ADAPTIVE PRE-EMPTIVE CONTROL OF VACUUM DEWATERING IN PAPER MANUFACTURING 1. Petar Bjegovic 3 Perry Y. Li 2

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1 Copyrgh 2002 IFC 5h Trnnal World Congrss arclona Span DPTIVE PRE-EMPTIVE CONTROL OF VCUUM DEWTERING IN PPER MNUFCTURING Par jgoc 3 Prry Y. L 2 Dparmn of Mchancal Engnrng Unrsy of Mnnsoa Church S. SE Mnnapols MN U.S.. bsrac: Papr manufacurng nols h squnal rmoal of war from pulp by mans of gray acuum dwarng mchancal prssng and hrmal dryng. Ths rsarch proposs a nw conrol opology for mosur conn usng surroga masurmns o nfr n-procss mosur conn and h prssur sngs n h acuum dwarng scon as acuaors. Th nw opology has h ponal o orcom h prformanc robusnss and nrgy ffcncy lmaons ha h currn procss conrol opology has du o a long dad-m and xcss us of h dryrs. prmp conrol law was prously proposd for h nw opology o dcoupl h upsram dsurbancs from propagang downsram n h procss. In hs papr w dlop an adap rson of hs pr-mp conrollr o alla h nd o know h dwarng coffcns prcsly. Th adap conrol problm has h nrsng aspcs ha h unknown paramrs ar nonlnarly mbddd n h rror dynamcs and hr s a m dlay bfor h rror n h smad paramrs can b obsrd. Kywords: Fdforward m dlay adap conrol nonlnar paramrzaon papr ndusry.. INTRODUCTION Papr manufacurng (Fg.) nols h squnal rmoal of war from pulp wh 99.5% mosur by mans of gray acuum dwarng mchancal prss and hrmal dryng n a papr machn o form h fnal produc wh mosur conn of 4 8%. Unform and accura mosur conn of h fnal produc s a ky qualy masur. Currn procss conrol srags mak us of an onln mosur conn scannr a h nd of h papr machn o adjus h sam prssur n h dryr scons. lhough a ary of conrol laws ha bn proposd (.g. P- Rsarch suppord by h Naonal Scnc Foundaon gran ENG/DMI Emal: pl@m.umn.du 3 Emal:par@m.umn.du I adap sochasc conrol slf-unng LQ nurofuzzy modl algorhmc conrol c.) bcaus of h long m dlay bwn h acuaors and h snsor hs conrol opology lms conrol prformanc. Moror h conrol sragy rls haly on h dryng scon whch s h mos nrgy nns of all h war rmoal procsss. n alrna conrol sragy s proposd n whch h prssur sngs of h 5-0 sucon boxs whn h acuum dwarng scon ar usd as procss conrol acuaors. Snc drc n-procss mosur conn masurmns ar no aalabl a h w nd of h procss h mosur conn s nfrrd from ar flow ra hrough h w sh (ar flow dcrass as mosur conn s ncrasd). Th xsnc of such a corrlaon s suppord by (rundr and ans 966; Washburn and uchanan 964; Ramaswamy

2 hadbox gray dranag ncomng slurry Crculang wr msh... acuum dwarng W prss Dryr scon Dryr cans sam Conrollr prssur Mosur snsor Fg.. Schmac of a war rmoal procsss n a papr machn and h currn procss conrol sragy al. 999; Cu al. 999; Pola al. 992; Pola al. 987; Tz and Schlundr 993). Th fundamnal quana rlaonshps bwn ar flow mosur conn and ohr facors ar currnly bng xprmnally nsgad. asd on h concp of surroga n-procss mosur conn masurmns a pr-mp dsrbud conrol approach was proposd n (L al. 200). In hs approach hr s a dsrd arg mosur conn for ach dwarng (or sucon) box. Th surroga mosur conn masurmns ar usd as local fdback for ha dwarng box and ar prodd for h downsram dwarng box o pr-mply lmna h mosur conn rror du o h daon of h ncomng mosur conn from nomnal alus. lhough h pr-mp conrol law n (L al. 200) dos ha som robusnss proprs du o h masurmns and acuaon bng co-locad for prmp rror lmnaon o b succssful h dwarng coffcns K of ach dwarng box