Abstract
In this paper, a robust model predictive control (MPC) technique is introduced to control MIMO linear parameter-varying (LPV) systems subject to input-output constraints. The LPV system is represented in input-output form, which is a common form obtained through LPV system identification. The method guarantees asymptotic stability of the closed-loop system and provides integral action for a given piecewise constant reference trajectory based on past measurements of the inputs and outputs. The technique offers computationally more efficient and less conservative design approach compared to previous works. A simulation example is given to demonstrate the effectiveness of the proposed technique.
| Original language | English |
|---|---|
| Journal | IFAC-PapersOnLine |
| Volume | 51 |
| Issue number | 26 |
| Pages (from-to) | 112-117 |
| Number of pages | 6 |
| DOIs | |
| Publication status | Published - 01.01.2018 |
Funding
A New Approach to Robust MPC Design A New Approach to Robust MPC Design AforNLewPVApSpyrsoteamchs tion RInopbuuts-tOMutPpuCt DFoesrimgn for LPV Systems in Input-Output Form for LPV Systems in Input-Output Form Hossam S. Abbas∗,†∗,† Jurre H∗∗∗anema∗∗∗∗ Roland∗∗∗∗Tóth∗∗∗∗ HossJamvadS.MAobhbaams maJduproreurH∗∗a∗nNemadaer MRoelsaknind∗T∗∗ó∗th HossJaamvadS.MAobbahams∗m,†adJuprreourH∗∗a∗nNaemdae∗r∗ MRoeslankind∗T∗∗ó∗th∗∗ Javad Mohammadpour∗∗∗ Nader Meskin∗∗∗∗ ∗ Javad Mohammadpour Nader Meskin ∗ Institute for Elec. Eng. in Medicine, Unifieristy of Lübeck, 23558, ∗Institute for Elec. Eng. in Medicine, Unifieristy of Lübeck, 23558, LüIbnesctki,tuGteerfmoranEyle,c(.eE-mnga.il:inhoMsesdaimcienlee,dUinn.ifaiebrbisatsy@oufniL-ülbuecekb,e2c3k5.5d8e,) L∗übeck, Germany, (e-mail: [email protected]) ∗L∗üCbeocnkt,roGleSrymstaenmys, (Ger-omuapi,l:Dheopsts.aomfeEleledci.nE.nagb.b,[email protected])y ∗L∗übeck, Germany, (e-mail: [email protected]) o∗∗f Technology, P.O. Box 513, 5600 MB Eindhofien, The Netherlands, of∗∗ Technology, P(.eO-m. aBiol:x{j5.1h3a,n5e6m0a0,rM.BtoEthin}[email protected],lT) he Netherlands, ∗o∗f∗Technology, P.O. Box 513, 5600 MB Eindhofien, The Netherlands, ∗∗∗School of Ele(ce.-manaild:C{omj.hanema,r.tothputer Eng., Col}[email protected])g., Unifiersity of SGcheooorglioaf, EAltehce.nasn,dGCAo3m0p6u0t2erUESnAg,.,(eC-molaleigl:ejoafvEandmg.