Control Synthesis for Stability and Safety by Differential Complementarity Problem

TL;DR

This paper introduces a control synthesis approach using Differential Complementarity Problems that ensures stability and safety for control-affine systems, avoiding some issues of existing quadratic programming methods.

Contribution

It presents a novel CLF-CBF-DCP controller framework that automatically relaxes constraints without tuning, improving safety and stability guarantees.

Findings

Controller certifies stability and safety simultaneously.

Avoids undesirable local equilibria with proper parameter choices.

Outperforms quadratic programming techniques in certain scenarios.

Abstract

This paper develops a novel control synthesis method for safe stabilization of control-affine systems as a Differential Complementarity Problem (DCP). Our design uses a control Lyapunov function (CLF) and a control barrier function (CBF) to define complementarity constraints in the DCP formulation to certify stability and safety, respectively. The CLF-CBF-DCP controller imposes stability as a soft constraint, which is automatically relaxed when the safety constraint is active, without the need for parameter tuning or optimization. We study the closed-loop system behavior with the CLF-CBF-DCP controller and identify conditions on the existence of local equilibria. Although in certain cases the controller yields undesirable local equilibria, those can be confined to a small subset of the safe set boundary by proper choice of the control parameters. Then, our method can avoid undesirable equilibria that CLF-CBF quadratic programming techniques encounter.