2005 Amercn Contro Conference June 8-10, 2005. Portnd, OR, USA WeB12.6 Perodc Lernng of B-spne Modes for Output PDF Contro: Appcton to MWD Contro H. Wng *, J. F. Zhng nd H. Yue Astrct Perodc ernng of B-spne ss functons mode for the output proty densty functon (PDF) contro of non-gussn systems s studed n ths pper usng the recursve est squre gorthm. Wthn ech contro nterv, the ss functons re fxed nd the contro nput desgn s performed tht contros the shpe of the output PDFs. However, etween ech contro nterv, perodc ernng technques re used to tune the shpe of the ss functons. hs hs een shown to e e to mprove the ccurcy of the B-spne pproxmton mode. As such, the over B-spne mode of the output PDFs ecomes du-mode reted to oth tme nd spce vres. he gorthm hs een pped to smuton study of the moecur weght dstruton (MWD) contro of styrene poymerzton process, edng to some nterestng resuts. I. INRODUCION IN recent yers, the contro of the whoe shpe of the output proty densty functon (PDF) hs een studed n response to the ncresed demnd from mny prctc systems. he typc processes ncude the we sod dstruton contro n ppermng [1], the prtce sze dstruton (PSD) contro n mng processes [2,3,4,5], nd the moecur weght dstruton (MWD) contro [6,7,8] nd prtce sze dstruton contro [9,10,11] n poymerzton processes. For ths type of systems, the ctu controed output s the shpe of the output proty densty functons nd the nputs re ony reted to tme (such s fow rte nd vve openng, etc). In ths regrd, the foowng prt dfferent equton (PDE) cn e genery used to represent the dynmc evouton of the output PDFs n n 1 m m 1 0,,,,,,, n n1 m m1 y y y t t t (1) where s gener nonner functon nd () denotes hs wor ws supported y Chnese NSFC proect 60128303 nd UK s Leverhume rust grnt F/00038/D *H. Wng, the correspondng uthor, s wth the Contro Systems Centre, he Unversty of Mnchester, Mnchester, M60 1QD, Unted Kngdom hong.wng@umst.c.u J. Zhng s wth the Insttute of Automton, Chnese Acdemy of Scences, Beng 100080, Chn nfng.zhng@m...cn H. Yue s wth the Insttute of Automton nd the Unversty of MAnchester, Mnchester M60 1QD, Unted Kngdom h.yue@umst.c.u the output PDF. hs PDE mode s gener expresson of mny poputon nce equtons such s the foowng wdey used prtcute system mode [12] W(, t) [ (, t) W(, t)] h(, t) (2) t where t s tme; s the ntern coordnte; W,t s the numer densty of prtces; (, t) s the prtce growth rte; nd h (, t) s the net creton of prtces. Drect use of the PDE mode s dffcut n prctce n tht ether such mode s dffcut to estsh through the frst prncpe pproches due to the compcted nture of the process, or the otned contro gorthms re too compcted to e pped n the re-tme stutons. o sove ths proem, the B-spne pproxmton to () hs een proposed snce 1998 s one of the mn groups of methods to contro the output PDFs for non-gussn stochstc systems [13,14,15]. he de s to use set of fxed ss functons together wth group of tme-vryng weghts to pproxmte the output PDFs t ech tme nstnt. he contro nput cn therefore e desgned to smpy contro the weghts n the tme-domn. hs s equvent to sovng PDE mode y usng the seprton of vres technque wth fxed set of ss functons. ht s, there re no spce reted dfferent equtons n terms of the evouton of the shpe of the ss functons. Sever B-spne modes hve een deveoped ever snce nd hve een shown cpe of controng the output PDFs to good ccurcy [16,17,18,19], et the numer of B-spne ss functons cn e qute hgh for compcted output PDF shpes nd the