Sufficient Conditions for Feasibility of Optimal Control Problems Using Control Barrier Functions

TL;DR

This paper introduces a new sufficient condition using a Control Barrier Function to ensure the feasibility of quadratic programs in optimal control, especially under tight constraints, demonstrated through an adaptive cruise control example.

Contribution

It proposes a single, compatible constraint via a Control Barrier Function that guarantees quadratic program feasibility in optimal control problems with safety and control bounds.

Findings

Guarantees QP feasibility under tight constraints

Compatible constraint increases overall feasibility

Effective in adaptive cruise control scenario

Abstract

It has been shown that satisfying state and control constraints while optimizing quadratic costs subject to desired (sets of) state convergence for affine control systems can be reduced to a sequence of quadratic programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). One of the main challenges in this approach is ensuring the feasibility of these QPs, especially under tight control bounds and safety constraints of high relative degree. In this paper, we provide sufficient conditions for guranteed feasibility. The sufficient conditions are captured by a single constraint that is enforced by a CBF, which is added to the QPs such that their feasibility is always guaranteed. The additional constraint is designed to be always compatible with the existing constraints, therefore, it cannot make a feasible set of constraints infeasible - it can only increase the overall feasibility. We illustrate the effectiveness of the proposed approach on an adaptive cruise control problem.