HJB Based Optimal Safe Control Using Control Barrier Functions

TL;DR

This paper introduces a method to design optimal safe controllers that minimize a cost functional while ensuring safety via control barrier functions, using a Hamilton-Jacobi-Bellman framework and a modified Galerkin approximation.

Contribution

It formulates a constrained optimal control problem with safety constraints as an HJB equation and provides a closed-form, continuous solution that guarantees safety and optimality.

Findings

The solution ensures the safe set is forward invariant.

The method achieves optimal stabilizability and safety simultaneously.

Implementation on a nonlinear system demonstrates effectiveness.

Abstract

This work proposes an optimal safe controller minimizing an infinite horizon cost functional subject to control barrier functions (CBFs) safety conditions. The constrained optimal control problem is reformulated as a minimization problem of the Hamilton-Jacobi-Bellman (HJB) equation subjected to the safety constraints. By solving the optimization problem, we are able to construct a closed form solution that satisfies optimality and safety conditions. The proposed solution is shown to be continuous and thus it renders the safe set forward invariant while minimizing the given cost. Hence, optimal stabilizability and safety objectives are achieved simultaneously. To synthesize the optimal safe controller, we present a modified Galerkin successive approximation approach which guarantees an optimal safe solution given a stabilizing safe initialization. The proposed algorithm is implemented on a constrained nonlinear system to show its efficacy.