NEURO FUZZY MODELING OF CONTROL SYSTEMS

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1 NEURO FUZZY MODELING OF CONTROL SYSTEMS Efré Gorrosteta, Carlos Pedraza Cetro de Igeería y Desarrollo Idustral CIDESI, Av Pe de La Cuesta 70. Des. Sa Pablo. Querétaro, Qro, Méxco gorrosteta@teso.mx pedraza@cdes.mx Abstract The aalyss of the models s carred out startg from expermetal data of a multvarable system MISO (May Iput Sgle Output). The models mplemetato was made usg fuzzy logc. I fuzzy logc, the cluster techque was used to decrease the umber of rules to use the detfcato. Ths techque s opposed to the covetoal method whch requres a cosderable umber of fuzzy ferece rules to approach the model. I the cosequece of fuzzy model, dfferet techques are used to mplemet Taag-Sugeo type rules. By other had, we mplemeted the Neuro-fuzzy modelg methods, whch let represet the o-lear system ad at the same tme a system wth some learg degree usg dfferet topologes. By comparso the goodess of each method s obtaed. 1. Itroducto INSIDE the cotrol of egeerg systems, oe of the most mportat tass s the cotrol systems modelg. At the preset tme, there are may forms to do ths modelg. Durg the past years, the mprovemet of the expert systems has allowed to corporate ew methods to covetoal techques, such le the eural etwors, fuzzy logc ad geetc algorthms. I the year of 1965 Loft Zadeh the Uversty of Calfora maes the frst publcato o fuzzy logc the Joural of Iformato ad Cotrol, ttled Fuzzy Set [1]. Ths publcato was the begg of may later wors related to the applcato of the fuzzy logc the cotrol systems. Wthout a doubt, oe of the most relevat wors was the oe of the Taag-Sugeo, whom drew up the drectve of the fuzzy systems modelg []. Most of the tme whe we face the modelg of a system, we couted oly wth the put-output data that descrbes the system. Hece, we must cosder a blac box wth several puts ad outputs. I order to obta the fuzzy modelg of dyamcal systems, we must remember that these systems are geerally modeled by a state fucto, le eq. 1 y( ) f ( y( ), u( )) (1) where f s deomated the trasto state fucto. Ths fucto s used by the fuzzy dyamc model. The most commo fucto utlzed ths case s the ARX (Auto Regressve exogeous), whch s show eq. ; where y() s the system output, u() represets the system puts, e() represets a whte ose sgal, ad s the model s order [3]. y( ) 1 a y( ) () The Taag-Sugeo model has a very partcular structure, maly the cosequet part; therefore ths structure ca be represeted by: If X s A the y =F (x) 0 b u( ) e( ) The left part correspods to the atecedet, ad for ths partcular case t s evaluated f the put varable X has the value assged by the dffuse varable A, whch ca be a lgustc varable that s used the fuzzy systems. However, the value of the Y varables ca t be a lgustc value. Fucto F whch correspods to the cosequet has a defed structure, as ca be see Proceedgs of the 16th IEEE Iteratoal Coferece o Electrocs, Commucatos ad Computers (CONIELECOMP 006)

2 eq. 3, whch s very smlar for all the rules ad the oly oe chagg values are the put vector X. Also, the F value ca be smply a costat. T F( x) a x b (3) As we ca see, the structure of the cosequet could correspod to eq., ad ths smlarty s a advatage to relate the ARX model wth the Taag-Sugeo techque. A coveece of usg ARX s that ths model represets a lear relatoshp, ad for the partcular case of a system that represets a o-lear relatoshp t s ecessary to bult may ARX structures. Fally, the last part of ths paper a Neuro-fuzzy model s proposed by usg a Neural Networ to approxmate the results of the model to the real system.. Fuzzy Model The fuzzy system of the type Taag- Sugeo[] s represeted by the eq. 4 y 1 (4) Aother way to express ths equato s by separatg the membershp fuctos of the deomator ad leave the correspodg for each umerator, remember that ths structure the chages are due to the varables that represet the preset puts, the outputs ad the prevous puts. O equato 5, the same fuzzy fucto s represeted wth ths ew order. ( x) a0 ( x) a 1 x1 ( x) a x y ( x) ( x) ( x) ( a x b ) (5) By other had we defe a part of the eq. 5 that does ot clude the costats ad the varables, as show eq. 6. By smplfyg the orgal fuzzy fucto to a more smple expresso we could defe the values of the costats to ther correspodg rules. ( x) (6) ( x) 1 T 1 The, the fuzzy fucto expressed wth the ew assgmet s represeted by the vector multplyg as show eq. 7 y T (x) (7) Where s defed by eq. 8 ad (x) s defed by eq. 9 a a, a ] (8) [ 11, 1 1 ( x) [ 1, x, x ] (9) To ow the values of the costats from the vectorthe least squares method s used due to the form the fucto adopts [4][5][6]. Oce the structure s defed, we face the problem of defg the umber of ecessary rules to obta the system model. The umber of ecessary rules s a fucto of the umber of fuzzy parttos made each lgustc varables. Ths umber of rules ca be obtaed from eq. 10. [7] v r f (10) Where r s the umber of rules, f s the umber of lgustc terms or parttos from the fuzzy varable ad v s the umber of the varables from the system. For stace, f we cosder a system wth 3 parttos or fuzzy subsets ad varables, the umber of rules wll be r=9, by creasg a fuzzy partto each varable, the umber of rules wll crease utl 16. If we wat our model to have a better approxmato to the real system, we have to crease the fuzzy parttos each varable ad cosequetly the umber of the rules wll crease expoetally accordg to eq. 10. To optmze the umber of rules, we must group the data obtaed the regos where the elemets show some smlar characterstcs. The theory groups (cluster) help us to prepare the data, ad t determes the umber of c parttos to use the model, whch must be less tha the calculated by eq. 10. The algorthm requred to fd the c parttos from the set of data correspodg to the umber of rules s show the ext 7 steps: Proceedgs of the 16th IEEE Iteratoal Coferece o Electrocs, Commucatos ad Computers (CONIELECOMP 006)