mus b known. Ths s no ralsc n applcaon bcaus K dpnds on pulp composon whch s a major uncrany n h procss. In hs papr an adap pr-mp conrol law s proposd o rlax h nd for knowng K prcsly. Th nrsng aspcs of hs adap conrol problm ar ha a) h unknown paramrs K s ar mbddd nonlnarly n h rror dynamcs; b) bcaus smaon rrors of h K s affc h rror dynamcs n h downsram dwarng boxs hr s a sgnfcan ranspor dlay bfor h ffc of K smaon rror and h m whn h ffcs ar obsrd. To orcom hs dffculs w ulz ) a smlar adap conrol sragy for nonlnarly paramrzd sysms proposd n (-Poh al. 999; nnaswamy al. 998) 2) a frs ordr sysm approxmaon of h m dlay o dr h adapaon law and 3) a unng funcon o ak car of any approxmaon rrors. Th proposd conrol law causs h mosur conn rror o conrg o s nomnal alu whn a prdfnd dsrd prcson. Th papr s organzd as follows. Th dlopmn of h conrol dsgn modl s gn n Scon 2. Scon 3 prsns h orgnal prmp conrol algorhm. Th robusnss of h prmp conrol schm o modl paramr uncrany s analyzd n scon 4. Th adap conrol algorhm s prsnd n scon 5. Scon 6 prsns h smulaon rsuls ha llusra h usfulnss of h approach. Scon 7 conans som concludng rmarks. 2. SYSTEM MODEL Th on-lump acuum dwarng box modl dlopd n (L al. 200; jgoc 200) s gn by: d d w whr w Ω w c ou τ () s h oal mosur conn n h doman of h N-h dwarng box s h ranspor spd c ou s h x mosur conn of h h box (akn o b h xng mosur conn a h nd of h gray scon whn ) τ s h ranspor dlay bwn h s and h h boxs K Ω : P (2) f wh f : K P (3) K dnos h dwarng coffcn assumd o b consan n m and n h doman of h -h dwarng box s h lngh of h dwarng slo lngh and P s h prssur dffrnc appld on a papr sh whch s h manpulad arabl. Th x mosur conn c ou c ou s rlad o w K P c ou K P f f w by (4) Th compl draon of h acuum dwarng box modl can b found n (jgoc 200). 3. PREEMPTIVE CONTROL USING SURROGTE MESUREMENTS 3. Conrol problm formulaon Th conrol objc s o conrol h prssur sngs P so ha for ach N w "! w whch s h arg mosur conn for ach acuum dwarng box. w ar dsgnd so ha war rmoal aks plac gradually whou saurang h conrol or causng caasrophc salng of h shs. W assum ha w N ha bn dsgnd so ha whr Ω : 0 ẇ Ω w c ou (5) K # P$ f$ f : K # P$ n whch P h nomnal oprang prssur and c ou K P f w f s h arg x mosur conn c 0 ou s akn o b h nomnal ncomng slurry mosur conn and P s h nomnal prssur sng. s

3 3.2 Prmp conrol law L : w % w b h mosur conn rror for h h dwarng box. I can b shown ha ė Ω Ω Ω w c ou τ (6) whr c ou τ : c ou τ & c ou. Moror c ou τ s aalabl from Eq.(4) bcaus w τ and P τ ar known. Th basc pr-mp conrol sragy s o choos Ω as Ω ' Ω ( λ f f c w ou τ ) or qualnly Ω ' λ f b λ f f c w ou τ λ f b +* λ f f c ou- (7) whr. λ f f. 0 and λ f b. 0 ar h prmp fdforward and fdback conrol gans rspcly. For snc masurmn of h ncomng mosur conn s no aalabl λ f f 0. Th rsulng rror dynamcs for / ar hn gn by: ė 0 Ω λ f b 2 λ f f c ou τ (8) Rmark: Gn chocs for Ω n (7) h acual prssur sngs P ar oband by nrng (2) whch s nrbl for Ω / 3 (unl prssur sauraon lms occur). From (8) s obous ha h conrol law Eq.