Min-Max Predictive Control of a Pilot Plant using a QP Approach

Min-Max P edic i e Con ol o a Pilo Plan using a QP App oach J.K. G ube , D.R. Rami ez, T. Alamo, C. Bo dons, E.F. Camacho Abs ac —The p ac ical implemen a ion o Min-Max MPC (MMMPC) con olle s is limi ed by he compu a ional bu den equi ed o compu e he con ol law. This p oblem can be ci cum en ed by using app oxima e solu ions o uppe bounds o he wo s possible case o he pe o mance index. In a p e ious wo k, he au ho s p esen ed a compu a ionally e icien MMMPC con ol s a egy in which a close app oxima ion o he solu ion o he min-max p oblem is compu ed using a quad a ic p og amming p oblem. In his pape , his app oach is alida ed h ough i s applica ion o a pilo plan in which he empe a u e o a eac o is con olled. The pilo plan is ope a ed wi h a Sima ic IT SCADA sys em. The con olle has been implemen ed in Ma lab and connec ed o he SCADA using OPC. Realis ic alues o he pa ame e s o he MMMPC con olle ha e been used. The beha io o he sys em and he con olle is illus a ed by means o expe imen al esul s. I. INTRODUCTION The idea behind min-max model p edic i e con olle s (MMMPC) is no new ([4]). In hese con olle s, he con ol signal is compu ed o he wo s case o a cos unc ion ha conside s he e ec o p ocess model unce ain ies and dis u bances in he con olle pe o mance. The main d awback o his app oach is he compu a ional bu den ha akes o compu e he con ol signal. This usually in ol es he solu ion o an NP-ha d min-max p oblem ([10], [16]). As a esul , he numbe o applica ions o hese con ol s a egies is e y small, e en when he e is e idence ha hey wo k be e han s anda d p edic i e con olle s in p ocesses wi h unce ain dynamics [5], [8]. Mul i-pa ame ic p og amming has been applied o show ha he MMMPC con ol law is piecewise a ine when a quad a ic ([14]) o 1-no m based c i e ion ([2], [7]) is usedas cos unc ion. Thus, explici o ms o he con ol law can be buil . Such explici o ms can be e alua ed e y as p o ided ha he complexi y o he s a e space pa i ion is mode a e, which is he case o many applica ions. Howe e , i he p ocess model o he con olle uning pa ame e s change, he compu a ion o he con olle has o be edone. Acommonsolu ion o hecompu a ionalbu denissueis o use an uppe bound o he wo s case cos ins ead o com- pu ing i explici ly. This uppe bound can be compu ed by using linea ma ix inequali ies (LMI) echniques such as in [9], [11]. Howe e , he LMI p oblems ha e a compu a ional bu den ha canno be neglec ed in ce ain applica ions. In [1] adi e en app oachbasedonacompu a ionallycheapuppe bound o he wo s case cos is p esen ed. In ha wo k, he The au ho s a e wi h he Dep . de Ingenie ´ıa de Sis emas y Au om´a ica, Escuela Supe io de Ingenie os, Uni e si y o Se ille,Spain {jg ube , dani , alamo}@ca uja.us.es, {bo dons, edua do}@esi.us.es min-max p oblem is eplaced by a quad a ic p og amming (QP) p oblem ha p o ides a close app oxima ion o he solu ion o he o iginal min-max p oblem. The compu a ional bu den is much lowe han ha o he min-max p oblem and is compa able o ha o a s anda d cons ained MPC based on a quad a ic cos unc ion. Thus, i can be easily imple- men ed in almos any pla o m capable o un a cons ained MPC. Also, s abili y o he p oposed app oach is gua an eed. In his wo k, he app oach p esen ed in [1] has been alida ed by means o i s applica ion o a pilo plan . The pilo plan is used o simula e an exo he mic chemical eac ion wi h nonlinea dynamics. This p ocess has been used in p e ious wo ks, hus he expe imen al esul s p esen ed can be compa ed wi h o he s a egies such as nonlinea and linea p edic i e con ol [6]. In he expe imen s, es ic ions in he con ol ac ion and he ou pu ha e been conside ed. The esul s ob ained p o e he alidi y o he con ol s a egy. The low compu a ional bu den o he con ol s a egy applied o he pilo plan allows ealis ic alues o he con ol and p edic ion ho izons (i.e., he