Identification of MIMO Hammerstein models using Singular Value Decomposition approach

Transcription

1 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March Idetfcato of MIMO Haerste odels sg Sglar Vale Decoposto approach Badredde Lohch Laboratory of Sceces ad echqes of Atoatc cotrol & copter egeerg. Natoal School of Egeerg of Sfax, Uversty of Sfax Ahed o Laboratory of Sceces ad echqes of Atoatc cotrol & copter egeerg. Natoal School of Egeerg of Sfax, Uversty of Sfax ABSRAC I ths paper, we preset a ew approach to detfy ltvarable Haerste systes based o the Sglar Vale Decoposto (SVD) ethod. he techqe allows for the deterato of the eoryless statc olearty as well as the estato of the odel paraeters of the dyac Ato-Regressve odel wth exogeos pt (ARX) part. Frst of all, a terato procedre s proposed to detfy the paraeters of Mlt-Ipt Mlt-Otpt (MIMO) Haerste odels by sg the Recrsve Least Sqares (RLS) algorth. Secodly, the obtaed paraeter estates of the detfcato odel clde the prodct ters of the paraeters of the orgal systes. So, to separate these paraeters of the orgal paraeters fro the prodct ters, the sglar vale decoposto ethod s dscssed. Fally, a slato stdy s perfored to deostrate the effectveess of the proposed ethod copared wth the exstg approaches. Geeral ers Haerste systes, Syste detfcato. Keywords MIMO Haerste systes, Paraeter estato, Sglar vale decoposto ethod, Recrsve least sqares algorth, No lear systes.. INRODUCION rasfer fcto odels are sed for desg of cotrol systes for ldly olear systes. However, for hghly olear systes, the cotroller desg based o the lear odel ay ot be adeqate. For sch cases, the lear odel based o a fxed cotroller wll ot gve a satsfactory respose. Ideed, a stable olear odel represetato wll be desrable [6]. wo of the ost freqetly stded olear systes are the Haerste ad Weer odels the olear bloc s statc ad follows or followed by a lear syste. A Haerste odel cossts of statc olear bloc followed by a lear dyac bloc, ad a Weer odel cossts of lear dyac bloc followed by a statc olear fcto. o detfy the Haerste odel, a varos syste detfcato ethods have bee proposed the lteratre. However, ost of the ethods focs o Sgle-Ipt Sgle- Otpt (SISO) processes [4, 3]. he frst wor whch developed a teratve detfcato procedre for Haerste odel s preseted by Naredra et al. [6]. Recetly, to garatee the global covergece of the odel paraeters a teratve aer [4] developed a pdatg algorth based o the Lyapov approach. I addto, several approaches have bee proposed to detfy Haerste odels a o-teratve fasho. For exaples Potta et al. [7] proposed a two-stage detfcato algorth to extract the odel paraeters. o separate the detfcato of the lear dyac part fro that of the statc olear part, Sg [9] sed a specal test sgal. Lasarayaa et al. [9] proposed ltvarate statstcal tools to detfy the Haerste odels. Al- Dwash et al. [] sed a hybrd odel cosstg of a eral etwor to detfy the statc olear part seres wth Ato-Regressve Movg Average (ARMA) odel for detfcato of SISO ad Mlt-Ipt Mlt-Otpt (MIMO) Haerste odels. Several other detfcato ad cotroller desg ethods for Haerste odels were developed by [,, 3]. Althogh there are soe ethods that ca hadle MIMO processes [], ay of the asse the strctre of the olearty to be separate [8],.e., the th otpt of the olear fcto s oly affected by the th pt. eral etwors or fzzy logc [,, 4, 5], ad polyoal wth cross-ters have ofte bee sed to deal