Asymptotic Stability of Economic NMPC: The Importance of Adjoints

Abstract

Recently, it has been shown in a sampled-data continuous-time setting that under certain regularity assumptions a simple linear end penalty enforces exponential stability of Economic (nonlinear) Model Predictive Control (EMPC) without terminal constraints. This paper investigates the same framework in the discrete-time case, i.e. we establish sufficient conditions for asymptotic stability of the optimal steady state under an EMPC scheme without terminal constraints. The key ingredient is a linear end penalty that can be understood as a gradient correction of the stage cost by means of the adjoint/dual variable of the underlying steady-state optimization problem. Although almost all stability proofs for EMPC focus on primal variables, our developments elucidate the importance of the adjoints for achieving asymptotic stability without terminal constraints. Moreover, we propose an adaptive gradient correction strategy which alleviates the need for solving explicitly the steady-state optimization. Finally, we draw upon two simulation examples to illustrate our results.