(7) ncrass h conrgnc ra usng local fdback and compnsas for any dscrpancy n h ncomng mosur conn o h prsn box. In parcular f λ f f s chosn o b and λ f b. 0 hn h ffc of any upsram dsurbancs wll b complly dcoupld from h -h box and conrgs o 0 xponnally. Th localzaon of h fdback nformaon o h prsn dwarng box and s nghbor can b concpualzd as a Smar Vacuum Dwarng ox whch conrols s own oal mosur conn and prods an sma of h xng mosur conn o s nghbors. Th compl conrol schm s mad up of h nrconncon of such Smar Vacuum Dwarng boxs. 4. EFFECT OF THE PRMETRIC UNCERTINTY W analyz h ffc of uncrany n h dwarng ranspor coffcns K. ssum ha h surroga mosur conn masurmns ar corrc so ŵ 4 w and h fdforward gan s: λ f f. L ˆK b h sma of K. W wll us h noaon ˆΩ o dno h xprsson for Ω n Eq.(2) wh K subsud by ˆK (ncludng nsd f ) and ĉ ou o dno h c ou n Eq.(4) wh K subsud by ˆK. Thn h compuaon of h conrol ffor P wll b basd on: w ˆΩ ĉ ou τ ) λ f b (9) Hr wh ˆα : ĉ ou ˆα 4 w ˆK P and ˆf ˆf ˆf (0) Th alus of ˆΩ ar gn by Eq.(2) and Eq.(3) wh K rplacd by ˆK. In h acual suaon h prssur P would b compud basd on ˆΩ n (9) bu appld o Ω n (2) and hn n urn o h rror dynamcs gn by (5) and (6). Snc ˆΩ and Ω ar pos hr xss s 5/ 0 such ha Ω s : K ˆf () ˆΩ ˆK Smlarly w can dfn α g : f s ˆα ˆf so ha ĉ ou ˆα w f s %6 K ˆK 7 # P 6 7 g c ou (2) Th rror dynamcs for 2 N hn bcom: ė 8 Ω s λ f b- s :9 s 6 K ; ˆK ; 7 # P ; 6 τ 7 < c ou τ (3) Undr normal oprang condons w xpc ha h xponns n f and ˆf ar n h rang > so s s gn approxmaly by: K s '? ˆK P ˆK K P (4) From Eq.(5) h rror bound can b oband as: C c! ou Ω s λ f b s s 6 K ; ˆK ; 7 # P ; (5) whr s D s h lowr bound of s E/ 0. From hs quaon w can conclud ha h fdback gan λ f b can b bnfcal n h cas of uncran procss paramrs. lso h uncrany n K only has a bngn ffc on h fdback acon by changng h ffc fdback gan from λ f b o s λ f b.

4 QO 5. DPTIVE CONTROL Th poor knowldg of h procss paramrs K drcly affcs h prformanc of h prmp conrol algorhm as shown n Eqs.(3) and (5). In h acuum dwarng procss hs coffcns ar xpcd o ary du o changs n papr bass wgh and pulp composon. Ths araons of h procss paramrs ar ry slow compard o h dynamcs of h procss n h acuum dwarng scon so w can assum ha ach K s unknown bu consan. In hs scon w dlop an adap conrol approach n ordr o cop wh h uncrany n h procss paramrs K. Th proposd adap conrol algorhm prods rror conrgnc whn a prdfnd dsrd prcson ε n h cas of unknown paramrs K. Th followng assumpons ar mad: ) masurmns of h sa arabl w ar corrc 2) h paramrc uncrany s boundd such ha ach consan bu unknown paramr K blongs o a known compac s K mn K max> 3) h compl pr-mp fdforward conrol s usd.. λ f f and 4) h sauraon lms for h prssur sngs ar suffcnly larg. From Eq.(3) h wo nrsng aspcs of hs adap conrol problm ar ha a) h unknown paramrs K ar nonlnarly mbddd n h rror dynamcs; and b) h rror ar affcd by h smaon rror K τ n h pas.. hr s a m dlay. Th adap conrol law s modfd from h prmp conrol Eq.(9) wh an addon of h unng gan θ : w ˆΩ ĉ ou λ f b ) θ (6) Th purpos of nroducng a unng gan whch s frs nroducd n (-Poh al. 999; nnaswamy al. 998) s o cop wh rror n h approxmaons o b nroducd lar. In Eq.(6) h calculaon of ĉ ou s basd on h currn sma ˆK. Th conrol npu P s hn compud basd on ˆΩ n (9) and sma ˆK. I s hn appld o Ω gn by Eq.(2). Whn hs conrol acon s appld o h modl gn by Eq.() and f w us h rlaon bwn h acual and smad mosur conn gn by Eq.