@,uUgan.ifeiedrus)ity of ∗∗∗ Georgia, Athens, GA 30602 USA, (e-mail: [email protected]) ∗∗∗S∗chool of Elec. and Computer Eng., College of Eng., Unifiersity of ∗∗∗∗Dept. of Elec. Eng., College of Eng., Qatar Unifiersity, PO Box De2p7t1.3oDf Eohleac,.QEantga.r,,C(eo-llmegaeil:ofnaEdnegr.,[email protected],aP)O Box ∗∗∗∗ 2713 Doha, Qatar, (e-mail: [email protected]) †† Dept. of Elec. Eng., College of Eng., Qatar Unifiersity, PO Box †Elec. Eng. Dept., Faculty of Eng., Assiut Unifiersity, 71515, Assiut, Elec. Eng. EDgeypptt., F(ea-cmulatiyl:ohfoEsnsga.m,[email protected], g7)1515, Assiut, †Elec. Eng.EDgeyptpt.,,F(ea-cmulailty:[email protected]., Assiut Unifieedursit.ey,g71515) , Assiut, Egypt, (e-mail: [email protected]) Egypt, (e-mail: [email protected]) Abstract: In this paper, a rˇbust moddeel preddiicctifie controoll (MfflC) techhnniique is intrˇduced tˇ cAˇbntsrtˇrlaMctI:MIOn tlihniesaprappaerra,maetreˇrb-fuiastryminogde(Ll fflprVed)iscytisftieΞcosnsturbojlec(Mt tfflˇCin)ptuetcΘhˇnuitqpuuet cisˇninstrˇadinutcse∆dThtˇe cˇntrˇl MIMO linear paraammeter-fiarying (LfflV) systeΞs subject tˇ inputΘˇutput cˇnstraints∆ The LcˇfflnVtrˇsyl sMteIΞMOis rlienperaersepnatreadmienteinr-pfuiatrΘyˇiuntgpu(LtfflfˇVrΞ) s,ywstheiΞchsisuabcjˇecΞtΞtˇˇinfpˇurtΞΘˇˇubtptauitnceˇdntshtrˇauingths∆LTfflhVe cˇntrˇl MIMO linear parameter-fiarying (LfflV) systeΞs subject tˇ inputΘˇutput cˇnstraints∆ The systeΞ identtiificatiˇn∆ The Ξethˇd guarantees asyΞptˇtic stability ˇf the clˇsedΘlˇˇp systeΞ LfflV systeΞ is represented in inputΘˇutput fˇrΞ, which is a cˇΞΞˇn fˇrΞ ˇbtained thrˇugh LfflV and prˇvides integral actiˇn fˇr a giveenn piecewise cˇnstant reference trajectˇry based ˇn past systeΞ identificatiˇn∆ The Ξethˇd guarantees asyΞptˇtic stability ˇf the clˇsedΘlˇˇp systeΞ ΞeasureΞents ˇf the inputs and ˇutputs∆ The techhnniique ˇffers cˇΞputatiˇnally Ξˇre efficient and less cˇnservaattive design apprˇach cˇΞpared tˇ previˇus wˇˇrrkks∆ A siΞulatiˇn exaΞple is gainvdenletsˇs dceˇΞnsˇenrsvtartaivte tdhesiegfnfecatpivpernˇeascshˇcfˇtΞhepaprreˇdpˇtsˇedprteevcihˇnuisquweˇ∆rks∆ A siΞulatiˇn exaΞple is ganivdenletsˇsdceΞˇnsˇenrsvtartaivteetdheesieffgnectapivpeernnˇesacshˇcfˇΞthepareprˇdpˇtsˇedpretevcihˇunisquwe∆ˇrks∆ A siΞulatiˇn exaΞple is given tˇ deΞˇnstrate the effectiveness ˇf the prˇpˇsed technique∆ g©iv2e0n18t,ˇ IFdAeΞC ˇ(Instterantaetiotnhael eFfefdeecrtaitvieonneosfsAˇuftothmeatpicr ˇCpoˇnsterodl)t Hecohstniniqgu bey∆ Elsevier Ltd. All rights reserved. Keywords: Mˇdel fflredictive Cˇntrˇl, Rˇbust Stability and fflerfˇrΞance, LMI and Cˇnvex Keyworddss: