Stabilization with guaranteed safety using Barrier Function and Control Lyapunov Function

Abstract The stabilization problem with guaranteed safety is motivated in the case that the systems might be at risk whenever certain unsafe states are reached. Consequently the design of controller must comply with state constraints and avoid unsafe states. This paper proposes a control method for stabilization with guaranteed safety for affine systems. Firstly, a novel Barrier Function (BF) is presented for bounded convex unsafe state set. The design of BF is only based on the unsafe state set and independent of Control Lyapunov Function (CLF). So besides its simpleness and ease of implementation, the presented BF needs weak constraint conditions and admits a larger class of functions. Based on the BF and a CLF, a necessary and sufficient condition is presented for stabilization with guaranteed safety for affine systems. Further, the proposed method can accommodate the case of multiple different sets of unsafe states immediately by only adding the corresponding BFs to the CLF and need not modify the design for the existing unsafe state sets. Finally, the obtained results are extended to the case that the unsafe set is a class of unbounded nonconvex set. Illustrative examples are given to show the proposed results.