Infinite Horizon MPC applied to batch processes. Part II

Transcription

1 Innte Hozon MPC appled to batch pocesses Pat II Aleando H González Edado J Adam Dac Odloa 2 and Jacnto L Machett Insttte o echnologcal Development o the Chemcal Indsty (INEC) CONICE - Unvesdad Naconal del Ltoal Santa Fe Agentna (alegoneadamlmach)@santae-concetgova 2 Depatment o Chemcal Engneeng Polytechnc School Unvesty o São Palo São Palo SP Bazl (e-mal: odloa@spb) Abstact: In the second stage o ths wo (pat II) a new nnte hozon model pedctve contolle (IHMPC) wth leanng popetes appled to batch pocesses s pesented When a batch pocess s attempted to be contolled two convegence analyses ae necessay: the convegence nto a gven teaton o batch n (nta-n stablty) and the convegence om n to n (nte-n stablty consdeng an nnte nmbe o batch ns) As was shown n González et al 2009 to accont o the st one the poposed stategy ses a vtal hozon that matches the tadtonal dea o nnte ecedng hozon o MPC wth the nte daton o the n batch o accont o the second convegence analyss a leanng scheme based on the closed-loop paadgm o the IHMPC s developed o evalate the poposed contolle a nmecal example coespondng to batch eacto s shown whee the leanng popetes o the algothm can be clealy seen INRODUCION A batch pocess s one that contnosly epeats a ntedaton pocede (n) along the tme hs nd o systems can be ond n seveal ndstal elds (Lee and Lee 2000; Bonvn 2006; Cel and Bodons 2008) Becase o ts chaactestc these epettve pocesses have two conte ndexes (some athos call them two tme scales): one o nte length beng the tme wthn a n o tal and the othe o nnte length dentyng the nmbe o ns As a conseqence o ths two deent tme scales handlng epettve systems eqes a contol stategy that acconts o two deent obectves: the st one s an on-lne o wthnbatch contol whch eects dstbances occng dng a gven n and no necessaly eman nmoded o the next n he othe s the n to n contol whch eect dstbances that eman almost constant om one n to the next and so the contolle can se nomaton om pevos opeatons In ths last case a contol scheme wth leanng popetes s desed As t was sad n the st stage (González et al 2009) the IHMPC poposed n ths wo s omlated nde a closedloop paadgm (Rosste 2003) he basc dea o a closedloop paadgm s to choose a stablzng contol law and assme that ths law (ndelyng npt seqence) s pesent thoghot the pedctons he dea hee s to consde an ndelyng contol seqence as a manplated npt canddate (npt eeence) o the peect tacng contol and to assocate ths npt eeence wth the leanng vecto (e the vecto that s pdated om one batch to the next to mpove the peomance) I thee s no addtonal nomaton (st teaton) the npt eeence cold be a constant vale hen by means o a leanng pocede (based on the tme convegence o each batch) t s ensed that t conveges teatvely to the peect tacng contol (n to n convegence) hs s the way the poposed contolle acconts o the typcal chaactestcs o batch pocesses e nte tme daton and events epetton he pape s oganzed as ollows In secton 2 the basc denton and notaton ae pesented hen n secton 3 t s ntodced the poposed MPC omlaton and some elated popetes he epettve leanng scheme (man eslt) s pesented n secton 4 Fnally a sccnct llstatve example and the conclson ae pesented n sectons 5 and 6 espectvely 2 PRELIMINARIES We assme hee the same pelmnaes denton consdeed n the pat I o the pape except o the batch ndex whch wll explctly appea n the omlaton n ode to denty each batch n So the qanttes y and d wll be eplaced by y and d he otpt dstbance d s assmed to be nown (t s assmed to eman constant o seveal batch ns) Hee the nomnal model s the same as the one pesented n the pat I o ths wo (González et al 2009) 2 Indexes o clay the notaton we dene the ollowng ndexes: s the teaton o n ndex whee =0 s the st batch n when any leanng pocede s appled s the tme nto a gven batch n Fo a gven teaton t goes om 0 to - (that s tme nstants) s the tme o the MPC pedctons Fo a gven batch n and a gven tme nstant nto the batch n t goes om 0 to - 22 Convegence analyss 477

