A Game-Theoretic Approach to Robust NMPC via Pontryagin’s Minimum Principle and Penalty Functions

In this paper, we propose a novel robust Nonlinear Model Predictive Control (RNMPC) strategy that combines the classic min-max formulation for robust optimization with game theory. The RNMPC control problem is defined as a zero-sum differential game, in which control input and disturbance act as opposing players. Such a problem is solved leveraging the Pontryagin's Minimum Principle (PMP), which recasts it as a two-point boundary value problem (TPBVP), which can be efficiently solved with a low computational burden. State constraints are incorporated within the RNMPC problem by including suitable penalty functions within the min-max stage cost. The optimal solution is computed as the Nash equilibrium (NE) of the differential game, of which we prove the existence and employ its structure to obtain a more numerically efficient version of the TPBVP. The effectiveness of the proposed RNMPC strategy is validated in simulation on the real-world case study of an unmanned ground vehicle (UGV), demonstrating its superiority over the non-robust case in both attaining the control task and delivering a more energy-efficient control action.