pa ame e s on which he compu a ional bu den depends). I is also no ewo hy ha he compu e on which he p edic i e con ol algo i hm is imple- men ed and simul aneously execu es he SCADA Sima ic-IT, does no ha e su icien calcula ion powe o implemen a con en ional min-max p edic i e con ol s a egy. The e o e acompu a ionallye icien s a egyas ha usedin hiswo k is a good choice i he use o his ype o con ol is desi ed. The pape is o ganized as ollows: sec ion II p esen s he MMMPC s a egy. Sec ion III p esen s he p oposed implemen a ion s a egy. In sec ion IV a de ailed desc ip ion o he used pilo plan is gi en. The s a egy is illus a ed by means o expe imen al esul s o he pilo plan in sec ion V. Finally, sec ion VI p esen s some conclusions. II. MIN-MAX MPC WITH BOUNDED ADDITIVE UNCERTAINTIES Conside he ollowing s a e space model wi h bounded addi i e unce ain ies ([3]): x( +1)=Ax( )+Bu( )+D θ ( +1)(1) wi h x( )∈Rdimx he s a e ec o , u( )∈Rdimu he inpu ec o and θ ( )∈{ θ ∈Rdim θ :∥ θ ∥∞≤ ε } he unce ain y, ha is supposed o be bounded. The sys em is subjec o p s a e and inpu ime in a ian cons ain s Fuu( )+Fxx( )≤g whe e Fu∈Rp×dimu and Fx∈Rp×dimx.I isassumedasemi- eedback app oach in which he con ol inpu is gi en by u( )=−Kx( )+ ( ),(2) P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. whe e he eedback ma ix Kis chosen o achie e some desi ed p ope y such as nominal s abili y o LQR op imali y wi hou cons ain s. The MMMPC con olle will compu e he op imal sequence o co ec ion con ol inpu s ( ).The s a e equa ion o sys em (1) can be ew i en as x( +1)=ACLx( )+B ( )+D θ ( +1),ACL =(A−BK).(3) The p oposed s a egy also wo ks wi hou semi- eedback app oach (i.e., u( )= ( )). All he compu a ional ad an ages o he s a egy emain he same. Fu he mo e i he p ocess is open-loop s able (as in he case o pilo plan used in his wo k) he s abilizing condi ions, which will be discussed la e , can be used wi hou p oblems. The cos unc ion is a quad a ic pe o mance index: V(x, , θ θ θ )= N−1 ∑ j=0 x( +j| )TQx( +j| )+ (4) N−1 ∑ j=0 u( +j| )TRu( +j| ) +x( +N| )TPx( +N| ) whe e x( | )=x,x( +j| )is he p edic ion o he s a e o +jmade a and u( +j| )=−Kx( +j| )+ ( + j| ).No e ha hese aluesdependon he u u e al- ues o he unce ain y. The sequence o u u e alues o θ ( )o e a p edic ion ho izon Nis deno ed by θ θ θ = ! θ ( +1)T,···, θ ( +N)T"T,andΘ Θ Θ={ θ θ θ ∈RN·dim θ :∥ θ θ θ ∥∞≤ ε }is he se o possible unce ain y ajec o ies. On he o he hand, =! ( | )T,···, ( +N−1| )T"Tis he con ol co ec ion sequence. Ma ices Q,P∈Rdimx×dimx and R∈ Rdimu×dimu a e symme ic posi i e de ini e ma ices used as weigh ing pa ame e s. Min-Max MPC ([4]) minimizes he cos unc ion o he wo s possible case o he p edic ed u u e e olu ion o he p ocess s a e o ou pu signal. This is accomplished h ough he solu ion o a min-max p oblem: ∗(x)= a gmin max θ θ θ ∈ΘV(x, , θ θ θ ) s. .Fuu( +j| )+Fxx( +j| )≤g, j=0,...,N,∀ θ θ θ ∈Θ, x( +N| )∈Ω,∀ θ θ θ ∈Θ, (5) A e minal egioncons ain x( +N| )∈Ω,whe eΩis a polyhed on, is included o assu e s abili y o he con ol law ([12]). The p edic ions x( +j| )and u( +j| )depend linea ly on x, and θ θ θ .Thismeans ha i ispossible o inda ec o d∈Rpand ma ices Gx,G and G θ ,such ha all he obus linea cons ain s o p oblem (5) can be ew i en as: Gi xx+Gi +Gi θθ θ θ ≤di,i=1...,p,∀ θ θ θ ∈Θ, whe e Gi x,Gi ,Gi θ deno e he i- h ows o Gx,G and G θ espec i ely and diis he i- h componen o d∈Rp.Deno e now ∥Gi θ ∥1 he sum o he absolu e alues o ow Gi θ . Taking in o accoun ha max θ θ θ ∈ΘGi θθ θ θ =max∥ θ θ θ ∥∞≤ ε Gi θθ θ θ = ε ∥Gi θ ∥1, he obus ul illmen o hecons ain sissa is ied i and only i Gi xx+Gi + ε ∥Gi θ ∥1≤di,i=1,...,p. The e o e, o gua an ee obus cons ain sa is ac ion, he ollowing se o linea cons ain s mus be sa is ied: Gxx+G ≤d ε . whe e he i- h componen o d ε is equal o di− ε ∥Gi θ ∥1.No e ha his is a necessa y and su icien condi ion. Taking in o accoun (3),(2) and (4), he cos unc ion can be e alua ed as a quad a ic unc ion: V(x, , θ θ θ )= TM + θ θ θ TM θθθ θ θ +2 θ θ θ TM θ +2xTMT +2xTMT θ θ θ θ +xTM x(6) whe e he ma ices can be ob ained om he sys em and he con ol pa ame e s ([3]). Due o he con exi y p ope ies o V(x, , θ θ θ ),p oblem(5)isequi alen o([3]) ∗(x)=a g min max θ θ θ ∈ e (Θ Θ Θ)V(x, , θ θ θ ) s. .Gxx+G ≤d ε (7) whe e e (Θ Θ Θ)is he se o e ices o Θ Θ Θ. The e minal egion Ωis assumed o sa is y he ollowing condi ions: •C1:I x∈Ω hen ACLx+D θ ∈Ω, o e e y θ ∈{ θ ∈ Rdim θ :∥ θ ∥∞≤ ε }. •C2:I x∈Ω hen u(x)=−Kx ∈U,whe eU!