Continuous-time nonlinear model predictive control based on Pontryagin Minimum Principle and penalty functions

Abstract

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 neighbourhood of the target state.

Keywords:

• Nonlinear control systems
• predictive control
• stability
• constrained control

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Henceforth, all signals defined on the predictive time interval are denoted by symbols with ˆ, e.g. $\hat{x},\hat{u}$ etc., in order to distinguish them from the functions defined on the actual time axis.

2 For simplicity, here we assume that the reference is constant. In practice, the controller can be also applied to slowly changing reference signals.

3 Proving or disproving the recursive feasibility it a highly non-trivial task even for convex MPC problems (Löfberg, 2012).

4 Notice that we do not guarantee the uniqueness of a minimizer in each OCP. In some situations, e.g. where the plant is linear and $\varkappa =0$, such a uniqueness is guaranteed by the strong convexity of the cost function. In practice, one has to choose the solution returned by the solver of TPBVP problems in use.

5 Notice that if the optimal trajectories stay far from the boundary of the state constraint set $X_C$, then typically $\varkappa (x(t))\ll 1$, and one can prove that the presence of penalty terms have small effect on the solution. We, however, omit the relevant analysis to keep things maximally simple.

6 Notice that if the optimal trajectories stay far from the boundary of the state constraint set $X_C$, then typically $\varkappa (x(t))\ll 1$, and one can prove that the presence of penalty terms have small effect on the solution. We, however, omit the relevant analysis to keep things maximally simple.

7 One can always get rid of weight matrices by a linear change of variables.

8 Recall that the minimum exists in view of Lemma 5.2

Additional information

Funding

M. Pagone is funded by the initiative ‘PNRRNextGenerationEU,’ Mission 4, ‘Sustainable Mobility Center (Centro Nazionale per la Mobilità Sostenibile – CNMS),’ Spoke n.2 – Sustainable road vehicle – with grant agreement no. CN 00000023. This manuscript reflects only the authors' views and opinions, neither the European Union nor the European Commission can be considered responsible for them.