Editing Model predictive control (section)

== Explicit MPC ==
Explicit MPC (eMPC) allows fast evaluation of the control law for some systems, in stark contrast to the online MPC. Explicit MPC is based on the [[parametric programming]] technique, where the solution to the MPC control problem formulated as optimization problem is pre-computed offline.<ref>{{Cite journal |last1=Bemporad |first1=Alberto |last2=Morari |first2=Manfred |last3=Dua |first3=Vivek |last4=Pistikopoulos |first4=Efstratios N. |title=The explicit linear quadratic regulator for constrained systems |journal=Automatica |volume=38 |issue=1 |pages=3–20 |doi=10.1016/s0005-1098(01)00174-1 |year=2002 }}</ref> This offline solution, i.e., the control law, is often in the form of a [[Piecewise linear function|piecewise affine function]] (PWA), hence the eMPC controller stores the coefficients of the PWA for each a subset (control region) of the state space, where the PWA is constant, as well as coefficients of some parametric representations of all the regions. Every region turns out to geometrically be a [[convex polytope]] for linear MPC, commonly parameterized by coefficients for its faces, requiring [[Quantization (signal processing)|quantization]] [[accuracy]] analysis.<ref>{{Cite book |doi=10.1109/CDC.2015.7402565 |arxiv=1509.02840 |isbn=978-1-4799-7886-1 |bibcode=2015arXiv150902840K |chapter=Explicit model predictive control accuracy analysis |title=2015 54th IEEE Conference on Decision and Control (CDC) |pages=2389–2394 |year=2015 |last1=Knyazev |first1=Andrew |last2=Zhu |first2=Peizhen |last3=Di Cairano |first3=Stefano |s2cid=6850073 }}</ref> Obtaining the optimal control action is then reduced to first determining the region containing the current state and second a mere evaluation of PWA using the PWA coefficients stored for all regions. If the total number of the regions is small, the implementation of the eMPC does not require significant computational resources (compared to the online MPC) and is uniquely suited to control systems with fast dynamics.<ref>{{Cite journal |last1=Klaučo |first1=Martin |last2=Kalúz |first2=Martin |last3=Kvasnica |first3=Michal |title=Real-time implementation of an explicit MPC-based reference governor for control of a magnetic levitation system |journal=Control Engineering Practice |volume=60 |pages=99–105 |doi=10.1016/j.conengprac.2017.01.001 |year=2017 }}</ref> A serious drawback of eMPC is [[exponential growth]] of the total number of the control regions with respect to some key parameters of the controlled system, e.g., the number of states, thus dramatically increasing controller memory requirements and making the first step of PWA evaluation, i.e. searching for the current control region, computationally expensive.