Chattering Free with Noise Reduction in Sliding-Mode Observers Using Frequency Domain Analysis

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1 Chaerig Free wih Noie Reducio i Slidig-Mode Oerver Uig Frequecy Doai Aalyi M. Hajaipour, M. Farrokhi * Depare of Elecrical Egieerig, Ira Uiveriy of Sciece ad echology, ehra 6846, IRAN hi paper pree aalyi ad regulaio of wichig i he lidig-ode oerver for oliear ye. Fir, he high gai propery of he lidig oerver i eployed i order o oai fa covergece of he eiaed ae o he ye oe. he, whe he ye i i he wichig codiio, igal are decopoed io wo ode: he low ode ad he fa ode. he oerver paraeer are deiged aed o he relay feedack ye uch ha he high gai propery i provided for he low ode operaio. hi eure fa covergece of he eiaed ae ad a he ae ie, y corollig he fa ode of he oerver, he high frequecy ocillaio (i.e. he chaerig pheoea) ca e rejeced y a iple low-pa filer. I addiio, he ehavior of he propoed oerver i aalyzed i he preece of he eauree oie. Moreover, a variale relay-equivale gai echique will e iroduced o ake he propoed oerver le eiive o eauree oie ad o aiai good eiaio of he ae. he propoed oliear oerver i eed hrough iulaio i a illuraive exaple ivolvig a ioreacor. Siulaio reul how good perforace of he propoed ehod a copared o he coveioal lidig-ode oerver. Keyword: Slidig-ode oerver; Relay-feedack ye; High-gai oerver; Noliear Sye; Swichig frequecy; Bioreacor. Iroducio Slidig-ode oerver are kow o e rou agai preece of diurace ad odelig uceraiie [ 3]. Alhough hee oerver are highly rou wih repec o oie i ye ipu, i ured ou ha he correpodig ailiy degrade i ye ha exhii oupu oie or ixed uceraiie [4]. Whe he eauree are oiy, ecaue he ie derivaive cao e accuraely calculaed, i ca lead o poor cloedloop perforace or iailiie i he eiaio procedure. Slidig-ode oerver are kow a high-gai oerver, which have appropriae ehavior i diurace rejecio [5 7]. Vailjevic ad Khalil have how ha a high-gai oerver ac a a differeiaor i he lii a he gai approache ifiiy [8]. Hece, i he preece of he eauree oie, i agifie he oie i he eiaed ae. herefore, here i a rade-off ewee he oerver gai ad he oie effec i he ae eiaio. I addiio, hey have howed ha a oud o he eiaio error exi ha deped o he axiu apliude of he eauree oie. Recely, Ahre ad Khalil have iroduced a high-gai oerver, where he gai arix wiche ewee wo value [9]. I hi ehod, whe he eiaio error reache a eady-ae hrehold, i wiche o a ecod gai o reduce he effec of he eauree oie. he idea of gai wichig i oerver ha ee eployed efore y oher reearcher for oie cacellaio i he ae eiaio []. Sice he ai ource of he lidig pheoea i he relay elee, reearcher ue relay feedack ye o aalyze lidig properie []. Due o he iereig characeriic of relay feedack ye, reearch i hi area i fa developig. A uer of aalyzig ehod for he odelig of relay feedack ye a well

2 a ehod for eiaig he wichig frequecie ad he apliude of ocillaio are dicued i [, 3]. I i well kow ha whe he relay feedack ye i operaig i he wichig ode, i ay have fiie or ifiie wichig frequecie, which are called he lii cycle ad he ideal lidig ode, repecively. For he fir ie, Boiko ad Frida applied he relay feedack cocep o he deig of liear oerver, where hey have ued he relay equivale gai echique [4, 5]. I coiuaio of heir work, a ovel approach for he oerver deig for oliear ye i preece of he eauree oie i preeed i hi paper. he propoed ehod ake advaage of he high-gai oerver i order o eforce he eiaed ae o ove quickly o he ye ae. herefore, i he fir ep, he codiio for forcig he oerver o go o he wichig zoe are aified. Nex, whe he oerver i operaig i he wichig or he lidig codiio, he igal are decopoed io wo par: he low-ode par ad he fa-ode par. For every ode, he correpodig deig for he appropriae operaio of he oerver will e coidered. I will e how ha he apliude ad frequecy of ocillaio have direc effec o he dyaic ad ailiy of he fa ad lowode of he ae eiaio whe he oerver i operaig i he wichig codiio. Moreover, i order o ake he propoed oerver rou agai he eauree oie, fir he perforace of he oerver i preece of he eauree oie i aalyzed. Nex, he effec of he eauree oie i ae eiaio will e coideraly reduced y iroducig a Variale Relay-Equivale Gai (VREG) echique. he ai feaure of hi paper ca e uarized a follow: Frequecy regulaio of he lidig-ode oerver wihou ay eed for liearizaio Preeig a relaiohip ewee he wichig frequecy ad ailiy ad he perforace of lidigode oerver Providig high gai propery i he wichig ode for lidig-ode oerver ad coequely oaiig fa covergece of he eiaed ae o he acual ae Propoig a variale gai echique i order o have le eiiviy o he eauree oie i lidigode oerver. Siulaio reul how good perforace of he propoed ehod a copared o he coveioal lidigode oerver i a ioreacor applicaio. hi paper i orgaized a follow. Secio give he prole aee, preliiary defiiio, ad aupio. Secio 3 provide he deig procedure of he propoed ehod. Secio 4 pree eauree oie aalyi. Secio 5 how illuraive exaple followed y cocluio i Secio 6. Noaio: hroughou hi paper, λax ( A) deoe he axiu eigevalue of arix A ad A deoe he - or of i.. Prole aee Coider he followig cla of ye: y = Cx (, u) λ (, u) x = Ax+ f x + E x () where x R i he vecor of ukow ae variale, u R i he ipu igal, y R i he eaured oupu, f : R R R i a oliear ooh fucio, E R ad λ : R R uceraiie ad diurace, repecively. R i he odelig