2 In the next sectons we wll consde two convegence analyses: Inta-n convegence: concens the deceasng o a Lyapnov ncton (assocated to the otpt eo) along the n tme that s Vy yvy y o n one specc batch I the contol algothm execton goes beyond wth and the otpt eeence emans constant at the nal eeence vale ( y y o ) then the nta-n convegence concens the convegence o the otpt to the nal vale o the otpt eeence taectoy ( y y as ) hs convegence was poved n González et al (2009) Inte-n convegence: concens the convegence o the otpt taectoy to the complete eeence taectoy om one batch to the next one that s consdeng the otpt o a gven n as a vecto o components (y y as ) 3 BASIC FORMULAION Fo the st poposed MPC omlaton we wll assme that an appopate npt eeence s avalable and the dstbance seqence d s nown he MPC mzaton poblem assocated to batch n s as ollows: Poblem P) 0 mn V e F x sbect to: e Cx d y 0 () x Ax B 0 (2) U 0 N (3) 0 (4) 0 N (5) whee y d 2 2 y y 2 he mpotance o the n the MPC algothm s descbed n González et al (2009) n the ema 0 Rema : hs poblem s the one pesented by González et al (2009) n the secton 3 (Poblem P) except that now t s assocated to a patcla batch n As a eslt all the popetes ae the same o both omlatons and they ae omtted hee o bevty Patclaly the convegence o the MPC cost (vtal hozon convegence) can be expessed as: V V e 0 4 IHMPC WIH LEARNING PROPERIES In the last secton we stded the wthn-n contol poblem We assmed that an npt eeence s avalable and the otpt dstbance s nown One way s by assocatng the (6) cent npt eeence and dstbance to the last batch ones (e the mplemented npt and the estmated dstbance dng the last n begnnng wth a constant seqence and a zeo vale espectvely) In ths way a dal MPC wth leanng popetes accontng o the n-to-n contol s obtaned Next we wll ty to elcdate ths pont Consde the poblem P (González et al 2009) o a gven batch n wth the ollowng vaaton: 0 d d G y d 2 d d 2 whee the dstbance as well as the states o pedcton ae obseve-based estmates he dea hee s to assocate the npt eeence and the dstbance coespondng to n wth the actal npt and dstbance mplemented at the n - (See Fge ) hat s and d d o =2 and 0 G y G y d 0 : 0 0 In addton t s possble to dene a vecto o deences between two consectve mplemented npt seqences as : - and t s nteestng to notce that ths vecto s gven by hs means that ths deence vecto s made o the st element o the solton o each mzaton poblem o =0 - sed n a ecedng hozon manne 4 New nte-n convegence constants o batch pocess Now n ode to acheve a n-to-n convegence we eplace the ognal constant (5) o poblem P by the ollowng one: 0 N s N s (8) whee N s = mn (HN) (9) In ths way a new shnng contol hozon N s s dened e o the last N tme steps (= -N -) o each n the contol hozon s edced as the tme steps nceases As wll be shown late ths modcaton allows the sccessve n costs to be matched Rema 2: he new shnng contol hozon allows the cost to be expessed by means o H H 0 (0) V e F x egadless o the vale o he next popety shows to be sel o the convegence poo: Popety : Assmng that a shnng contol hozon s sed then Eq (6) holds te o the last N MPC costs o a gven n Fthemoe the last cost o a gven n ae gven by: V e F x (7) 478

