Automatic Semiqualitative Analysis: Application to a Biometallurgical System

Au oma ic Semiquali a i e Analysis: Applica ion o a Biome allu gical Sys em R. M. Gasca, J. A. O ega, M. Ta o Depa amen o de Lenguajes y Sis emas In o ma icos, Facul ad de In o ma ica y Es adis ica A da. Reina Me cedes s/n, Se illa (Espana) e-mail:{gasca,jao ega,m o o }@lsi.us.es Abs ac . The aim o his wo k is he ep esen a ion and analysis o semiquali a i e models. Thei quali a i e knowledge is ep esen ed by means o quali a i e ope a o s and en elope unc ions. A semiquali a i e model is ans o med in o a amily o quan i a i e models. In his pape he analysis o a model is p oposed as a cons ain sa is ac- ion p oblem. Cons ain sa is ac ion is an umb ella e m o a a ie y o echniques o A i icial In elligence and ela ed disciplines. In his pape a en ion is ocused on in e als consis ency echniques. The semiquali a- i e analysis is au oma ically made by means o consis ency echniques. The p esen ed me hod is applied o a indus ial biome allu gical sys em in o de o show how inc ease he capaci y o p oduc ion. 1 In oduc ion In enginee ing and science, he models made up o he s udy o dynamical sys ems a e no mally composed o quan i a i e and quali a i e knowledge. This knowledge is composed by bo h o hem. I is known as semiquali a i e know- ledge. Real models con ain quan i a i e, quali a i e and semiquali a i e know- ledge. All his knowledge mus be conside ed when hese models a e s udied. The echniques de eloped o analyze and simula e quan i a i e models a e well known. A g ea a ie y o echniques has been s udied o he ep esen a ion and he manipula ion o quali a i e knowledge, such as algeb a o signs, in e al a i hme ic, uzzy se s, and o de o magni ude easoning. In o de o analyze indus ial models, i is necessa y some imes o sol e con- lic s on he eques o accu acy and lexibili y. The models o dynamical sys ems should p o ide di e en le els o nume ical abs ac ion o hei elemen s. These le els may be a pu ely quali a i e desc ip ion [8), semiquali a i e [2), [6), nume- ical based on in e als [11), quan i a i e and mixed o all le els [7]. On o he hand, he sys ems dynamics ob ains he di e en ial equa ions o a sys em om i s s uc u e. This echnique could ob ain di e en quali a i e beha io s o a gi en s uc u e. The analysis o hese beha io s cons i u es he quali a i e analysis o dynamical sys ems. The ma hema ical quali a i e heo y o dynamical sys ems in ol ed s udying quali a i ely he beha iou (e.g. asymp- o ic beha iou ) o ime e ol ing sys ems. 322 In o de o au oma e he quali a i e analysis o dynamic sys ems se e al ap- plica ions ha e been de eloped. They combine echniques o nume ical me hods wi h symbolic compu a ion, and me hods p oceeding om he knowledge o he science and he ma hema ics. These applica ions begin wi h he op-le el speci i- ca ions o physical model. They p epa e simula ion expe imen s, and accomplish hem. Also hey in e p e he nume ical esul s, and hey o mula e he esul s in quali a i e e ms. Among hem, we can ci e PLR [9], bi u ca ion in e p e e [1], KAM [12], POINCARE [10], and MAPS [13]. In his pape , a me hod o ca y ou he analysis o dynamical sys ems au oma ically is shown. The semiquali- a i e analysis is p oposed as a se o in e al cons ain sa is ac ion p oblems. They a e sol ed applying consis ency echniques [5]. 