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

TL;DR

This paper develops a novel predictive control barrier function approach for piecewise affine systems with non-smooth constraints, ensuring safety through set-valued derivatives and explicit approximations, suitable for real-time control.

Contribution

It introduces a new method using Clarke derivatives for safety guarantees in non-smooth, control-non-affine systems, with an explicit PSF approximation for computational efficiency.

Findings

Guarantees safety for piecewise affine systems with non-smooth constraints.

Provides an explicit PSF approximation suitable for real-time implementation.

Demonstrates effectiveness through numerical examples.

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.