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1 Semiário do Programa de Pó-Graduação em Egeharia Química MODELING AND IDENTIFICATION OF NONLINEAR SYSTEMS USING SISO LEM- HAMMERSTEIN AND LEM-WIENER MODEL STRUCTURES P. Bologee Ferade, D. Schlipf, J. O. Trierweiler Group of Itegratio, Modelig, Simulatio, Cotrol ad Optimizatio of Procee (GIMSCOP Departameto de Egeharia Química, Uiveridade Federal do Rio Grade do Sul (UFRGS R. Eg. Lui Eglert, /. Campu Cetral. CEP: Porto Alegre - RS - BRASIL, pedro@eq.ufrg.br, david@eq.ufrg.br, jorge@eq.ufrg.br Palavra Chave: Sytem Idetificatio; Noliear Model; Liearizatio; Wieer Sytem; Hammertei Sytem. Reumo: Thi paper dicue the applicatio of the cocept of liearizatio aroud the equilibrium maifold (LEM already preeted i the literature i order to cotruct model tructure that ca be viewed a exteio of the covetioal Wieer ad Hammertei model. Itead of liear time-ivariat ubytem i aociatio with tatic oliearitie, thee exteio exhibit variable dyamic character ad ca therefore model a broader cla of ytem tha the origial approache. Moreover, the idetificatio trategy already ued with LEM ytem ca be applied i order to cotruct uch model from experimet, ad the techique detied for aalyi ad cotrol of Wieer ad Hammertei ytem ca be applied promptly. To applicatio of thee cocept to the modelig ad idetificatio i demotrated with a umerical example, coiderig a heat exchage ytem. INTRODUÇÃO I order to cotrol atifactorily a oliear plat, two mai approache exit: either the ue of iheret oliear cotrol techique or the ue of robut liear method to guaratee tability ad adequate performace eve i the preece of oliear effect. I the firt approach, it i eceary that oliear dyamic model are available, what i very ofte ot the cae. Thi i maily due to the cot of oliear modelig ad/or idetificatio, but alo to the fact that uiveral ad fail-free method allowig for the idetificatio of accurate oliear model are till miig. I order to decribe the oliear characteritic that are ecoutered i the practice, it i ofte adequate to coider a give dyamic ytem a the compoitio of a liear dyamic block followed by a tatic oliearity, the o-called Wieer ytem. By reverig the order of the block, the reult i the Hammertei model. There i a plety of literature o pecific method for idetificatio of either Hammertei or Wieer model, or both. A good urvey o thee model tructure ca be foud i (Pearo, 995. Although iteretig from the practical poit of view, thee approache may be too imple if the decriptio of a oliear dyamic i ought. Therefore, the cocept of liearizatio aroud the equilibrium maifold (LEM ca be ued to iclude uch characteritic i the model repreetatio. The advatage of the LEM ytem i that they ca be cotructed i a traightforward maer ad reult i imple, traparet model tructure. Thi paper i orgaized a follow: Sectio A review the cocept of LEM ytem already dicued i the literature, which i the bai for the two propoed model tructure. A exteded Hammertei tructure i how i Sectio B, that

2 Semiário do Programa de Pó-Graduação em Egeharia Química i, a ytem compoed by a oliear tatic gai fuctio followed by a dyamic block with oliear dyamic. Sectio C preet a tructure baed o the Wieer model reultig by revertig the order of the elemet cited before. The model are the applied i Sectio 3 i the modelig ad idetificatio of a oliear ytem i a umerical example. Cocludig remark ca be foud i Sectio 4. MATERIAIS E MÉTODOS A. LEM SYSTEMS Coider a cotiuou SISO oliear dyamic ytem of the form r( x, y h( ( where r: X U R i at leat oce cotiuouly differetiable o X R, U R, ad h: X R i at leat oce cotiuouly differetiable. The output equatio will be frequetly omitted i the equel for horte. The equilibrium maifold of ( i defied a the family of cotat equilibrium poit ( {( x, u, y Ξ R R R :. r( x, u 0, y h( x, u } Similarly, the family of liearizatio of ( at the et of equilibrium poit determied by ( i give i the uual way a r( x, x x, u (3 r( x, ( x x + u x, u ( u u ad imilarly for the output equatio. Uder the coditio that the rak of [ r(x,u / x] i for all poit i the et Ξ (Wag ad