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

TL;DR

The paper tackles safe and stable controller synthesis for nonlinear, control-affine systems by introducing a quadratic-programming-based filter that jointly enforces safety via a CBF and local stability via a CLF. A key novelty is the relaxed compatibility condition, which guarantees feasibility even when CLF and CBF constraints would otherwise conflict, and the resulting $u^*(x)$ is locally Lipschitz. The authors develop a sum-of-squares-based design pipeline to construct polynomial CBF/CLF pairs for polynomial dynamics and semi-algebraic safe sets, including an iterative algorithm to handle bilinearities. Simulations benchmark the approach against existing methods, showing larger regions of attraction, elimination of interior equilibria, and competitive control effort. Overall, the framework enables reliable, online-safe, and locally stable control without requiring a stabilizing nominal controller, with practical SOS-based tools for synthesis.

Abstract

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.

Figures (5)

• Figure 1: Trajectories of the closed-loop system with different methods. The black points are initial points, the green set is the obstacle. The penalty parameter $p_d$ is set to $100$ for the method of tan2021undesired, ames2016control, and our method. For the method of mestres2022optimization, we set $\epsilon=0.01$ . Our method, tan2021undesired and mestres2022optimization achieve stabilization from all the initial points except for the top one, while ames2016control only achieves inexact convergence. From the very top initial point, all the trajectories converge to a point on the boundary of the obstacle. Safety is ensured by every method.
• Figure 2: Trajectories near the origin, the red ones correspond to our method, while the green dotted ones correspond to the method of ames2016control. All the trajectories of our methods converge to the origin, while these of ames2016control converge to the black dots, which are equilibrium points away from the origin.
• Figure 3: Comparison of filter performance by 100 Monte-Carlo experiments. The vertical axis represents $\log(||u_{\mathrm{o}}^*(x)||^2/||u^*(x)||^2)$, while the horizontal axis represents number of experiments.
• Figure 4: Phase portrait of system \ref{['eq:case2nonlin']} using a CLF $V(x)$ and a CBF $b(x)$ that satisfy the relaxed compatibility condition. The controller $u^*(x)$ is synthesized by solving \ref{['eq:ourfilter']}, using the designed CLF and CBF. The green set represents the obstacle. The control invariant set $\mathcal{B}$ is filled in red while its boundary curve $\partial \mathcal{B}$ is highlighted in black. The red arrows represent the vector field $f(x)+g(x)u^*(x)$. Two trajectories start from the black rectangle, avoid $\mathcal{B}^c$, and finally converge to the origin.
• Figure 5: Phase portrait of system \ref{['eq:case2nonlin']} using a CLF $V(x)$ and a CBF $b(x)$ that satisfy the strict compatibility condition (Definition \ref{['def:compatible']}), designed by the algorithm proposed in schneeberger2024advanced. The green set represents the obstacle, while the control invariant set $\mathcal{B}$ is filled in red. The red arrows represent the vector field $f(x)+g(x)u^*(x)$.