Control Barrier Functions-based Semi-Definite Programs (CBF-SDPs): Robust Safe Control For Dynamic Systems with Relative Degree Two Safety Indices

TL;DR

This paper develops a convex semi-definite programming approach for robust safe control of dynamic systems with relative degree two safety indices, accounting for parametric uncertainties and enabling online solutions.

Contribution

It introduces SDP formulations for robust control barrier functions with relative degree two, using duality and lossless s-procedure methods, for systems with parametric uncertainty.

Findings

SDP formulations are equivalent via duality and lossless s-procedure.

The approach handles parametric uncertainties in real-time control.

Provides a tractable convex optimization framework for safety-critical systems.

Abstract

In this draft article, we consider the problem of achieving safe control of a dynamic system for which the safety index or (control barrier function (loosely)) has relative degree equal to two. We consider parameter affine nonlinear dynamic systems and assume that the parametric uncertainty is uniform and known a-priori or being updated online through an estimator/parameter adaptation law. Under this uncertainty, the usual CBF-QP safe control approach takes the form of a robust optimization problem. Both the right hand side and left hand side of the inequality constraints depend on the unknown parameter. With the given representation of uncertainty, the CBF-QP safe control ends up being a convex semi-infinite problem. Using two different philosophies, one based on weak duality and another based on the Lossless s-procedure, we arrive at identical SDP formulations of this robust CBF-QP problem. Thus we show that the problem of computing safe controls with known parametric uncertainty can be posed as a tractable convex problem and be solved online. (This is work in progress).