Editing Model predictive control (section)

===Theory behind MPC===

[[Image:MPC scheme basic.svg|thumb|A discrete MPC scheme.]] MPC is based on iterative, finite-horizon optimization of a plant model.  At time <math>t</math> the current plant state is sampled and a cost minimizing control strategy is computed (via a numerical minimization algorithm) for a relatively short time horizon in the future: <math>[t,t+T]</math>.  Specifically, an online or on-the-fly calculation is used to explore state trajectories that emanate from the current state and find (via the solution of [[Euler–Lagrange equation]]s) a cost-minimizing control strategy until time <math>t+T</math>. Only the first step of the control strategy is implemented, then the plant state is sampled again and the calculations are repeated starting from the new current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called '''receding horizon control'''.  Although this approach is not optimal, in practice it has given very good results. Much academic research has been done to find fast methods of solution of Euler–Lagrange type equations, to understand the global stability properties of MPC's local optimization, and in general to improve the MPC method.<ref>Nikolaou, Michael; "Model predictive controllers: A critical synthesis of theory and industrial needs", ''Advances in Chemical Engineering'', volume 26, Academic Press, 2001, pages 131-204</ref><ref>{{Cite journal |last1=Berberich |first1=Julian |last2=Kohler |first2=Johannes |last3=Muller |first3=Matthias A. |last4=Allgöwer |first4=Frank |date=2022 |title=Linear Tracking MPC for Nonlinear Systems—Part I: The Model-Based Case |url=https://ieeexplore.ieee.org/document/9756294 |journal=IEEE Transactions on Automatic Control |volume=67 |issue=9 |pages=4390–4405 |doi=10.1109/TAC.2022.3166872 |arxiv=2105.08560 |s2cid=234763155 |issn=0018-9286}}</ref>