{u: Fuu+Fxx≤g}. Mo eo e , ma ix P ha cha ac e izes he e minal cos is assumed o sa is y •C3:P−AT CLPACL >Q+KTRK. The s abili y o ACL gua an ees he exis ence o a posi i e de ini e ma ix Psa is ying C3. The maximum cos o a gi en xand is deno ed as V∗(x, )= max θ θ θ ∈ e (Θ Θ Θ)V(x, , θ θ θ )=V(x, ,0)(8) +max θ θ θ ∈ e (Θ) θ θ θ TH θ θ θ +2 θ θ θ Tq(x, ) whe e H=M θθ ,q(x, )=M θ +M θ xand V(x, ,0)= TM +2xTMT +xTM xis he pa o he cos ha does no depend on he unce ain y. Wi h his de ini ion, p oblem (7) can be ew i en as ∗(x)= a gmin V∗(x, ) s. .Gxx+G ≤d ε ,(9) and he sys em is con olled by KMPC(x( )) = −Kx( )+ ∗( | ),whe e ∗(x( )) = ! ∗( | )T,···, ∗( +N−1| )T"T. III. A QP APPROACH TO MIN-MAX MPC In his sec ion he main esul s o [1] a e p esen ed b ie ly. In ha wo k, i is shown how he min-max p oblem (9) can be eplaced by a ac able QP p oblem which p o ides a close app oxima ion o he solu ion o he o iginal p oblem. This can be accomplished wi h he ollowing s eps: 1) Ob ain an ini ial guess o he solu ion o (9), deno ed ˜ ∗.Asseenla e , hiscanbeachie edbysol ingaQP p oblem. 2) Using ˜ ∗,ob ainaquad a ic unc iono ha bounds he wo s case cos . P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. 3) Compu e he con ol law. This in ol es he solu ion o aQPp oblem. All hese s eps will be de ailed in he ollowing. A. Compu ing ˜ ∗ Gi en Hde ined as in equa ion (8), deno e Ti= ∑N·dim θ j=1|Hij|,whe eHij deno es he (i,j)- h componen o ma ix H.Then,de ine hediagonalma ixTas T=diag(T1,···,Tn)(10) Because o how ma ix Tis de ined, T−His a symme ic diagonally dominan eal ma ix wi h nonnega i e diagonal en ies, hus T−H≥0whichimplies ha T≥H.Le ˜ V(x, , θ θ θ )be: ˜ V(x, , θ θ θ )=V(x, ,0)+ θ θ θ TT θ θ θ +2qT(x, ) θ θ θ (11) F om he inequali y T≥Hi is in e ed ha ˜ V(x, , θ θ θ )≥ V(x, , θ θ θ ).Themaximumo ˜ V(x, , θ θ θ )can be compu ed as ˜ V∗(x, )=max θ θ θ ∈Θ Θ Θ ˜ V(x, , θ θ θ ) =V(x, ,0)+ ace(T) ε 2+2 ε ∥q(x, )∥1 =V(x, ,0)+∥H∥s ε 2+2 ε ∥q(x, )∥1(12) whe e ∥H∥sdeno es he sum o he absolu e alues o he elemen s o H.Thenanini ialguesso hesolu iono (9) can be ob ained as ˜ ∗(x)=a gmin ˜ ˜ V∗(x,˜ ) s. .Gxx+G ˜ ≤d ε ,(13) This p oblem can be cas ed as a QP p oblem by making use o slack a iables o deal wi h he 1-no m e m in ˜ V∗(x, ). B. Ob aining an uppe bound o he wo s case cos The uppe -bound o he maximum will be ob ained in wo s eps. In he i s one we compu e a se o pa ame e s om ˜ ∗ ha allows us la e , in he second s ep, o compu e he bound as a quad a ic unc ion o . 1) Compu ing he pa ame e ec o α ( ):No e ha : V∗(x, )= max θ θ θ ∈ e (Θ Θ Θ)# θ θ θ 1$T#Hq(x, ) qT(x, )V(x, ,0)$# θ θ θ 1$ =max ∥z∥∞≤1 zTM( )z(14) wi h z=% θ θ θ T ε 1 &T ,M( )= # ε 2H ε q(x, ) ε qT(x, )V(x, ,0)$∈Rn×n, whe e n=N·dim θ +1. The ollowing p ocedu e p o ides an uppe bound o he wo s case cos o a gi en .I compu es α ( )= [ α 1( ),..., α n−1( )]Tand a diagonal ma ix Γ( )≥M( ) such ha i s ace is an uppe bound o he wo s case cos o (see p ope y 1 o [1]). P ocedu e 1: Compu a ion o α ( )= [ α 1( ),..., α n−1( )]Tand Γ( ). 1) Le S(0=M( )∈Rn×n. 2) Fo k=1 on−1 3) Le M(k−1 sub =[S(k−1 ij ] o i,j=k···n. 4) Ob ain he pa i ion M(k−1 sub =#ab T bM $,whe e a∈R,b∈Rn−kand M ∈R(n−k)×(n−k). 5) Make α k( )='∥b∥1. 6) I α k( )=0 henS(k=S(k−1,elseS(k=S(k−1+ (0T k−1,1 α k( )−bT α k( ))T(0T k−1,1 α k( )−bT α k( )). 