Model Predictive Control Barrier Functions: Guaranteed Safety with Reduced Conservatism and Shortened Horizon

In this study, we address the problem of safe control in systems subject to state and input constraints by integrating the Control Barrier Function (CBF) into the Model Predictive Control (MPC) formulation. While CBF offers a conservative policy and traditional MPC lacks the safety guarantee beyond the finite horizon, the proposed scheme takes advantage of both MPC and CBF approaches to provide a guaranteed safe control policy with reduced conservatism and a shortened horizon. The proposed methodology leverages the sum-of-square (SOS) technique to construct CBFs that make forward invariant safe sets in the state space that are then used as a terminal constraint on the last predicted state. CBF invariant sets cover the state space around system fixed points. These islands of forward invariant CBF sets will be connected to each other using MPC. To do this, we proposed a technique to handle the MPC optimization problem subject to the combination of intersections and union of constraints. Our approach, termed Model Predictive Control Barrier Functions (MPCBF), is validated using numerical examples to demonstrate its efficacy, showing improved performance compared to classical MPC and CBF.