Recursive Identification of Hammerstein Systems with Polynomial Function Approximation

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1 Ieraioal Joural of Maagee ad Fuzzy Syses 7; 3(6): hp:// doi:.648/.ifs.736. ISSN: (Pri); ISSN: (Olie) Research/echical Noe Recursive Ideificaio of Haersei Syses wih Polyoial Fucio Approxiaio Wag Jia-hog, ag De-zhi, Jiag Hog, ag Xiao-u School of Elecroic Egieerig ad Auoaio, Jiagxi Uiversiy of Sciece ad echology, Gazhou, Chia School of Elecrical ad Iforaio Egieerig, Ahui Uiversiy of echology, Ma-a-sha, Chia Eail address: (ag Xiao-u) o cie his aricle: Wag Jia-hog, ag De-zhi, Jiag Hog, ag Xiao-u. Recursive Ideificaio of Haersei Syses wih Polyoial Fucio Approxiaio. Ieraioal Joural of Maagee ad Fuzzy Syses. Vol. 3, No. 6, 7, pp doi:.648/.ifs.736. Received: Sepeber 5, 7; Acceped: Ocober 7, 7; Published: Noveber, 7 Absrac: Noliear syse ideificaio is cosidered, he oliear saic fucio was approxiaed by a uber of polyoial fucios. I is based o a piecewise-liear Haersei odel, which is liear i he paraeers. he ideificaio procedure is divided io wo seps. Firsly we adop he exeded sochasic gradie algorih o ideify soe ukow paraeers. Secodly usig sigular value decoposiio (SVD), we propose a ew ehod o ideify oher paraeers. he basic idea is o replace u-easurable oise ers i he iforaio vecors by heir esiaes, ad o copue he oise esiaes based o he obaied paraeer esiaes. he applicabiliy of he approach is illusraed by a siulaio. Keywords: Noliear Syse, Haersei Syses, Polyoial Fucios Approxiaio, Recursive Ideificaio, Sigular Value Decoposiio. Iroducio Modelig, ideificaio ad predicio are hree ai ubiquious pheoea i our daily lives. hrough our ideas ad seses, we collec iforaio abou he world, he we ierpre, predic ad reac acios accordig o our percepios. I aural sciece, los of experies or observaios guide us o forulae laws of aure, which describe differe aspecs of he world ad le us predic all sors of higs, like plae ovees or weaher forecas. Also i oder echology, odelig ad ideificaio have uch beefi o offer us oe descripio correspodig o he physical obec. Every ad everyhig aroud us, here is a eed for auoaic corol echaiss such as i aero-plaes, cars, cheical process plas, obiles phoes, heaig of houses ec. However o be able o corol a syse, oe eeds o kow a leas soehig abou how i behaviors ad reacs o differe acios ake o i. Hece we eed a odel of he syse. A syse ca iforally be defied as a eiy which ieracs wih he res of he world hrough ore or less well defied ipu ad oupu daa. A odel is he a approxiae descripio of he syse. A ideal odel should be siple, accurae ad geeral. his approxiae descripio of he syse ca be cosruced by syse ideificaio sraegy, as he goal of syse ideificaio is o build a aheaical odel of a dyaic syse based o soe iiial iforaio abou he syse ad he easuree daa colleced fro he syse. Accordig o [], he process of syse ideificaio cosiss of desigig ad coducig he ideificaio experie i order o collec he easuree daa, selecig he srucure of he odel ad specifyig he paraeers o be ideified ad eveually fiig he odel paraeers o he obaied daa []. Fially he qualiy of he obaied odel is evaluaed hrough odel validaio process. Geerally syse ideificaio is a ieraive process ad if he qualiy of he obaied odel is o saisfacory, soe or all of he lised phases ca be repeaed i order o obai oe saisfied odel. May oliear syses ca be odeled by a Haersei odel (liear ie-ivaria (LI) block followig soe saic oliear block), Wieer odel (LI block precedig soe saic oliear block), or Haersei-Wieer odel (LI block sadwiched by wo oliear blocks) [3]. he Haersei odel is a special kid of oliear syses