2 Semiquali a i e models A dynamical sys em can be conside ed as he cons ain s <P(x, x,p), x( o) = xo, <Po(p,xo) ( 1) being x he s a e a iables o he sys em, p he pa ame e s, x he a ia ion o he s a e a iables wi h he ime, <Po he cons ain s among pa ame e s and ini ial condi ions, and <P he cons ain s on x, x and p. The dynamical sys em ep esen ed in (1) can symbolically be ans o med in o a se o con ain s wi h a iables, pa ame e s and in e als. In his pape , we only s udy sys ems ha can be ans o med as x = l(x,p), x( o) = Xo, <Po(p, xo) (2) The ec o ield may be composed o quan i a i e and quali a i e a iables, cons an s, a i hme ic ope a o s, unc ions and en elope unc ions, exp essed as i is indica ed in ou p e ious pape [4], whe e quali a i e a iables and en elope unc ions a e ans o med o in e al exp essions. I we ake in o accoun he s ablished concep s in ha pape , he dynamical sys em (2) is ans o med in x = (x, ,p), x( 0) = xo, <Po (p, , xo) (3) whe e E II a e new pa ame e s, p E II, xo E II, and does no con ain en elope unc ions, being II he se o closed in e als o JR. These unc ions ep esen a dynamical sys ems amily depending on p, x0 and . I is deno ed as semiquali- a i e model and i is ep esen ed u he on x = (x,p), x( o) = xo, <Po(p,xo) (4) whe e p and ha e been joined in an unique pa ame e s ec o p. 323 3 Semiquali a i e analysis Quali a i e analysis o a dynamical sys em in ends o analyze he phase po ai o phase space o he sys em. The phase space o he dynamical sys em is cons i- u ed by he a iables o s a e x, and he ex ended phase space by a iables and pa ame e s x, p. The phase po ai is o med by he p ojec ion o he ajec o- ies o he dynamical sys em in he ex ended phase space. The phase po ai is in e p e ed as a co espondence be ween he di e en ial equa ions and he ec- o ield. In his pape semiquali a i e sys ems ha hey a e s able s uc u ally a e s udied. In hem, li le pe u ba ions keep hei quali a i e beha io s. The i s s ep o semiquali a i e analysis o a dynamical sys em ( 4) is he de e mina ion o he equilib ium egions. They a e de ined by he cons ain s Equilib ium(x,p)::: { (x,p) = 0, (5) The s udy o solu ions o (5) le us know he s uc u e o he phase po ai . Each s able equilib ium egion is an a ac o egion. The s abili y o each equilib ium egion is ela ed o he eal pa o he eigen alues o he Jacobian o he sys em. I has been demons a ed in he bibli- og aphy ha in he s able ixed poin s he eal pa o he eigen alues is nega i e. In o de o apply he s abili y c i e ia, i is necessa y o cons uc he ollowing de e minan s. They a e o med wi h he coe icien s o he cha ac e is ic polyno- mial Pn o he Jacobian ma ix A o he dynamical sys em. The Jacobian ma ix o (4) is A= Dx (x,p), and Pn is de ined as Pn(, ) = de (A- , I)= aoAn + a1, n- 1 + ... + an-1, +an (6) In o de o de e mine he s abili y condi ions, he ma ices a e de ined ( a1 a3 as ... a2i-1) . _ d ao a2 a4 ... a2;- 2 b . . _ 1 g, -e emg z - , ... , n ... ... ... ... . .. 0 0 0 0 a; (7) The elemen s ak o g; a e he coe icien s o Pn o k > n, and 0 o k ::S n. Bo h ak and g; a e symbolic exp essions dependen on x, p. We can apply wo s abili y c i e ia. Fi s is he Rou h-Hou wi z c i e ion. Fo his c i e ion he p edica e S able_Pol is de ined as S able_Pol(Pn(, )) =: { g1 > 0, ... ,gn > 0 (8) Second is he Linea d-Chipa d c i e ion. I de ines he p edica e S able_Pol as S able_Pol(Pn(, )) := { a 1 > O, .... ,an> O, gn-1 > O,gn-3 > 0, ... The e o e he cons ain s ha de ine he s able equilib ium egions a e S able(x ) = { Equilib ium(x,p), A= Dx (x,p), ,p - Pn = Pc(A), S able_Pol(Pn(, )) (9) (10) 324 whe