(2) h rror dynamcs bcom: ė Ω s whr s λ f b - F f τ G c ou τ & s θ f τ 6 ˆK ; 6 τ 7 K ; 7 # P ; 6 τ 7 (7) Followng (nnaswamy al. 998) dfn h dsrd conrol prcson ε / 0 of h masurd oal mosur conn w so ha whn H HIJH w w HLK ε hs rror s akn o b accpabl. Inroduc h synhc rror (Fg.2) ε 4 ε S 9 ε < (8) Hr S x s h sauraon funcon dfnd as: P S x NMO x. x H xhrk x No ha ε 0 f h masurd rror H HRK ε. (9) To cop wh h m dlay ssu w nroduc h approxmaon: ˆK τ? ˆK old (20) whr h ffc of h m dlay s approxmad by h frs ordr dynamcs: ˆK old λ s λ ˆK (2) Hr h rlaon bwn ranspor m dlay τ and consan λ s dfnd as: τ λ. Equaon (2) wll yld: ˆK (22) ˆK old 8 λ 0 ˆK old 4 Dfn h Lyapuno funcon candda for h h dwarng box: whr V K 2 2 ε ) 2γ K 2 ˆK old 4 K and γ / 0 s an adapaon gan. Snc h dsconnuy a H HS ε s of h frs knd and snc ε E 0 f H HRK ε h dra of V wh rspc o m xss and s gn by: Ω V 8 s λ f b- ε F 0 s f τ 4 s 2 ĉ ou τ ε s θ ε " ˆK old (23) γ K Now approxma h rm wh h nonlnar paramrzaon by a lnar paramrzaon rm and ulz h approxmaons n Eq.(20) and ha s 5?. L D b h rror assocad wh hs approxmaons: : 0 s f τ 4 s 2 T9 K P τ Th alu of whr 2 < (24) dnos h suprmum of h magnud of h argumn has o b smad. Th bound on xss snc ) h uncran paramrs blong o a known boundd s and 2) h rror du o approxmang h ffc of h known

5 ERROR COORDINTE TRNSFORMTION ε ε ε ε ε Θ TUNING GIN: Fg. 2. Error coordna ransformaon and h unng gan usd for adap algorhm m dlay by a frs ordr ransfr funcon s fn. Eq.(23) wll hn bcom: V 8 9 K Ω P s λ f b - s θ ε ) ε < ĉ ou γ K τ ε ˆK old (25) Paramrc adapaon algorhm (P) s dfnd as follows. In ordr o cancl ou h lnarzd rm n Eq.(25) w nd ˆK old γ ĉ ou τ P τ " ε If w us h rlaon gn by Eq.(22) P hn consss of (22) and h fdback ˆK ˆK old 4 γ ĉ λ ou τ P τ ε (26) Fnally h unng gan θ s dsgnd so ha dscrpancy du o h gn approxmaon rrors (for usng a lnar paramrzaon and for usng a frs ordr sysm o modl a m dlay) dos no adrsly affc h sgn of V. Ths condon s sasfd f: 4 ĉ ou τ 4 s θ 0 (27) Thrfor w can dfn h unng gan o b: θ % ĉ s ou τ S 9 < (28) ε whr s dnos h lowr bound on s D wh rspc o h wors-cas paramrc uncrany. Thorm. Consdr h sysm wh h boundd paramrc uncrany and h ranspor m dlay τ bwn conrol scons modld by Eq.(). For h compl fd-forward basd conrol gn by Eq.(6) procss paramr updang quaon (26) and wh h unng gan dfnd as n (28) lm U ε 0 manng ha h masurd oal mosur conn says whn a dsrd prcson ε abou s nomnal alu. PROOF. W wll consdr h wo dsnc cass: ) Frs consdr h cas whr H HS. ε. Th P s dsgnd o cancl ou h lnarzd rm n Eq.(25). Whn P s appld Eq.(25) can b rwrn as: Ω V s λ f b- 4 ε F 4 ĉ ou ε 4 s θ ε (29) Wh rspc o h sgn of h rror ε h unng gan s dfnd such ha Eq.(27) s sasfd and hrfor: Ω V ) s λ f b- ε Snc sgn ' sgn ε mpls ha V 0. 2) Consdr now h cas whn CK ε. I mpls ha ε V 0 and hrfor V 0 snc ˆK old 4 0 n hs cas. Ths pros ha V 0 and hrfor V s a propr Lyapuno funcon whch lads o global bounddnss of boh ε and K. Ths furhr mpls ha ė ε gn by Eq.(7) s boundd whch from arbala s lmma pros ha h rror dynamcs ar globally asympocally conrgn: lm U H ε HR 0. Thrfor h rror n h oal mosur conn conrgs o a rgon ε around h orgn. Ths compls h proof. W 6. SIMULTIONS Th proposd conrol sysm s smulad for h papr machn wh N6 dwarng boxs of slo lngh 0 05m and wh machn spd of 20m/s. Th ranspor coffcn K dcrass from K 0 8s Pa X 2 o K 6 0 3s Pa X 2 and h dsrd oal mosur conns whch ar dsgnd basd on ncrasng nomnal prssurs from 8000Pa o 56000Pa dcrass from w 0 68 o w Sauraon lms of Y 0% of h nomnal prssurs ar mposd on npu prssurs. Th nomnal ncomng mosur conn c ou 0 s 5kg/kg. In all suaons λ f f λ f b 3 400sc. Th ranspor dlay bwn dwarng boxs ar τ sc. W consdr h suaon whn hr s a 20rad3 s snusodal dsurbanc a h ncomng mosur conn wh h magnud of kg3 kg. Th nomnal prmp conrollr wh prfc knowldg of K (Fg. 3) nabls h mosur conn w o conrg o h dsrd alus owards h las acuum box. h ohr boxs h ffc of h dsurbancs has bn gradually anuad. Conrgnc dos no occur a ach box bcaus of h sauraon lms. To s h adap conrol algorhm 25% daon n h nomnal procss paramrs K s ar also assumd. Th acuaor sauraon lms ar rlaxd o 50% for hs cas. Fg.4 shows h dsrd and acual oal mosur conn wh and whou h adapaon afr h 4-h acuum box. Tm rajcory of mosur conn wh adapaon conrgs o h nomnal

