Application of a Stochastic-Fuzzy Approach to Modeling Optimal Discrete Time Dynamical Systems by Using Large Scale Data Processing

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1 Applicaion of a Sochasic-Fuzzy Approach o Modeling Opimal Discree Time Dynamical Sysems by Using Large Scale Daa Processing AA WALASZE-BABISZEWSA Deparmen of Compuer Engineering Opole Universiy of Technology Opole POLAD a.walaszekbabiszewska@gmail.com Absrac: - In he paper he proposed dynamical model of he sysem is defined by fuzzy saes and a ransiion funcion. The ransiion funcion is represened by condiional probabiliies of respecive fuzzy evens. The crieria of opimizaion as well as consrains consiue fuzzy ses. The dynamical model has a form of a sochasic-fuzzy knowledge base where he rules and weighs of rules have been buil by using large-scale daa ses. ey-words: - Fuzzy conrol nowledge base Sochasic sysem wih fuzzy saes Inroducion In he revoluionary conribuion eniled Decision making in a fuzzy environmen by R.E. Bellman and L.A. Zadeh a very imporan mahemaical ool o opimal conrol of dynamic sysems has been formulaed []. According o ha work a fuzzy environmen represens fuzzy consrains and fuzzy opimizaion crieria. During nex years many works on he field of fuzzy decision making and fuzzy conrol has been iniiaed. In he 98s many of J. acprzyk s works have been dedicaed o mulisage fuzzy conrol problems concerning deerminisic sochasic or fuzzy dynamic objecs [][3][4]. The class of conrol models presened in he menioned above works is known as prescripive in opposie o descripive models given by e. g. linguisic rules. In he works by Mamdani and Assilian also Tong and many oher auhors he conrol models were called linguisic o accenuae relaionships beween descripive models and he linguisic way of formulaing problem and finding soluions specific for human beings [5][6]. The srucure of his paper is as follows: he fuzzy opimal conrol problem of he discree ime sochasic sysem is formulaed according o he objec lieraure in secion. In secion 3 he proposiion of he sochasic-fuzzy conrol model in a form of he knowledge base is presened. The weighs of rules represen empirical condiional probabiliies of he sae ransiion. Fuzzy Mulisage Conrol Models For he ransparency le us firs consider he fuzzy conrol problem where an objec an objecive funcion and consrains are very well deermined. Opimal soluion can be find on he way of mahemaical ransformaions. In his chaper an example concerning he mulisage fuzzy conrol of a sochasic dynamic objec is presened. A dynamic sysem deermined in he discree domains of is variables is considered. Le he sae space be a finie se Ξ = { x xm }; le s s+ Ξ mean he saes a momens + respecively. The sae variable is idenified wih he oupu variable. Also le he space of a conrol (inpu) variable be a finie se Λ = { u u... u } ; le c Λ denoes he conrol variable applied a momen. A dynamics of he sysem is described by a ransiion law condiional probabiliy funcion p( s + / s =... () Transiion law () deermines a probabiliy of he occurrence he sae s + Ξ a momen + if a ime he sysem sae was s Ξ and conrol c Λ was applied. The sae s + Ξ does no depend on he earlier saes (Markov propery ). E-ISS: Volume 8 9

2 Transiion law () saisfies he condiions of he disribuion funcions: p s / s c ) = () s+ Ξ ( + s = cons c = cons. Condiional probabiliy funcions () realize a mapping he caresian produc of he corresponding spaces ino he uniary inerval of real numbers [ ] R : p : Ξ Λ Ξ []. (3) Le us noe ha he ransiion law () as condiional probabiliy funcions of discree variables can be presened in a form of a marix (able). A formalizaion of a mulisage fuzzy conrol ask when an objec is a Markov sochasic chain given by a ransiion funcion in a form of a condiional probabiliy p( s / s c ) =... + of he occurrence he sae s + Ξ a momen + providing s Ξ and c Λ a ime mus be compleed as follows: - he number of sages is given =.. ; - here are consrains Κ = - on he conrol variable c Λ in a form of fuzzy subses of a conrol space Λ ; consrains are deermined by heir membership funcions µ ( c ) such ha Κ µ c ) : Λ [ ] R ; = -; Κ ( - here is a fuzzy objecive funcion on he final sae s Ξ ; is a fuzzy subse of he sae space Ξ given by is membership funcion µ ( s ) such ha µ ( s ) : Ξ [ ] R. Hence he saes {s } represen a realizaion of a sochasic process he fuzzy objecive funcion sands for a fuzzy even. Probabiliy of any fuzzy even according o Zadeh s definiion is equal o he expeced value of is membership funcion P( ) = E[ µ ( s )] [] and can be wrien as follows [8]: P( ) = µ ( s = xi ) p[( s = xi )/ s c ] xi Ξ (4) The sequence of opimal decisions c... c (opimal conrol) a sages = - maximizes he probabiliies of reaching he goal and (in he sense of fuzzy logic) fulfils he consrains []. This opimizaion crierion can be wrien by using dynamic programming in he recurren form [] [7]: µ i ( s i ) = maxc [ µ i ( c ) [ + ( + )]] i E µ i s i i (5) Example. This simple example shows paricular seps of he opimal fuzzy conrol ask according o Bellman and Zadeh s approach given above. Suppose he sochasic Markov objec deermined by: sae space Ξ = { x x x3} conrol space { u } condiional probabiliy funcions of a sae ransiion p( s + / s s s + Ξ c Λ given in able. Also suppose wo sages (=) for achieving he fuzzy objecive funcion ( s) =.8/ x + / x +.5/ x3. (6) Fuzzy consrains a sages = and = are given as follows: Κ = / u +.5/ u (7) Κ =.5/ u + / u. (8) Tab.. The condiional probabiliy funcions p s + / s c ) of he sae ransiion s c ( s + x x x 3 x u x u 3 u x u x u 3 u Le us sar from he final sage = o compue he condiional probabiliies of he achieving he fuzzy goal depend on he sae s and conrol c a he previous sage. According o (4) and (6) also aking ino accoun he firs raw in able we have P ( ( s / x u)) = =.8 Similarly aking ino accoun he second raw in able we obain: P ( s / x u )) = =.86 ( The possible six values of condiional probabiliies P( (s /s c )) have been presened in able. Taking ino accoun he resuls in able and consrain (8) he following rules of conrol a sage = can be formulaed: -saring from he sae s =x apply - saring from he sae s =x apply c =u c =u E-ISS: Volume 8 9