Mˇdel fflredictive Cˇntrˇl, Rˇbust Stability and fflerfˇrΞance, LMI and Cˇnvex OKKepytwiΞooirrzddasst::iˇMnˇ, dLeilnl efflfflarrreddffliiaccrttaiivΞeeeCteˇˇrnΘtVrraˇˇrll,yiRnˇgbb(uuLssffltt VSStt)aSbbyiilsiiteyyΞaasnndd fflerfˇrΞance, LMI and Cˇnvex OpKeytwiΞoirzdats:iˇnMˇ,deLilnefflarredifflaraΞctiveeCteˇrΘntVraaˇrrl,yiRˇngbus(LffltVSt)aSbiylsitteyΞasnd fflerfˇrΞance, LMI and Cˇnvex OptiΞizatiˇn, Linear fflaraΞeterΘVaryying (LfflV) SysteΞs OptiΞizatiˇn, Linear fflaraΞeterΘVarying (LfflV) SysteΞs 1∆ INTRODUCTION and applied successfully tˇ several applicatiˇns, e∆g∆, BachΘ 1∆ INTRODUCTION and applied successfully tˇ several applicatiˇns, e∆g∆, BachΘ 1∆ INTRODUCTION anads eatppall∆ie(d20su14cc)∆esHsfˇuwlleyvteˇr,sΞevˇesrtalˇaf pMpffllicCaΘtLiˇfflnVs, ael∆ggˇ∆,rBitahcΞhsΘ 1∆ INTRODUCTION nandas eatpplal∆ie(d2014)succ∆esHsfˇulwleyvteˇr,sΞevˇsertalˇfapplMfflicCaΘtLiˇfflnsV,ale∆gˇrig∆, BtahcΞhsΘ Moddeel preddiicctifie controoll (MfflC), see, e∆g∆, Mayne et al∆ suacsheastsailn∆ (L2u01an4n)d∆ AHrkˇkwuenve(r2000),0Ξ00ˇ)shtaˇvfeMbffleeCnΘdLefflvVelˇpaplgeˇdrbitashsΞeds (MM2o0od0e0l),pirseddaiiccΞttiiffuiielticvoanrittarroblle(McˇfflntCrˇ)),lsteereea,,teeeg∆∆gy∆∆,tMhaatycnnaeeneecttˇaple∆∆ sˇunaschLefflast ValiΘn∆S(LS2014)uΞanˇdd∆eAHlsrkˇ,wwuenhvie(cr,2000)h ΞˇfˇstehtnaˇfvreeMlybffleeˇCnnΘdLtefflhveVelaˇpalvgˇriaeilad bbtashilΞietdys (2000), is a Ξultivaarriiable cˇntrˇl strategy that can cˇpe ˇn LfflVΘSS Ξˇdels, which ˇften rely ˇn the availability Model predictifie control (MfflC), see, e∆g∆, Mayne et al∆ ˇf tLhffleVsyΘSstSeΞˇsdtaetlse,swinhircehalˇfttieΞne∆reTlyhiˇsnintthreˇdauvcaeislaebxiltirtya e((22ff00ic00i00e))n,,tliiyss awiΞthuultsivgaanrraiial bcllˇeencsˇˇtnrnattirrnˇˇtllss∆strrTaahtteggyyMtfflhhCaatt pccaarnadcciˇgpΞee ˇn thLffle VsyΘSsteS ΞΞˇsdtaetels,s winhircehalˇftiteΞne∆reTlyhiˇsnintthreˇdauvcaeilas ebxiltritya ie(s2000)fficbiaesne,tdliys ˇawniΞtshuˇlltvsiiivgnangraiablˇnclˇleinncseˇtraintrˇlnctˇsn∆stsratTtrhaeeingyeMdtfflhˇCatptpicΞaaanrrizaaaddctiˇpigˇΞne ˇˇf ΞthpelexsyitsytetΞˇ ΞsteaatseusreinˇrretˇalestitΞiΞea∆ tTe hthiseΞin∆trMˇdˇruecˇevsere,xtthrae is based ˇn sˇlving ˇnline a cˇnstrained ˇptiΞizatiˇn cˇΞplexity tˇ Ξeasure ˇr tˇ estiΞate theΞ∆ Mˇreˇver, the efficiently with signal cˇnstraints∆ The MfflC paradigΞ ˇcˇfΞthpelexsyitsytetΞˇΞstaeatesusreinˇrretˇalestitΞiΞea∆tTehthiseΞin∆trMˇdˇrueˇcevsere,xttrhae tphreˇbbfaluestΞeudrˇeˇvnevrˇsalˇulvsteiinqˇnugeˇnˇfcnetlinhˇefecsyˇanstcerˇΞˇnlsaftˇccrrattiinˇanegssdivtthˇenpaattiΞpˇepriztiˇiiaΞdtiˇizˇnef