Including Qualitative Knowledge in Semiqualitative Dynamical Systems

Including Quali a i e Knowledge in Semiquali a i e Dynamical Sys ems J. A. O ega, R. M. Gasca, M. Ta o Uni e sidad de Se illa 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:{jao ega,gasca,m o o }@lsi.us.es Abs ac . A new me hod o inco po a e quali a i e knowledge in semi- quali a i e sys ems is p esen ed. In hese sys ems quali a i e knowledge may be exp essed in hei pa ame e s, ini ial condi ions and/o ec o ields. The ep esen a ion o quali a i e knowledge is made by means o in e als, con inuous quali a i e unc ions and en elope unc ions. A dynamical sys em is de ined by di e en ial equa ions wi h quali a i e knowledge. This de ini ion is ans o med in o a amily o dynamical sys- ems. In his pape he semiquali a i e analysis is ca ied ou by means o cons ain sa is ac ion p oblems, using in e al consis ency echniques. 1 In oduc ion In enginee ing and science, knowledge abou dynamical sys ems may be ep e- sen ed in se e al ways. The models cons uc ed o s udying hem a e no mally composed o quali a i e as well as quan i a i e knowledge. Models which only in- co po a e quan i a i e knowledge (quan i a i e models) ha e been well s udied. The echniques de eloped o analyse and simula e a e well known, oo. On he o he hand, a g ea a ie y o echniques ha e been s udied o ep e- sen a ion and 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. The knowledge composed o quan i a i e and quali a i e knowledge is known as semiquali a i e. Real models con ain quan i a i e, quali a i e and semiquali- a i e knowledge, and all o hem need o be conside ed when hey a e s udied. The e o e, some imes i is necessa y 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 pu ely quali a- i e desc ip ions [7], semiquali a i e [1], [5], nume ical based on in e als [14], quan i a i e and mixed o all le els [6). In he six ies, he me hodology o sys em dynamics was p oposed. I inco - po a ed quali a i e knowledge o models by means o a iables and quan i a i e unc ions sui ably chosen. Bu , i is no un il he eig hies when he in e es o s udying quali a i e knowledge independen ly o i s quan i a i e ep esen a ion 330 eme ges. This in e es appea s a ound quali a i e simula ion [7], and quali a- i e analysis [11]. Ma hema ical concep s o quan i a i e analysis o dynamical sys ems ha e been applied in o quali a i e simula ion and analysis (see [8]). Quali a i e me hods o s udying dynamical sys ems began a he end o he las cen u y, by he F ench ma hema ician Hen i Poinca e. The subsequen e o- lu ion o hese wo ks has o igina ed he quali a i e heo y o dynamical sys ems. In [10] he echniques o ca y ou he analysis o he quali a i e models we e in oduced, ha is, he s udy o equilib ium egions, s abili y and bi u ca ion poin s. In his pape , he quali a i e knowledge is ep esen ed by means o eal in- e als, con inuous quali a i e unc ions and en elope unc ions. The in e als include all he eal alues whe e he quali a i e label o such magni ude is ound. A con inuous quali a i e unc ion s ands o a amily o unc ions de ined by means o landma ks. An en elope unc ion s ands o a amily o unc ions in- cluded be ween a eal supe io unc ion and an in e io one. I is also p esen ed A me hod o ans o m semiquali a i e models in o a cons ain ne wo k is p esen ed, as well. The in e al cons ain sa is ac ion p o- blems a e sol ed applying consis ency echniques [3]. 2 Semiquali a i e models A semiquali a i e model is ep esen ed by «P(:i:, 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, :i: he a ia ion o he s a e a iables wi h he ime, «Po he cons ain s in he ini ial condi ions, and «P he cons ain s depending on :i:, x and p. They a e exp essed by means o he ope a o s de ined in he nex sec ion. They a e composed o a iables, cons an s, a i hme ic ope a o s, unc ions, quali a i e unc ions and/o en elope unc ions. The e o e he equa ions (1) s and o a amily o dynamical sys ems depending on p and xo. The in eg a ion o quali a i e knowledge is made by adding cons ain s o he ne wo k. They a e cons ain s combined wi h 'and' and 'o ' ope a o s. This ep esen a ion will help us o ob ain he beha io o he sys em i we apply an app op ia e algo i hm o sol e he esul ing cons ain ne wo k (see [3]). 