Vector Control Lyapunov and Barrier Functions for Safe Stabilization of Interconnected Systems

In this paper, we investigate the safety and stabilization problems of interconnected nonlinear systems. In particular, different subsystems may involve potentially conflicting control objectives and distributed safety constraints. For this purpose, we first propose a novel vector control Lyapunov function (VCLF) to deal with the stabilization problem and vector control barrier functions (VCBFs) to resolve the safety problem. Both VCLF and VCBFs not only extend the control Lyapunov function and control barrier function from the scalar case to the vector case, but also allow for the decentralized controller design to reduce the conservatism caused by the existing methods based on scalar control functions. To address the safety and stabilization problems simultaneously, the proposed VCLF and VCBF are combined into a quadratic programming, which is in a decentralized manner and results in a decentralized optimization-based control approach. The decentralized controllers are derived explicitly in a closed form, which facilitates the analysis and shows the simultaneous guarantee of the safety and stabilization objectives. Finally, the derived results are illustrated via an example from the formation control problem of multirobot systems.