2 Ieraioal Joural of Maagee ad Fuzzy Syses 7; 3(6): which has applicaios i ay egieerig probles ad herefore, ideificaio of Haersei odels has bee a acive research opic for a log ie. Exisig ehods i he lieraure ca be roughly divided io six caegories: he ieraive ehod, he over-paraeerizaio ehod, he sochasic ehod, he oliear leas squares ehod, he separable leas squares ehod ad he blid ehod [4]. he Haersei odel is a oliear syse a oliear block is followed by a liear dyaic block. he Haersei odel has bee widely used i ay areas icludig oliear filerig, acuaor sauraios, audio-visual processes. here exiss a large uber of papers o he opic of he Haersei odel ideificaio. Referece [5] cosidered a exeded sochasic gradie ideificaio algorih for Haersei-Wieer ARMAX syses. o iprove he ideificaio accuracy, a exeded sochasic gradie algorih wih a forgeig facor is give. Referece [6] proposed hree ehods of separaig he paraeer esiaes. he hree ehods are he averaged ehod, peruaio ad cobiaio ehod ad sigular value ehod. Referece [7] cosidered wo ideificaio algorihs, a ieraive gradie ad a recursive sochasic gradie based, for a Haersei oliear ARMAX odel.. Proble Forulaio I his paper we focus o he ideificaio of Haersei odel which cosiss of a oliear eory-less elee followed by a liear dyaical syse [8]. he rue oupu u x ( ) ad he ier variable u are ieasurable, is he syse ipu, y is he easuree of x ( ) bu is corruped by he disurbace w( ), he oupu of N ( z ) drive by a addiive whie oise v wih zero ea, G ( z) is he rasfer fucio of he liear par i he odel, N ( z) is he rasfer fucio of he oise odel []. Figure. he discree ie SISO Haersei syse. Assue ha he liear dyaical block i Figure is described by a ARMAX odel, which has he = + y x w B z x ( ) = G ( z) u ( ) = A z D z w = N ( z) v ( ) = A z u v Here A( z), B ( z ) ad D ( z) are polyoials i he shif operaor z z y = y ( ) wih A z = + az + a z + a z bz B z = bz + b z + D z = + d z + d z + d z d d Noice ha i he characerizaio of he Haersei G z are acually o uique. Ay pair odel, f ( u ) ad, af u G z a for soe ozero ad fiie cosa a would produce ideical ipu ad oupu easurees. I oher ohers, ay ideificaio schee ca o disiguish bewee af ( u), G ( z) a ad, () () f u G z. herefore o ge a uique paraeerizaio, oe of he gais of f ( u) ad G ( z) has o be fixed. here are several ways o oralize he gais. A oliear saic fucio ca be approxiaed by a uber of basis fucios, coeced o each oher ad forig a piecewise-liear fucio. u = F ( u) i C (3) C is a vecor of paraeers represeig values c of he piecewise-liear fucio u ( ) i kos. C = c c c c (4) ( + ) he kos are ois of lie seges defied i ers of ipu sigalu. saisfyig u = u u u u (5) ( + ) u < u < < u < u < < u + Furherore F ( u) is a vecor of polyoial fucio.

