Editing Model predictive control (section)

==Nonlinear MPC==

Nonlinear model predictive control, or NMPC, is a variant of model predictive control that is characterized by the use of nonlinear system models in the prediction. As in linear MPC, NMPC requires the iterative solution of optimal control problems on a finite prediction horizon. While these problems are convex in linear MPC, in nonlinear MPC they are not necessarily convex anymore. This poses challenges for both NMPC stability theory and numerical solution.<ref>An excellent overview of the state of the art (in 2008) is given in the proceedings of the two large international workshops on NMPC, by Zheng and Allgöwer (2000) and by Findeisen, Allgöwer, and Biegler (2006).</ref>

The numerical solution of the NMPC optimal control problems is typically based on direct optimal control methods using Newton-type optimization schemes, in one of the variants: [[Shooting method|direct single shooting]], [[direct multiple shooting method]]s, or [[Collocation method|direct collocation]].<ref>{{cite journal|first=John D. |last=Hedengren |first2=Reza |last2=Asgharzadeh Shishavan |first3=Kody M. |last3=Powell |first4=Thomas F. |last4=Edgar |title=Nonlinear modeling, estimation and predictive control in APMonitor |journal=Computers & Chemical Engineering |year=2014 |volume=70 |issue=5 |pages=133–148 |doi=10.1016/j.compchemeng.2014.04.013 |s2cid=5793446 |url=https://scholarsarchive.byu.edu/facpub/1667 |url-access=subscription }}</ref> NMPC algorithms typically exploit the fact that consecutive optimal control problems are similar to each other. This allows to initialize the Newton-type solution procedure efficiently by a suitably shifted guess from the previously computed optimal solution, saving considerable amounts of computation time. The similarity of subsequent problems is even further exploited by path following algorithms (or "real-time iterations") that never attempt to iterate any optimization problem to convergence, but instead only take a few iterations towards the solution of the most current NMPC problem, before proceeding to the next one, which is suitably initialized; see, e.g.,..<ref>{{cite journal |last=Ohtsuka |first=Toshiyuki |title=A continuation/GMRES method for fast computation of nonlinear receding horizon control |journal=Automatica |year=2004 |volume=40 |issue=4 |pages=563–574 |doi=10.1016/j.automatica.2003.11.005 }}</ref> Another promising candidate for the nonlinear optimization problem is to use a randomized optimization method. Optimum solutions are found by generating random samples that satisfy the constraints in the solution space and finding the optimum one based on cost function.<ref>{{cite journal |last1=Muraleedharan |first1=Arun |title=Real-Time Implementation of Randomized Model Predictive Control for Autonomous Driving |journal=IEEE Transactions on Intelligent Vehicles |year=2022 |volume=7 |issue=1 |pages=11–20 |doi=10.1109/TIV.2021.3062730 |s2cid=233804176 |doi-access=free }}</ref>

While NMPC applications have in the past been mostly used in the process and chemical industries with comparatively slow sampling rates, NMPC is being increasingly applied, with advancements in controller hardware and computational algorithms, e.g., [[preconditioning]],<ref>{{cite book |doi=10.1109/ACC.2016.7526060 |arxiv=1512.00375 |isbn=978-1-4673-8682-1 |chapter=Sparse preconditioning for model predictive control |title=2016 American Control Conference (ACC) |pages=4494–4499 |year=2016 |last1=Knyazev |first1=Andrew |last2=Malyshev |first2=Alexander |s2cid=2077492 }}</ref> to applications with high sampling rates, e.g., in the automotive industry, or even when the states are distributed in space ([[Distributed parameter system]]s).<ref>{{cite journal |first=Míriam R. |last=García |first2=Carlos |last2=Vilas |first3=Lino O. |last3=Santos |first4=Antonio A. |last4=Alonso |title=A Robust Multi-Model Predictive Controller for Distributed Parameter Systems |journal=Journal of Process Control |year=2012 |volume=22 |issue=1 |pages=60–71 |doi=10.1016/j.jprocont.2011.10.008 |url=http://www.hamilton.ie/miriam/publications/(2011)_Garcia_Vilas_Santos_Alonso_JJPC_bw.pdf }}</ref> As an application in aerospace, recently, NMPC has been used to track optimal terrain-following/avoidance trajectories in real-time.<ref>{{cite journal |first=Reza |last=Kamyar |first2=Ehsan |last2=Taheri |title=Aircraft Optimal Terrain/Threat-Based Trajectory Planning and Control |journal=Journal of Guidance, Control, and Dynamics |year=2014 |volume=37 |issue=2 |pages=466–483 |doi=10.2514/1.61339 |bibcode=2014JGCD...37..466K }}</ref>