TY - JOUR

T1 - Average constraints in robust economic model predictive control

AU - Bayer, Florian A.

AU - Müller, Matthias A.

AU - Allgöwer, Frank

N1 - Funding information: The possibly non-convergent closed-loop behavior leads to thyestsecmhsemweithprgoepnoesreadl i(npotshsiisbplyapneornclianneabre) apvpelriaegdetcoonnosntrlaininetasr. The possibly non-convergent closed-loop behavior leads to avstemseragecwithonstraintsgenerabyl(pousingssiblyamodifienonlinedar)auxaviliaeragryeocutputonstrafuncints.- the fact that besides classical pointwise-in-time constraints, Tyosttehmissewndit,hwgeenuesreala(npoaspspirbolyprniaotnellyinetiagrh)taevneerdagvercsoionnstroafinthtse. The possibly non-convergent closed-loop behavior leads to Ttioon.thisThisend,allweowsuseusantoagppropuaranteriatelye satisftigachtetionnedofversionthe avoefrathgee which are typically considered in MPC, also constraints on tion. This allows us to guarantee satisfaction of the average tahvheircaffhaacgceattsretthaohfatyttsptbbeiacetassideselildayensdcocclassicnilnaspsidsuietcraalevldarppiioointwise-inanbinlMetwsPiCsbee,-ciaonlm--timestoiemceoofnccsioonstratnrntaesitrnreatssiints,tnotisnn, atiovonstraintsen.rageThisconstraintsabyllowthseuclosed-loopsbytousingguaraantemodifiesysteme satisfddespiteacauxtioniliatheryofothpreutputeseavnceefuncragofe-averages of state and input variables become of interest in constraints by the closed-loop system despite the presence of wevchoeirncaohgmeaaresirceoMfttypyPspCtiicactaally(eAllaynngdccoeolinnsideinsepitduaetlrre.ev[dda2r0iiinan1b2lMMPC,]e,sPMCbu¨e,lclaaeolsolrmsoeetcocaoolf.nstraintsn[si2tnr0tae1irn4et]ss)t.ooIinnn tioconstraintsn. Thisnceas.bylloInwthadseuditionclosed-loops to gtouarathenteabsystemoevsatisfe,wedespiteacdisctionusstheoffurththpreeerseavrenceesuragltsofe economicMPC(Angelietal.[2012],Mu¨lleretal.[2014]).In disturbances. In addition to the above, we discuss further results aapvvcaoeertrraniacoggesumelasicr,oofMsfuPsstacCthatet(ecAoanannnsgtdderaliinpuiinnepttsuatthl.avv[vaa2err0iiaa1toblesb2l]be,seMbecbdu¨eelcaleomeoltrmweetitaofohlf.o[iinn2nl0ttereie1nr4ee,]sst)ti..eIiinnn., cdisturbaonstraintsarenceos.byfinInthteadreeditionstclosed-loopwhetonconthesiderinabsystemove,gwedaespitevdiscerageussthecofurthnstraints,preerserencesuforltsof particular,suchconstraintshavetobedealtwithonline,i.e., which are of interest when considering average constraints, for ecwithoninomictheMPCoptimiza(Antiogelineprotal.blem[201th2],atMisu¨¨lsolelvreetdal.at[2ea014]).ch timeIn disturbawxaihstimple,uchrbaarncenecheoos.sfw.iInInntheseteadardeditiondsittcwionstraintsohnetotno cthethoensabaibcadooenvvre,eibn,egwewuseaevdiscdeidrsactogussuesenforcescofurthfunrstthrerearicnrerotesnsus,vufeltslotrs-r epcaortnicoumlaicr,MsuPcCh(cAonnsgteraliinetsahl.a[v2e01to2]b,eMduelaleltrwetitahl.o[n2l0i1n4e,])i..eIn., example,howtheseconstraintscanbeusedtoenforceconver-withpartiinculthear, sucoptimizah constraintstion prohblemave tothatbeisdealtsolvwithed atonlineache,timei.e., whichnceofaretheofcinlotesedrest-loowhepsystenconm.sideringaverageconstraints,for pstaeirtpthi.icnuTlhtahirse, siosupcithnimciooznansttitroraansitnpttsorohbsatlaevnmedtaotrhdabtestiadsbesialolitzlviwnegdithMatoPneClaic,nhew,thiiem.er.ee, example, how these constraints can be used to enforce conver-step. This is in contrast to standard stabilizing MPC, where example,howtheseconstraintscanbeusedtoenforceconver- swithatseiytphm.iinnpTtohtthehtiiseciaooptimizasvpeitnirmagcioezsanttiotoirofannssttapropttoreoabblemsntlaednmidnaptthrhudaatttsvtiisaasrbiassoilobizllvlveineesgddarMaatetdPeeaeCatecch,rhmwttimehiinemerdee gexancmple,eofthehowclothesesed-looconstraintspsystem.canbeusedtoenforceconver- asymptotic averages of state and input variables are determined The remainder of the paper is organized as follows. In Section asymptotic averages of state and input variables are determined 2, we will recapitulate the concept of robust economic MPC. asymptotic average constraints can be considered offline (when The proposed robust economic MPC scheme including aver-by their value at the set-point to be stabilized, and hence The proposed robust economic MPC scheme including aver-determining the set-point). The concept of average constraints age constraints as well as an analysis of the resulting closed-dseytemrmptiontiincgavtheerasgeet-cpoonisnttr)a.iTnthseccaonnbcepcot nosfidavereerdagoeffclionnest(rwaihnetns The proposed robust economic MPC scheme including aver-dreeattecetrrminingomr,inwinhgerttheehethsset-point).eeta-vpeoriangt)e.fTTheheeedcconfolonwcceesppthtooouffldaavvbeeraeracgegoenccosotrnnstraintsasintreadinitns aloopge csyonstraintsstem willasbeweproll asvideandanainlySsisectionof the3. Someresultinextensiog closedns-can be interesting for different applications such as a chemical age constraints as well as an analysis of the resulting closed-ordeanbertointerestingmeetstoraforgecdapaiffecities.rentapplicationssuchasachemical lareogoepstacsoyntesstdtermaininwtSecsilalstionbweepll4.roavsAidaenndumericalaninalSyseicstiooefnxampt3h.e Srleeosmuillustraletinexgtectenlosssioethends-reaanctboer,inwtehreersetitnhgefaovredriafgfeerefenetdapflpolwicasthioonusldsubcehcaosnastcrahienmedicianl loop system will be provided in Section 3. Some extensions reactor, where the average feed flow should be constrained in proposed approach in Section 5, and the paper is concluded in AsAs mostmost ofof thethe practicalpractical aapppplicalicationtionss areare aaffffeectedcted byby distudisturr--proposedapproachinSection5,andthepaperisconcludedinarreopsotaseteddapinprSoeaccthioinnS4e.cAtionu5m,earnicdatlheexpaampperleisilclounstcrlautdesedthine bances, some effort in taking disturbances into account within preocptioosned6.approach in Section 5, and the paper is concluded in bances, some effort in taking disturbances into account within Notation: By I=0, we denote the set of non-negative integers Aasncmeos,stsoomfetheeffoprrtacintictaalkianpgpdliicsatutirobnasncaerseianftfoecatcecdoubnytdwiistthuirn- Section 6. =0 The authors would like to thank the German Research Foundation (DFG) for anNodtabytioIn[:a,Bb]ytheI=se0,tweof adell intenotegerstheinsethet ofintenorn-neval [gaa,tbiv]einteR.geThrse bances, some effort in taking disturbances into account within and by I the set of all integers in the interval [a, b] R. The baTnhceeasu,thsoorms weoeufldfolirkteitno tthaaknikntghedGiesrtmurabnaRnecseeasrcihnFtoouancdcatoiounn(tDwFGit)hfionr Notation:[a,bBy] I , we denote the set of non-negative integers financialsupportoftheprojectwithintheClusterofExcellenceinSimulation relatioand bynIoperathtoe==rss00eatreofmeallantintecompgers inonethent-wiseintervwheal [an, bapp] liedR. Thto ea [a,b] The authors would like to thank the German Research Foundation (DFG) for relation operators are meant compnonent-wise when applied to a TecThhneoalougthyo(rEsXwCou3ld10li/k2e)taottthheanUkntihveerGsietrymoafnSRtuetstgeaarrct.hFoundation(DFG)for vaenelcdattobiory.nIFo[oa,bpre]ertaxhtaoemrsspealteroe,ffmoalrelain,tbecgoemrRspinonn,teahnet=-iwnbtiesmrevweaalhn[easn,tbha]patplaiReid.=Ttohbeai Technology (EXC 310/2) at the University of Stuttgart. relation operators are meant compnonent-wise when applied[a,b]i to ai financial support of the project within the Cluster of Excellence in Simulation relation operators are meant compnonent-wise when applied to a Technology(EXC310/2)attheUniversityofStuttgart. vector. For example, for a, b Rn, a = b means that ai = bi CTeocphynroilgohgyt ©(EXC 201310/2)5 IFACat the University of Stuttgart. 44 vector. For example, for a, b R , a = b means that ai = bi Copyright © 2015 IFAC 44 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. CPoepery rriegvhietw © u 2n0d1e5r rIFesApConsibility of International Federation of Automat4ic4 Control. Copyright © 2015 IFAC 44 10.1016/j.ifacol.2015.08.155