Rugh, 987, Ferade 005, the equilibrium maifold ad coequetly the family of liearizatio of ( will be pecified by oe amog the + variable (x,. Therefore, if thi matrix i full rak, the iput fully parameterize both familie of equilibrium poit ad liearizatio. Callig the teady-tate map Ω: R R, uch that r(ω(, 0 (that i, the fuctio Ω give the teady-tate x correpodig to the cotat iput u, the iput-parameterized liearizatio aroud the equilibrium maifold (LEM of ( i defied a the ytem (Ferade 005, Ferade ad Egell, 005. A( ( x Ω( (4 where A( repreet the evaluatio of the Jacobia matrix [ r(x,/ x] o (Ω(,. The output equatio ca be liearized i a aalogou way, coiderig the tatioary output mappig Ψ: R R. The output fuctio Ω( ca be obtaied o the bai of the family of parameterized liearizatio by itegratio of dω( A( B(. (5 du where A ad B are the Jacobia matrice of r(x, with repect to x ad u, repectively, evaluated o the equilibrium maifold. The model (4 ha to be iterpreted a a (tate-affie oliear ytem that poee the ame family of equilibrium poit ( ad the ame liearizatio family (3 a the oliear ytem (. Followig the dicuio i (Ferade, 005, the LEM ytem ca cotitute alo a good approximatio of ( i traiet regime away from the equilibrium maifold, depedig o the degree of oliearity of the origial ytem. Obviouly, other repreetatio that are equivalet o the equilibrium maifold ca be cotructed o the bai of a igle parameter. Moreover, thee repreetatio ca be eaily iterchaged, provided that the ivere of the correpodig elemet i Ω( ad Ψ( exit. The focu o iput parameterizatio i due to the fact that idetificatio experimet are carried out by excitig the plat with a deiged iput igal. I thi ee, if oe aume that the local model ca be idetified by perturbig the plat aroud iolated equilibrium poit, it i atural to ue the iput i order to parameterize the liearizatio family. Therefore, a approximatio to ( ca be cotructed by mea of a fiite umber of liear local model that are coidered a member of it liearizatio family, obtaied by mea of a few local idetificatio experimet. Sice the exact LEM ytem (4 ivolve the ifiite family of liearizatio ad of the equilibrium poit of (, decribed by the matrix fuctio A( ad Ω(, i the idetificatio cotext jut a fiite ad probably mall umber of the member of thee familie are kow, but oe ca till ue approximatio or iterpolatio method i order to recotruct thee fuctio from the kow member. I order

3 Semiário do Programa de Pó-Graduação em Egeharia Química to olve the problem of cotructig a tate-pace repreetatio from local model obtaied from iput-output experimet, thee ca be traformed to a liear caoical ormal form prior to the cotructio of approximate fuctio A ( u ad Ω ( u (Ferade ad Egell, 005. I the abece of the umerical value of the teady-tate, the lat fuctio ca be obtaied by itegratio of A( B( (Ferade, 005. B. SISO LEM-HAMMERSTEIN MODELS The LEM cocept ca be ued to cotruct a Hammertei-like model of ( i which the dyamic deped o the operatig poit itead of the LTI dyamic ecoutered i the uual Hammertei tructure (Fig.. Static Noliearity No-liear Dyamic u ( t w (t y ( t q (u G Fig.. LEM-Hammertei model A geeric model with thi tructure ca be defied i tate-pace form by f ( + bw f ( + bq( y cx (6 uch that the overall family of trafer fuctio correpod to that of (. The LEM ytem (8 ca be cotructed with realizatio of the parameterized trafer fuctio of Eq. (7 i a uitable choe coordiate bai, a for example a caoical or ormal form. Obviouly, Eq. (8 deped o the ew iput w, but a equivalet tate- or output-parameterizatio ca be eaily cotructed, a dicued above. Thee are everthele dyamically wore tha the iputparameterized verio (Ferade, Egell ad Trierweiler, 004. Thi model ca be obtaied from experimet uig the LEM approach a follow: idetificatio of local liear model aroud ome iolated operatig poit; traformatio of the family of local model ito a family of uit-gai liearizatio; itegratio of k(u C(u A(u B(u i order to obtai q(; iterpolatio of A ad B i ome uitable caoical form ad itegratio of i A( w B( w order to geerate Ω ( w. Alteratively, ice the local gai i the