(665c) Lyapunov Stability Analysis of Economic NMPC

The ultimate goal of any operation strategy for a process plant is to make profit. Traditionally this goal could be achieved by a two-layer control system, where the upper layer solves an optimization problem aiming at maximizing profit to get the optimal setpoints for the controlled variables in the layer below. The lower layer then keeps the controlled variables at their given setpoints. However, there are some problems with this two-layer structure. For example, the setpoints calculated by the upper layer might not be reachable by the lower layer. One of the solutions is to combine these two layers and include the economic criterion directly into the cost function of the lower layer controller when an optimization-based controller such as MPC is used. This approach is often referred to as Economic MPC, and has been gaining increased interest in recent years.

In this study we would like to analyze Lyapunov stability properties of Economic NMPC. Lyapunov stability properties of setpoint tracking NMPC problems have been well established with the assumption that the stage cost is minimal at the desired equilibriums points so that the objective function could be used as a Lyapunov function. However, since Economic NMPC controller generally does not attain a minimum at the steady state optimal operation point, the objective function cannot be used as a Lyapunov function directly. 

As described in previous studies [1, 2], we consider a modified analysis through a transformed rotated stage cost. This is generated by first subtracting the optimal steady state from the variables and stage cost, and then augmenting the transformed cost to form the rotated stage cost. As shown in [1, 2] the modified problem has the same solution as the original problem, if the modified stage and terminal costs are strongly convex. Moreover, strong convexity can always be ensured through the addition of quadratic regularization terms. In this talk we establish stability results for three different terminal criteria: (1) fixed terminal point with finite cost function, (2) fixed terminal point with infinite horizon cost function, and (3) terminal set and a terminal cost for a finite horizon cost function. Moreover, we provide a simple constructive method based on Gershgorin bounds for regularization that ensure strongly convex stage costs where stability is guaranteed. We demonstrate these concepts on a detailed case study with two distillation columns in sequence.

1. M. Diehl, R. Amrit, J. B. Rawlings, A Lyapunov function for economic optimizing model predictive control, IEEE Transactions on Automatic Control 56 (2011) 703–707

2. J. Jäschke, X. Yang and L. T. Biegler, Fast Economic Model Predictive Control Based on NLP-Sensitivities, Journal of Process Control, in press (2014)