Safe-by-Design: Approximate Nonlinear Model Predictive Control with Real Time Feasibility

This paper establishes relationships between continuous-time, receding horizon, nonlinear model predictive control (MPC) and control Lyapunov and control barrier functions (CLF/CBF). We show that, if the cost function "behaves well" for points in the terminal set, then the optimal value function and the feasible set, respectively, define a compatible CLF/CBF pair on the MPC's region of attraction. We then proceed to prove that any approximation of the value function and the feasible set also define a CLF/CBF pair, as long as those approximations satisfy the same "well behavedness" condition; and that a feasible state feedback can be computed by solving an infinitesimal version of the MPC problem. This methodology permits the formulation of continuous-time small-sized quadratic programs for feedback and enables approximate solutions of the nonlinear model predictive controller with theoretical safety and convergence guarantee. Finally, we demonstrate the effectiveness of the proposed approach when compared to other constrained control techniques through numerical experiments for nonlinear constrained spacecraft control.