3 89 Wag Jia-hog e al.: Recursive Ideificaio of Haersei Syses wih Polyoial Fucio Approxiaio = ( ) F u F u F u F u F u Elee of he vecor F ( u) are defied as follows: F u = u i i i + subsiuig io equaio (3), he he oliear par i he Haersei odel is i he followig for: wih u = f u = u c f u (6) f u I ca be easily see ha: f u i = i = δi = c = i i if i = if i ( ) (7) f u = u (8) herefore he pla () ca also be represeed by he followig odel: = + A z y B z u D z v = = u f u u c f u f u = i = i i A regressive for ca be derived as follows: = + = bu ( i) c f ( u( i) ) + D( z) v A z y B z u D z v i = Fro he las equaliy, we ca easily ge he followig recursive equaio: = + ( ) + + y a y i bu i c f u i dv i v i i i i = = + ( ) + + y a y i u i f u i dv i v i i i i = d d (9) = bc i =, =, () i i Defie he paraeer vecor θ ad iforaio vecor as φ ( ) a ϕ θ = R, φ = R v ( d ) d v = + + a a a = R, a = R d d d d = R d ϕ a d d ϕ.. = R = R ϕa ϕ ( + ) = ϕ R ϕ ( ) y y = R y ( ) ( ) ( ) ( ) ( ) u f u u f u ϕ = R, =, u ( ) f ( u ( ) ) he we have = φ θ + y v Noice ha ϕ i = i ( ) v ( i)( i, d ) deoes he esiae ofθ. Sice zero ea, he () () φ is available bu φ are uavailable. Le θ v is a whie oise wih φ y = θ (3)

4 Ieraioal Joural of Maagee ad Fuzzy Syses 7; 3(6): I is he bes oupu predicio. Cosider he quadraic oupu predicio error crierio ( θ ) = = φ θ (4) J y i y i y i he quadraic error fucio i (4) is oe of he os coo cos fucios i he ideificaio lieraure [9]. May well-kow Haersei odel ideificaio ehods belog o his class ad he differeces lie oly i he φ. Le forulaio of he iforaio vecor Hece y ( ) ( ) ( ) φ y φ Y ( ) =, Φ ( ) = y ( ) φ ( ) = Φ Φ = Φ J θ Y θ Y θ Y θ (5) Provided ha Φ is persisely exciig, iiizig J ( θ ) gives he leas-squares esiae: Y θ = Φ Φ Φ However a difficuly arises because of v( i), Φ ( ) (6) he expressio o he righ-had side of (6) coais ukow oise ers v( i)( i =, d ), so i is ipossible o copue he esiae θ by (6), our approach is based o he ieraive ideificaio priciple []. Le, v i are replaced by heir k = he ukow variables correspodig esiae v ( i) a ieraiok, ad φ are replaced by φ ( k ) ofθ. hus he esiae of v is give by k i []. Le k θ be he ieraive soluio φ ( i) v i = y i θ (7) k k k φ k ϕ v k = R v k ( d ) k ( ) ( ) φ φk Φ k ( ) = φk ( ) Based o (6) ad replace Φ ( ) by Φ soluio k θ of θ ay be copued by: k (8) (9), he ieraive θk = Φk Φk Φ k Y, k =, () o iiialize he algorih, we ake θ = or soe sall real vecor, e.g. 6 θ = I wih I beig a -diesioal colu vecor, whose elees are. As θ coes i liearly, algorih () urs ou o be adequae paraeerizaio o ge esiaes of he paraeers a i, iad d i, usig he ieraive leas squares algorih. 3. Basic Forulaios for Pla Model Ideificaio Fro esiaes ( ) of ( i, ) θ ofθ, oe has o ge esiaes ( b i, c ) b c fori =, =,. We will firs cosruc a procedure o go back fro relaios o ge ( i, ) i ' s o b ' s ad c ' s. he b c fro i will be esablished. Observe ha () ca be rewrie as follows: b M = = c c c b i i () Noice ha sice M is a rak- arix, bi ' s ad ci ' s ca o be deeried uiquely fro i ' s, uless exra codiios are iposed o bi ' s ad ci ' s. Uiqueess of he soluio of () ca be achieved by iposig he followig couple of codiios: ρ ( b b) b b bi = ad ρ ( b b ) > () deoes he firs copoe of he vecor ha saisfies: ([ ]) ρ b b = sup b i.e he firs copoe wih a grea absolue value. Based o he above observaios [], a procedure is desiged usig sigular value decoposiio (SVD) []. ( ) Proposiio.. le M R + be ay rak- real arix. he is SVD decoposiio has he followig for: σ M = Γ Σ (3) ( ) ( ) Γ R, Σ R + + ad σ is he uique

