relaxed compatible pair of CLF and CBF. The proposed approach is compared with other methods in the literature using numerical examples, exhibits superior filter performance and guarantees safety and local stability.