Tracking MPC Tuning in Continuous Time: A First-Order Approximation of Economic MPC

Abstract

Economic MPC (EMPC) optimizes closed-loop performance by directly minimizing a given objective function, as opposed to Tracking MPC (TMPC) which instead penalizes deviations from a precalculated optimal reference. The main difference between the two approaches can be observed during transients, as the former always acts optimally, while the latter is only optimal when the reference is accurately tracked. Unfortunately, stability for EMPC is in general difficult to prove, as opposed to TMPC which builds on a rich theory. Additionally, many efficient algorithms are available for TMPC, while solving the EMPC problem can be much harder. In prior works 1, 2, a family of discrete-time TMPC schemes that provide approximate economic optimality has been developed in order to partially overcome these issues. In this paper, we aim at extending such a family of TMPC schemes to the continuous time case. Similarly to the discrete-time case, also in continuous-time we obtain a first-order approximation of the EMPC control law. We demonstrate the theory with a numerical example that confirms the first-order approximation and we show that our continuous-time formulation can be made equivalent to the discrete-time one.