Economic MPC without terminal constraints: Gradient-correcting end penalties enforce asymptotic stability

In recent years, economic MPC (EMPC) has gained popularity due to the promise of increasing performance by directly optimizing the performance index rather than tracking a given steady state. Moreover, EMPC formulations without terminal cost nor constraints are appealing for the simplicity of implementation. However, the stability and convergence analysis for such formulations is rather involved and so far only practical stability (in discrete time), respectively, practical convergence (in sampled-data continuous time) has been proven; i.e., convergence to a horizon-dependent neighborhood of the optimal steady state. In this paper, we prove that, whenever the cost has a non-zero gradient at the optimal steady-state and the MPC formulation satisfies a regularity assumption, nominal stability to the economic optimum cannot be achieved. Consequently, the average performance of EMPC is bound to be worse than that of tracking MPC. We propose to solve this problem by introducing a linear terminal penalty correcting the gradient at steady state. We prove that this simple correction enforces uniform exponential stability of the economically optimal steady state. We illustrate our findings in simulations using three examples.