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Multi-stage robust optimizing control

Multi-stage nonlinear model-predictive control

Nonlinear Model-Predictive Control (NMPC) is a model-based strategy to achieve economic or tracking objectives for nonlinear systems while satisfying the process constraints. If the model does not describe the plant behavior accurately, there is the necessity of a robust control scheme which is not overly conservative. Multi-stage NMPC is a non-conservative real-time-implementable robust control scheme which guarantees constraint satisfaction for nonlinear systems under uncertainty. The evolution of the uncertainty is modeled as a scenario tree (see the figure.). The important aspect of the approach is that it considers explicitly that new measurement information will be available at every stage of the prediction and that the decisions taken at every stage can be adjusted accordingly and thus can act as recourse to counteract the effects of the uncertainties. This improves the performance and reduces the conservativeness compared to traditional robust NMPC schemes.

When the state vector is not measured at each sampling interval but only noisy measurements of some outputs are available, additional uncertainty about the current state as well as inexact information about the future states must be taken into account. The presence of estimation error poses an interesting challenge of not only predicting the evolution of the plant under uncertainty but also the evolution of the plant under the control actions which result due to the presence of current and future estimation errors. With the assumption that the plant is observable and the innovations sequence is bounded, the samples of the innovations are modeled as new scenarios in the scenario tree in addition to the parametric uncertainties and the observer equations such as the EKF/UKF are used to predict the evolution of the future states/estimates. The proposed approach is shown to be robust to plant-model mismatch and to estimation errors.
If the values of a subset of the uncertain parameters can be estimated online using the state or parameter estimation techniques, the scenario tree can be adapted. Instead of having a global bound on the uncertain parameters, a less conservative local estimate of the bound of the (possibly time-varying) parameters can be obtained. The resulting innovations caused by the estimation errors in addition to the innovations generated by the possible parameter changes then have to be considered. The observer equations in the controller can be used to predict the future parameter estimates in addition to the states. Our approach considers the fact that not only the states but also the parameters are estimated at every sampling time and the future control inputs can act as a recourse for the effect of the changes in the parametric uncertainties along with the evolution of the plant at those times, resulting in further improvement in controller performance.
The dyn group has developed a modular and numerically efficient implementation of multistage NMPC in the framework of DO-MPC. It is an open source free software and can be downloaded from: https://github.com/do-mpc/do-mpc.


Lucia, S., Andersson, J., Brandt, H., Diehl, M., Engell, S.: Handling uncertainty in economic nonlinear model predictive control: a comparative case-study. J. of Process Control, 24(8), 2014, 1247–1259.

Subramanian, S., Lucia, S., Engell, S.: Economic multi-stage output nonlinear model predictive control. In: Proc. of the 2014 IEEE Multi-Conference on Systems and Control, 2014.

Lucia, S., Tatulea-Codrean, A., Schoppmeyer, C., Engell, S.: An Environment for the Efficient Testing and Implementation of Robust NMPC. In: Proc. IEEE Multi-conference on Systems and Control, Antibes, 2014.

Subramanian, S., Lucia, S., Engell, S.: Adaptive Multi-stage Output Feedback NMPC using the Extended Kalman Filter for time varying uncertainties applied to a CSTR. Proc. of the 5th IFAC Conference on Nonlinear Model Predictive Control, Sevilla, 2015.