Control Barrier Function-Based Safety Filters: Characterization of Undesired Equilibria, Unbounded Trajectories, and Limit Cycles

Abstract

This paper focuses on safety filters designed based on Control Barrier Functions (CBFs): these are modifications of a nominal stabilizing controller typically utilized in safety-critical control applications to render a given subset of states forward invariant. The paper investigates the dynamical properties of the closed-loop systems, with a focus on characterizing undesirable behaviors that may emerge due to the use of CBF-based filters. These undesirable behaviors include unbounded trajectories, limit cycles, and undesired equilibria, which can be locally stable and even form a continuum. Our analysis offer the following contributions: (i) conditions under which trajectories remain bounded and (ii) conditions under which limit cycles do not exist; (iii) we show that undesired equilibria can be characterized by solving an algebraic equation, and (iv) we provide examples that show that asymptotically stable undesired equilibria can exist for a large class of nominal controllers and design parameters of the safety filter (even for convex safe sets). Further, for the specific class of planar systems, (v) we provide explicit formulas for the total number of undesired equilibria and the proportion of saddle points and asymptotically stable equilibria, and (vi) in the case of linear planar systems, we present an exhaustive analysis of their global stability properties. Examples throughout the paper illustrate the results.

Figures (12)

• Figure 1: Closed-loop system that is the subject of this paper. A nominal controller $k$ is used as input to the safety filter, that finds the controller closest to $k$ that satisfies the CBF condition.
• Figure 2: Control-affine systems with a safety filter with (a) half-plane and (b) circular obstacles. The plots show the trajectories from random initial conditions, the undesired equilibria (colored in blue), and the desired equilibrium (the origin, colored in black).
• Figure 3: Sketch of the setting considered in item \ref{['it:limit-cycle-proof-third']} of the proof of Proposition \ref{['prop:no-limit-cycles-general']}. The connected components comprising $\mathbb{R}^2\backslash\mathcal{C}$ are depicted in green, whereas the origin is represented by the blue dot.
• Figure 4: Depiction of the setting considered in Example \ref{['ex:existence-limit-cycle-higher-dimensions']}, with parameters $x_c = 2$, $r=1$, $p_1 = 1$, $p_2=6$, $p_3=1$. The obstacle is depicted in green, the limit cycle $\hat{{\mathbf{x}}}$ in red, and the origin in black.
• Figure 5: Plot of different trajectories of \ref{['eq:v-linear-system']}. Trajectories are depicted in orange. The unsafe set is colored in green. Black crosses denote initial conditions, the black dot denotes the origin, and the yellow region denotes a continuum of undesired equilibria.

Theorems & Definitions (56)

• Definition 2.1: Control Barrier Function
• Lemma 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Remark 4.4
• Proposition 4.5
• proof