7) end o 8) Make Γ( )=S(n−1. No e ha in he p e ious p ocedu e, 0m,ndeno es a (m×n) ma ix o ze os. P ope y 1 o [1] shows ha he ace o Γ( )cons i u es an imp o ed uppe bound o V∗(x, ).Tha is, V∗(x, )≤ ace(Γ( )) ≤˜ V∗(x, ). 2) Ob aining he bound as a quad a ic unc ion on : The diagonaliza ion p ocess shown in p ocedu e 1 can be used o ob ain a ma ix deno ed by ˆ Γ( ),whichallowsone o ob ain a bound o he maximum ha can be compu ed as aquad a ic unc iono .Thisisachie edbymeanso he ollowing p ocedu e: P ocedu e 2: Ob aining he ma ix ˆ Γ( ). 1) Ob ain ˜ ∗ om he QP p oblem de ined in eq. (13). 2) Compu e α (˜ ∗)by p ocedu e 1. 3) Le ˆ S(0( )=M( )∈Rn×n. 4) Fo k=1 on−1 5) Le ˆ Msub( )=[ˆ S(k−1 ij ( )] o i,j=k···n. 6) Ob ain he pa i ion ˆ Msub( )=#a( )bT( ) b( )M ( )$, whe e a( )∈R. 7) I α k(˜ ∗)=0 hen ˆ S(k( )= ˆ S(k−1( ),else ˆ S(k( )= ˆ S(k−1( )+ (0T k−1,1 α k(˜ ∗)−b( )T α k(˜ ∗))T(0T k−1,1 α k(˜ ∗)−b( )T α k(˜ ∗)). 8) end o 9) Make ˆ Γ( )= ˆ S(n−1( ). Deno e ha ˆ V∗(x, )= ace (ˆ Γ( )).Theo em1o [1] shows ha ˆ V∗(x, )is a quad a ic unc ion on and also an uppe bound o he o iginal wo s case cos V∗(x, ). C. Compu ing he con ol law The alue o he con ol signal is ob ained by sol ing he ollowing QP op imiza ion p oblem ˆ ∗(x)=a gmin ˆ ˆ V∗(x,ˆ ) s. .Gxx+G ˆ ≤d ε ,(15) and he sys em is con olled by ˆ KMPC(x( )) = −Kx( )+ ˆ ∗( | ),whe eˆ ∗( | )is he i s elemen o ˆ ∗(x). The compu a ional bu den o he p oposed s a egy is much lowe han ha o he exac MMMPC. This com- pu a ional bu den is mos ly due o he QP p oblems ha mus be sol ed o ob ain he ini ial guess and he p oposed solu ion i sel . No e ha in bo h cases, he complexi y o each p oblem is he same as ha o a s anda d cons ained MPC using a quad a ic cos unc ion. In con as o e alua e he maximum cos V∗(x, )i is necessa y o e alua e he unc ion o all he 2N∗dim θ e ices o Θ.No e ha hisisa well known NP-ha d p oblem. P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. Fig. 1. Pilo plan used o apply he MMMPC. Hea exchange Wa e ank 8 Fj Fj Tj,ou Tj,in F F TT2 T Fig. 2. Diag am o he pilo plan wi h i s ou main elemen s: eac o , hea exchange , cooling jacke and al e. IV. PROCESS DESCRIPTION A ealp ocess ep esen edbyapilo plan hasbeenchosen o he applica ion o he p oposed algo i hm. The p ocess has been s udied p e iously by se e al au ho s [6], [17]. A. Labo a o y p ocess The pilo plan (see Fig. 1) is used o simula e exo he mic chemical eac ions based on empe a u e changes. I has been used as a benchma k o con ol pu poses by se e al esea che s [13], [6]. The main elemen s o he pilo plan a e he eac o , he hea exchange , he cooling jacke and he al e o manipula e he low a e h ough he cooling jacke (see Fig. 2). Acoolingjacke isused o educe he eac o empe a u e. The hea dissipa ion can be egula ed by he al e 8which manipula es he low a e Fj h ough he cooling jacke . The cooling luid, wa e , en e s he cooling jacke wi h a cons an empe a u e. The eac i e is supplied o he eac o by he eed F ,in o keep he chemical eac ion ac i e. Be o e en e ing he eac o , he eed passes h ough a hea exchange in o de o adop he empe a u e o he eac o con en . The ou low F ,ou is used o keep he olume o he eac o con en cons an . To simula e exo he mic eac ions, he eac o possesses an elec ical esis ance in o de o supply calo ic ene gy. The ene gy o be supplied by he 14.4kW elec ical esis ance is calcula ed by means o a ma hema ical model o he simula ed eac ion. The use o a esis ance means ha no chemical eac ion akes place in he eac o , ins ead he eac ion is emula ed on basis o empe a u e changes, as done in [15]. B. Ma hema ical model Al hough i is no necessa y o ha e a ma hema ical model o he design o he min-max p edic i e con olle , his sec ion shows he p ocess model o emphasize i s nonlinea cha ac e . The ma hema ical model also jus i ies he way o emula e he hea gene a ed by he chemical eac ion wi h he aid o he esis ance. The emula ed chemical eac ion, ep esen ing a e inemen p ocess, was used p e iously