(684b) On the Robustness of Economic Nonlinear Model Predictive Control

Model predictive control (MPC) has seen a variety of applications, and its advantages include a natural way of handling inequality constraints and multiple-input-multiple-output systems due to the optimization formulation of the problem. A survey of industrial uses of MPC is given in [10], and a thorough treatment of MPC is given in [11]. Nonlinear model predictive control (NMPC) has the added advantage of being able to use a detailed first-principles dynamic model in order to provide accuracy across a wide range of states, and the fundamentals of NMPC are given in [4]. Furthermore, if a sensible initialization strategy is used, an exact solution of the associated nonlinear programming (NLP) problem is not required, as shown in [9,15,13,16].

Input-to-state stability (ISS) is used to extend stability analysis to systems with uncertainty. The property was originally described for continuous time (CT) systems in [12] and was extended to discrete time (DT) systems in [7]. Furthermore, ISS has been proposed as a framework for NMPC [8], and it provides a very convenient and natural way of thinking about robust stability. Previous work [3] provides a method for calculating ISS bounds. Input-to-state practical stability allows for a slight modification to ISS in order to be more widely applicable.

Economic NMPC allows for the use of more general objective functions in the controller. The difficulty is that standard stability results of tracking MPC cannot be used. Originally, asymptotic stability was shown by using strict dissipativity of the rotated cost function [2], but this property is not easy to show in general. However, it is possible to show a stronger property, strong convexity of the rotated cost function, by including regularization terms in the objective function. It has been shown that a sufficiently large regularization term (e.g., a tracking term) may be added to the objective in order to force the rotated cost function to be strongly convex, which leads to asymptotic stability [6]. The drawbacks of this method are that regularization weights may be difficult to calculate and very conservative, limiting economic gains. In [14] it is proposed to replace the regularization term in the objective with a stabilizing inequality constraint derived from the inherent robustness margin of an auxiliary, asymptotically stable MPC controller.  It is demonstrated that this approach (that we call eMPC-sc) provides flexibility to optimize economic performance. Moreover, it is shown that this approach is related to regularization-based Lyapunov-based MPC approaches, but the cumbersome calculations required to find regularization terms are not required.  The eMPC approach differs from the Lyapunov-based approach [5] in that feasibility of the stabilizing constraint can be guaranteed directly while in the Lyapunov approach the feasible set for the states needs to be adjusted. Also, a formulation for the tracking case with uncertainty with a similar stabilizing constraint is shown in [1]. Here, the stabilizing constraint is required to be feasible for a list of possible plant realizations, while we only require feasibility of the terminal constraint for the nominal plant.

In this work, we extend eMPC with a stabilizing constraint to the nonlinear case with uncertainty and state constraints. We show that a system to which this controller is applied is ISpS, which requires a slight modification to existing theory on ISpS. We show simulation results on a CSTR, as well as on a system of two distillation columns in series.

References

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