Hybrid Lyapunov and Barrier Function-Based Control with Stabilization Guarantees

TL;DR

The paper tackles safety-critical control by addressing deadlock in standard CLF-CBF-QP methods through a hybrid CLF-CBF framework that guarantees global asymptotic stabilization while enforcing safety over a polytopic avoidance domain. It introduces a switching mechanism among active safe half-spaces and a sequence of safe stabilization subproblems, with a backstepping extension to higher-order dynamics via a joint CLF-CBF design and Gaussian-centroid smoothing for continuity. The approach is validated through simulations on first- and second-order systems, showing robust deadlock avoidance and improved decisiveness compared to existing hybrid methods. The work holds practical significance for real-time safety-critical control in systems with complex unsafe regions and paves the way for extensions to time-varying, multi-polytope, and experimental scenarios.

Abstract

Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs) can be combined, typically by means of Quadratic Programs (QPs), to design controllers that achieve performance and safety objectives. However, a significant limitation of this framework is the introduction of asymptotically stable equilibrium points besides the minimizer of the CLF, leading to deadlock situations even for simple systems and bounded convex unsafe sets. To address this problem, we propose a hybrid CLF-CBF control framework with global asymptotic stabilization and safety guarantees, offering a more flexible and systematic design methodology compared to current alternatives available in the literature. We further extend this framework to higher-order systems via a recursive procedure based on a joint CLF-CBF backstepping approach. The proposed solution is assessed through several simulation examples.

Figures (5)

• Figure 1: System trajectories obtained using the three discussed strategies for two polytopic unsafe sets. Blue trajectories indicate cases where the system successfully avoids the unsafe set and reaches the desired equilibrium point. Meanwhile, orange trajectories denote cases in which the system incurs in a deadlock situation. Dashed lines represent the boundary of the safe set resulting from each approximation. The initial state is labeled as and the desired equilibrium point as .
• Figure 2: Illustration of the switching mechanism when a jump occurs.
• Figure 3: Examples of system trajectories and the corresponding temporal profiles under the hybrid control law from Section \ref{['Sec:HybridController']} for three different polytopes while considering a fixed desired equilibrium point. The plots on the left display trajectories obtained with $\mu = 0.2$ (blue and orange) and $\mu = 1$ (purple and yellow) for a fixed $\sigma = 0.1$. The plots on the right display the respective time evolution of the state, the input, and the index of the active half-space for the blue and orange trajectories. The initial state is denoted as and the desired equilibrium point as .
• Figure 4: Comparison between the system trajectories generated using the hybrid feedback strategy detailed in Section \ref{['Sec:HybridController']} (left) and the ones obtained with the approach proposed in marley2024hybrid (right) across different polytopes and desired equilibrium points. Blue trajectories correspond to cases where the system successfully avoids the polytope and reaches the desired equilibrium point. Meanwhile, orange trajectories indicate cases in which the system incurs in a deadlock situation. The initial state is denoted as and the desired equilibrium point as .
• Figure 5: Examples of top-level system trajectories and the corresponding temporal profiles for the double-integrator system under the hybrid control law from Section \ref{['Sec:Backstepping']} for three different polytopes while considering a fixed desired equilibrium point. The plots on the left display trajectories obtained with $\mu = 0.2$ (blue and orange) and $\mu = 1$ (purple and yellow) for a fixed $\sigma = 0.1$. The plots on the right display the respective time evolution of the top-level state, the input, and the index of the active safe half-space for the blue and orange trajectories. The initial top-level state is denoted as and the desired equilibrium point as . For all trajectories, the system starts at rest.

Theorems & Definitions (19)

• Definition 1: Class-$\mathcal{K}\,$/$\,\mathcal{K}_\infty$ Function
• Definition 2: Extended Class-$\mathcal{K}\,$/$\,\mathcal{K}_\infty$ Function
• Definition 3: Positive-Definite Function Around a Point
• Definition 4: Asymptotic Stability
• Definition 5: Forward Invariance
• Definition 6
• Theorem 1: Stabilizing Control sontag1989universal
• Definition 7: ames2016control
• Theorem 2: Safeguarding Controller ames2016control
• Remark 1