wth ore geeral oleartes. Recetly, several approaches have bee proposed to detfy MIMO Haerste odels. For exaples, L et al. [] exteded Sg s [9] decoplg ethod to MIMO systes, whch however reqres olear optzato. Kwog et al. [8] exteded Raos s [8] ethod to MIMO odels. hey proposed a approach based o ltvarable cardal cbc sple fctos to odel the statc oleartes. he proposed ethod s effectve odelg processes wth hard ad/or copled oleartes. he athors [, 3] preseted a ethod for the detfcato of Haerste odels based o Least Sqares Spport Vector Maches (LS-SVMs). It wll be show that the lear odel paraeters ad the statc olearty ca be obtaed by solvg a set of lear eqatos wth sze the order of the ber of observatos. Hlag et al. [] proposed a geeralzed Haerste odel cosstg of a statc polyoal fcto seres wth a te-varyg lear odel order to odel the Haerste-le ltvarable processes whose lear dyacs vary over the operatg space. A terato procedre s proposed to detfy the geeralzed Haerste odel by sg the Jst-I-e Learg (JIL) techqe. 9

2 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March For Haerste systes, the paraeters fro the detfcato odel clde the prodcts of the orgal syste paraeters [3, 5], so separatg the orgal paraeters fro the obtaed paraeter estates of the prodct ters s reqred. he athors [6 9] proposed a sple average ethod of separatg paraeters for Haerste odels. Aother separatg paraeter ethod s the sglar vale decoposto doe by Ba [3]. hs paper presets a algorth for detfcato of MIMO Haerste olear systes by sg the Recrsve Least Sqares (RLS) wth a expoetal forgettg factor ad ses the sglar vale decoposto ethod to separate the syste estated paraeters. he paper s orgazed as follows. Secto descrbes the syste forlato related to the MIMO Haerste odels. Secto 3 presets a detfcato algorth to estate the paraeters of the syste. Secto 4 trodces a separatg paraeter ethod. he a reslts are gve secto 5. Secto 6 gves soe coclsos.. PROBLEM FORMULAION wo possble strctres as depcted fgres ad ca be sed to descrbe a MIMO Haerste odel depedg o whether the oleartes are separate or cobed [, 9]. he cobed olearty case s ore geeral, bt t ca case a very challegg paraeter estato proble becase of the large ber of paraeters to be estated []. herefore, the MIMO Haerste odel wth separate oleartes wll be cosdered ths paper (fgre ). As etoed prevosly, the syste cossts of a statc part whch cotas all the olearty followed by a Lear e Ivarat (LI) odel Hq ( ) whch cotas all the dyacs of the process. () () Fg : MIMO Haerste odel wth cobed oleartes. () f(.) f (.) v () v v () v () H q y () y () y y y () are the syste pt, y otpt ad whte ose wth zero ea at te respectvely. f (.) are polyoals of a ow order the pt as follows : ad E e e f () paraeters.,,, preset the olear syste Asse that the LI syste has the Ato-Regressve wth exogeos pt (ARX) strctre (bt other strctre le Ato-Regressve Movg Average wth exogeos pt (ARMAX), Ato-Regressve Itegrated Movg Average wth exogeos pt (ARIMAX) etc. are also possble). he pt otpt relatoshp s the gve by : A B () Y A Y B V E r r v f r v r r V v v f r v v f r r r a a b y b x A ad B ay a y b y y b y deote the lear syste paraeter atrces. (3) y y y x Asse that for, () =, y () = ad e () = for {,, y } ad A, B, for {,, }, y ad represet the order of the otpt, the order of the pt, the order of the olearty, the ber of otpts ad the ber of pts respectvely. hs paper presets a detfcato algorth to estate the paraeters a, b ad of the syste () fro gve pt otpt data { (), y ()} ad to evalate the accracy of the estated paraeters by slato o copters. r () f(.) v () H q y () f(.) v y y Fg : MIMO Haerste odel wth separate oleartes. y, U Y y y 3

