Editing Model predictive control (section)

== Robust MPC ==

Robust variants of model predictive control are able to account for set bounded disturbance while still ensuring state constraints are met. Some of the main approaches to robust MPC are given below.

* ''Min-max MPC''. In this formulation, the optimization is performed with respect to all possible evolutions of the disturbance.<ref>{{cite journal |doi=10.1109/9.704989 |title=Min-max feedback model predictive control for constrained linear systems |journal=IEEE Transactions on Automatic Control |volume=43 |issue=8 |pages=1136–1142 |year=1998 |last1=Scokaert |first1=Pierre O. M. |last2=Mayne |first2=David Q. }}</ref> This is the optimal solution to linear robust control problems, however it carries a high computational cost. The basic idea behind the min/max MPC approach is to modify the on-line "min" optimization to a "min-max" problem, minimizing the worst case of the objective function, maximized over all possible plants from the uncertainty set.<ref>{{Cite journal |date=1996-06-01 |title=Robustness of MPC-Based Schemes for Constrained Control of Nonlinear Systems |url=https://www.sciencedirect.com/science/article/pii/S1474667017586127 |journal=IFAC Proceedings Volumes |language=en |volume=29 |issue=1 |pages=5823–5828 |doi=10.1016/S1474-6670(17)58612-7 |issn=1474-6670 |last1=Nevistić |first1=Vesna |last2=Morari |first2=Manfred |url-access=subscription }}</ref>
*  ''Constraint Tightening MPC''. Here the state constraints are enlarged by a given margin so that a trajectory can be guaranteed to be found under any evolution of disturbance.<ref>{{cite journal |last=Richards |first=Arthur G. |last2=How |first2=Jonathan P. |title=Robust stable model predictive control with constraint tightening |journal=Proceedings of the American Control Conference |year=2006 }}</ref>
* ''Tube MPC''. This uses an independent nominal model of the system, and uses a feedback controller to ensure the actual state converges to the nominal state.<ref>{{cite journal |last=Langson |first=Wilbur |first2=Ioannis |last2=Chryssochoos |first3=Saša V. |last3=Raković |first4=David Q. |last4=Mayne |title=Robust model predictive control using tubes |journal=Automatica |year=2004 |volume=40 |issue=1 |pages=125–133 |doi=10.1016/j.automatica.2003.08.009 }}</ref> The amount of separation required from the state constraints is determined by the robust positively invariant (RPI) set, which is the set of all possible state deviations that may be introduced by disturbance with the feedback controller.
* ''Multi-stage MPC''. This uses a scenario-tree formulation by approximating the uncertainty space with a set of samples and the approach is non-conservative because it takes into account that the measurement information is available at every time stage in the prediction and the decisions at every stage can be different and can act as recourse to counteract the effects of uncertainties. The drawback of the approach however is that the size of the problem grows exponentially with the number of uncertainties and the prediction horizon.<ref>{{cite journal |last1=Lucia |first1=Sergio |last2=Finkler |first2=Tiago |last3=Engell |first3=Sebastian |title=Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty |journal=Journal of Process Control |date=2013 |volume=23 |issue=9 |pages=1306–1319 |doi=10.1016/j.jprocont.2013.08.008 }}</ref><ref>{{cite journal |last1=Lucia |first1=Sergio |last2=Subramanian |first2=Sankaranarayanan |last3=Limon|first3=Daniel |last4=Engell |first4=Sebastian |title=Stability properties of multi-stage nonlinear model predictive control|journal=Systems & Control Letters |date=2020 |volume=143 |issue=9 |pages=104743 |doi=10.1016/j.sysconle.2020.104743 |s2cid=225341650 }}</ref>
* ''Tube-enhanced multi-stage MPC''. This approach synergizes multi-stage MPC and tube-based MPC. It provides high degrees of freedom to choose the desired trade-off between optimality and simplicity by the classification of uncertainties and the choice of control laws in the predictions.<ref>{{cite journal |last1=Subramanian |first1=Sankaranarayanan |last2=Lucia |first2=Sergio |last3=Paulen |first3=Radoslav |last4=Engell |first4=Sebastian |title=Tube-enhanced multi-stage model predictive control for flexible robust control of constrained linear systems |journal=International Journal of Robust and Nonlinear Control |date=2021 |volume=31 |issue=9 |pages=4458–4487 |doi=10.1002/rnc.5486 |arxiv=2012.14848 |s2cid=234354708 }}</ref><ref>{{cite journal |last1=Subramanian |first1=Sankaranarayanan |last2=Abdelsalam |first2=Yehia |last3=Lucia |first3=Sergio |last4=Engell |first4=Sebastian |title=Robust Tube-Enhanced Multi-Stage NMPC With Stability Guarantees |journal=IEEE Control Systems Letters |date=2022 |volume=6 |pages=1112–1117 |doi=10.1109/LCSYS.2021.3089502 |s2cid=235799791 }}</ref>