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In this draft article, we consider the problem of achieving safe control of adynamic system for which the safety index or (control barrier function(loosely)) has relative degree equal to two. We consider parameter affinenonlinear dynamic systems and assume that the parametric uncertainty is uniformand known a-priori or being updated online through an estimator/parameteradaptation law. Under this uncertainty, the usual CBF-QP safe control approachtakes the form of a robust optimization problem. Both the right hand side andleft hand side of the inequality constraints depend on the unknown parameter.With the given representation of uncertainty, the CBF-QP safe control ends upbeing a convex semi-infinite problem. Using two different philosophies, onebased on weak duality and another based on the Lossless s-procedure, we arriveat identical SDP formulations of this robust CBF-QP problem. Thus we show thatthe problem of computing safe controls with known parametric uncertainty can beposed as a tractable convex problem and be solved online. (This is work inprogress).

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