ccurcy to the PDF trcng my not e gurnteed. Snce the PDF of process cn vry wdey over operton, t my e dffcut to cpture the ehvour over n extended opertng perod wth fxed ss functons. As resut, t woud e de f the ss functons cn e regury updted ccordng to the output PDF chnges durng the contro process. In ths pper, the perodc ernng nd repettve contro re comned to perform the tunng of the ss functons for the output PDF contro. In ths context, the contro horzon s dvded nto numer of ntervs ( -1)( ), ( ) ( 1,2, ) wth eng the contro nterv ength nd eng the tme perod to tune 0-7803-9098-9/05/$25.00 2005 AACC 955
the B-spne ss functons. Wthn ech nterv ( -1)( ), ( 1), the ner B-spne functons wth FIXED ss functons re used to generte the requred contro nputs tht contro the output PDF shpe. In the nterv -1,, the ss functons re updted to otn etter pproxmton ccurcy to the output PDFs. Such set of updted ss functons w e used s the fxed ss functons for the next contro nterv. hs mens tht the ss functons re tuned perodcy nd the foowng fgure shows such tunng phse. fxed ss functons nput to the system; G nd H re the prmeter mtrces whch represent the system dynmcs for ( -1)( ), ( 1). As presented n [15], et B ( y, ) ( = 1, 2,, n) stnd for the fxed ss functons for the th contro nterv stsfyng B y, dy then n equton (3) B Ly, 1, 2,... n; 1, 2,... (4) y, mode Cy, updte contro B y, 1 B y 2 1 n,, B2 y, Bny,, ntervs n n n 1, Bn 1 y, B, n y (6) n n n (5) 0 Fg. 1. Iustrtve tunng prncpe o strt wth, n the perod of [0,] the contro nputs re desgned wth set of FIXED B-spne functons, wherey the contro s rezed v the contro of the weghts n the B-spne pproxmton. When the smpe tme reches, the tunng of the ss functons s ctvted. hs w st for perod of, durng whch the contro w stys the sme s tht of [0, ]. hs enes the tunng to e focused on the ss functons nd the prmeters of the weghts dynmcs. Once the tunng s competed, the second contro nterv w strt from the smpe nstnt + y usng the updted ss functons nd the mode prmeters. hs process w repet unt the end of contro horzon s reched. Assume tht the output PDF of the consdered stochstc systems, ( y, ), s defned n nown nterv denoted y [, ] (.e. y [, ]). In ths pper, the foowng ner B-spne functon mode [15] w e used to represent the dynmc retonshp etween the nputs nd the output PDFs for ech contro nstnt ( -1)( ), ( 1) 2 2( ) II. MODEL PRESENAION V G V 1Hu1 y, Cy, V Ly, 1, 2, ; 1, 2, (3) n 1 where V R s the weghts vector tht groups the ndependent weghts n the B-spne mode; n s the numer of ss functons chosen for pproxmton; u 1 s scr f y, = y, Ly (, ) (7) then the mode n (3) cn e further expressed n one-step-hed nput nd output form to red f 1y, 1 f y, n 1fn2y, Cy, D0 u Cy, D1u 1 C y, Dn2un2 (8) ( 1,, n1 ) nd D ( 0,, n 2 ) re the prmeters nd prmeter vectors formuted from the stte spce mode of equton (3) usng nformton of G nd H for ( -1)( ), ( 1). Smr to G nd H, nd re fxed wthn ech contro nterv. However, o smpfy the foowng expresson, denote D ther vues re updted smutneousy wth the tunng of the ss functons. he foowng performnce functon s used to mesure the functon dstnce etween the output PDF nd the trget PDF g y(so defned on [, ]) J 1 y, gy 2 dy (9) where 1 y, s the output PDF of the stochstc system t tme nstnt 1 for the th contro nterv. o mnmze J, u cn e otned y sovng J 0 (10) u to gve the foowng feedc formt where u C y, D0 g y, dy C y, D dy 2 0 (11) 956