PY - 2015/7/1

Y1 - 2015/7/1

N2 - In this paper, we extend a previously introduced methodology for tube-based robust economic MPC to consider nonlinear average constraints, i.e., constraints on system states and inputs that need to be satisfied on average. A specifically defined integral stage cost takes the disturbance into account when considering the performance. The key idea is to use an appropriately tightened version of the average constraints by using a modified auxiliary output function in their formulation. By means of the tightened constraints, satisfaction of the original average constraints can be guaranteed despite disturbances acting on the system. For some special cases, we provide concepts to simplify the tightening (which might in general be difficult to determine). In addition, we discuss how average constraints can be used in order to enforce convergence of the closed-loop system to an invariant set. Finally, the proposed approach is illustrated with a numerical example.

AB - In this paper, we extend a previously introduced methodology for tube-based robust economic MPC to consider nonlinear average constraints, i.e., constraints on system states and inputs that need to be satisfied on average. A specifically defined integral stage cost takes the disturbance into account when considering the performance. The key idea is to use an appropriately tightened version of the average constraints by using a modified auxiliary output function in their formulation. By means of the tightened constraints, satisfaction of the original average constraints can be guaranteed despite disturbances acting on the system. For some special cases, we provide concepts to simplify the tightening (which might in general be difficult to determine). In addition, we discuss how average constraints can be used in order to enforce convergence of the closed-loop system to an invariant set. Finally, the proposed approach is illustrated with a numerical example.

UR - http://www.scopus.com/inward/record.url?scp=84992500009&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2015.08.155

DO - 10.1016/j.ifacol.2015.08.155

M3 - Conference article

AN - SCOPUS:84992500009

VL - 28

SP - 44

EP - 49

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8963

IS - 8

T2 - 9th IFAC Symposium on Advanced Control of Chemical Processes, ADCHEM 2015

Y2 - 7 June 2015 through 10 June 2015

ER -