A Model predictive control scheme with ultimate bound for economic optimization

Abstract

Abstract-This paper presents a Model Predictive Control (MPC) scheme for nonlinear continuous-time systems where an economic stage cost, which is not a measure of the distance to a desired set point, is combined with a classic stabilizing stage cost. The associated control strategy leads to a closedloop behavior that compromises, in a seamless way, between the convergence of the closed-loop state trajectory to a given steadystate and the minimization of the economic cost. More precisely, we derive a set of sufficient conditions under which the closedloop state trajectory is ultimately bounded around the desired steady-state, with the size of the bound being proportional to the strength of the economic cost. Numerical results show the effectiveness of the proposed scheme on a target estimation and tracking control problem.

Economic and Distributed Model Predictive Control: Recent Developments in Optimization-Based Control

“…In [75], [114], [115] it was shown how Problem 1 can be modified such that the resulting closed-loop system satisfies these constraints. The articles [116], [117] consider combined economic and tracking cost functions and establish closed-loop convergence or ultimate boundedness depending on how the economic and tracking part are weighted. For the case where steady-state operation is optimal, a tracking MPC problem is formulated in [118], [119] which is shown to be (locally) equivalent to economic MPC.…”

An energy efficient trajectory tracking controller for car-like vehicles using Model Predictive Control

“…ECONOMIC MPC The EMPC is designed by augmenting the stabilizing stage cost l s (·) of the stable MPC obtained in Section IV with an economic stage cost l e (·). In order to use the results of [9], reported in Section II-C, and thus guarantee ultimate boundedness of the error trajectories, we need to fulfill Assumption 6. Observe that Assumptions 1-3 are satisfied with the designed stabilizing stage cost (7), terminal cost (8) and terminal set (13).…”

“…The following assumption and theorem are taken from [9]. Assumption 6 (Bound on the Economic Stage Cost): The norm of the economic stage cost l e (·), evaluated along the closed-loop state and input trajectories, is uniformly bounded by a strictly positive constant value, i.e.…”

“…Thereafter the results from [9] are reported in order to combine the stage cost designed using [10] with an economic stage cost, whilst still guaranteeing ultimate boundedness of the closedloop state trajectories.…”

“…This paper adopts a dual target approach, with the aim to jointly minimize the vehicle consumption and the tracking error. The design is performed considering an under-actuated car-like vehicle and exploiting the result [9] to guarantee convergence of the tracking error to an ultimate bound with size proportional to the desired energy saving.…”