Control Barrier Functions for Linear Continuous-Time Input-Delay Systems with Limited-Horizon Previewable Disturbances

TL;DR

This paper proposes a novel limited preview control barrier function (LPrev-CBF) that avoids both ends of the spectrum and provides strong safety guarantees in a less conservative manner than standard CBF approaches while considering a more realistic setting with limited preview and input delays.

Abstract

Cyber-physical and autonomous systems are often equipped with mechanisms that provide predictions/projections of future disturbances, e.g., road curvatures, commonly referred to as preview or lookahead, but this preview information is typically not leveraged in the context of deriving control barrier functions (CBFs) for safety. This paper proposes a novel limited preview control barrier function (LPrev-CBF) that avoids both ends of the spectrum, where on one end, the standard CBF approach treats the (previewable) disturbances simply as worst-case adversarial signals and on the other end, a recent Prev-CBF approach assumes that the disturbances are previewable and known for the entire future. Moreover, our approach applies to input-delay systems and has recursive feasibility guarantees since we explicitly take input constraints/bounds into consideration. Thus, our approach provides strong safety guarantees in a less conservative manner than standard CBF approaches while considering a more realistic setting with limited preview and input delays.

Figures (3)

• Figure 1: Angular error (left) and input (right) trajectories: (i) Without CBFs (exceeds black dashed bounds), (ii) standard CBF ames2016control with $u_m=1.119$, (iii) Prev-CBF pati2023preview with $u_m=1.119$, (iv) LPrev CBF with $u_m=1.119$, (v) standard CBF ames2016control with $u_m=1.8$, (vi) Prev-CBF pati2023preview with $u_m=1.8$, and (vii) LPrev CBF with $u_m=1.8$. Furthermore, an amplified segment in the (right) plot elucidates the initial intervention disparities for all three conditions when $u_m=1.8$.
• Figure 2: Intervention times (left) and stopping times (right, shown as box plots) vary with $u_m$ between 1.119 and 2. The standard CBF ames2016control typically acts sooner and has extended stopping times compared to LPrev-CBF. Conversely, Prev-CBF pati2023preview shows late intervention and exhibits shorter stopping times than LPrev-CBF. However, these differences become less pronounced as $u_m$ rises
• Figure 3: Lateral displacement trajectories $y(t)$ (left) and trajectories of CBF intervention of $u(t)$ (right) given by $\Delta u(t)\triangleq u(t)-k(x(t),t)$, where $k(x(t),t)$ is the legacy controller.

Theorems & Definitions (15)

• Definition 1: Safe Sets
• Definition 2: Controlled Invariant Set
• Definition 3: Limited Preview Safe Set
• Definition 4: Limited Preview CBF
• Theorem 1: Safety with Limited Preview
• proof
• Definition 5: Worst-Case Stopping Time
• Lemma 1: Closed-Form Candidate Limited Preview CBF
• proof
• Lemma 2: Worst-Case Stopping Time