3 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March he th otpt eqato fro syste () ca be wrtte as follows A y A Y A Y b A : b + b b b b + b b b B b B B B b B b B B B b B + b B b b b B B b A a a a, for {,,, y A } ad for {,,, y }. he, we ca wrte all the paraeters of the syste (4) as a vector for : A B A A B B (5) B b b b, for {,,, B } ad for {,,, y }. Before closg ths secto, we observe that the paraeterzato of the MIMO Haerste odel s r actally ot qe. For stace, ay par B, for soe ozero ad fte costats δ provdes a detcal syste as the oe (4). I the other words, ay detfcato experet caot dstgsh betwee the paraeter vector r sets B, ad r B,. herefore, to obta a qe paraeterzato, wthot loss of geeralty, oe of the r eleets of B, has to be fxed. So, we adopt the followg asspto : r Asspto : For syste (4), asse that zero ad r or (4) B are ot B ( stads for the -or) r =,,,, =,,, y ad =,,, B [, 6]. 3. IDENIFICAION OF HE MIMO HAMMERSEIN MODEL Defe the paraeter vector of the syste (4) as : ad forato vector A A b b b b A B B b b x, Ay B (7) (6) f f B B xa y A xy y y y,,,..., y A f f x B x,,,..., f he, eqato (4) s rewrtte as : y e (8) Eqato (7) s forlated a stadard state space forat for the olear MIMO Haerste ARX syste (4). Note that are avalable. ad the forato vector We defe the predcto error expresso : y ˆ by the followg (9) he RLS ethod s a effectve approach ole detfcato. hs techqe s to dscot old easreets so that the odel adapts to the chagg stato dyacally. he coplete algorth [7, 9, 3] of the RLS ethod for ARX odelg s gve as follows [4] : ˆ ˆ P P P P P P () ˆ y P x s the paraeter estato error covarace atrx wth P I x, α s a postve scalar. Also, λ() s a expoetal forgettg fcto to dscot old easreets ad ca be detered by the followg frst-order dfferece eqato : (), ad l. For the prposes of coparso, three dfferet perforace crtera have bee copted. Naely, the Mea Sqare Error (MSE), the Varace Accoted For (VAF) ad the best FI crtero who are gve by the followg expressos, respectvely : M MSE y yˆ () M ˆ Va r y y VAF ax,x% Va r y y deotes the real otpt, yˆ (3) deotes the otpt of the odel, M s the ber of valdato data ad 3

4 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March Va r. deotes the varace of a qas-statoary sgal [4, 5]. Y Y v FI x Y y ea (4) Y s a vector cotag the otpt of the odel whe t s slated wth the valdato pt data, Y v s a vector wth the valdato otpt data ad y ea s the ea vale of the otpt y [6]. 4. SEPARING PARAMEERS : SINGULAR VALUE DECOMPOSIION MEHOD (SVD MEHOD) After gettg the estates of the paraeter vector (for {,, y }) by the above RLS algorth, the followg step s to obta the estated paraeter vector ˆ of fro the paraeter vector. ˆ Frstly, the estates ˆ A of A (for {,, A }) ca be read fro the frst A. y etres of the paraeter vector ˆ. Secodly, to obta the other paraeters whch clde the estates of the eleets prodcts of the paraeter vector ethod s dscssed. ˆ, the sglar vale decoposto Uder asspto wth (for {,, }), the sglar vale decoposto ethod [5] s appled to of the th decopose the paraeter vector Haerste syste. o splfy the atrx expresso, we ot () ad deote r b r r r by b of ˆ to, ad rearrage the A. y + to etres B b b b B ˆ ˆ. ˆ b b b B b b b ˆ x ˆ ˆ ˆ ˆB b b b ad for {,, }. ad he proble s how to estate the paraeter vectors B ˆ (5) fro the estate ˆ. It s clear that the closest estates ˆ ad ˆ, the -or sese, are those that solve the followg optzato proble : ˆ ˆ ˆ, arg (6), heore : Let ecooy-sze SVD of ˆ x have ra > p, ad let the be gve by : ˆ R W r w (7) s a dagoal atrx cotag the ozero sglar vales,,..., of ˆ ocreasg order, x the atrces x R r r ad W w w cota oly the frst cols of the tary atrces R ad W s provded by the fll SVD of ˆ, ˆ R W, respectvely. he, the x x atrces aˆ p ad bˆ p that ze the or ˆ ab, are gve by : x x ˆ ˆ aˆ, b arg a b R, W (8) ab, x p R, x p W ad dag,,..., p are gve by the followg partto of the ecooy-sze SVD (8), ˆ W R R W ad the approxato error s gve by : p (9) ˆ ˆ ˆ ab () Usg ths pt, we ca prove the lear part of the odel. So, ths algorth ca be sarzed as follows: Algorth : Step : Copte the least sqares estate as (), ad the atrx ˆ sch that : ˆ blocvec ˆ () Step : Copte the ecooy-sze SVD of ˆ as heore, ad the partto of ths decoposto as Eq. (9). Step 3: Copte the estates of the paraeter atrces ad ˆ R as respectvely : ˆ W ˆ () he solto to ths optzato proble s provded by the SVD of the atrx ˆ. he reslt s sarzed the followg theore [4]. 3

5 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March 5. SIMULAION EXAMPLES AND DISCUSSIONS o llstrate the proposed detfcato approach, we trodce two slato exaples. Wthot loss of geeralty, a ltvarable process wth two pts ad two otpts wll be tlzed to detal the proposed detfcato procedre. 5. Exaple Cosder a process wth two pts ad two otpts descrbed by the followg eqatos : y.5. y.3. y y.8.7 y.9.5 y (3).7.5v e v e the oleartes are gve by : v v he pts, whch show Fg. 3, ad, (4) are a lt-step sgal seqece seqece wth zero ea ad varace resposes, e e are a whte ose.. he otpt y y ad ther estates are show Fg. 4 ad Fg. 5 respectvely. re olearty ad ea estated olearty are show Fg. 6 ad Fg. 7 respectvely Idetfcato of the er syste otpt er Syste otpt er Estate syste otpt te() Fg 5. he d otpt ad ts estate. v (re) v (estete) Estato of the er olearty pt he syste pts Fg 6. he st olearty ad ts estate..5 pt v (re) v (estete) Estato of the ee olearty te() Fg 3. he trag pt seqece for ad 6 4 Idetfcato of the er syste otpt er Syste otpt er Estate syste otpt te() Fg 4. he st otpt ad ts estate Fg 7. he d olearty ad ts estate. By sg the proposed ethod, the atrces the ARX odel ad the vectors of the oleartes are coverged to the followg : 33

6 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March ˆ ˆ A ; A ; ˆ B ˆ ; ˆ It s clear that the atrces  ad  are closed to the sae atrces the ARX plat odel. Bt the atrx ˆB ad the vectors ˆ ad ˆ are dfferet fro the odel. hs certaty s related to the SVD approach. However, the ltplcato ad the otpt of the syste coverged to the oleartes at the pts whch are show Fgs. 4, 5, 6 ad 7. he dfferet perforace crtera wth the SVD ethod betwee the actal ad detfed oleartes are show able. st syste d syste MSE 4.384e e-7 VAF (%).. FI (%) he syste pts Idetfcato of the er syste otpt pt pt te() Fg. 8. he trag pt seqece for ad er Syste otpt er Estate syste otpt. able. he dfferet perforace crtera - 5. Exaple Cosder a olear process descrbed by the Haerste syste as follows : y.7. y.8. y y..5 y.5.4 y (5).3. v.6. v e.5.6 v..6 v e the oleartes are gve by :.5 v.94.8 v he pts, whch show Fg. 8, ad, 3 are a lt-step sgal seqece seqece wth zero ea ad varace resposes, e e are a whte ose (6).. he otpt y y ad ther estates are show Fg. 9 ad respectvely. re olearty ad ea estated olearty are show Fg. ad Fg. respectvely te() Fg. 9. he st otpt ad ts estate. Idetfcato of the er syste otpt er Syste otpt er Estate syste otpt te() Fg.. he d otpt ad ts estate. 34

