Explicit Solutions for Safety Problems Using Control Barrier Functions

TL;DR

This paper develops an explicit, piece-wise Lipschitz continuous safe control law for nonlinear control-affine systems using control Barrier functions, transforming online optimization into an offline parameterized problem.

Contribution

It introduces a method to explicitly synthesize safe control laws by converting online quadratic programs into offline parameterized optimizations, enabling explicit control design.

Findings

Explicit safe controllers are piece-wise Lipschitz continuous over state space partitions.

The approach effectively handles infeasible cases with adaptive Barrier functions.

Simulation results validate the controller properties and state-space partitioning.

Abstract

The control Barrier function approach has been widely used for safe controller synthesis. By solving an online convex quadratic programming problem, an optimal safe controller can be synthesized implicitly in state-space. Since the solution is unique, the mapping from state-space to control inputs is injective, thus enabling us to evaluate the underlying relationship. In this paper we aim at explicitly synthesizing a safe control law as a function of the state for nonlinear control-affine systems with limited control ability. We propose to transform the online quadratic programming problem into an offline parameterized optimisation problem which considers states as parameters. The obtained explicit safe controller is shown to be a piece-wise Lipschitz continuous function over the partitioned state space if the program is feasible. We address the infeasible cases by solving a parameterized adaptive control Barrier function-based quadratic programming problem. Extensive simulation results show the state-space partition and the controller properties.