3 - saring from he sae s =x 3 apply c =u. Tab.. The condiional probabiliies of achieving he fuzzy goal a = P( (s /s c )) c s x x x 3 u u.6.88 Coninuing calculaions a sage = assuming ieraion i= in (5) we derive he relaionship for he fuzzy objecive funcion a =: µ ( s) = maxc [ µ ( c ) E[ µ ( s)]]. (9) I is easy o noe ha (9) expresses a fuzzy composiion of he max-min ype and can be wrien in he form: [.5 ] = [.6.88] () The righ side of () represens he membership coefficiens of he fuzzy objecive funcion ha is: ( s) = / x +.6/ x +.88/ x3. () Hence o opimize conrol a sage = he condiional probabiliies P ( ( s / s c)) should be calculaed. The fuzzy even deermined by () and he condiional probabiliies given in able serve as daa o calculaion according o (4). Resuls of calculaion are presened in able 3. Tab. 3. The condiional probabiliies of achieving he fuzzy goal a = P( (s /s c )) c s x x x 3 u u The fuzzy objecive funcion a sage = can be deermined on he basis of (5) for i= as he resul of a fuzzy max-min composiion : µ s ) = max [ µ ( c ) E[ ( )]] () ( c µ s [.5] = [.74.84] Tha is ( s) = / x +.74/ x +.84/ x3. (3) Thus here are he following rules of conrol c a sage =: - saring from he sae s =x apply c =u - saring from he sae s =x apply c =u - saring from he sae s =x 3 apply c =u. The sequence of opimal decisions (opimal conrol) c c a sages = and = respecively maximizes he probabiliy of reaching he fuzzy objecive funcion a final sage and (in he sense of fuzzy logic) fulfils he consrains a sages = and =. 3 Large Scale Daa and nowledge-based Models Big daa reposiories and specialized mehods and algorihms of knowledge discovery became sources of creaing more meaningful knowledge bases. A knowledge base as a par of compuer exper sysems usually has a form of logic condiional rules IF THE given by expers. Such se of rules can represen knowledge for e.g. diagnosic ask predicion problem or conrol problem considered in his paper. According o mahemaics and logic operaions applied in creaing he knowledge model we can hink of deerminisic sochasic fuzzy or sochasic-fuzzy knowledge bases. Consider conrol of a sochasic dynamic sysem wih fuzzy saes. Assume discree sae space Ξ = { x xm } and disinguished fuzzy saes X X... X of he sysem. Saes are defined by respecive membership funcions µ X ( x) µ X ( x)... µ X ( x) x Ξ Borel measurable and fulfilling relaionship [9]: k= µ ( x) = x Ξ (4) X k Moreover i is assumed he exising probabiliies of non-fuzzy elemenary evens x x M Ξ equal o p ( x ) p( x)... p( x M ) [ ] and fulfilling he condiion of disribuion funcion E-ISS: Volume 8 9