5 9 Wag Jia-hog e al.: Recursive Ideificaio of Haersei Syses wih Polyoial Fucio Approxiaio ozero sigular value of M. Furherore M ca be uiquely decoposed as follows: wih b M = [ c c c ] (4) b [ b b ] = sig ( r) Γ Γ [ σ ] [ σ ] [ c c c ] sig ( r) [ σ ] [ ] = Γ Σ ( [ ]) r = ρ Γ σ (5) he vecor b b hus obaied is he oly soluio of (3) ha saisfies: Esiaes ( i, ) bi = ad ρ ( b b ) > b c of ( i, ) b c ca be recovered fro '. i s Followig closely proposiio.[], oe firs cosiders he arix: M = (6) Proposiio.: le he sigular values decoposiio of M ( ) be wrie as follows: σ σ M = Γ Σ σ he proposiio. suggess he followig esiaes for b, c he paraeers b i i σ Γ sig ( r ( = )) Γ σ b c c = sig r Γ Σ c (7) (8) ( ) σ [ ] ( ) ρ σ r = Γ (9) he esiaes hus obaied are he oly oes ha saisfy he codiios i b = ( b b ) ρ > Noice ha he sigular values σ σ have o bee accoued for i he rules (7), (8) ad (9). I has o effec whe he i ' s coverge o heir rue values, because he rak of arix M coverges o. his is ade precisely i he ex proposiio. M be he real arix sequece { } Proposiio.3: le defied by (6). Le b ( ) b ( ) ad c c be he vecors obaied fro M ( ) accordig o he rules (7), (8) ad (9). If he i ' s coverge o heir rue values i, he he esiaes (, i ) b c will coverge o heir rue values ( i, ) b c. 4. Recursive Ideificaio Algorih I his secio, we derive a recursive ideificaio algorih which ca be o-lie ipleeed. We rewrie he equaio. φ θ y = + v (3) Equaio.(7) is a pseudo-liear regressio ideificaio φ odel for he Haersei syse. Noe ha ϕ i is available bu v ( i), i =, d i φ ( ) are uavailable. Le E deoe he expecaio operaor, θ ( ) he esiae of θ a ie, x = r xx he or of he arixx. Sice v is a whie oise, forig a quadraic cos fucio J ( θ ) = E y φ θ iiizig J ( θ ) leads o he followig sochasic gradie algorih of esiaig θ φ θ = θ ( ) + y φ θ ( ) r φ (3) r = r +, r = (3)

6 Ieraioal Joural of Maagee ad Fuzzy Syses 7; 3(6): However he algorih i (3) ad (3) is ipossible o φ o he righ-had realize because he iforaio vecor side coais ukow oise ers v( k ). he soluio is o replace he ukow variables v( i) correspodig esiaes v ( k ), ad furher defie: = ϕ ( ) wih heir φ v v d (33) Fro (33), we have = φ v y replacig φ ( ) ad θ i he above equaio wih φ ( ) θ ( ), he esiaed residual ca be copued by θ φ θ replacig φ ( ) i (3) ad (3) wih φ ( ) ad v = y (34), we ca obai he exeded sochasic gradie [ ESG] ideificaio algorih of esiaig θ φ θ = θ + y φ θ ( ) r φ φ r r = r +, = (35) ϕ v = v ( d ) θ a = (36) d o iiialize his ESG algorih θ ( ) is geerally ake o 6 be soe sall real vecor, e.g. θ = I wih I beig a -diesioal colu vecor whose elees are. he ESG algorih has low copuaio, bu is covergece is relaively slow. I order o iprove he rackig perforace of he ESG algorih, we iroduce a forgeig facor λ i he ESG algorih o ge he ESG algorih wih a forgeig facor, which is referred o he EFG algorih. φ θ = θ + y φ θ ( ) r r r = λr + φ, =, < λ < (37) whe λ =, he EFG algorih reduces o he ESG algorih. 5. Siulaio A siulaio is give o deosrae he effeciveess of he proposed algorihs. Cosider he followig syse: A( z) y = B ( z) u + D ( z) v = + + = A z az a z z z = + = + B z bz bz.8z.6z = = + =.64 D z d z z = + + = u.f ( u) +.4f ( u) +.6f ( u) u u c f u c f u c f u F u = u i i i θ = [ ] σ =., ad { }.6,.8,.6,.4,.3,.4,.48,.36,.6 u is ake as a persise exciaio sigal sequece wih zero ea ad ui variace u { } v as a whie oise sequece wih zero ea ad cosa variace σ v. Apply he proposed algorihs i (8)-(34) ad (5)-(6) respecively, o esiae he paraeers of his syse. he paraeer errors δ versus are show i Figure, he dae legh is, he forgeig facor λ =.98, he δ = θ θ, ad he absolue paraeer esiaio error oise-o-sigal raio 7.35% s δ =. Whe chagig.9 λ =, he paraeer error δ versus is show i Figure 3.

