Convex Co-Design of Mixed-Relative Degree Control Barrier Functions and Feedback Controllers for Linear Systems

Control Barrier Functions (CBFs) have been proposed as an efficient tool for safe control design. Given an initial set, a safe set and system dynamics, designing a candidate CBF is known to be NP-hard in general. In this paper, we propose a convex design method for linear dynamical systems under mild assumptions on the forms of the initial and the safe set. A CBF and an associated safe feedback controller can be co-designed by solving a single semi-definite program. Our method can handle mixed- (high-) relative degree problems directly, without the need to use backstepping or other methods. The efficacy of the proposed method is demonstrated on two different numerical examples.