6 Toal mosur conn sold ln wh conrol dod ln no conrol for h conrol of acuum dwarng n papr manufacurng. Th wo ky aspcs of hs problm ar h nonlnar paramrzaon and h m dlay. oh ssus ar dal wh analycally. Th proposd conrol law has bn aldad n smulaons n n h prsnc of npu sauraon. Snc h aalably and accuracy of h mosur conn smas from ar-flow masurmn s crcal o h succss of h proposd conrol approach our currn rsarch ams o xprmnally dlop such a modl. Toal mosur conn from h 4h box K 4 acual nomnal and smad alu Tm (s) Fg. 3. Dsrd and acual oal mosur conn (w 6) wh snusodal mosur conn dsurbanc from hadbox wh and whou h nomnal prmp conrol. K ar assumd wll known sold ln wh adapaon dod ln whou adapaon Tm (s) Fg. 4. Nomnal and achd oal mosur conn whn 25% rror n h procss modl - paramr K s nroducd wh and whou adapaon acual K 4 sma K 4 Nomnal K Tm (s) Fg. 5. Nomnal acual and smad K 4 alu whn h dfnd prcson ε 0 4 kg3 kg alhough h sma of h procss paramr dos no conrg o s acual alu (Fg.5). Ths s bcaus for h h box uncrans n boh K and K affc bu only K s adapd basd on. I can also b shown ha f h m dlay s no akn no consdraon h adap conrollr would b unsabl. 7. CONCLUSION In hs papr w prsnd an adap rson of h pr-mp conrol law prsnd n (L al. 200) REFERENCES -Poh Loh al. (999). dapaon n h prsnc of gnral nonlnar paramrzaon-an rror modl approach. IEEE Transacons on uomac Conrol 44(9) nnaswamy. M. Frdrk P. Skanz and -Poh Loh (998). dap conrol of connous m sysms wx conx/conca paramrzaon. uomaca 34() jgoc Par (200). Modlng and conrol of mosur conn n papr manufacurng usng acuum dwarng boxs as acuaors. Masr s hss. Dparmn of Mchancal Engnrng Unrsy of Mnnsoa. Plan. rundr E. and D.W. ans (966). Th flow of ar hrough w papr. Tapp Journal. Cu Y. Shr Ramaswamy and C. Tourgny (999). Through ar dryng of ssu and owl. Tapp Journal 82(44) L Prry Y. Shr Ramaswamy and Par jgoc (200). Pr-mp conrol of mosur conn n papr manufacurng usng surroga masurmns. Transacons of h Insu of Masurmns and Conrol. Spcal Issu on Papr Manufacurng Indusry. In prss. Pola O. R. H. Croogno and W. J. M. Douglas (992). Transpor phnomna analyss of hrough ar dryng of papr. Indusral Engnrng Chmsry Rsarch 3(3) 736. Pola O. W. J. M. Douglas and R. H. Croogno (987). Exprmnal sudy of hrough ar dryng of papr. In: Procdngs of Dryng 987. p Ramaswamy Shr Y. Cu and C. Tourgny (999). nalyss of conc ha and mass ransfr n hrough ar dryng of papr. In: IChE Symposum Srs on Fundamnal dancs and Innoaons n h Pulp and Papr Indusry. Vol. 95. pp Tz M. and E. U. Schlundr (993). Through ar dryng of papr Par : Sucon drum dsgn from consan ar flow daa. Chmcal Engnrng and Procssng Washburn O.V. and J.G. uchanan (964). Th surfac and nsl fracurs of groundwood handshs as obsrd wh scannng lcron mcroscopy. Pulp Papr Magazn of Canada 65(2)

t=0 t>0: + vr - i dvc Continuation

t=0 t>0: + vr - i dvc Continuation hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM