Reformulating Nonrobust Nonlienar Model Predictive Control, D. Griffith and L.T. Biegler
Model predictive control (MPC) is an optimization based form of control that functions by solving successive optimization problems online, and it is particularly suited to multiple-input-multiple-output systems with inequality constraints due to the mathematic programming formulation of the problem. Nonlinear model predictive control (NMPC) has the added benefit of being able to take advantage of a fully nonlinear dynamic model in order to provide accuracy across a wider range of states. These traits make NMPC very attractive to chemical engineers, since chemical processes with many states, many controls, operational constraints, and significant nonlinearities are common.
Furthermore, applying NMPC to very large problems with significant nonlinear programming (NLP) problem solve time is possible thanks to advances in NLP sensitivity techniques and the development of advanced step NMPC (asNMPC)  and advanced multi-step NMPC (amsNMPC) . These methods rely on solving the NLP problems one or more time steps in advance and updating their solutions with sensitivity information. In fact, the sensitivity updates are nearly instantaneous, so systems of arbitrary size may be controlled efficiently if enough processors are used.
Much academic research has centered on showing robust stability of the various forms of NMPC. This is typically done by using the optimal objective function value of the controller as a Lyapunov function and subsequently applying a discrete-time (DT) Lyapunov theorem to show that NMPC is input-to-state stable (ISS). The main assumptions of this analysis are on the regularity conditions of the NLP problems. If the NLP problems satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ), and the general strong second order sufficient condition (GSSOSC), then it can be shown that the Lyapunov function is uniformly continuous in the state, which is the main feature necessary for showing ISS. Similar results also exist for asNMPC and amsNMPC.
However, examples exist in the literature where the necessary conditions on the NLP are violated and NMPC becomes nonrobust. This can occur even if the NLP problems are always feasible and the disturbances are arbitrarily small, so such instances are large obstacles to the implementation of NMPC. In order to resolve this, we use a reformulation  of the NLP problems utilizing soft constraints penalized in the objective function in place of hard inequality constraints. It can be shown that this reformulation satisfies the necessary conditions of the NLP problems, and the system therefore becomes robustly stable.
In this work, we consider three pathological examples from literature . We explicitly show the traits of the NLP that lead to nonrobustness, reformulate the problem with soft inequality constraints and objective function penalties, and show that the reformulation satisfies the necessary assumptions on the NLP problems. Smooth reformulations and approximations are necessary in some cases to make the problems amenable to NLP. Furthermore, simulation results are shown for each example under multiple conditions, including asNMPC and amsNMPC. These results include comparisons of control horizon length and slack variable weight.
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