teriˇrateclˇsedΘlˇˇpperfˇrΞanceinterΞsˇfinputdisturΘcˇseΞˇpflexˇbisteyrtvˇerΞstˇeaseusrtieΞˇartetˇthesetisΞtaatetestchaeΞns∆igMniˇfirceˇanvtelry,tdheeΘ tphreˇbflueΞtureˇveevrˇˇallusteqiˇnuenˇfcethˇef cˇsynstterΞˇl afˇrctiˇangisvthenatpˇepritˇiΞdizeˇf teriˇrateclˇsedΘlˇˇpperfˇrΞanceinterΞsˇfinputdisturΘ tihΞeefuctaullreedetvhˇˇelluptrieˇˇdnnicˇtifotnhehosryizsotenΞ∆ MffˇfflrCaahggaiisvvebnneeppneerrwiiˇidelˇˇyff tearniˇcrearteejcelcˇtsiˇednΘlwˇhˇpenpeinrfpˇurtΞcaˇnncsetrinaitnetrsΞbseˇcˇfΞinepuatctdivisattuerdΘ tiΞe called the preddiicction horizon∆ MfflC has been widely bearniˇcerartejecectlˇsiˇedΘn lwˇhˇpenpeinrfpˇurtΞcˇancnsetrinaitnetrsΞbsecˇˇfΞinpute actdiivsatturedΘ atipΞpeliecdalilnedththeeinpdrueddstiiccrtyi,osneehhooQrriiinzzooannn∆dMMBfflfflaCdghhwaaesllb(e2e0n03w)i∆dely as shˇwn in Wang and Yˇung (2006)∆ Alternatively, MfflC aptiΞpeliecdalleindththeeinpdruedsticrtyion,seehorQiiznonan∆dMfflBadCghawesllb(e2003)enwi∆dely as shˇwn in Wang and Yˇung (2006)∆ Alternatively, MfflC appliedintheindustry,seeQinandBadgwell(2003)∆ asschseΞhˇwesnhianveWbaeenng andedveYlˇˇpuedngi(n2006)HaneΞ∆ Aalteternaatl∆i(v2e0ly1,6M) afflnCd Tapˇplfiuredtheinrthenhae inncduestrtyhe,seaeppQieanlianngdfeBaadturgewselˇlf(2003)the Mffl∆ C Abbas et al∆ (2016) directly based ˇn LfflVΘIO represenΘ aTpˇprfuˇratchhe,raegnnahhiaannnΘscccehetdhueleadppMeaffllCingscfheeaΞtuerehsasˇfbethhenee iMntfflrˇCΘ AscbhbeΞasesethaalv∆e(b2016)eendedvierelˇcptledy binasHedanˇneΞaLffletVaΘlI∆O(2re01p6re)saenndΘ apprˇach, a gainΘscheduled MfflC schheeΞe has been intrˇΘ tabtibˇanss eftˇralw∆ h(2ic0h16ˇ)ndlyirepcatslty vbaalsueeds ˇˇnf tLhffleVsΘyIsOteΞrepirnepsuentsΘ Tˇ further enhance the appealing features ˇf the MfflC tatiˇns fˇr which ˇnly past values ˇf the systeΞ inputs ducedinLuandArkun(2000)fˇrlinearparaammeter-fiarying taaanttdiiˇˇnsnˇustfputˇrswhiarcehreˇqnluiyrepadsˇtnlvianelue∆sHˇˇwfetvheer,sythesteΞdiffiinputcultys apprˇach, a gainΘscheduled MfflC scheΞe has been intrˇΘ and ˇutputs are required ˇnline∆ Hˇwever, the difficulty (LfflV) systeΞs∆ These systeΞs are capable ˇf describing aˇfndthˇeutfˇputrΞserarsechreeΞquierisedthˇanltineit∆aHssˇuwΞeevsert,hetheavdiailaffibculilittyy duced in Lu and Arkun (2000) fˇr linear parameter-fiarying ˇf the fˇrΞer scheΞe is that it assuΞes the availability nonlineaarr/time-fiarying (NL/TV)behaviˇrsinterΞsˇfa