7 v (re) v (estete) v (re) v (estete) Estato of the er olearty Fg.. he st olearty ad ts estate. Fg.. he st olearty ad ts estate Estato of the ee olearty Fg.. he d olearty ad ts estate. By sg the proposed ethod, the atrces the ARX odel ad the vectors of the olearty are coverged to the followg : ˆ ˆ A ; ; A ˆ ˆ B ; B ; ˆ ˆ he reslts were obtaed sg the sae algorth that the atrces ˆB ad ˆB are closed to the Â, Â, sae atrces the ARX plat odel. I the sae way, the otpt of the syste coverged to the oleartes at the pts whch are show Fgs. 9,, ad. he dfferet perforace crtera wth the SVD ethod betwee the actal ad detfed oleartes are show able. st syste d syste MSE.988 e e-7 VAF (%).. FI (%) able. he dfferet perforace crtera Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March 5.3 DISCUSSIONS I ths paper, MIMO Haerste odel detfcato based o RLS algorth ad SVD ethod for decoposto of the lear ad olear paraeters have bee deostrated for two slato exaples. hs approach offers the advatage of beg ore geeral tha the other approaches to preset other papers sch as the LS-SVM approach preseted by [3] ad JIL approach preseted by [] of ade that these two approaches cosder that the atrces are selected eqal to the dettes, o the other had or approach we cosder the case ore geeral B s specfed. It s portat to ow the codtos der whch the RLS/SVD algorth wll coverge. he RLS/SVD s a cobato of the RLS ad SVD algorths. Hece, the covergece propertes of the RLS/SVD algorth are drectly assocated to the covergece propertes of the RLS ad SVD algorths. For deterstc systes, t s well ow that the RLS prodces based estates of the paraeters provded that the process order s ow ad the pt s persstetly exctg [3]. O the other had, the lear odel paraeters ad the statc olearty ca be obtaed sltaeosly by solvg a set of lear eqatos followed by the sglar vale decoposto (SVD). he, by recrrg to SVD ad ra redcto, optal estates of the paraeter atrces characterzg the lear ad olear parts ca be obtaed. Coparg the oleartes Fgs. 6 ad 7 the frst exaple ad Fgs. ad the secod exaple, we ca cosder that the approach s satsfactory by detfyg the oleartes. he pts of the frst exaple are show Fg. 3 ad that of the secod exaple are show Fg. 8. he coparso betwee real ad estate otpt crves of the syste are gve Fgs. 4 ad 5 for the frst exaple ad Fgs. 9 ad for the secod exaple. Fro ables ad ad Fgs. 3 to, we ca fer the followg coclsos: he paraeter estates provded by the detfcato algorth coverge to ther tre vales. he resposes of the orgal syste ad the detfed syste are very slar. It s clear that the errors are becog saller as creases. hs rear cofrs the proposed algorth. For the sae data legth, the recrsve algorth gves good estated paraeters. he covergece of the estated paraeters to ther tre vales. he MIMO Haerste odel otperfors the estates odel for each of the three cosdered crtera. hs state shows that the proposed algorth s effectve. 6. CONCLUSION Exhastve slatos have bee sed to stdy the covergece propertes of the RLS/SVD algorth ad to stdy the effects of the type of statc olearty, ad ose o the behavor of the RLS/SVD algorth. I all cases, strog covergece s observed provded that soe gde les are followed. 35

