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1、February15,200622:41InternationalJournalofControlFoRM˙IJCInternationalJournalofControlVol.00,No.00,DDMonth200x,1–25ParametricOptimizationandOptimalControlusingAlgebraicGeometryMethodsIoannisA.Fotiou∗,1,PhilippRostalski∗,PabloA.Parrilo†andManfredMorari∗∗AutomaticControlLaboratorySwissFederalIn
2、stituteofTechnology(ETH),8092Zurich,Switzerland.†LaboratoryforInformationandDecisionSystemsMassachusettsInstituteofTechnology,Cambridge,MA02139-4307,USA.(v2.0releasedFebruary2006)Wepresenttwoalgebraicmethodstosolvetheparametricoptimizationproblemthatarisesinnonlinearmodelpredictivecontrol.Wec
3、onsiderconstraineddiscrete-timepolynomialsystemsandthecorrespondingconstrainedfinite-timeoptimalcontrolproblem.Thefirstmethodisbasedoncylindricalalgebraicdecomposition.ThesecondusesGr¨obnerbasesandtheeigenvaluemethodforsolvingsystemsofpolynomialequations.Bothmethodsaimatmovingmostofthecomputati
4、onalburdenassociatedwiththeoptimizationproblemoff-line,bypre-computingcertainalgebraicobjects.Then,anon-linealgorithmusesthispre-computedinformationtoobtainthesolutionoftheoriginaloptimizationprobleminrealtimefastandefficiently.Introductorymaterialisprovidedasappropriateandthealgorithmsareaccomp
5、aniedbyillustrativeexamples.1IntroductionModelpredictivecontrolisaveryactiveareaofresearchwithbroadindustrialapplications(QinandBadgwell,2003).Itisamongthefewcontrolmethodologiesthatprovidesasystematicwaytoperformnonlinearcontrolsynthesisunderstateandinputconstraints.Thisabilityofdealingwithc
6、onstraintsisoneofthemainreasonsforthepracticalsuccessofmodelpredictivecontrol(MPC)(Garciaetal.,1989).MPCuseson-lineoptimizationtoobtainthesolutionofanoptimalcontrolprobleminrealtime.Thismethodhasbeenprovenmosteffectiveforapplications.Typically,theoptimalcontrolproblemcanbeformulatedasamathemat
7、icalprogram,whosesolutionyieldsasequenceofcontrolmoves.Basedonthesystemmodel,thesystemevolutionoverafinitenumberoffuturetimesteps(thepredictionhorizon)ispredictedandaperformancecriterionoverthesametimeperiodisminimized.Outofthesecontrolmoveson