3 1) Select the umber of parttos from the sample uverse A. The possble values of the partto are betwee two ad the maxm umber of data cotaed the sample. The value of c s defed. ) Italze the Mfc matrx wth radom data. 3) Choose the allowed lmt of the error e betwee the matrx Mfc (0) ad Mfc ( r). A value betwee 0.01 ad s desrable. 4) Select the m value startg from 1. We cosder a value of m=. 5) Calculate the ceters {v ( 0) } for each group defed the step 1). 6) Geerate matrx Mfc ( r) usg the ext formato ad the eq. 11 ( r) c d 1 d (0) ( r) 1 m1 (11) 7) If the comparso betwee Mfc (0) ad Mfc ( r) s less tha the stpulated error o step 3, the we have foud the groups that belogs to the gve data; otherwse, t s ecessary to update the matrx Mfc (0) ad go bac to step 5). Wth the above descrbed algorthm, we ca fd the c parttos of the set; but ths lead us to the ext questo: whch oe of all the parttos s the best oe to carry out the model?. O eq. 1, the selected crtera s show o [8] ad [9], to fd the umber of ecessary rules to buld our fuzzy model. To aalyze ths value, t s ecessary to revew; the m weght (tally cosdered m=), the ceters of the group v, the membershp degree the data ad the average value of the data. S( c) c 1 1 m ( ) ( x v v x ) (1) Therefore, we must fd the mmum value of eq. 1 varyg m to ow whch s the best umber of rules [9][10]. 3. Neuro- Fuzzy Model Applcato. Three possbltes are avalable to buld the Neurofuzzy model, order to o the fuzzy system wth the eural etwor. I ay case, the ma obectve s to fd a advatage accordg to the problem The frst Neuro-fuzzy method uses a Neural Networ the frst part of the structure to get the weght requred to multply the ext layers of the et. Fgure. 1 Neural etwor appled the frst part of the fuzzy system. The data requred to tra the et was obtaed from the fuzzy group the Mfc array, where the data correspods to each ext value of the etwor. The secod Neuro-fuzzy model cosder a Neural Networ the ext level. To tra ths etwor, the elemets beloggs oly to oe group were used; to do ths model three Neural Networs were traed accordace wth the model showed [5]. The fuzzy process was the same tha the prevous secto. The structure s show equato 13. If x1, x, x3 ad x4 belogs to S, Y=NN(x1,x,x3) I ths structure, the data group requred to tra the etwor does ot requre a smlar treatmet, due to the short sze of t; ths s the reaso why t s possble to apply a more smple eural etwor tha the oe appled a eural model. I the thrd procedure (eural fuzzy procedure), the structure apples the bas of a fuzzy model, where t s possble to fd the three compoets of a fuzzy system. The correspodg parameters to every fucto to the structure are the uow varable, ad the same codto exsts to the correspodg costats to each oe of the ext values; uder ths codto a leal relatoshp betwee put ad ext values ca be structured. Frst, we do a treatmet to the whole fuzzy system a smlar way to a Neural Networ, usg some method to tra the et ad defe the fucto parameters 13]. Next, a bac propagato method was appled [1] to fd the ceter ad sze of the membershp fuctos. The most used fucto was the Gauss fucto.[5]. Proceedgs of the 16th IEEE Iteratoal Coferece o Electrocs, Commucatos ad Computers (CONIELECOMP 006)