derivative of the tatioary mappig with repect to u at a give operatig poit, q ca be directly obtaied by mea of obervatio of the tatioary output. Thi procedure ca alo be ued iteratively, that i, value of y ca be ued to refie the iterpolatio of k ad vice-vera. where b ad c are vector of proper dimeio. A poibility of cotructig a model of the form of Eq. (6 o the bai of the LEM model i to eparate the tatic oliear gai fuctio from the family of trafer fuctio (Pearo ad Pottma, 000, that i, β ( ( ( ; ( δ + + β δ + δ δ G k K α ( δ + α ( δ + K+ α( δ + (7 where δ i a calar parameterizig the et of equilibrium poit/liearizatio (u i thi cae. The reultig LEM-Hammertei ytem i of the form (8 with A( w( x Ω( w, w q( y cx q ( u ( δ dδ u k (9 C. SISO LEM-WIENER MODELS I parallel to the Hammertei-type tructure coidered above, it i alo poible to defie a exteded Wieer model by replacig the liear block with a elemet poeig variable dyamic (Fig.. No-liear Dyamic Static Noliearity u ( t x ( t y(t G h ( x Fig.. LEM-Wieer model Note that thi model i ot obtaied by imply reverig the order of the block i Fig., ice the fuctio h i a calar valued fuctio of argumet wherea q i a -valued fuctio of oe argumet (that i, a collectio of calar fuctio. Thi model ca be defied i the tate-pace i the ame fahio a i Eq. (6. Neverthele, due to the

4 V h V V c Vh V Vc UNIVERSIDADE FEDERAL DO RIO GRANDE DO SUL Semiário do Programa de Pó-Graduação em Egeharia Química oliear depedece of h o x, the iputparameterized LEM model would exhibit a direct feedthrough characteritic, what i ot deirable for imulatio (Ferade, 005. I thi cae, it i poible to cotruct a output-parameterized LEM ytem, provided that the adequate coditio hold (Wag ad Rugh, 987; i the SISO cae, for example, thi implie that there i o chage of the ig of the tatioary gai. I ay cae, the LEM- Wieer model i give by where x M x f( + g( x ( u φ( x y h( f( a j ( x x j j h( x + b j ( x x j j (0 ( where Φ( i uch that Φ(x 0 ad [ Φ(/ x] x 0. I particular, oe poibility for Φ( i T ( x Ω( x H ( x Ω( x Φ( (4 where the matrix H ha to be adjuted from experimet, ad Ω(x [ x 0 0 ] T. The advatage i that H doe ot affect the dyamic of (0 ad coequetly doe ot caue problem of ubouded repoe, for example. Moreover, ice output deped liearly o H, it ca be adjuted by mea of computatioally imple method (leat-quare, for example.. 3 RESULTADOS E DISCUSSÕES The model tructure preeted i the previou ectio will be teted i the modelig ad imulatio of the heat exchage ytem (Duraiki, 00 depicted i Fig. 3. Fci, Tci where the fuctio a i (x, b j (x, j, i 0,,, j 0, correpod to the coefficiet of the parameterized trafer fuctio Fi, Ti Fhi, Thi Vh Uh V Uc Vc G( ; δ ( b + a ( δ + K + g ( δ ( δ + K+ a ( δ 0 Fc, Tc F, T Fh, Th where δ i a calar parameterizig the et of equilibrium poit/liearizatio (x, i thi cae, ad a 0 (x g (x dφ(x /dx. The advatage of thi form for idetificatio i that all ivolved fuctio are calar ad ca be therefore idetified by mea of the variatio of oe igle parameter. Moreover, ice the teady-tate of thi repreetatio are of the form x, y, x j, 0, j,,, thee fuctio ca be obtaied by mea of local liear model parameterized by the output. Aother advatage of the LEM-Wieer model tructure i that it ca be further exteded by icludig a ecod-order term i the output equatio, i order to improve the accuracy of the model away from the equilibrium maifold, that i, h ( x x + b j ( x x j + Φ( (3 j Fig. 3. Heat exchage ytem Thi ytem i cotituted by a iulated tak divided i three eparate chamber that are allowed to trafer heat but ot ma. The cetral chamber i i cotact with both hot (h ad cold (c chamber, but thee are i cotact jut with the cetral oe. The volume of the chamber V h, V ad V c, are cotat, ad all chamber are well-mixed. Water i fed to ad removed from each chamber eparately. Uder thee aumptio, the ytem ca be decribed by mea of the followig differetial equatio: dth Fhi hi dt Vh dtc Fci ci dt Vc dt Fi i dt V 4 ( T T ( T T ( T T h U h Ah ( Th T Vh Cp ρ ( c Uc Ac ( Tc T Vc Cp ρ U h Ah U A ( T T c + c h + ( Tc T V Cp ρ V Cp ρ

