Feedback min‐max model predictive control using a single linear program: robust stability and the explicit solution

Abstract

SUMMARYIn this paper we introduce a new stage cost and show that the use of this cost allows one to formulate a robustly stable feedback min-max model predictive control problem that can be solved using a single linear program. Furthermore, this is a multi-parametric linear program, which implies that the optimal control law is piecewise affine, and can be explicitly pre-computed so that the linear program does not have to be solved on-line. We assume that the plant model is known, is discrete-time and linear time-invariant, is subject to unknown but bounded state disturbances and that the states of the system are measured. Two numerical examples are presented; one of these is taken from the literature, so that a direct comparison of solutions and computational complexity with earlier proposals is possible. This is a preprint of an article published in International Journal of Robust and Nonlinear Control

Model-based multi-parametric programming strategies towards the integration of design, control and operational optimization

“…In mp-MPC, e orts have been made to develop a multi-parametric model-based controller that guarantees performance while satisfying the constraints, termed robust mp-MPC. While early approaches focused on (i) additive disturbances [6,259,286,306], (ii) model uncertainties [35,279], (iii) Min-max robust mp-MPC [6,35,81,172] and (iv) linear input/output models [82,259], recent advances point towards a more general approach for robust mp-MPC [188,279]. The key idea is thereby (i) a dynamic programming reformulation of the original problem, (ii) the formulation of the robust counter-part and (iii) the solution of the resulting mp-P problem.…”

Predictive control of demand side units participating in the primary frequency reserve market

“…While this formulation takes the future disturbances into account in the optimization, it suffers from often being conservative [26]. The reason for this conservatism is that this strategy is open-loop within the horizon, in the sense that the controller does not take into account that at the next time sample, more information will be available and the optimization will be redone including this new information.…”

“…This means that we do not commit to a certain control input sequence for the whole control horizon; instead we choose a control policy which will allow different control sequences depending on the realizations of the future disturbances. Hereby the controller will achieve a closed-loop behavior, where we allow recourse as more information becomes available (see, e.g., [26], [22], [27]). Note that the terminology of open-loop MPC vs. closed-loop MPC is adopted from the literature, e.g., the references above.…”

Robust Model Predictive Controller Design: Finite and Infinite Horizon

“…They are based on the infinite prediction or optimization horizon with the on-line LMI optimization [Kothare et al, 1996], the min-max optimization with a terminal constraint in an invariant set Mayne, 1998, Lofberg, 2003], the off-line LMI optimization to calculate a sequence of output feedback laws and the on-line selection of the appropriate law [Ding et al, 2008] or the robust tube-based control design approach [Mayne et al, 2005]. The problems of the computational complexity and the robust stability are solved in the robust explicit MPC [Kerrigan and Maciejowski, 2004]. The dynamic output feedback robust model predictive controller for a system with both polytopic uncertainty and bounded disturbance is addressed in the paper [Ding, 2013].…”