4 M m= p ( x ) =. m Sochasic dynamic sysem wih fuzzy saes is deermined by he following condiional probabiliies of he ransiion of fuzzy saes: - condiional probabiliy funcions of he occurrence he fuzzy sae S+ { X X... X } a momen + providing fuzzy sae S { X X... X } and conrol c Λ a momen P( S + / S =... (5) such ha P S / S c ) (6) ( + = S+ { X... X } S = cons c = cons. Also an iniial probabiliy disribuion funcion of fuzzy evens exiss P( S ) = { P( X k )} k =... where P ( X k ) [] P( X k ) = µ X ( xm) p( xm) k =... (7) k xm Ξ k = P ( X k ) =. Moreover condiional probabiliies P(S + /S c ) do no depend on earlier saes (Markov propery of sochasic process wih fuzzy saes) [9]. Example. Le he space of numerical values of he esed objec be Ξ = { x x5}. Le he probabiliy p(x)=. be he same for every elemen of he space. Suppose ha hree fuzzy saes have been deermined by expers: X = / x +.5/ x X =.5/ x + / x3 +.5/ x4 X 3 =.5/ x4 + / x5. Fuzzy saes can have some linguisic meaning as e.g. low middle high ec. Iniial probabiliy disribuion of fuzzy saes calculaed according o (7) is : P S ) { P( )} {.3.4.3}. ( = X k k = 3 = The values of condiional probabiliies P ( S + / S c ) can be calculaed on he basis of empirical daa from daa reposiories as i is shown in [9] []. In able 4 here are some exemplary values. Tab. 4. Exemplary values of condiional probabiliies of he fuzzy sae ransiion S / S c ) P ( + S c + S X X X 3 X u X u..6. X 3u X u X u X 3 u Table 4 consiue he sochasic-fuzzy conrol model of he esed process. This model can be wrien in a form of condiional weighed rules in fuzzy logic as follows: R : If ( S is X ) And ( c is u ) Then R : If ( S is X ) And ( c is R 6 : If ( S is X 3 ) And ( u ) Then Also ( + Also ( c is u ) Then ( S + is X )(P=); ( S + is X )(P=.) S is X )(P=.6) S is X 3 )(P=.); ( S + is X )(P=.) Also ( S+ is X )(P=.7) Also ( S + is X 3 )(P=.). To make decision concerning c a momen expers can use some measure deermining a coefficien of similariy beween fuzzy ses: cerain fuzzy goal and fuzzy saes S + =S included ino aneceden of he given rule. Le he goal be given as: =./x +.3/x +/x 3 +./x 4 +/x 5. Taking normalized Hamming s measure as he similariy coefficien E-ISS: Volume 8 9

5 I η ( S ) = µ S ( xi ) µ ( x i ) (8) I i= we have: η( X )=.54 η( X )=.64 η( X 3 )=.46. Consider fuzzy sae S + =X which is raher close o S. Probabiliy of he achieving his sae a momen + is equal o.7 if S =X 3 (rule R 6 ) and he conrol value c =u was applied. Also probabiliy of S + =X is equal o.6 if a momen S =X (rule R ) and he conrol value c =u was applied. In he oher rules probabiliies of ransiions o S + =X are very low. 4 Conclusion Two presened approaches o fuzzy conrol can be summarized as follows: -when he model of a sochasic objec is given in he precise mahemaical form he sequence of opimal decisions (opimal conrol) a paricular sages maximizes he probabiliy of reaching he fuzzy objecive funcion a final sage and (in he sense of fuzzy logic) fulfils he consrains a he sages; - when he model of he objec has a form of a sochasic-fuzzy knowledge base he choice of conrol is raher subopimal because he model depends on he hisorical daa concerning observed siuaions. When he collecion of daa consiues a large scale se and concerns many siuaions hus he selecion of he conrol value can be opimal. [6] Tong R.M. Synhesis of Fuzzy Models for Indusrial Processes Some Recen Resuls Inernaional Journal of eneral Sysems pp [7] acprzyk J. Mulisage Fuzzy Conrol (in Polish) WT Warsaw. [8] Zadeh L.A. Fuzzy Ses Informaion and Conrol pp [9] Walaszek-Babiszewska A. Fuzzy Modeling in Sochasic Environmen. Theory nowledge Bases Examples LAP LAMBERT Academic Publishing Saarbrucken. [] Walaszek-Babiszewska A. and Rudnik. Sochasic-Fuzzy nowledge-based Approach o Temporal Daa Modeling in: Time Series Analysis Modeling and Applicaions pp Pedrycz W. and Shyi-Ming Chen (Eds) Springer Berlin 3. References: [] Bellman R.E. and Zadeh L.A. Decision Making in a Fuzzy Environmen Managemen Science 7 97 pp [] acprzyk J. and Saniewski P. A ew Approach o he Conrol of Sochasic Sysems in a Fuzzy Environmen Archiwum Auomayki i Telemechaniki XXV 98 pp [3] acprzyk J. Conrol of a Sochasic Sysem in a Fuzzy Environmen wih Yager s Probabiliy of a Fuzzy Environmen Busefal 98 pp [4] acprzyk J. Mulisage Decision Making under Fuzziness Verlag TUV Rheinland Cologne 983. [5] Mamdani E.H. and Assilian S. An Experimen in Linguisic Synhesis wih a Fuzzy Logic Conroller Inernaional Journal of Man- Machine Sudies pp. -3. E-ISS: Volume 8 9