in [6]. Wi h F =F ,in =F ,ou and a cons an olume, he model o he chemical eac ion can be de ined as: dT d =−Fj V(Tj,in −Tj,ou ) +(−∆H)·V MCp k0e−E/(RT)C2 A(16) dCA d =F V(CA,in−CA)−k0e−E/(RT)C2 A(17) deno ing Fj,Tj,in and Tj,ou he low a e h ough he jacke and he empe a u e o he wa e en e ing and lea ing he cooling jacke , espec i ely. CAand CA,in ep esen he eac- i e concen a ion in he eac o and in he eed, espec i ely. The eed passes h ough he hea exchange and en e s he eac o nea ly wi h he empe a u e o he eac o con en . Thus i is assumed ha no hea is nei he emo ed o supplied due o he eed. The hea exchange in he cooling jacke is gi en by he ollowing empi ical model: Fj·(Tj,ou −Tj,in)=T− α β (1−e− γ Fj)(18) wi h α =292.19K, β =14.94s/land γ =13.18s/l. The chemical eac ion is nonlinea in he dynamics o he empe a u e and he concen a ion due o he quad a ic e ms o he concen a ion in he model equa ions (16) and (17). Fo u he de ails on he model pa ame e s see [6]. V. EXPERIMENTAL RESULTS The s a egy desc ibed in sec ion III has been applied o he e inemen p ocess. In his sec ion he expe imen al esul s will be exposed and discussed. CARIMA ype p edic- ion models wi h bounded addi i e unce ain ies we e used in he expe imen s. This ype o model ex ends he concep o noise in adi ional CARIMA models so ha an unce ain y is conside ed: A(z−1)y( )=z−dB(z−1)u( −1)+C(z−1) θ ( ) ∆(19) P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. wi h ∆=1−z−1, θ ( )∈{ θ ∈Rdimy :∥ θ ∥∞≤ ε },anddimy he dimension o y( ).Theuseo his ypeo p edic ion models esul s in a con ol law wi hou e o in s eady s a e. The e a e ew di e ences be ween implemen ing he algo i hm o sec ion III o a s a e space model and a CARIMA model wi h bounded addi i e unce ain ies. The main di e ence is he me hod used o ind he ma ices o he p edic ion equa ion [3]. The cos unc ion is he same as in (6). Appendix A p esen s p ac ical aspec s ela ed o he use o a CARIMA model (19) o p edic ion. In he ollowing sec ions he con ol sys em in he pilo plan will be desc ibed and he necessa y s eps o ob ain ap edic ionmodelwillbep esen ed.Finally,expe imen al esul s will be exposed. A. Desc ip ion o he con ol sys em The senso s and ac ua o s in he plan a e connec ed o a PMC-10 con ol uni . The PMC-10 is connec ed by ARCne o a pe sonal compu e ha uns he con ol and moni o ing sys em Sima ic-IT.Thecon olalgo i hmhas been implemen ed di ec ly in Ma lab and he communica ion wi h Sima ic-IT is done using he OPC p o ocol (OLE o P ocess Con ol). Bo h Sima ic-IT and he con olle un on he same pe sonal compu e , based on a Pen ium II p ocesso a 300 Mhz. This compu e does no ha e enough compu a ional powe o sol e exac ly he min-max p oblem o a ypical MMMPC, bu can compu e he con ol ac ion using he p oposed s a egy. B. Iden i ica ion o he p edic ion model APRMSS (Pseudo-Random Mul ile el S ep Sequence)has been applied o he eci cula ion al e wi h he objec i e o collec ing da a o he pa ame e iden i ica ion o he p edic ion model. The pe iods o he PRMSS ha e been chosen su icien ly long o obse e he eac ion o he pilo plan o changes in he inpu (see Fig. 3). I can be seen ha he empe a u e o he ank eaches s eady s a e in each s ep in some hing mo e han wo hou s, al hough he a ia ions in s eady s a e a e o se e al deg ees. The eagen concen a ion also su e s a ia ions in s eady s a e. I can be obse ed ha he inpu –ou pu gain is nega i e and clea ly a iable (g ea e gain o low openings o 8). A i s o de ans e unc ion model wi h delay is p oposed as p edic ion model. This low o de model canno co ec ly desc ibe he dynamics o he plan , bu i is a good app oach o check he obus ness o he con olle in p esence o unce ain ies and dis u bances. Using he da a o Fig. 3 he ollowing model has been iden i ied: G(s)= −0.975 950s+1e−31.25s(20) This model was disc e ized wi h a sampling ime o Ts=60s. The delay was ounded o 1 sampling ime in o de o a oid app oxima ions o he ime delay, e.g. Pad´e app oxima ion. The eby, he ollowing CARIMA model was ob ained: y( +1)=0.939y( )−0.0597u( −1)+ θ ( ) ∆(21) wi h he noise polynomial C(z−1)=1. 