3 Rep esen a ion o quali a i e knowledge We shall ocus ou a en ion in dynamical sys ems whe e he e may be quali a- i e knowledge in hei pa ame e s, ini ial condi ions and/o ec o ield. They cons i u e he semiquali a i e di e en ial equa ions o he sys em. Fi s , we need o ake in o accoun ha he ep esen a ion o he quali a i e knowledge is ca ied ou by means o ope a o s. They ha e associa ed eal in e - als. This ep esen a ion p o ides he ollowing ad an ages: easy in eg a ion o 331 quali a i e and quan i a i e knowledge [2]; and i makes possible he de ini ion o he ange o quali a i e a iables and pa ame e s o he sys em. This de ini- ion is p o ided by expe s, and i allows o echniques de eloped on in e als analysis and cons ain sa is ac ion p oblem o be used [4], [13] and [3]. 3.1 Quali a i e pa ame e s and ini ial condi ions The quali a i e ep esen a ion o pa ame e s and/o ini ial condi ions o dyna- mical sys ems may be ca ied ou by means o he quali a i e ope a o s U and B. They ep esen , espec i ely, he se o una y and bina y quali a i e ope a o s. Fo example, U = { e y nega i e, mode a ely nega i e, sligh ly nega i e, sligh ly posi i e, mode a ely posi i e, e y posi i e } and B = {much less han, mode a- ely less han, sligh ly less han, much g ea e han, ... }. Each ope a o op E U o op E B has associa ed a eal in e al op. This eal in e al deno es a quan i y. I s ands o hose alues whe e he magni ude has he quali a i e label. The bina y quali a i e ope a o s a e classi ied in wo classes acco ding o hei ypes: • Ope a o s ela ed o he di e ence. They can be exac ly equal o =, smalle o equal o ~, and la ge o equal o 2::. • Ope a o s ela ed o he quo ien . They can be much less han «, mo- de a ely less han - <, sligh ly less han ~<, app oxima ely equal o :::i, sligh ly g ea e han >~, mode a ely g ea e han > -, and much g ea e han » 3.2 En elope unc ions These unc ions es ablish a possible ange o alues o i s image o each gi en alue. They ep esen he amily o unc ions included be ween wo de ined unc- ion, a supe io one g : IR --+ IR and ano he in e io one g : IR --+ JR. Le be y = g(x) an en elope unc ion (see igu e l.a). I is ep esen ed by means o (£ (X) , g( X) , J) , ' :/x E I: £(x) ~ g(x) (2) whe e I is he de ini ion domain in he eal line o g, and x is a a iable. 3.3 Quali a i e con inuous unc ions Le be y = h(x) a quali a i e con inuous unc ion. I ep esen s a unc ional ela ionship wi h x as independen a iable, and y as dependen a iable. y = h(x), h ={PI, s1, P2, ... sk-I, Pk} wi h Pi= (di, ei) (3) whe e each Pi is a poin . I s ands o an impo an quali a i e landma k o h. Each Pi is ep esen ed by means o a pai ( di, ei) whe e di is he associa ed qua- li a i e landma k o he a iable x and ei o y. Poin s a e sepa a ed by he sign Si o he de i a i e in he in e al be ween a poin and he ollowing. The sign Si is + i he unc ion is s ic ly mono onic inc easing in ha in e al, -i i is s ic ly mono onic dec easing, and 0 i i is cons an . The de ini ion o a unc ion 332 g y a) b) X X - - Fig. 1. Quali a i e unc ions is always comple ed wi h he landma ks which deno e he cu poin s wi h he axes, and he poin s whe e he sign om he de i a i e changes (a maximum o a minimum o h). Quali a i e in e p e a ion o h (see igu e l.b) o each P; is: { x = d;::::} y = e; y- h(x) = 0 = { s; : +::::} e; < y < ei+l di < x < di+l ::::} si = - ::::} e;_! y > ei+l si- 0 => y- e, A special case o con inuous unc ion is ha whe e he sign o all in e als is he same, ha is, s 1 = ... = Bk-l = s. I is a s ic ly mono ononic unc ion. This unc ion can be exp essed in a sho way by h = M s { P 1, P 2, .•. , Pk} 4 F om quan i a i e and quali a i e knowledge o cons ain ne wo ks The quali a i e knowledge is added by means o a se o cons ain s. They a e combined wi h 'and' and 'o ' ope a o s. The cons ain s ob ained om quali a- i e knowledge a e: • Quali a i e pa ame e s and ini ial condi ions Fo each ope a o , i is ob ained a cons ain acco ding o i s ype. Le be a new a iable gene a ed. Iu, h a e he in e als associa ed o he una y and bina y ope a o s. In e als Iu a e s ablished in acco dance wi h [12], and in e als h wi h [9]. * Le u be an una y ope a o u E U. Le e be an a i hme ic exp ession. The esul ing cons ain s a e u(e):={e- =O, Eiu * Le b be a bina y ope a o b E B. Le op he bina y ope a o , and le e1, e2 be wo a i hme ic exp essions. I b is an ope a o ela ed o he di e ence ( =, ~, 2:) hen he esul ing cons ain s a e 333 and i b is an ope a o ela ed o he quo ien , • En elope unc ions Fo each en elope unc ion y = g ( x) he cons ain ( 4) is ob ained g(x) = O: !