7 Wag Jia-hog e al.: Recursive Ideificaio of Haersei Syses wih Polyoial Fucio Approxiaio.5 paraeer esiaio error Solid lie: he ESG algorih; dos: he EFG algorih. ( λ =.98 ) Figure. he paraeer esiaio error δ versus. - Error value Figure 3. he paraeer esiaio error δ versus of he EFG algorih.( λ =.9). 6. Coclusio We have cosidered syse ideificaio based o Haersei odel, he oliear elee is defied by a piecewise-liear fucio. We have desiged a ideificaio schee ha deeries precisely he ukow b, c i =, =, ad oliear paraeers ( i ) hose of he liear rasfer fucio. A ieraive algorih based o replacig u-easurable oise variables by heir esiaes are derived for Haersei oliear odels. Ackowledgees his work was suppored by he Gras fro he Naioal Sciece Foudaio of Chia (o. 6563), he Jiagxi

8 Ieraioal Joural of Maagee ad Fuzzy Syses 7; 3(6): Provicial Naioal Sciece Foudaio (o. 4BAB6, o. GJJ5889), ad he Naioal Sciece Foudaio of he Colleges ad Uiversiy i Ahui Provicial (o.kj6a94). Refereces [] Aa Hageblad, Lear Lug, Adria Wills. Maxiu likelihood ideificaio of Wieer odels[j]. Auoaica, 8.44(): [] Lear Lug. Syse Ideificaio: heory for he user [M], Preice-Hall, Upper Saddle River, 999. [3] Mari Eqvis, Lear Lug. Liear approxiaios of oliear FIR syses for separable ipu processes. Auoaica [J], 5. 4(3): [4] J. J. Bussgag. Cross correlae o fucios of apliude-disored Gaussia sigals. echical Repor echical repor 6, MI Laboraory of Elecroics, 95. [5] Bai E-W. Frequecy doai ideificaio of Haersei odels[j]. IEEE rasacios o auoaic corol (4): [6] Bai E-W. A rado leas-ried-squares ideificaio algorih [J]. Auoaica (9): [7] Bai E-W. Ideificaio of liear syses wih hard ipu olieariies of kow srucure [J]. Auoaica,. 38(5): [8] Lear Lug. Esiaig liear ie ivaria odels of o-liear ie-varyig syse [J]. Europea Joural of Corol,. 7 ():3-9. [9] R. Pielo, J. Schoukes. Fas approxiaio ideificaio of oliear syses [J]. Auoaica, 3. 39(7): [] Dig F, ogwe Che. Ideificaio of Haersei oliear ARMAX syses [J]. Auoaica, 5. 4 (9): [] Dig F, ogwe Che. Perforace aalysis of uli-iovaio gradie ype ideificaio ehods [J]. Auoaica, 7. 43():-4. [] Wag DQ, Dig F. Exeded sochasic gradie ideificaio algorihs for Haersei-Wieer ARMAX syses [J]. Copuers & Maheaics wih Applicaios, 8. 56():