This paper presents an approach to dual robust nonlinear model predictive control (NMPC). Dual control is traditionally formulated as a technique that seeks to solve the trade-off between probing actions, which result in a better estimation of the unknown parameters, and the optimal operation of the uncertain dynamic system. We propose a dual robust NMPC method based on the multi-stage approach that represents the uncertainty as a scenario tree of its possible realizations. We achieve a dual control formulation by taking explicitly into account the reduction of the uncertainty that future probing actions would provide over the prediction horizon. Using such formulation, our approach does not require any a priori decision on the relative importance of the probing actions with respect to the optimal operation of the system, as proposed in some recent approaches. The results obtained for a chemical engineering example show the advantages of using a dual NMPC approach based on multi-stage NMPC over the sequential use of parameter estimation with a posteriori approximation of the covariance of the parameter estimates.

We address the question of how to reduce the inevitable loss of performance that is incurred by robust multi-stage NMPC due to the lack of knowledge compared to the case where the exact plant model (no uncertainty) is available. Multi-stage NMPC in the usual setting over-approximates a continuous parametric uncertainty set by a box and includes the corners of the box and the center point into the scenario tree. If the uncertainty set is not a box, this augments the uncertainty set and results in a performance loss. In this paper, we propose to mitigate this problem by two different approaches where the scenario tree of the multi-stage NMPC is built using sigma points. The chosen sigma points help to capture the true mean and covariance of the uncertainty set more precisely. The first method computes a box over-approximation of the reachable set of the system states whereas the second method computes a box over-approximation of the reachable set of the constraint function using the unscented transformation. The advantages of the proposed schemes over the traditional multi-stage NMPC are demonstrated using simulation studies of a simple semi-batch reactor and a more complex industrial semi-batch polymerization reactor benchmark example.