e Dx s ands o he Jacobian, and Pc s ands o he se o cons ain o cha- ac e is ic polynomial. I cons ain s (10) a e sa is ied by an equilib ium egion, i is s able. O he wise i is no s able. The s udy o he bi u ca ions poin s o a sys em in ends o di ide he pa am- e e s space in egions. The sys em has he same numbe and ype o a ac o s in hese egions. The on ie s o hese egions a e o med by bi u ca ion poin s. An a ac o appea s, disappea s o changes o ype, when we c oss a de e mined on ie . The mos elemen al classi ica ion o bi u ca ion poin s dis inguishes hem in o s a ics and dynamics. The s a ics bi u ca ion poin s a e he simples . They appea in hose poin s whe e he numbe o a ac o s poin s a ies. The de- e minan o he Jacobian ma ix is annuled in hem, ha is, he cha ac e is ic polynomial has a null oo . The dynamic bi u ca ion poin s in ol e limi cycles o s ange a ac o s. We s udy he Hop bi u ca ion, whe e an a ac o poin is con e ed in o a limi cycle o ice e sa. In hese bi u ca ion poin s he cha ac e is ic polynomial o he Jacobian ma ix has a pai o oo s wi h eal pa equal o ze o. { E.quilib imn(x,p), { l~quilib ium(x,p), A= Dx (x,p), A= Dx (x,p), S a_Bi (x,p) := Pn = Pc(A), Din_Bi (x,p) := Pn = Pc(A), (11) Pn = , Qn-1, Pn = (, 2 + w2) Qn-2, S able_Pol(Qn-1) S able_Pol(Qn-2) I is in e es ing o no ice ha all he p edica es de ined be ween (5) and (11) a e o muled as in e al cons ain sa is ac ion p oblems. They a e sol ed by adequa e consis ency echniques [5]. 4 A biome allu gical sys em 4.1 Desc ip ion and de e mina ion o he model Fo a long ime, i has been obse ed na u al ans o ma ions o he sulphu and i on compounds. They a e o igina ed om he dissolu ion o mine als. P es- ence o i on-oxiding bac e ia in mining a eas and hei acid d ainages has been epo ed epea edly. Thiobacillus Fe ooxidans is conside ed o be he mos impo an o ganism o he bac e ial leaching o mine als. In indi ec leaching he bac e ia gene a e e ic i on by oxidizing soluble e ous i on. The global eac ion is (11) This me hod o p oduc ion o acidi ied e ic solu ions is used because e ic i on in u n oxidizes o he me als in mine al, ans o ming hem in he soluble o m, and because i a oids ecological con amina ion p oblem o indus ial ex ac ion o me als om he ocks. 325 I he equa ion o Michaelis-Men ion is applied o he eac ion ( 11), hen oxida ion a e V is calcula ed as ollows [5] V = Vmax km + [5] (12) whe e Vmax is maximum a e ha i can be eached by inc easing in he subs a e concen a ion, [5] is subs a e concen a ion, and km is Michaelis cons an . This cons an s ands o he concen a ion which he eac ion a e is hal o he max- imum a e. This equa ion has wo p oblems: he concen a ion bac e ian is no cons an and i canno be applied o he bac e ian g ow h because i is exponen- cial. Due o he complexi y o he ac o s ha ake pa in he bac e ia oxida ion o Fe(II) in Ro a ing Biological Con ac o s ( RBC), as shown in igu e (1). I has no been possible o de e mine a gene al ma hema ical model o his p ocess. Howe e , i has been p o ed ha he bioxida ion eac ion con inues a kine ic o i s o de wi h espec o he subs a e concen a ion. In he expe imen a ion he e a e wo in e connec ed RBC. In hem i is in oduced a low Q wi h an e ous i on concen a ion. Disk Di isio ~--~---------.---. D i e sha Ou luen +-C: :J +- In luen L-~--~--~--~--~ La e al Raised G ound Fig. 1. A Ro a ing Biological Con ac o s ( RBC) The equa ions o he model o his dynamical sys em a e (13) 4.2 Expe imen al da a Acco ding o he expe imen al esul s [3] i has been de e mined he quasi- equilib ium poin s o he sys em. They ha e been ob ained s udying di e en in luen lows p1 and alues o i on concen a ion in such lows P2. 