Universal Barrier Functions for Safety and Stability of Constrained Nonlinear Systems

TL;DR

This work introduces Universal Barrier Functions (UBFs) to jointly enforce safety and stability for constrained nonlinear systems, including those with higher relative degrees. By encoding unions/intersections of state and input constraints through smooth log-sum-exp constructions, UBFs provide an inner, convergent representation of the complex safe set and yield a feasible UBF-QP that preserves forward invariance. A key result is that a UBF exists under mild stabilizability and safety assumptions, and that the UBF-QP remains feasible and Lipschitz, enabling well-posed closed-loop dynamics. The framework is extended to High-Order UBFs (HO-UBF) to handle higher relative degrees, and validated through simulations on single/double integrators and a quadrotor, demonstrating safety, input-constrained feasibility, and stability in practice.

Abstract

In this paper, we address the problem of synthesizing safe and stabilizing controllers for nonlinear systems subject to complex safety specifications and input constraints. We introduce the Universal Barrier Function (UBF), a single continuously differentiable scalar-valued function that encodes both stability and safety criteria while accounting for input constraints. Using the UBF, we formulate a Quadratic Program (UBF-QP) to generate control inputs that are both safe and stabilizing under input constraints. We demonstrate that the UBF-QP is feasible if a UBF exists. Furthermore, under mild conditions, we prove that a UBF always exists. The proposed framework is then extended to systems with higher relative degrees. Finally, numerical simulations illustrate the effectiveness of our proposed approach.

Figures (14)

• Figure 1: Illustration of state and input constraint specifications. The set $\mathcal{P}_{{\boldsymbol{x}}} = \{1\}$ represents a union operation applied after the state constraint set $\mathcal{S}_1$, while $\mathcal{Q}_{{\boldsymbol{x}}} = \{2, 3, 4\}$ represents intersection operations applied after the state constraint sets $\mathcal{S}_2$, $\mathcal{S}_3$, and $\mathcal{S}_4$. Similarly, the set $\mathcal{P}_{{\boldsymbol{u}}} = \{5\}$ represents a union operation applied after the input constraint set $\mathcal{U}_1$, and $\mathcal{Q}_{\boldsymbol{u}} = \varnothing$ indicates that there are no further input constraints following $\mathcal{U}_2$. Note that the subscript ${\boldsymbol{x}}$ in $\mathcal{P}_{{\boldsymbol{x}}}$ or $\mathcal{Q}_{{\boldsymbol{x}}}$ indicates that there is a union or intersection operation after the state constraint set. Similarly, for $\mathcal{P}_{{\boldsymbol{u}}}$ and $\mathcal{Q}_{{\boldsymbol{u}}}$.
• Figure 2: The depiction of sets $\mathcal{A}_s$\ref{['eqn:specification_with_stability']}, $\mathcal{A}$\ref{['eqn:state_and_input_constraint_set_complex']}, and $\mathcal{S}_V$ characterized by the CLF $V$ condition \ref{['eqn:clf_condition']}.
• Figure 3: The set $\mathcal{S}_h$ characterized by the UBF $h$ provides an inner approximation for the safe set $\mathcal{A}_s$.
• Figure 4: The depiction of sets $\mathcal{A}_s$, $\mathcal{A}^M_s$ and $\mathcal{A}_{h^o}$ discussed in Section \ref{['subsec:high_order_ubf']}
• Figure 5: Trajectory of single integrator system

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Theorem 1
• Definition 3
• Definition 4
• Theorem 2
• Example 1
• Definition 5
• Lemma 1
• proof