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

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).