n1 1 1 1 g y, f y, Cy, D u L y, 2 g y f y (12) 1, hs contro w cn e used together wth the mesured output PDFs so s to formute set of necessry nformton for the updte of the ss functons s we s nd D wthn the th tunng perod -1,. III. UPDAE BASIS FUNCIONS AND {, D } Assume tht the contro functon (11) s pped to the system for ( -1)( ), ( 1), then ccordng to fgure 1 the updte of the ss functons nd {, D } shoud te pce n -1,. For such n updte, the nformton ve shoud e { f y,, f y,, D u, D u,, D u 2} 1 1 0 1 1 n2 n (13) It s so mportnt to note tht n -1,, the contro nputs re st ccuted usng (11) wth the sme set of ss functons nd prmeters {, D } for the th contro nterv ( -1)( ), ( 1), where shoud stsfy equton (8). Denote n1 11 y, f 1y, f 1y, R (14) 1 1 1 2 n R n (15) y, Cy, 1 n Ly, R (16) D0 u D1 u1dn2 un2, 1, 2,, n 1 (17) n 1 (18) where D ( 0,1,, n 2) s the th component of vector D. Usng the ove nottons nd y fxng {, D }, equton (8) ecomes 11 y, y, R (19) hs s ner mode where the updte of y, shoud te pce y usng the dt coected durng [(-1) (+ ), + (-1) ]. Smr to the scnnng prmeter estmton technque used n [15], set of yp re seected from the [, ] nterv for p = 1, 2,., M, so tht the foowng equton hod for ech y p y, y,, p 1,2,..., M p p (20) where M s pre-specfed postve nteger. As y nd re ve t th nterv, p, y, p 1 for the (+1)th contro nterv cn e drecty updted y usng stndrd est squre dentfcton, edng to the foowng recursve est squre gorthm: Pe, yp yp, 1 1 yp, 1 (21) 1 P e, y y, y, 1 (22) p p p P 1 P P 1 P1 P 1 1,, N ; p 1, 2, M (23) where N s the numer of smpng ponts ong the tme xs [ 1( ), ( -1) ]; yp, 1 s the updted vue of y, 1 p t smpe nstnt. he nt vue of yp, 1 s evuted from the most recent 3 6 fxed ss functons nd P0 10 I n. he procedures for the updte of the B-spne functons y, 1 p ) s therefore gven y:- (nmey n the form of 1. At smpe tme, coect y, nd t the contro nterv; 2. Use equtons (21)-(23) to ccute y, 1 p p 1, 2,... M ; th wth 3. Increse y 1 nd go c to step 1 unt N. Here N s the numer of dt prs smped t ech contro nterv. Once the ss functons re updted, the next scn for the th nterv shoud e mpemented to updte the mode prmeters {, D }. hs cn so e rezed y the recursve est squre gorthm. For ths purpose, denote 1 n 1 D0 D0 n Dn2,,, (1),, ( 1),, (1),, 2 1 2( 1) n D n n n R (24) y,, f y,,, f y,, u C y,,, u C y,, n2 1 n1 2 n n1, u C 2 1 y,,, u C y, R n n2 n1 (25) where y,, s composed of the updted ss functons. As resut, the modfcton of s crred out usng the foowng gorthm: P 1 1 yp,, ep, y p 1 1 y,, P y,, p P p (26) 957
1, 1,,, e y f y y p p p p 36 0 10 Inn 1 (27) P 1 yp,, y,, Pp I P 1 (28) 1 yp,, P 1 yp,, where Pp vue of used n the, the nt vue of th contro nterv. 1 s the he procedures used to updte the prmeter vector 1 cn e summrzed s foows: 1. At smpe nstnt (=1, 2,, N ), formute f 1 yp, y,, p ; nd 2. 1 Ccute for p 1,2,..., M wth equtons (26)-(28); 3. Increse y 1 nd go c to step 1 unt N. he contro nd tunng of ss functon nd prmeters cn e ustrted n fgure 2. J, u 1 1 2 2 J, u 3 3 J, u IV. A SIMUAION SUDY OF MWD CONROL he proposed gorthm s pped to smuton exmpe of n MWD contro system. he process of nterest s styrene u poymerzton recton n pot-pnt contnuous strred tn rector (CSR) s shown n fgure 3, n whch styrene s the monomer for poymerzton nd zossoutyrontre s used s the nttor. hese two fows re nected nto the CSR wth the rto dusted y pump. he energy for the recton s provded y the heted o n the CSR s cet nd the o temperture s controed to e constnt. he tot fow