Predictive control barrier functions for piecewise affine systems with non-smooth constraints

TL;DR

This work addresses safety for constrained nonlinear systems, focusing on continuous-time piecewise affine (PWA) dynamics with non-smooth state and input constraints. It introduces a predictive safety filter (PSF) framework that uses generalized Clarke derivatives to enforce safety across all derivative elements, plus an Aumann-sensitivity-based implementation for tractable flow derivatives. An explicit PSF approximation is proposed to render online computation feasible, and the theory extends classical, smooth-CBF safety guarantees to Lipschitz, non-smooth CBFs with general nonlinear dynamics. The approach is validated on an inverted pendulum and a multi-room temperature-control case that show reduced conservatism and substantial computational savings, highlighting practical viability for real-time safety-critical control. The method positions PSFs as an online counterpart to Hamilton-Jacobi reachability, with potential extensions to broader Lipschitz systems and probabilistic safety under uncertainty.

Abstract

Obtaining control barrier functions (CBFs) with large safe sets for complex nonlinear systems and constraints is a challenging task. Predictive CBFs address this issue by using an online finite-horizon optimal control problem that implicitly defines a large safe set. The optimal control problem, also known as the predictive safety filter (PSF), involves predicting the system's flow under a given backup control policy. However, for non-smooth systems and constraints, some key elements, such as CBF gradients and the sensitivity of the flow, are not well-defined, making the current methods inadequate for ensuring safety. Additionally, for control-non-affine systems, the PSF is generally nonlinear and non-convex, posing challenges for real-time computation. This paper considers piecewise affine systems, which are usually control-non-affine, under nonlinear state and polyhedral input constraints. We solve the safety issue by incorporating set-valued generalized Clarke derivatives in the PSF design. We show that enforcing CBF constraints across all elements of the generalized Clarke derivatives suffices to guarantee safety. Moreover, to lighten the computational overhead, we propose an explicit approximation of the PSF. The resulting control methods are demonstrated through numerical examples.

Figures (10)

• Figure 1: Graphical representation of the paper structure.
• Figure 2: The evolution of the PWA system given in Example 1. The black solid lines refer to the boundaries of the polyhedra. The critical set $\mathcal{C}$ is the positive part of the $x_1$-axis, represented by the red line. The backward reachable set $\Phi_{\text{back }}(\mathcal{C})$ is the union of the red and blue lines. According to Proposition \ref{['proposition1']}, the solution $\phi_\mathrm{b}$ is not differentiable at all points on the red and blue lines. The black dashed curves represent some trajectories starting near the blue line.
• Figure 3: Evolution of the sensitivity function $Q_\mathrm{A}$ over time for any initial state at the blue line in Fig. \ref{['example1']}. The components $Q_1$ to $Q_4$ represent individual elements of $Q_\mathrm{A}$. The sensitivity branches when the solution $\phi_{\mathrm{b}}(\bar{x}_0, \tau)$ enters each critical set. In Example 1, this occurs at $\tau_{k_1}$ when the trajectory reaches $(0,0)$ and at $\tau_{k_2}$ when the trajectory reaches $(2,0)$.
• Figure 4: Diagram of the inverted pendulum interacting with an elastic wall.
• Figure 5: Time responses of the closed-loop system with the PSF \ref{['safety filter pwa']} and the standard safety filter. The initial state is $[0.1\;\;0.1]^T$. “SF” means “safety filter”. Both SFs successfully control the system without recording any constraint violations. The predictive SF is less conservative than the standard SF because the trajectory regulated by the predictive SF is closer to the reference.

Theorems & Definitions (26)

• Definition 1: Control barrier functionames2019control
• Definition 2: Constrained reachable set
• Lemma 1: Safety of backup CBFs molnar2023safetygurriet2020scalable
• Remark 1
• Definition 3: Generalized Clarke derivative clarke1976inverse
• Theorem 1: Feasibility
• proof
• Lemma 2
• proof
• Remark 2