8 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March I ths paper, we have proposed a ew techqe for the detfcato of MIMO Haerste ARX systes. he ethod s based o recrsve least sqares (RLS) approxato ad allows to detere the eoryless statc olearty as well as the lear odel paraeters fro a lear set of eqatos. he obtaed estated paraeters of the detfcato odel clde the prodcts of the orgal syste paraeters. o separate the estated paraeters to the orgal paraeters, the sglar vale decoposto (SVD) ethod s dscssed. Moreover, the proposed ethod s applcable to MIMO systes wth separate or cobed oleartes. he recrsve algorth s a ovel cobato of RLS ad SVD algorths. Slato reslts reveal the robstess ad effectveess of the proposed ethod. 7. ACKNOWLEDGMENS We tha the stry of hgher edcato ad scetfc research of sa for fdg ths wor. 8. REFERENCES [] Aboy, J., Babsa, R., Botto, M.A., Szefert, F. ad Nagy, L., () "Idetfcato ad cotrol of olear systes sg fzzy Haerste odels". Id Eg Che Res, 39 pp [] Al-Dwash, H. ad Kar, M.N., (997) "A ew ethod for the detfcato of Haerste odel". Atoatca 33 pp [3] Ba, E. W., (998) "A optal two-stage detfcato algorth for Haerste Weer olear systes". Atoatca, 34 (3), pp [4] Ba, E.W., (4) "Decoplg the lear ad olear parts Haerste odel detfcato", Atoatca 4 (4) pp [5] Chao, F.Z., Gr, F., Rochd, Y., Haloa, M., ad Natal, A., (5) "Syste detfcato based o Haerste odel", Iteratoal Joral of Cotrol 78 (6) pp [6] Dg, F. ad Che,., (5) "Idetfcato of Haerste olear ARMAX systes", Atoatca 4 (9) pp [7] Dg, F., Sh, Y. ad Che,., (6) "Gradet-based detfcato ethods for Haerste olear ARMAX odels", Nolear Dyacs 45 (_) pp [8] Dg, F., Sh, Y. ad Che,., (7) "Axlary odel based least-sqares detfcato ethods for Haerste otpt-error systes", Systes & Cotrol Letters 56 (5) pp [9] Dg, F., Che,. ad Iwa, Z., (7) "Adaptve dgtal cotrol of Haerste olear systes wth lted otpt saplg", SIAM Joral o Cotrol ad Optzato 45 (6) pp [] Dg, F., Peter, L. X., ad L, G., () "Idetfcato ethods for Haerste olear systes", Dgtal Sgal Processg pp [] Hlag Y.M., Ch M.-S. ad Lasharayaa S., (7) "Modelg ad cotrol of ltvarable process sg geeralzed Haerste odel". Checal Egeerg Research ad Desg ras, Vol 85 (A4) pp [] Ha Lag ad Bol Wag, (7) "Idetfcato of MIMO Haerste Models Usg Spport Vector Mache". ISNN 7, Part III, LNCS 4493, pp Sprger-Verlag Berl Hedelberg. [3] Iva Goethals, Krstaa Pelcas, Joha A.K. Syes, Bart De Moor, (5) "Idetfcato of MIMO Haerste odels sg least sqares spport vector aches". Atoatca 4 pp [4] Ja, L., Ch, M.S. ad Ge, S.S., (5) "Iteratve detfcato of ero-fzzy based Haerste odel wth global covergece". Id Eg Che Res, 44 pp [5] Ja L., Ch M.S., Ge S.S., (5) "A oteratve erofzzy based detfcato ethod for Haerste processes", Joral of Process Cotrol 5 (7) pp [6] Ja C. Goez, Erqe Baeyes, (4) "Idetfcato of bloc-oreted olear syste sg orthooral bases", Joral of Process Cotrol 4 pp [7] Jyoth S. N., Chdabara M., () "Idetfcato of Haerste odel for boreactors wth pt ltplctes". Boprocess Egeerg 3 pp Sprger-Verlag. [8] Kwog Ho Cha, Je Bao, Wlla J. Whte, (6) "Idetfcato of MIMO Haerste systes sg cardal sple fctos". Joral of Process Cotrol 6 pp [9] Lasharayaa, S., Shah, S. L., & Nadaar, K. (995) "Idetfcato of Haerste odels sg ltvarate statstcal tools". Checal Egeerg Scece, 5 (), pp [] Lee, M.W., Hag, H.P. ad Jeg, J.C., (4) "Idetfcato ad cotroller desg for olear processes sg relay feedbac". J Che Eg Japa, 37 pp [] Lee Y.J., Sg S.W., Par S., (5) "Ipt test sgal desg ad paraeter estato ethod for the Haerste Weer processes", Idstral ad Egeerg Chestry Research 43 (3) pp [] L, Y., Sheg, J., ad Dg, R., () "Covergece of stochastc gradet estato algorth for ltvarable ARX-le systes", Copters ad Matheatcs wth Applcatos 59 pp [3] Lg, L. (987) "Syste Idetfcato heory for the User". Pretce-Hall, Eglewood Clffs, NJ. [4] Lohch B., o A., (8) "Idetfcato of Greehose paraeters sg ARMAX ad NARMAX strctres", Iteratoal Revew of Physcs (I. R. E. PHY.) vol., N., pp [5] Lohch B., o A., () "Idetfcato of olear Haerste Weer ARMAX systes", th Iteratoal coferece o Sceces ad echqes of Atoatc cotrol & copter egeerg (SA ), Moastr, se. [6] Naredra, K. S. ad Galla, P. G. (966) "A teratve ethod for the detfcato of olear systes sg a Haerste odel". IEEE rasactos o Atoatc Cotrol,, pp

9 Iteratoal Joral of Copter Applcatos ( ) Vole 4 No.9, March [7] Potta, M., Ubehae, H. ad Seborg, D.E., (993) "Applcato of geeral ltodel approach for detfcato of hghly olear processes". It J Cotrol, 57 pp. 97. [8] Raos J.A., Drad J.F., (999) "Idetfcato of olear systes sg a B-Sples paraetrc sbspace approach". Proceedgs of the Aerca Cotrol Coferece, vol. 6, pp [9] Sg S.W., () "Syste detfcato ethod for Haerste processes". Idstral ad Egeerg Chestry Research 4 (7) pp AUHORS PROFILE Badredde LOUHICHI was bor Sfax (sa) o Agst 976. He receved the Electrcal Egeerg Dploa fro Sfax Egeerg Natoal School (ENIS), Uversty of Sfax, sa,. He prepared collaborato wth the Polytechqe School, Uversty of Nates, Frace, hs dploa of DEA. He receved the DEA (Master) atoatc cotrol fro ENIS. He s crretly preparg the Ph.D. the feld of atoatc cotrol. He s a eber of the Laboratory of Sceces ad echqes of Atoatc cotrol & copter egeerg (Lab-SA) of the Uversty of Sfax. He s crretly a Assocate Professor of electrc egeerg the Faclty of the Sceces of Sfax, sa. Ahed OUMI was bor Sfax (sa) o Noveber 95, he s Professor the Sfax Egeerg Natoal School (ENIS), receved the Electrcal Egeerg Dploa fro (ENIS/sa), the DEA (Master) Istretato ad Measreet fro Uversty of Bordeax-/Frace 98 [3] S, H.. ad McAvoy,.J., (993) "Itegrato of ltlayer percepto etwors ad lear dyac odels: a Haerste odelg approach". Id Eg Che Res, 6 pp [3] Wag, D.Q. ad Dg, F., (8) "Exteded stochastc gradet detfcato algorths for Haerste Weer ARMAX systes", Copters ad Matheatcs wth Applcatos 56 pp [3] Zh Y., () "Idetfcato of Haerste odels for cotrol sg ASYM", Iteratoal Joral of Cotrol 73 (8) pp ad the Doctoral hess fro the Uversty of s 985. He oed the Sfax Egeerg Natoal School (ENIS), as a Assocate Professor of Electrc Egeerg, sce October 98. I, he obtaed the Uversty Habltato (HDR) fro the Sfax Egeerg School (ENIS). He s at preset, Professor o Atoatc Cotrol, ad the Drector of the Electrcal Egeerg Departet ENIS. he a research area cocers the odellg, the stablty of the electrc aches, the electrcal etwors, the fzzy logc cotrol. He s a eber of the Laboratory of Sceces ad echqes of Atoatc cotrol & copter egeerg (Lab-SA) of the Uversty of Sfax. He s the Head of the Research ea of Idstral Processes Cotrol (CPI) of the Sfax Uversty. Sce, he s the Presdet of the teratoal coferece o Sceces ad echqes of Atoatc cotrol (SA) whch have tae place every year a ber of torst ctes of sa. 37