ˇˇffttheheffutˇrurΞeerscshcheduleΞeingisvtahraiatblitesa,swsuhiΞchesisthunce aˇvΞailaΞˇbnilitiny (LfflV) systeΞs∆ These systeΞs are capable ˇf describing ˇf the future scheduling variables, which is uncˇΞΞˇn in linear dynaΞic structure, see, e∆g∆, HˇffΞann and Werner pfratchteicfeu,tuwrheesrcehaesdtuhleinglavttaerriasbclhese,ΞwehifcˇhrΞisuluantecsˇΞstΞabˇnilitiny nonlinear/time-fiarying (NL/TV) behaviˇrs in terΞs ˇf a practice, whereas the latter scheΞe fˇrΞulates stability (2015)∆ThelinearstrurtucredependsˇnsˇΞeΞeasurable pguraactraince,teewshberaseaedsˇnthealabtiltinereascrhmeΞaterixfˇrinΞeuqlautalesitysta(BbiMlitIy) linear dynaΞic structure, see, e∆g∆, HˇffΞann and Werner guarantees based ˇn a bilinear matrix inequality (BMI) signals called schedduuling fiariables that cˇrresˇnd tˇ the cgˇunadraitniˇtene,swbhaicsehdisˇvneray cbˇilninsearvratmivaetrainxdincˇeqΞupaulittyati(ˇBnMallIy) (2015)∆ The linear strurtucre depends ˇn sˇΞe Ξeasurable cˇnditiˇn, which is very cˇnservative and cˇΞputatiˇnally ˇperatingpˇintˇftheΞˇdelledsysteΞandusedtˇschheedΘ cdeˇndiΞatndiiˇn,ngw∆ hich is very cˇnservative and cˇΞputatiˇnally uˇsˇigpleenrˇaanlsttliiinncgaellepcˇdirnrtsecsˇˇhpffeˇttdnhhudeeliniΞnggˇddcfiˇeearlnlliabteerddˇllsselyessrtsthe∆ΞLatfflaaVnncddˇrΞuuressˇsedˇdenltsdˇctˇssaccnhhtheebddeeΘΘ decˇndiΞanditiˇn,ngw∆hichisverycˇnservativeandcˇΞputatiˇnally ulˇpeerˇanltiingnepcˇˇirnrtesˇpfˇtndiheΞngˇdecˇlnlterdˇlslyesrtse∆ΞLfflaVndΞusˇededltsˇcsacnhebdΘe deΞanding∆ fuˇlreΞˇunllainteedciˇnrrsetaspteˇ-nnspddaiinncegg(cLˇfflntVrˇΘSllSe)rsˇ∆rLinfflpVutΞ-oˇuddtpeelustc(aLnfflVbeΘ IneΞthainsdpianpge∆r,weˇvercˇΞethedifficultiesˇfHaneΞaetal∆ ule ˇnline cˇrrespˇnding cˇntrˇllers∆ LfflV Ξˇdels can be In this paper, we ˇvercˇΞe the difficulties ˇf HaneΞa et al∆ IfˇO)rΞfuˇrΞlatesd,sineestTatˇˇett-hspeatceal(L∆(ffl2012)VΘSS∆)Tˇhreinlpuattte-rouˇntpuetis(LˇfffltVΘen (2016) and Abbas et al∆ (2016) by Ξˇdifying fˇrΞulatiˇns used in the cˇntext ˇf LfflV identificatiˇn frˇΞ data, which L2016)ˇfflVprΘˇIOpanˇsΞdeˇAadnbeblsiasΞspuerbtˇjalveec∆dt(2016)trˇbIuOstbcyˇMnΞfflsˇtCrdaiisfnycthinsegΞwfeiˇtrhtΞˇsutclaatˇbniitlˇnirtˇyslt2ˇ0pr1r6ˇ)paˇnsdeAanbbiaΞsprertˇavle∆d(2r0ˇ1bus6u)stbyMfflΞfflˇCdisfychinegΞfeˇrtΞˇuclaˇtnitˇrnˇsl huasesdbiencˇtΞheecwˇˇennltlesxutpˇˇpffˇLrtffleVd bidyepntˇiwfiecraftuiˇl nidferrnˇˇtΞΞifidcatiaaˇ,,nwthˇiˇclhs LfflVΘIO Ξˇdels subject tˇ IO cˇnstraints with stability huasesdbiecˇn tΞheecwˇenltlesxutpˇfpˇLrtfflVed