5 Semiário do Programa de Pó-Graduação em Egeharia Química where T h, T c ad T are the temperature of each chamber, C p ad ρ are the pecific heat ad pecific ma of water (coidered to be idepedet of the temperature, U h /U c ad A h /A c are repectively the overall heat exchage coefficiet ad heat exchage area betwee the correpodig chamber. A more detailed decriptio of thi ytem ca be foud i (Duraiki, 00. I thi example, the iput variable i coidered to be the feed flowrate of hot water, F h,i, which ha cotat temperature T h,i. The output i the temperature of the cetral chamber, T. The value coidered for the phyical parameter ad other iflow ca be foud i the Appedix. The variatio of the dyamic character i obviou from the aalyi of Fig. 3 ad Fig Cotructig a approximated model i LEM, LEM-Hammertei ad LEM-Wieer form The origial LEM, LEM-Hammertei ad LEM-Wieer model decribed i the previou ectio ca be cotructed aalytically o the bai of the model (4. I the firt cae, we have a ytem i the form of Eq. (4 with u A( u Ω( 469.u u u (5 ad y x 3. For the LEM-Hammertei model, it i firt eceary to covert the matrice above to a ormal form i order that the idividual trafer fuctio from w to y i Fig. have uit gai. The ytem i of the form: A( u w Ω( w 0.54w ( u u where u ha to be ubtituted by q (w for implemetatio, with 469.u w q(. (7 0.7u The LEM-Wieer model (0-( ca be cotructed imilarly, givig ( x + 7. f ( x ( x ( x x3 ( x. g ( x x x φ x h( x +.5x (8 The ytem decribed above were imulated i Matlab with repect to the iput fuctio how i Fig. 4; the repoe are plotted i Fig. 5. The repoe of the liearized model at the operatig poit determied by u i alo how for compario. Exceptig thi ytem, the other curve are practically iditiguihable. u (L/ time ( Fig. 4. Tet iput igal y (K Noliear Sytem Liear Sytem LEM Sytem LEM-Hammertei LEM-Wieer time Fig. 5. Repoe of the everal ytem to the igal i Fig. 3

6 Semiário do Programa de Pó-Graduação em Egeharia Química 3. Cotructig the approximated model with idetified local model Approximated verio of the model derived i the previou ectio ca be cotructed with local model obtaied either from liearizatio or from idetificatio experimet; oly the lat approach i exemplified here. The followig procedure wa adopted: three liear local model correpodig to the operatig poit defied by u, L/ (y, K, u, 8 L/ (y, K, u,3 5 L/ (y, K were idetified by mea of local experimet, that i, with idetificatio igal of mall amplitude aroud thee operatig poit. No pecial methodology wa employed to elect the umber or the locatio of thee poit; they were imply ditributed over a deired rage of the maipulated iput. For each operatig poit, a idetificatio igal u id of the form depicted i Fig. 6 wa deiged. The witchig period σ of the igal wa determied a t 63 /0, where t 63 i the time eeded from the tep repoe to reach 63% of it teadytate value, what wa obtaied previouly for each poit by mea of a tep tet with the oliear model (poitive tep of 0. L/ i. The amplitude of the idetificatio igal wa fixed to 30% of u,i. A iput equece of the form u,i u id wa employed with validatio purpoe. u,i + i u (L/ u,i u,i i 0 σ σ 3σ 4σ 5σ time ( Fig. 6. Idetificatio iput igal The repoe of the oliear model (4 wa imulated i Matlab for the idetificatio igal u id. I order to imulate the effect of meauremet error, a white-oie, Gauia equece with zero mea ad tadard deviatio of 0.0 K wa added to the output. A typical plot of the oiy output meauremet i give i Fig. 7. The imulated igal were ampled with a coveiet ample period i order to be ued with the idetificatio algorithm. y (K time ( Fig. 6. Noiy ad filtered output Sice a accurate repreetatio of the local liearizatio i eceary, the followig procedure wa adopted. Firt, a et of two ru wa performed with u id for each operatig poit ad the average of the correpodig output y id wa take a the idetificatio data; thi ha the objective of reducig the effect of oie. Secod, the data wa filtered by mea of a leat-quare moothig cubic plie (Matlab fuctio pap. The bet parameter et of the plie fuctio wa determied iteratively i fuctio of the reult of the idetificatio