3 and snce cent and one steps pedcton ae concdent wth the actal vales (Rema 4) t ollows that: V e F x () Poo Smla to the poo o theoem n the st stage o the wo (González et al 2009) t s possble to dene a easble solton to the mzaton poblem at tme based on the solton at tme - hen showng that the cost coespondng to these soltons s not geate than the mal cost at tme - neqalty (6) holds 42 Popetes o the poposed algothm One nteestng pont hee s to answe what happens the MPC contolle eceves as npt eeence taectoy a contol seqence that t s nected to the system podces a nll otpt eo Snce the MPC contolle does not add the npt eeence to the compted npt (as typcal coecton) bt to pedcted npts some cae mst be taen Popety 2 above asses that o ths npt eeence the MPC cost s nll Wthot lose o genealty we wll consde the nomnal case (no deence between plant and model) o smplcty n what ollows Denton : Let s consde the ollowng peect contol npt taectoy pe pe pe 0 whch epesents the contol seqence that s nected nto the system podces a nll otpt eo taectoy e e e 0 0 It s assmed at ths pont that the otpt eeence s desgned n sch a way (smooth shape) and the dstbances ae sch that the peect contol s possble Notce that the peect contol wold be physcally possble an nnte nmbe o teatons ae peomed Popety 2: I the MPC cost penalzaton matces Q and R ae dente postve (Q>0 and R>0) and peect contol npt pe taectoy s a easble taectoy then V 0 o =0 -; whee H V e F x H 0 Poo ) Let s assme that V 0 o =0 - hen the mal pedcted otpt eo and npt wll be gven by e 0 o =0 and 0 o =0 - espectvely I pe e 0 and 0 smltaneosly t ollows that o =0 - snce t s the only npt seqence that podces nll pedcted otpt eo (othewse the mzaton wll necessaly nd an eqlbm sch that e 0 and 0 povded that Q>0 and R>0 by hypothess) Conseqently pe ) Let s assme that pe Becase o the denton o the peect contol npt the mzaton poblem wthot any npt coecton wll podce a seqence o nll otpt eo pedctons gven by e 0 pe e Cx y 0 CAx B y e Cx y pe pe C A x AB B y 0 Conseqently the mal seqence o decson vaables (pedcted npts) wll be 0 o =0 - and =0 - snce no coecton s needed to acheve nll pedcted otpt eo hs means that V 0 o =0-43 Inte-n convegence Let s consde the ollowng mzaton poblem: Poblem P2) mn V sbect to: ()-(4) (7)-(9) When we say that the algothm conveges om n to n t means that both the otpt eo taectoy e and the npt deence between two consectve mplemented npts = - - conveges to zeo as Followng an Iteatve Leanng Contol nomenclate ths means that the mplemented npt conveges to the peect contol npt pe o a scently lage nmbe o teatons o show ths convegence we wll dene a cost assocated to each n whch penalzes the otpt eo As t was sad MPC mzaton poblems ae solved at each n that s om =0 to = - So a canddate to descbe the n cost s as ollows: J : V (2) whee 0 V epesents the mal cost o the on-lne MPC mzaton poblem at tme coespondng to the n Notce that ths MPC cost once the mzaton poblem P2 s solved and an mal npt seqence s obtaned s a ncton o only e y y e heeoe t maes sense sng (2) to dene a batch cost snce t epesents a (nte) sm o postve penalzatons o the cent otpt eo that s to say a postve ncton o e Howeve snce the new batch ndex s made o otpts pedctons athe than o actal eos some caes mst be taen nto consdeaton Fstly as occs wth sal ndexes we shold demonstate that nll otpt eo vectos podce nll costs (whch s not tval becase o pedctons) Secondly we shold demonstate that the peect contol npt coesponds to a nll cost Popety 3: I the MPC cost penalzaton matces Q and R ae dente postve (Q>0 and R>0) and peect contol npt 479

