Computing Safe Control Inputs using Discrete-Time Matrix Control Barrier Functions via Convex Optimization

TL;DR

This work tackles the challenge of enforcing safety in discrete-time control systems when safe sets are nonconvex, which makes discrete-time CBF-based safety filters nonconvex and hard to solve in real time. It introduces exponential discrete-time matrix CBFs (DTE-MCBFs) and a convex-projection-based safety filter (PDTE-MCBF) that reduces the safety input computation to two convex optimization problems per step, even for nonconvex unsafe sets. The authors define safe-set concepts including SDTE-MCBFs, PDTE-MCBFs, and indefinite/multi-eigenvalue extensions, and prove a zeroing property ensuring exponential convergence of negative eigenvalues toward zero. Numerical simulations on double-integrator systems and a bicopter demonstrate that PDTE-MCBF achieves comparable safety and tracking performance to nonconvex approaches while offering an order-of-magnitude or more improvement in computation time, validating its practicality for real-time safety-critical control.

Abstract

Control barrier functions (CBFs) have seen widespread success in providing forward invariance and safety guarantees for dynamical control systems. A crucial limitation of discrete-time formulations is that CBFs that are nonconcave in their argument require the solution of nonconvex optimization problems to compute safety-preserving control inputs, which inhibits real-time computation of control inputs guaranteeing forward invariance. This paper presents a novel method for computing safety-preserving control inputs for discrete-time systems with nonconvex safety sets, utilizing convex optimization and the recently developed class of matrix control barrier function techniques. The efficacy of our methods is demonstrated through numerical simulations on a bicopter system.

Figures (8)

• Figure 1: Illustration of our proposed method to maintain convexity of computing safe control inputs for discrete-time systems. (A): The current state is projected onto the unsafe set. (B): A halfspace is formed which excludes the obstacle, and a control input keeping the next state in the halfspace is computed. (C): The process is repeated iteratively. (D): The proposed method can be applied to scenarios where the unsafe set is a subset of a union of convex regions.
• Figure 2: Typical obstacle geometries. The figure shows a) circle (\ref{['ssec:circle']}), b) ellipse (\ref{['ssec:ellipse']}), c) convex polytope (\ref{['ssec:polytope']}), and d) spectrahedron (\ref{['ssec:spectra']}) shaped obstacles.
• Figure 3: Discretized double integrators with PDTE-MCBF formulation. The figure shows the trajectory of an agent modeled by double integrators for all $k \in [0, k_{\rm end}]$ with $k_{\rm end} = 1200,$ the unsafe sets given by the obstacles, and the buffer sets, which are obtained by setting the buffer distance $\epsilon = 0.4.$
• Figure 4: Discretized double integrators with nonconvex CBF formulation. This figure shows the trajectory of an agent modeled by double integrators for all $k \in [0, k_{\rm end}]$ with $k_{\rm end} = 1200,$ the unsafe sets given by the obstacles, and the buffer sets, which are obtained by setting the buffer distances $\epsilon_{\mathfrak{C}} = \epsilon_{\mathfrak{El}} = \epsilon_{\mathfrak{P}} = 0.4$ and the buffer ratio $\epsilon_{\mathfrak{Sp}} = 0.25.$
• Figure 5: A bicopter system whose motion is constrained in a vertical plane.

Theorems & Definitions (39)

• Theorem 1: Weyl's Inequality horn2012matrix
• Definition 1: Projection Mapping
• Definition 2: Normal Cone rockafellar1998variational
• Theorem 2: rockafellar1998variational
• Definition 3
• Definition 4
• Theorem 3: agrawal2017discrete
• Lemma III.1
• proof
• Definition 5