rte to the system, F, s composed of the fow of monomer, F m, nd the fow of nttor, F,.e., F Fm F. he monomer nput rto s Fm defned s C. In ths wor, dustment of C s F consdered to e the mens to contro the MWD of the poymer. he mode of ths system cn e seen n [20]. F m F nttor monomer CSR Hetng o 0 2 2( ) 3 2 Fg. 2. he contro nd prmeters modfcton tme seres he compete updtng gorthm cn e summrzed s foows: Step1: Durng [0, ], the cosed-oop system uses set of fxed ss functons nd prmeter vector to reze the contro cton s descred n (11), where the dt (nmey u nd f 1 y ) re stored for the updtng operton of the ss functons; Step2: From the tme nstnt to +, the sved dt re used to ccute the B-spne ss functons wth equtons (21)-(23). Step3. Usng the updted B-spne ss functons n step 2, the sved dt of [0, ] re used gn to tune the mode prmeters v equtons (26)-(28). Durng step 2 nd step 3, the system s controed wth the sme mode prmeters nd B-spne functons s those n [0, ]; he procedure w crry on unt the pre-specfed contro horzon ends. hs consttutes perodc ernng process, whch regury updtes the ss functons nd the mode prmeters for the weght dynmcs. c PDF controer product MWD estmton Fg. 3. Styrene poymerzton system n pot CSR For the ove poymerzton process, seven thrd-order poynom B-spne functons re chosen for the MWD pproxmton. he B-spne mode s estshed sed on the dt provded y prevousy deveoped frst-prncpe MWD mode. he contro nput s desgned so tht the output MWD w foow desred MWD. he perodc ernng of the B-spne functons s crred out s ustrted n the prevous secton. Smuton resuts re shown n fgure 4 to fgure 7. Fg. 4 shows the trget MWD, the nt MWD nd the output MWD of the system t the end of the contro horzon, where t s cer tht the output MWD cn foow the shpe of the trget MWD. Fg. 5 dspys the responses of the MWD n terms of 3D mesh formt, showng the perodc ernng nd tch-to-tch process. In ths fgure there re ten contro ntervs, ech conssts of ffteen MWD responses. In fg. 6, the responses of the contro nput ccuted from equton (11) re gven, from whch the 958
perodc ernng operton cn e seen. For ths system, the re contro nput to the process s mted n rnge from 0.2 to 0.8. In fg. 7, the cosed-oop performnce, nmey Js 2 1 t ( 1) Js y, g y dydt, 1,, 10 s dspyed, ndctng the consecutve mprovement of the contro resuts. As such, t cn e seen from these fgures tht when ppyng the perodc ernng gorthm to the MWD contro of the poymerzton system, the process hs n mprovement n terms of tch-to-tch operton. Fg. 6. he contro nput durng the contro process Fg. 4. he desred MWD nd the output MWD t the end of contro Fg. 7. he ntegr of performnce functon durng ech contro perod Fg. 5. he output MWD durng the whoe contro process V. CONCLUSIONS A perodc ernng gorthm s proposed for the output PDF contro sed on the B-spne pproxmton mode. Wth ths contro strtegy, the output PDF mode not ony retes to tme ut so retes to spce. Wth the updte of the B-spne functons, the vrton of PDF durng operton cn e consdered nd therefore the mode ccurcy of the PDF pproxmton cn e mproved. hs gorthm s pped to the smuton study of tch-to-tch MWD contro system. Smuton resuts show the convergence nd effectveness of the gorthm. he current method s ony perodc ernng gorthm, n whch the ss functons re updted y est squre dentfcton rue. For more effectve updte of the ss functons, certn tertve ernng rues shoud e consdered to modfy the wdth nd heght of the s functons. In the future wor, the modfcton of the ss functons nd the mode prmeters w e studed further to form gener expresson for the output PDF contro. 959
