Prediction-Based Control Barrier Functions for Input-Constrained Safety Critical Systems

TL;DR

The paper addresses safety-critical control for systems with input constraints by extending control barrier functions (CBFs) with a prediction-based margin. It introduces prediction-based CBFs (PB-CBFs) where a term $\delta h$ is propagated forward using a nominal control $\mathbf{u}_0$ to ensure feasibility and invariance of the safe set, yielding a new barrier $h_P(x) = h(x) + \delta h(x)$. Theoretical conditions guarantee that an admissible controller exists to keep the system within the PB-CBF safe set $C_P = \{x : h_P(x) > 0\}$ under input constraints, and practical implementation is cast as a QP-based safety filter. The approach is demonstrated on a simple double integrator and a complex aircraft stall-prevention model, showing PB-CBFs can be less conservative and react sooner than basic CBFs while maintaining safety. Overall, PB-CBFs provide formal, implementable safety guarantees for nonlinear, input-constrained systems with potential practical impact across aerospace and autonomous systems.

Abstract

Control barrier functions (CBFs) have emerged as a popular topic in safety critical control due to their ability to provide formal safety guarantees for dynamical systems. Despite their powerful capabilities, the determination of feasible CBFs for input-constrained systems is still a formidable task and a challenging research issue. The present work aims to tackle this problem by focusing on an alternative approach towards a generalization of some ideas introduced in the existing CBF literature. The approach provides a rigorous yet straightforward method to define and implement prediction-based control barrier functions for complex dynamical systems to ensure safety with bounded inputs. This is accomplished by introducing a prediction-based term into the CBF that allows for the required margin needed to null the CBF rate of change given the specified input constraints. Having established the theoretical groundwork, certain remarks are subsequently presented with regards to the scheme's implementation. Finally, the proposed prediction-based control barrier function (PB-CBF) scheme is implemented for two numerical examples. In particular, the second example is related to aircraft stall prevention, which is meant to demonstrate the functionality and capability of the PB-CBFs in handling complex nonlinear dynamical systems via simulations. In both examples, the performance of the PB-CBF is compared with that of a non-prediction based basic CBF.

Figures (7)

• Figure 1: CBF control signal safety filtering.
• Figure 2: Modules of the prediction-based control barrier function.
• Figure 3: Graph of the safe set, the CBF super-level set, and the PB-CBF super-level set in the $r\text{–}\dot{r}$ phase plane for $\mu = 1.5$ and $u_{\max} = 1$.
• Figure 4: Simulation results for the double integrator system: (a) the path, (b) the inputs for scenarios with (i) no CBFs, (ii) the base CBF, and (iii) the PB-CBF ($\gamma = 2$). The pre-modified control signals are displayed in dashed red.
• Figure 5: Simulation results for the double integrator system: (a) the path, (b) the inputs for scenarios with (i) no CBFs, (ii) the base CBF, and (iii) the PB-CBF ($\gamma = 10$). The pre-modified control signals are displayed in dashed red.

Theorems & Definitions (8)

• Definition 1: Forward Invariant Set
• Definition 2: Control Barrier Function O5
• Theorem 1: O5O10
• Definition 3: Prediction-Based Control Barrier Function
• Theorem 2
• proof
• Proposition 1
• proof