bidyepntˇiwfieecrratfuiˇnl idferˇΞntifidcaattia,ˇnwthˇiˇclhs LˇfflVprΘˇIOpˇsΞeˇadnelsiΞsprubˇjveecdt trˇˇbusIOtcˇMfflnstCraisncthseΞweithtˇstcaˇbnitlirtˇyl hasbecˇΞewellsuppˇrtedbypˇweerrfulidentificatiˇntˇˇls LufflaVraΘInOteeΞsˇbdaeslesdsuˇbnjelicnteatrˇmIOatrciˇxnsintreaqiunatlsitwyit(hLMstIa)bcilˇitnyΘ guarantees based ˇn linear matrix inequality (LMI) cˇnΘ ⋆⋆haTshbisewcˇoΞrkehwaseblleesnupsuppˇprotretdedbpyarptiˇawllyerbfyulthideeEnutriofpiceaatniˇRnesteˇaˇrclhs guitaiˇrannstaeensdbwaistehdˇuˇtnilninfˇeraΞramtiaˇtnriaxbiˇnuetqufaultiutyre(LscMheId)uclˇinngΘ ⋆ TThhiiss wwoorrkk hhasas bbeeeenn ssuupppporortteedd ppararttiialalllyy bbyy tthhee EEuurropopeeanan RReesseeaarrchch gutaiˇransntaeends bwasitehˇd utˇn ilnfinˇeraΞr amtiatˇnrixabinˇutequfalutiturye(LscMIhedul) cˇingnΘ This work has been supported partially by the European Research daitriiˇanbsleas∆ndSuwchithΞˇˇudt iifnicfaˇtriΞˇnatrieˇdnuacbesˇustigfnuiftiucraentslcyhethdeulidnegΘ C⋆ouncil (ERC) under the European Unions Horizon 2020 research variables∆ Such Ξˇdificatiˇn reduces significantly the deΘ aCnoTduhninicsinlwo(voEartkRioChnas)pubrnoedgeernrasmtuhp(epgrorEauntretodappgearraenetimUalelnyitobnNysotHh7eo1r4Ei6zuo6rn3op).2e0an20Rreesseeaarrchch ditiˇns and withˇut infˇrΞatiˇn abˇut future scheduling and innovation program (grant agreement No 714663). variables∆ Such Ξˇdificatiˇn reduces significantly the deΘ anCoudnincinlo(vEatRionC)purnogrderamth(egrEaunrtopagreaneemUennitonNsoH714663)orizon.2020 research variables∆ Such Ξˇdificatiˇn reduces significantly the deΘ 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2018 IFAC 392 CPoeepry rreigvhietw © u 2n0d1e8r rIFesApConsibility of International Federation of Automa3ti9c2 Control. Copyright © 2018 IFAC 392 10.1016/j.ifacol.2018.11.159 Copyright © 2018 IFAC 392 sign conservatisˇ and enhances the overall perfflorˇance. Furtherˇoreff in contrast with the previous scheˇesff the proposed approach can handle biproper MIMO ˇodels and output constraints. In additionff a worst-case cost is considered to cope with the uncertaintΞ offl the scheduling variables over the prediction horizon taking into account their rates offl variation. The proposed MPC provides also integral action to achieve refflerence tracking and handles constant disturbances. An application to a MIMO cheˇi-cal process is presented to illustrate the proposed ˇethod.
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