8[%] CA[mol/l] T[oC] [min] 0 0 0 100 100 100 200 200 200 300 300 300 400 400 400 500 500 500 600 600 600 30 40 40 50 60 60 70 80 80 0 0.1 0.2 0.3 0.4 Fig. 3. Expe imen o he p edic ion model iden i ica ion. F om op o bo om: Tank empe a u e (T), al e opening ( 8)y eagen concen a ion (CA). e o [min] 100 200 300 400 500 600 −0.5 −0.25 0 0 0.25 0.5 Fig. 4. One s ep ahead p edic ion e o du ing he expe imen o he model iden i ica ion. C. Expe imen al esul s o he con olle The p oposed con ol s a egy was applied o he pilo plan desc ibed in sec ion IV-A using (21) as a p edic ion model. Fo he p edic ion and con ol ho izons alues o N= 15 and Nu=12 ha e been used1.The e o e hep edic ion ho izons includes app oxima ely one ime cons an o he p ocess, a common alue o his pa ame e in p edic i e con ol. The weigh ing ac o o he con ol e o has been chosen equal o Rj=2. Based upon he one s ep ahead p edic ion e o (see Fig. 4) he pa ame e ε has been chosen o ε =0.25. As a esul , in 97% o he samples he one s ep ahead p edic ion e o is bounded by he chosen alue. Finally, in o de o es ic he sys em inpu and ou pu in he expe imen s, he ollowing cons ain s ha e been used: 30 ≤ˆy( +j| )≤70,j=2,...,16,∀ θ θ θ ∈ e (Θ Θ Θ) 5≤u( +j| )≤100,j=0,...,11 −20 ≤∆u( +j| )≤20,j=0,...,11 No e ha in he ou pu es ic ions he e ec o he unce - ain y has o be conside ed. In o de o analyse he sys em beha iou , se e al ex- pe imen s wi h e e ence changes and dis u bance ejec ion ha e been made using he p oposed con ol s a egy. Fig. 5shows he esul so he ackingexpe imen wi h e - e ences di e en enough o esul in con ol ac ions in a la ge in e al. A e he i s e e ence change no o e shoo 1The model delay implies ha he p edic ion ho izon s a s in +2and ends in +16. P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. 8[%]T[oC] mol/ [min] 0 0 150 150 35 40 40 45 50 50 50 55 60 60 65 20 80 100 100 100 TT2[oC]CA[mol/l] [min] 0 0 150 150 50 50 100 100 0.1 0.15 0.2 0.25 0.3 17 17.5 18 18.5 Fig. 5. Re e ence acking expe imen . F om op o bo om: Tank empe a u e (T), al e opening ( 8), eagen concen a ion (CA)andcold wa e empe a u e (TT2). appea s in spi e o a qui e as con olle eac ion. A e he second e e ence change a small o e shoo (o abou −0.5oC), jus i ied by he nonlinea p ocess beha iou , can be obse ed. In s eady s a e he con olle shows small changes in he con ol ac ion necessa y o s abilize he ou pu on he e e ence in p esence o a ia ions in he gene a ed hea and he cold wa e empe a u e. In second place a dis u bance ejec ion expe imen was ca ied ou . The esul s o a dis u bance in he sys em inpu , he opening o he al e 8,a ep esen edinFig. 6. As can be seen, a e app oxima ely 70min a cons an dis u bance in he inpu o ∆ 8=15% was applied. The con olle eac s apidly and ejec s he pe u ba ion in abou 20 minu es. A e he disappea ance o he pe u ba ion in = 101min he con olled sys em shows he same beha iou and eaches s eady s a e in app oxima ely 20 minu es. Nei he he empe a u e no he con ol ac ion show oscilla ions a e he pe u ba ion. The hi d expe imen , using an addi i e dis u bance in he eeding F ,isshowninFig.7.In =70min a change in he eeding low o ∆F =0.0125l/s, which co esponds o an e o o 25%, has been applied. Wi h an inc easing e o , he con olle educes he opening o he al e and eaches acompensa iono hedi e gencea e 15minu es.In his expe imen an o e shoo o −0.50oCcanbeobse ed.The oscilla ion in he empe a u e and he con ol ac ion is qui e small and seems accep able due o he s ong dis u bance. The e e ence acking expe imen was epea ed wi h a linea cons ained p edic i e con olle (GPC) o allow he compa ison be ween he p oposed s a egy and a s anda d MPC me hod. The GPC is based on he linea model (21) 8[%]T[oC] [min] 0 0 120 120 40 40 40 45 50 50 55 60 60 60 60 65 70 20 20 80 80 100 100 30 u[%] TT2[oC]CA[mol/l] [min] 0 0 0 120 120 120 40 40 40 40 50 60 60 60 60 70 20 20 20 80 80 80 100 100 100 0.05 0.1 0.15 0.2 17 17.5 18 18.5 30 Fig. 6. Expe imen wi h inpu dis u bance ejec ion. F om op o bo om: Tank empe a u e (T), al e opening( 8), con olle ou pu (u), eagen concen a ion (CA)andcoldwa e empe a u e(TT2). 