326 Acco ding o he da a supplied by he expe s P5 is simila o p9 and hei o de o absolu e magni ude is mode a ely posi i e, P7 is e y posi i e, and p3 is sligh ly g ea e ha P7. The e o e, i i is associa ed he co esponding in e als o he p e iously exp essed quali a i e ope a o s, i is ob ained P1 = 0.61ljh,p2 = 3.96gjl, P3 = [5.6,5.8),p4 = 0.741, P5 = [0.4,0.5), P6 = 0.015, P7 = [5.3, 5.6), Ps = 0.781, pg = [0.4, 0.5), Plo = 0.01 Using hese da a and applying he exposed echniques, we ca y ou he semi- quali a i e analysis o hese dynamical sys ems. 4.3 Semiquali a i e analysis The semiquali a i e analysis o his sys em is ca ied ou o s udy how o inc ease he capaci y o p oduc ion, when sys ems pa ame e s a e a ied. The equilib ium egions o he sys em a e de e mined sol ing he ne wo k o con ain s ! (P1P2- P3P4x,"'+ps- P1x1) P6 = 0, (Pl X1- P7 Ps x2 ".[:P• - P1X2) PlD = 0, Equilib ium(x,p) ::::= 0.4:::; p5 :::; 0.5, 0.4:::; p9 :::; 0.5, 5.6 :=:; P3 :=:; 5.8, 5.3 :S P7 :S 5.6, P1 = 0.61, P2 = 3.96, P4 = 0.74, P6 = 0.015, Ps = 0.78, P10 = 0.01 I i is applied in e al a i hme ic he esul s ob ained a e oo wide. Ne e heless i is applied in e al consis ency echniques de eloped in [5] and we will ob ain a na owing equilib ium egion Equilib ium(x,p) = {[0.397,0.49], x [0.0198,0.034]} This solu ion includes all expe imen al esul s ob ained om di e en expe ience da a. The Jacobian ma ix o his model is The cha ac e is ic polynomial o A is Pn(.. ) = ao.. 2 + a1.. + a2 = .. 2 + (-au- a22).. + aua22- a12a21 and acco ding o he Liena d-Chipa d c i e ion, S able_Pol(Pn(.. )) ::::= {(-au- a22) > 0, aua22- a12a21 > 0 Subs i u ing Pi o hei alues and simpli ying, he cons ain s ha de ine he s abili y a e 327 These cons ain s a e sa is ied wi h he ob ained equilib ium egion and he e o e i is conclude ha he egion is s able. The cons ain s ha de ine he bi u ca ions a e S B . ( ) = {Equilib ium(x,p), D. B" ( ) = {Equilib ium(x,p), a_ z x, p _ 0 0 zn_ z x, p _ 0 0 a1 > , a2 = a1 = , a2 > When i is applied cons ain sa is ac ion echniques o hese cons ain s, he e a e no solu ions, and hence he sys em has no bi u ca ions. 5 Conclusions This pape p oposes a me hod o ca y ou au oma ically he semiquali a i e analysis o dynamical sys ems by in e al consis ency echniques. Quali a i e knowledge is ep esen ed by in e als, and hey a e quali a i e ope a o s and en elope unc ions. I has been applied he p oposed app oach o sys ems appea ed in he bibli- og aphy and he ob ained esul s a e qui e simila o hem. In his pape , i has been s udied a eal biome allu gic sys em. The achie ed esul s ha e allowed o know how o inc ease he capaci y o p oduc ion. In he u u e, we a e going o apply he p e ious echniques o o he eal p ob- lems. We also wan o ex end he analysis p ocess wi h he s udy o o he ypes o a ac o s, dynamic bi u ca ions, and he inco po a ion o mul iple scales o ime, and delays. Re e ences 1. Abelson H., The bi u ca ion in e p e e : a s ep owa ds he au oma ic analysis in dynamical sys ems. In . J. Compu e s Ma h. Applic. Vol 20, n9,8 13-35, 1990. 2. D o ak D.Moni o ing and diagnosis o con inuous dynamic sys ems using semiquan- i a i e simula ion Ph.D. Disse a ion, Uni e si y o Texas. Tech. Repo AI92-170, 1992. 3. 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