4 taectoy s a easble taectoy cost (2) whch s an mplct ncton o e s sch that e 0 0 Poo ) Let s assme that e =0 hs means that e 0 o =0 Now assme that the npt eeence vecto s pe deent om the peect contol npt and consde the otpt eo pedctons necessay to compte the MPC cost : V 0 e e Cx y C Ax B B y Snce s not an element o the peect contol npt then CAx B y 0 Conseqently (assmng that * CB s nvetble) the npt necessay to mae e 0 wll be gven by: * CB y C Ax B whch s a non nll vale Howeve the mzaton wll necessay nd an eqlbm solton sch that e 0 * and snce Q>0 and R>0 by hypothess hs mples that e e 0 contadctng the ntal assmpton o nll otpt eo Fom ths easonng o sbseqent otpt eos t ollows that the only possble npt eeence to acheve e =0 wll be the peect contol npt ( = pe ) I ths s the case t ollows that V 0 o =0 (Popety 2) and so J =0 ) let s assme that J =0 hen V 0 whch mples that e 0 o =0 and o =0 Patclaly e 0 o =0 whch mples e =0 Coollay I the MPC cost penalzaton matces Q and R ae dente postve then pe J 0 Othewse pe J 0 Poo It ollows om Popety 2 and Popety 3 Now we establsh the n to n convegence wth the ollowng theoem heoem Fo the system (5)-(7) o González et al 2009 by sng the contol law deved om the on-lne execton o poblem P2 n a ecedng hozon manne togethe wth the leanng pdatng (7) and assmng that a easble peect contol npt taectoy thee exsts the otpt eo taectoy e conveges to zeo as In addton conveges to zeo e Note that o the nomnal case s e J as whch means that the eeence taectoy conveges to pe Poo See Appendx A Rema 3: In most eal systems a peect contol npt taectoy s not possble to each (whch epesents a system lmtaton athe than a contolle lmtaton) In ths case the costs V wll convege to a non-nll nte vale as and then snce the opeaton cost J s deceasng (see Appendx A) t wll convege to the smallest possble vale Rema 4: In the same way that the nta-n convegence can be extended to detemne a vaablty ndex n ode to establsh a qanttatve concept o stablty ( -stablty) o nte-n systems (Rema 9 o González et al 2009); the nte-n convegence can be extended to establsh stablty condtons smla to the ones pesented n Snvasan and Bonvn 2007 N y s H Pedcton Hozon s y y Vtal Hozon 0 +N + 2 Fg MPC dagam coespondng to each batch 5 ILUSRAIVE EXAMPLE In ode to evalate the poposed contolle peomance we assme a te and nomnal pocess gven by (Lee and Lee ) G(s)=/5s 2 +8s+ and G(s)=08/2s 2 +7s+ espectvely he samplng tme aded to develop the dscete state space model s = and the nal batch tme s gven by =90 he poposed stategy acheves a good contol peomance n the st two o thee teatons wth a athe edced contol hozon he contolle paametes ae as ollows: Q=500 R=005 N=5 Fg 2 shows the otpt esponse togethe wth the otpt eeence and the npts and o the st and thd teaton At the st teaton snce the npt eeence s a constant vale ( 0 ) and ae the same and the otpt peomance s qte poo (manly becase o the model msmatch) At the thd teaton howeve gven that a dstbance state s estmated om the pevos n the otpt esponse and the otpt eeence ae ndstngshable As expected the batch eo s edced dastcally om n to n 3 whle the MPC cost s deceasng (as was establshed n heoem ) o each n (Fg 3) Notce that the MPC cost s nomalzed tang 480