_(x) + (1- o:)g(x) o:E[0,1] (4) This cons ain s ands o a amily o unc ions included be ween g and g. I is in e es ing o no ice ha i o: = 0 :::::? g(x) = g(x) and i o: = 1 =?-g(x) = g(x) and any o he alue o o: in [0,1] s ands o any included alue be ween g(x) -and ~x). - • Quali a i e unc ions Fo each quali a i e unc ion y = h(x) de ined as (3) is ca ied ou he ollo- wmg: * Fo each landma k d; o e; appea ed in he de ini ion o h, i is added o he se o a iables o he model a new a iable wi h a domain ( -oo, +oo ). * The ollowing linea cons ain s due o he de ini ion o h a e added o he cons ain ne wo k { d1 < d2 < ... < dk, e1 o e2 o ... o ek-1 o ek, y _ h(x) = O = ((x = d1, y = e1); ... ; (x = dk, y = ek); (d1 < x < d2,e1 01 yo1 e2); ... ; (dk-1 <X< dk, e1 Ok-1 y Ok-1 ek)) (5) whe e (5) is a se o linea cons ain s combined wi h and and o ope a o s, espec i ely deno ed by comma (, ) and semicolon (; ) . The ope a o o is de ined as { x > y i s; -1 = s; = + X 0; y = X < y i Si -1 = Si = - X= y i Si-1 = Si = 0; Si-1 =F Si The se o cons ain s ob ained by he inclusion o quali a i e knowledge o he model is o muled as an in e al cons ain sa is ac ion p oblem. As we ha e indica ed, a cons ain -based easoning me hod by means o in e al consis ency echniques is applied (see [3]). The esul s ob ained a e a se o eal in e als by he a iables and cons ain s o de among hem. 5 An example Le be wo in e connec ed anks (see igu e 2). The semiquali a i e model is Va iables: V = {x1,x2, 1, 2,s,p} Landma ks: L = {a, b} Func ions: h1 := M+ {(0, 0), a, b} 91 =< 2x, x, [0, oo] > Cons ain s: 1 = h1(s), 2 = 91(x2), s = x1- x2 ~ - ~ -" " d" ( ) d - P- 1, d - 1 - 2, pos1 1 e me mm p 334 Fig. 2. The in e connec ed anks sys em The cons ain ne wo k ob ained applying he p oposed echniques is V = {xl,x2, l, 2,s,p,a,b} 0 <a, 0 < b, ( (s < 0, 1 < 0); (s = 0, 1 = 0); (0 < s <a, 0 < 1 <b); (s=a, 1=b); (s>a, l>b)), 2 = a(x2) + (1- a)2x2, a E [0, 1], u- ~- E (3 5] s = x1- x2, d - p- 1, d - 1- 2, p , (6) The concep s in oduced in (10] ha e been applied in o de o ca y ou he analysis o his model. The cons ain ne wo k ha s ands o he equilib ium egions o he model a e eplacing he exp essions dxl/ d , dx2/ d by ze o in (6). This cons ain sa is ac ion p oblem has an unique solu ion, hence he e is an equilib ium egion whe e i is sa is ied ha x1 > x2 (7) being lx 2 a eal in e al. I i is applied in e al a i hme ic o ( 6), he in e als ob ained o x1 ,x 2 a e oo wide. The equilib ium egion is [O.,oo] X [O.,oo]. In o de o na ow his solu ion i is applied he consis ency echniques [3], and hen i is ob ained o x2 he in e al (1.06487, 5.0]. The in e als o x1, s a e posi i es, and using he cons ain s = x1 -x2, hen esul s (7) a e concluded. The e o e he solu ion o lx 2 is closed o he eal solu ion [1.5, 5.0]. The esul s (7) may be in e p e ed as: he sys em has an unique equilib ium whe e he heigh o he i s ank is highe ha he second one. The heigh o second ank is in he eal in e al [1.5, 5.]. In a simila way, i is ob ained he ne wo k ha s ands o he s abili y o such egion. This ne wo k is also sa is ied, hence i is an s able equilib ium egion. The cons ain ne wo k ha de ine he bi u ca ions poin s a e no sa is ied. The e o e i is concluded ha he e a e no bi u ca ions. 6 Conclusions This pape p o ides a me hod o including quali a i e knowledge in semiqua- li a i e dynamical sys ems. Quali a i e knowledge is ep esen ed by means o in e als, con inuous quali a i e unc ions and en elope unc ions. This know- ledge helps us o make analysis o ha kind o sys ems. 335 We ha e applied he me hod p oposed o se e al examples. The ob ained esul s ha e been sa is ac o y. The echnique p esen ed is app op ia e o p e- dic i e p oblems in indus ial p ocesses whe e he e is quali a i e in o ma ion o hei componen s. A he momen , we a e applying he me hod o s udy a eal biome all u gic sys em. Re e ences 1. 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) 2. Gasca R.M., Ta o M., O ega J.A., P opagaci6n de es icciones in eg ando conocimien o cuali a i o y cuan i a i o Bole {n de AEPIA num 6 (1996) 23-30 3. Gasca R.M. Razonamien o y Simulaci6n en Sis emas que in eg an conocimien o cuali a i o y cuan i a i o Ph.D. Thesis, Uni e sidad de Se illa (1998) 4. Hy iinen E. Cons ain easoning based on in e al a i hme ic: he ole ance p opa- ga ion A i icial In elligence no. 58 (1992) 71-112 5. Kay H., Kuipe s B., Nume ical beha io en elopes o quali a i e models. 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