8[%]T[oC] [min] 0 0 40 40 40 40 45 50 50 55 60 60 60 60 65 70 20 20 80 80 80 100 100 30 TT2[oC]CA[mol/l] [min] 0 0 40 40 60 60 20 20 80 80 100 100 0.05 0.1 0.15 0.2 0.25 17 17.5 18 18.5 Fig. 7. Expe imen wi h dis u bance ejec ion in he eed low. F om op o bo om: Tank empe a u e (T), al e opening( 8), eagen concen a ion (CA)andcoldwa e empe a u e(TT2). P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. MMMPC MPC e MMMPC MPC 8[%]T[oC] [min] [min] 20 80 100 35 40 40 45 50 55 60 60 65 0 0 25 25 50 50 75 75 100 100 125 125 150 150 Fig. 8. Re e ence acking esul s o he MMMPC and he GPC. F om op o bo om: Tank empe a u e (T), al e opening ( 8). and was used wi h he same pa ame e s as he MMMPC. I can be obse ed in he esul s (see Fig. 8) ha he p ocess con olled by he GPC exhibi s signi ican oscilla ions in he empe a u e and he con ol ac ion a e he e e ence changes. The compa ison o he esul s shows ha he MMMPC s abilizes he empe a u e mo e e icien ly and wi h less oscilla ions in he opening o he al e. Finally, i is impo an o men ion ha he calcula ion o he con ol signal ook place wi hou p oblems wi hin he chosen sampling ime (60 seconds). Du ing he expe imen s he a e age compu a ion ime was 5.64 seconds, wi h a maximum o 9.90 seconds and a minimum o 1.863 seconds. VI. CONCLUSIONS In his pape an MMMPC based on an ac able QP p ob- lem was applied o a pilo plan . The esul s showed a good sys em beha iou and he s abilisa ion o he plan empe- a u e a ound he ope a ion poin . A e e e ence changes he con olle quickly compensa es de ia ions. Fu he mo e, he MMMPC showed i s capaci y o compensa e e o s caused by he dis u bances. The applica ion o a p ocess shown in his wo k joins he small numbe o MMMPC applica ions epo ed in specialised li e a u e. The low compu a ional equi emen s o he p oposed con ol s a egy allowed he use o app op ia e sampling imes and ealis ic p edic ion and con ol ho izons. The eby i is shown ha he use o p oposed s a egy allows he applica ion o his kind o con olle s o a la ge numbe o p ocesses. REFERENCES [1] T. Alamo, D.R. Ram´ı ez, D. Mu˜noz de la Pe˜na, and E.F. Camacho. Min-max MPC using a ac able QP p oblem. Au oma ica,43:693– 700, 2007. [2] A. Bempo ad, F. Bo elli, and M. Mo a i. Min-max Con ol o Con- s ained Unce ain Disc e e-Time Linea Sys ems. IEEE T ansac ions on Au oma ic Con ol,48(9):1600–1606,2003. [3] E.F. Camacho and C. Bo d´ons. Model P edic i e Con ol.Sp inge - Ve lag, second edi ion, 2004. [4] P.J. Campo and M. Mo a i. Robus Model P edic i e Con ol. In P oc. Ame ican Con ol Con e ence,pages1021–1026,June10-121987. [5] D. Mu˜noz de la Pe˜na, D.R. Ram´ı ez, E.F. Camacho, and T. ´ Alamo. Applica ion o an explici min-max mpc o a scaled labo a o yp ocess. Con ol Enginee ing P ac ice,13:1463–1471,2005. [6] J.K. G ube and C. Bo dons. Con ol P edic i o no Lineal Basado en Modelos de Vol e a. Aplicaci´on a una Plan a Pilo o. Re is a Ibe oame icana de Au om´a ica e In o m´a ica Indus ial (in spanish), 4(3):34–45, 2007. [7] E.C. Ke igan and J.M. Maciejowski. Feedback min-max Model p edic i e Con ol Using a Single Linea P og am: Robus S abili y and he Explici Solu ion. In e na ional Jou nal o Robus Nonlinea Con ol,14:395–413,2004. [8] Y.H. Kim and W.H. Kwon. An Applica ion o Min-Max Gene al- ized P edic i e Con ol o Sin e ing P ocesses. Con ol Enginee ing P ac ice,6:999–1007,1998. [9] M.V. Ko ha e, V. Balak ishnan, and M. Mo a i. Robus cons ained model p edic i e con ol using linea model inequali ies. Au oma ica, 32(10):1361–1379, 1996. [10] J.H. Lee