5 nto accont the maxmal vale ( V / Vmax ) whee V 6 max 0 and V max 2865 hs shows that the MPC cost J decease om one n to the next as was stated n heoem 2 Fnally Fg 4 shows the nomalzed nom o the eo coespondng to each n Fg 2 Otpt and npt esponses Fg 3 Eo and MPC cost Fg 4 Nom o the teaton eo 6 CONCLUSIONS In ths pape a deent omlaton o a stable IHMPC wth leanng popetes appled to batch pocesses s pesented Fo the case n whch the pocess paametes eman nmoded o seveal batch ns the omlaton allows a epettve leanng algothm whch pdates the contol vaable seqence to acheve nomnal peect contol peomance wo extenson o the pesent wo can be consdeed he ease one s the extenson to lnea-tmevaant (LV) models whch wold allow epesentng the non-lnea behavo o the batch pocesses bette A second extenson s to consde the obst case (eg by ncopoatng mlt model ncetanty nto the MPC omlaton) hese two sses wll be stded n te wos APPENDIX A Poo o heoem he dea hee s to show that V V o =0 - and so J J - Fst let s consde the case n whch the seqence o mzaton poblems P2 do nothng at a gven n hat s we wll consde the case n whch 0 0 o a gven n So o the nomnal case the total actal npt wll be gven by and the n cost coespondng to ths (cttos) npt seqence wll be gven by J : V whee 0 Ns H V e e 0 F x 0 Ns H e 0 Fx H (A) 0 HH Snce the npt eeence poblems s gven by = eo wll be gven by e that ses each mzaton then the esltng otpt o =0 H In othe e wods the open loop otpt eo pedctons made by the MPC mzaton at each tme o a gven n wll be the actal (mplemented) otpt eo o the past n - Hee t mst be notced that e ees to the actal eo o the system that s the eo podced by the mplemented npt Moeove becase o the poposed nte n convegence constant the mplemented npt wll be o H Let now consde the mal MPC costs coespondng to =0 - o a gven n - Fom the ecsve se o (6) we have 48

6 0 0 0 V e V V e V hen addng the second tem o the let hand sde o each neqalty to both sdes o the next one and eaangng the tems we can wte V e e V Fom () the cost V n - wll be gven by V e F x (A2) whch s the cost at the end o the (A3) heeoe by sbstttng (A3) n (A2) we have F x e e e V (A4) Now the psedo cost (A) at tme =0 V can be wtten as 0 0 e Fx V e F x 0 (A5) and om the compason o the let hand sde o neqalty (A4) wth (A5) t ollows that V V Repeatng now ths easonng o = - we conclde that 0 V V heeoe om the denton o the n cost J we have 2 0 J J (A6) he MPC costs V s sch that V V snce the solton 0 o =0 H s a easble solton o poblem P2 at each tme hs mples that J J (A7) Fom (A6) and (A7) we have 0 J J J (A8) whch means that the n costs ae stctly deceasng at least one o the mzaton poblems coespondng to the n - nd a solton 0 As a eslt two ons ase: pe I) Let s assme that hen by Coollay J0 and ollowng the easonng sed n the poo o Popety 3 0 o some hen accodng to (A8) wth 0 0 J J J - he seqence J wll stop deceasng only addton 0 o some 0 0 In 0 then pe whch mples that J =0 heeoe: lm J 0 whch by Popety 3 mples that lm e 0 Notce that the last lmt mples that lm 0 and pe conseqently lm II) Let s assme that and accodng to (A8) by Popety 3 e =0 REFERENCES pe hen by Coollay J=0 J J J 0 Conseqently Cel J R and C Bodons (2008) Iteatve nonlnea model pedctve contol Stablty obstness and applcatons Contol Engneeng Pactce González A H D Odloa and O A Sotomayo (2008) Stable IHMPC o nstable systems IFAC 2008 González A H E J Adam D O Odloa and J L Machett Innte hozon MPC appled to batch pocesses Pat I Renón de Pocesamento de la Inomacón y Contol (RPIC XIII) Rosao Santa Fe Agentna (2009) Lee K S and J H Lee (997) Model pedctve contol o nonlnea batch pocesses wth asymptotcally peect tacng Compte chemcal Engng Lee J H K S Lee and W C Km (2000)Model-based teatve leanng contol wth a qadatc cteon o tme-vayng lnea systems Atomatca Rosste JA (2003) Model-Based Pedctve Contol CRC Pess Snvasan B and D Bonvn (2007) Contollablty and stablty o epettve batch pocesses Jonal o Pocess Contol Notce that the n mplements the manplated vaable =0 - and 0 o some ; then accodng to (A6) J J Unnatally to have ond a non nll mal solton n the n - s scent to have a stctly smalle cost o the n 482