ACKNOWLEDGMEN hs wor s supported y the Chnese NSFC proect 60128303 nd the UK s Leverhume rust grnt F/00038/D. hs s grety cnowedged. REFERENCES [1] G. A. Smoo, Hndoo for Pup nd Pper echnoogsts, Vncouver: Angus Wde Puctons Inc., 1992. [2] G.. M. Cmpe nd C. We, "On predctng roer mng performnce, Prt I: the rege equton," Powder echnoogy, vo. 115, 2001, pp. 234-242. [3] G.. M. Cmpe, P. J. Bunn, C. We nd S. C. W. Hoo, "On predctng roer mng performnce, Prt II: the rege functon," Powder echnoogy, vo. 115, 2001, pp. 243-255. [4] R. A. Ee, nd O. H. Bosgr, Controty of prtcute processes n reton to the sensor chrcterstcs, Powder echnoogy, vo.108, 2000, pp.137-146. [5] H. J. C. Gommeren, D. A. Hetzmnn, J. A. C. Mooenr nd B. Scrett, Modeng nd contro of et m pnt, Powder echnoogy, vo. 108, 2000. pp.147-154. [6]. L. Cre-Prnge nd J. F. McGregor, "Optmzton of moecur-weght dstruton usng tch-to-tch dustments," Ind. Eng. Chem. Res., vo.37, 1998, pp.3660-3669. [7]. J. Crowey nd K Y. Cho, "Ccuton of moecur weght dstruton from moecur weght moments n free rdc poymerston," Ind. Eng. Chem. Res., vo.36, 1997, pp. 1419-1423. [8] M. Vcente, S. BenAmor, L. M. Gugott, J.R. Lez nd J. M. Asu, "Contro of moecur weght dstruton n emuson poymerston usng on-ne recton cormetry," Ind. Eng. Chem. Res., vo.40, 2001, pp.218-227. [9] F. J. Doye III, C. A. Hrrson,. J. Crowey, Hyrd mode-sed pproch to tch-to-tch contro of prtce sze dstruton n emuson poymerzton, Comp. Chem. Eng., vo. 27, 2003, pp. 1153-1163. [10] C. D. Immnue nd F. J. Doye III, Open-oop contro of prtce sze dstruton on sem-tch emuson copoymerston usng genetc gorthm, Chemc Engneerng Scence, vo.57, 2002, pp. 4415 4427. [11] J. Fores-Cerro, J. F. McGregor, Contro of prtce sze dstrutons n emuson semtch poymerzton usng md-course correcton poces, Ind. Eng. Chem. Res., vo. 41, 2002, pp. 1805 1814. [12] M. A. Henson, Dstruton contro of prtcute systems sed on poputon nce equton modes, Proc. Amercn Contro Conf., Denver, Coordo, June 4-6, 2003, pp. 3967-3972. [13] H. Wng, Roust contro of the output proty densty functons for mutvre stochstc systems, Proc. 37th IEEE conf. Decson nd Contro, Vo. 2, 1998, pp. 1305-1310. [14] H. Wng, Roust contro of the output proty densty functons for mutvre stochstc systems wth gurnteed stty, IEEE rns. on Automtc Contro, Vo. 41, 1999, pp. 2103-2107. [15] H. Wng, Bounded Dynmc Stochstc Systems: Modeng nd Contro, Sprnger-Verg London Lmted, 2000. [16] H. Wng, Mode reference dptve contro of the output stochstc dstrutons functons for unnown ner stochstc systems, Int. J. Syst. Sc., vo. 30, 1999, pp.707-715. [17] H. Wng, H. B nd P. Kore, Contro of ounded dynmc stochstc dstrutons usng squre root modes: n ppcty study n ppermng system, rns. Insttute of Mesurement nd Contro, vo. 23, 2001, pp. 51-68. [18] H. Wng nd H. Yue, A rton spne mode pproxmton nd contro of output proty densty functon for dynmc stochstc systems, rns. Insttute of Mesurement nd Contro, vo. 25, 2003, pp. 93-105. [19] J. L. Zhou, H. Yue nd H. Wng, Shpng of output proty densty functons sed on the rton squre-root B-spne mode, ACA AUOMAIC SINICA. (ccepted) [20] H. Yue, J. F. Zhng, H. Wng nd L. L. Co, Shpng of moecur weght dstruton usng B-spne sed predctve proty densty functon contro, Proc. 2004 Amercn Contro Conf., 2004, Boston USA, June 30-Juy 2, 2004, pp.3587-3592. 960