procedure achieved i the ubequet tep. The liear local model i dicrete form were idetified through the combied ue of ubpace (Matlab fuctio 4id/ubid ad tate-pace predictio error method (Matlab fuctio pem. The ubpace method gave the iitial etimate for the predictio error method ad were alo ued for determiig the order of the tate-pace model. A already uggeted i the literature (Ferade, 005, the author foud that a good local idetificatio i geerally achieved whe the order of the model i clearly evideced by the igular value tet provided by the ubpace routie. Moreover, a frequet idicatio of exceive model order ad poor idetificatio i the geeratio of utable pole, complex zero, etc by thee method. The idetificatio procedure wa a follow: firt, the filtered data wa ued i the ubpace method; the model order wa elected ad the etimate were paed to the pem routie. Thi reult wa the imulated ad validated agait the idetificatio ad validatio data. If eceary, the parameter of the moothig plie fuctio were chaged, the data wa filtered agai ad the

7 Semiário do Programa de Pó-Graduação em Egeharia Química local model idetified oce more; thi procedure wa repeated util a good reult wa foud. The local model idetified i thi maer were ued i the cotructio of the model tructure preeted i the ectio. The LEM ad LEM- Hammertei model were cotructed with local model i obervability form. The lat oe differ from the aalytical cae becaue the liear traformatio to ormal form deped o the relative degree which i ot a robut quatity to be obtaied from experimet. I all cae, proper plie or ratioal iterpolatio of the eceary fuctio wa performed (the reult are omitted due to the pace limitatio. The repoe of the three tructure with idetified local model for the iput igal i Fig. 4 are how i Fig 8. The mot igificat differece with repect to the aalytical cae refer to the Wieer model, due to the idetificatio/iterpolatio of the b i parameter that appear i the output fuctio. 4 CONCLUSÕES Thi paper preeted ew model tructure baed o the cocept of liearizatio aroud the equilibrium maifold (LEM. Thee model exted the covetioal Hammertei ad Wieer ytem, i the ee that they allow for the icluio of variable dyamic. Thee repreetatio ca be cotructed o the bai of the local model; a poibility of obtaiig them i for by idetificatio. A umerical example (biliear ytem howed that thee tructure are almot equivalet if the model are obtaied aalytically, but the effect of the error i the etimated parameter ca affect differetly the ditict model clae. y (K Noliear Sytem Liear Sytem LEM Sytem LEM-Hammertei LEM-Wieer time ( Fig. 8. Repoe of the everal ytem cotructed o the bai of idetified local model 5 AGRADECIMENTOS The author thak FINEP ad PETROBRAS for the fiacial upport. The firt ad ecod author thak repectively the Germa Academic Exchage Service (DAAD ad the Ladetiftug Bade-Württeberg. REFERÊNCIAS Duraiki, R. G (00. Cotrole Preditivo Não-Liear Utilizado Liearização ao Logo da Trajetória. Mater Thei, Federal Uiverity of Rio Grade do Sul, Brazil (i Portuguee. Bologee Ferade, P., S. Egell, J. O. Trierweiler (004. A ew approach to the local model etwork techique. Proceedig of the Brazilia Cogre of Egeharia Química, COBEQ04, Curitiba, Brazil. Bologee Ferade, P. (005. The iputparameterized liearizatio aroud the equilibrium maifold approach to modelig ad idetificatio. Phd Thei, Uiverity of Dortmud (to be publihed. Bologee Ferade, P., S. Egell (005. Cotiuou Noliear SISO Sytem Idetificatio uig Parameterized Liearizatio Familie. Proc. of the XVI IFAC World Cogre, Prague, Tchech Republic. Pearo, R. K (995. Gray-box idetificatio of blockorieted oliear model. Joural of Proce Cotrol, 0, Pearo, R. K. ad M. Pottma (000. Gray-box idetificatio of block-orieted oliear model. Joural of Proce Cotrol, 0, Wag, J. ad J. W. Rugh (987. Parameterized Liear Sytem ad Liearizatio Familie for Noliear Sytem. IEEE Traactio o Circuit ad Sytem, 34, APPENDIX Parameter value ued i the example: ρ 000 kg/m 3, C p 480 J/kg/K, V V c V h 0.3 m 3 U h. A h J/K/, U c. A c J/K/ F ci 0 m 3 /, T ci 80 K, F i 0.00 m 3 /, T i 300 K T hi 370 K