Continuous-Time Nonlinear Model Predictive Control based on Pontryagin Minimum Principle and Penalty Functions

Pontryagin's Minimum Principle (PMP) is a powerful tool for solving Nonlinear Model Predictive Control problems (NMPC), enabling the handling of time-varying input constraints and cost functions. However, applying PMP encounters challenges when state constraints must be satisfied. This arises because the optimal trajectory often requires a blend of unconstrained and constrained arcs with unknown junction points. To address this issue, relaxation methods are frequently explored, where state constraints are replaced with penalty functions. The contributions of this paper are as follows. First, a method of penalty functions allowing for coping with soft state constraints is examined. We prove the recursive feasibility of this method and demonstrate its efficiency in a numerical example. Second, the finite-time practical stability for the optimal reference tracking NMPC problem is addressed. By appropriately choosing the terminal cost, one can guarantee the convergence of the state vector to a predefined neighborhood of the target state.