and Zhenghong Yu. Wo s -case o mula ions o model p edic i e con ol o sys ems wi h bounded pa ame e s. Au oma ica, 33(5):763–781, 1997. [11] Y. Lu and Y. A kun. Quasi-Min-Max MPC Algo i hms o LPV sys ems. Au oma ica,36(4):527–540,2000. [12] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokae . Con- s ained model p edic i e con ol: S abili y and op imali y. Au oma ica, 36:789–814, 2000. [13] D.R. Ram´ı ez, M.R. A ahal, and E.F. Camacho. Min-Max P edic i e Con ol o a Hea Exchange using a Neu al Ne wo k Sol e . IEEE T ans. on Con ol Sys ems Technology,12(5):776–786,2004. [14] D.R. Rami ez and E.F. Camacho. Piecewise A ini y o Min-Max MPC wi h bounded addi i e unce ain ies and a quad a ic c i e ion. Au oma ica,42:295–302,2006. [15] L.O. San os, P.A.F.N.A. A onso, J.A.A.M. Cas o, N.M.C. Oli ei a, and L.T. Biegle . On-line implemen a ion o nonlinea MPC: an expe imen al case s udy. Con ol Enginee ing P ac ice,9(8):847–857, 2001. [16] P.O.M. Scokae and D.Q. Mayne. Min-max eedback model p e- dic i e con ol o cons ained linea sys ems. IEEE T ansac ions on Au oma ic Con ol,43(8):1136–1142,1998. [17] F. Szei e , T. Cho an, and L. Nagy. P ocess dynamics and empe a u e con ol o ed-ba ch eac o s. Compu e s & Chemical Enginee ing, 19(1):447–452, 1995. APPENDIX A. PRACTICAL ASPECTS RELATED TO THE INPUT/OUTPUT DESCRIPTION This appendix p esen s he main di e ences be ween he implemen a ion o he p oposed s a egy wi h a s a e space model (1) and an inpu /ou pu model like (19). In his case, he e olu ion o he ou pu along he p edic ion ho izon can be desc ibed in condensed o m as [3]: y=Guu+G θθ θ θ +Fxx(22) The ma ices Gu,G θ and Fxcan be ob ained om he o iginal model using se e al di e en me hods, e.g. he Diophan ine equa ion [3]. A mo e in ui i e me hod is he use o he s ep esponse coe icien s [3] o o m ma ix Gu.The ee esponse ec o Fxxcan easily be compu ed by i e a ing wi h he p ocess model wi h he assump ion o a cons an con ol signal along he p edic ion ho izon. Ma ix G θ is compu ed by ea ing he unce ain y as an addi ional sys em inpu weigh ed by a uni polynomial in he model. Thus, using he me hod o he Diophan ine equa ion, G θ can be compu ed as Gu,bu assuming ha B(z−1)=1. Also, G θ can be o med om he coe icien s o he sys em esponse o a s ep in he unce ain y θ ( ).The ec o ucon ains he u u e con ol inc emen s i a CARIMA model is used [3]. In his case s a e ec o xcon ains he p esen and pas ou pu alues as P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. well as he pas con ol inc emen s: x=[y( ),...,y( −na),∆u( −1),...,∆u( −nb)]T deno ing naand nb he polynomial o de s o (1−z−1)A(z−1) and B(z−1), espec i ely. The di e en model implies changes in he cos unc ion, bu , as shown in he ollowing, i can be ew i en as in (6). The usual o m o he cos unc ion wi h a p edic ion model like (19) is: V(x,u, θ θ θ )= N ∑ j=1 e( +j| )TQe( +j| )+ Nu−1 ∑ j=0 ∆u( +j)TRm∆u( +j)(23) being e( +j| ) he p edic ed e e ence acking e o a ime +j: e( +j| )=(y( +j| )−w( +j)) whe e w( +j)is he e e ence a ime +j.Wi h he p edic ion equa ion (22) and he assump ion Q=1 hecos unc ion can be w i en as: V(x,u, θ θ θ )=(Guu+G θθ θ θ +Fxx−w)T ·(Guu+G θθ θ θ +Fxx−w) +uTRu(24) wi h Radiagonalma ixo he o m: R=diag(Rm,...,Rm)(25) The ec o wcon ains he u u e alues o he e e ence ajec o y and is de ined as w=[w( +1),...,w( +N)]T. I a ze o e e ence is used, i.e. w( +j)=0 o j=1,...,N, he cos unc ion can be exp essed as in (6) being he ma ices Muu =GT uGu+R,M θθ =GT θ G θ ,M θ u=GT θ Gu, Mu =GT uFx,M θ =GT θ Fxand M =FT xFx.In hecaseo anonze o e e ence, hecompu a iono hecos unc ion can be ca ied ou in an analogous way and does no change he op imisa ion algo i hm. F om his poin , he p oposed s a egy can be applied in he same way as wi h s a e space models. P ep in submi ed o 47 h IEEE Con e ence on Decision and Con ol. Recei ed Ma ch 10, 2008. View publica ion s a sView publica ion s a s