4 4. Study Case To prove the structure usg several cases, a combusto system s proposed. Here we have 7 varables at the eter ad gve that the ature of the burer elemet, we ca easly ote that t has a olear behavor Fgure Neuro-fuzzy model 3 structure It s preferable to assg short trag values the calculator layer order to avod the system dvergece. The trag of ths fuzzy system s doe a smlar way that the trag of a Neural Networ uder supervsory codtos. Ths codto mples that the put data the frst layer of the et must be show, ad the output data wll be assged to evaluate the parameters alog the Neuro-fuzzy layers Fgure 3 The Neuro-fuzzy model Equato 13 brg us the output fucto, whch have the Taag-Sugeo structure: f ( u / ) y' R b ( u) R u c exp( ( ) ) 1 u c exp( ( ) ) (13) where b ca tae two types of codtos; the frst oe ca be a costat ad the secod oe a leal relatoshp betwee the varables at real tme ad the prevous momets. I ths codto b adopts the form used the eq.. The crtera to mmze the error was expressed equato 6; ths s the reaso why e m s expressed for 14 [3][1] as ext: Fgure 4 The Neuro-fuzzy error. The Neuro-fuzzy model ad ther respectve error plots are show o fgures 3 ad 4; ths case a varato of 10 degrees s mataed respect to the 900 or 1000 from the burer operato. e m 1 m m [ f ( u / ) y ] (14) Proceedgs of the 16th IEEE Iteratoal Coferece o Electrocs, Commucatos ad Computers (CONIELECOMP 006)

5 case, was called Fuzzy, also the result obtaed for a detfcato model usg the tradtoal least mea squares (lms) method was cluded. Model W Neuro-fuzzy1 7.1 Neuro-fuzzy 5.9 Neuro-fuzzy3 7. lms 1.8 Table 1 Comparso of the models Fgure 5 The Neuro-fuzzy3 model.. 6. Coclusos O secto 5, we have some results from the system modelg usg dfferet modelg techques. It ca be observed that a frst order classc model presets a bgger error tha the fuzzy models. The fuzzy modelg by cluster reduces great umber the rules of ferece whch are used the system modelg. By other had, the fuzzy logc model combato wth the eural etwors could represet a very good opto the case of o-lear systems. Fgure 5 The Neuro-fuzzy3 error. I fgure 4, the Neuro-fuzzy 3 output s show. I ths graph a data -dfferet from the oes used to buld the model- s used to buld the model. The goal s to valdate the model performace. I fgure 5, the error the output data s show. 5. Results The performace of the w models s calculated accordg to eq. 15 [5] where y* s the data from the real output, y s the data from the output of the model ad s the umber of total data. * 1 y y w *100 (15) * 1 y The comparso that was made s about the performace of each oe model respect to the w varable. O table 1, the values for w are show for each Neuro-fuzzy model; the used model the frst techque descrbed was called Fuzzy1, for the secod The w value was calculated for the frst order models wth smlar characterstcs. Cosderg the reported values, the most recommedable model s the Neurofuzzy wth the smallest value the performace, whch meas that geeral has a better approxmato to the real system. Oe good cocluso s the ferece of the most relevat varable, whch correspods to the prevous values of temperature. Also the models where all the varables were cluded wth the model were compared, later t was costructed from ths varable ad the results were detcal. 7. Refereces [1] Taaa K., A troducto to Fuzzy Logc for Practcal Applcatos, Ed. Sprger, U.S.A, 1991, pp [] Bera R. ad Trubatch S.. Fuzzy System Desg Prcples, Ed. IEEE press, U.S.A, pp [3] Aguado B. Alberto, Temas de Idetfcacó y Cotrol Adaptable. Isttuto de Cberétca, Matemátcas y Físca. La Habaa,000. Proceedgs of the 16th IEEE Iteratoal Coferece o Electrocs, Commucatos ad Computers (CONIELECOMP 006)

6 [4] Gorrosteta, E, Pedraza, C ad Rosete J. Fuzzy Modelg of Systems, Methos ad Models Robotcs 9 Agust 1 September 005, Medzyzdroe, Polad. pp.100 [5] Passo K. ad Yurovch S. Fuzzy Cotrol, Ed. Addso-Wesley, U.S.A., 1998 pp.1-86 [7] Bortolet P, Dpassaquay, S. A Ttl. Iteractve Fuzzy Modelg ad Cotrol of o lear System Usg Multdmesoal Fuzzy Set Laboratore D aalyse et D archtectura des Systemes LASS, Fraca., 1998 [6] Wolehauser O, Fuzzy System Idetfcato Taag-Sugeo Modellg ad Idetfcato. [8] Cao S.G. Rees, N.W y Feg G, Proceedgs of the Ffth IEEE Iteratoal Coferece Fuzzy Systems, Vol 1, 1996, pp [9] Sugeo M. ad Yasuawa T., IEEE Trasactos o Fuzzy System, Vol1, [10] Nguye H. ad Sugeo M.. Fuzzy System Modelg ad Cotrol, Ed. Kluwer Academc Publshers, U.S.A, 1998.pp [11] Tsoualas L. ad Rober E., Fuzzy ad Neural Approaches Egeerg, Ed. Wley ad So, U.S.A., pp [1] Gorrosteta E., y Vargas E., A Neuro PD Cotrol Appled for Free Gat o a Sx Legged Robot, 3er. WSEAS Iteratoal Coferece o Sgal Processg, Robotcs ad Automato (ISPRA 004) ISBN , 004,February 13-15, Salzburg, Austra, [13] Zhag J. ad Morrs J, Recurret Neuro Fuzzy Networs for Nolear Process Modelg Trasactos o Neural Networs, vol 10 No March. IEEE, 1999 Proceedgs of the 16th IEEE Iteratoal Coferece o Electrocs, Commucatos ad Computers (CONIELECOMP 006)