3 Aupio : A eiaio gai L ca e deiged uch ha (A LC) i ale. Aupio : he uceraiy fucio λ i aued o e ouded; ha i, λ( x,u ) upper oud. λ, where λ i he Aupio 3: he oliear fucio f( x, u) i Lipchiz wih repec o x ad uiforly for u A, where A i a adiile corol e. ha i, here exi a coa γ > uch ha (, ) ( ˆ, ) f x u f x u γ x x ˆ () he followig oerver i propoed for ye (): xˆ = Axˆ + f( xˆ, u) + L( y ˆ y) + Εu G OBS ( ): = ˆ y = Cxˆ (3) where ˆx deoe he eiaed ae vecor, L i he oerver arix, ad u i a dicoiuou igal wih he followig for: i which d i he relay gai ad ( σ y ( )) u = d g, (4) σ y ( ) = y ( ) y ( ) (5) ( ) ˆ ( ) ( ) Y jω = Y jω H jω, (6) where Y ( jω ) ad Yˆ ( jω ) are he Laplace rafor of igal y ( ) ad ŷ ( ) H ( jω ) i deiged uch ha i he frequecy doai where Φ ( j ) H for Ω Ω Ω Φ ( j Ω ) for Ω>Ω ( j ), repecively. Aue ha Ω i defied a he adwidh frequecy of he low-ode of he oerver i he wichig ode, Ω will e deiged laer o. Defiig he ae eiaio error a σ := x x, ˆ he error dyaic ecoe ( ˆ) ( (, ) ( ˆ, )) λ(, ) ( A LC) σ + ( f( x, ) f( xˆ, )) Eλ( x, ) E σ = Aσ L y y + f x u f x u + E x u Eu = u u + u u he propoed oerver i deiged for wo differe ae: he raie ae ad he wichig ae. I he raie ae, he oerver i a high-gai oerver. ha i, he high gai of he relay (d) guaraee fa covergece of he eiaio error o zero ad forcig he oerver o eer he wichig regio. Moreover, he high gai ca provide eer diurace rejecio. I hould e oed ha, efore he wichig ae (i.e. whe he eiaio error i large) he apliude of he eauree oie i egligile a copared o he apliude of he eiaio error. Hece, he oie reducio doe o play a ipora role whe he eiaio error i large. I he wichig ae, however, a ovel approach will e propoed where he igal are decopoed io he low ad fa ode, each ode requirig a differe deig. I oher word, he apliude (7) (8)

4 ad frequecy of ocillaio are defied uch ha he low-ode of he oerver i he wichig ode ehave like a high gai oerver. hi provide very quick covergece of he low-ode eiaio error o zero. Moreover, y regulaig he wichig frequecy, ocillaio ca iply e reoved fro he eiaed ae uig a low-pa filer. I addiio, a Variale Relay-Equivale Gai (VREG) echique will e propoed o diiih oie effec o he eiaed ae. Hece, he ojecive i o deig d, H( ), ad L o achieve hee goal. Defiiio : Due o he preece of he dicoiuiy i he relay, igal i he wichig ode ca e decopoed io wo par: ) he fa-ode par aociaed wih he periodical oio acro he wichig urface ad ) he low-ode par aociaed wih he oio alog he wichig urface. Le defie he low frequecy doai ad Ω Ω a Ω>Ω a he high frequecy doai. herefore, every igal i he ye ad i he oerver ca e decopoed io wo par: ) he low-ode par ad ) he fa-ode par; i.e. x x x, σ ( ) σ ( ) σ ( ) = + = +, σ = σ + σ, y = y + y, yˆ = ˆ y + yˆ, y = y + y, where ucrip y y y ad idicae he low ad fa ode of he correpodig igal, repecively. hi eparaig idea ha ee ued efore for liear ye y oe reearche uch a [6] ad [7]. Reark : A i will e coidered laer, ice he frequecy of he fa-ode ca e deiged high eough ad far fro he adwidh of he low-ode of he oerver Ω, he y paig igal hrough a low-pa filer wih a adwidh greaer ha Ω u aller ha he wichig frequecy Ω, which i high eough, he high frequecy ode ca e reoved fro he eiaed ae. hi idea wa propoed efore y oher reearcher uch a [8]. Hece, he ai ye ju oerve he low-ode of he eiaed ae variale, which ha frequecy i he doai of Ω Ω. Wihou lo of geeraliy, le aue ha he ai par of he ye igal, which are goig o e eiaed, are i he frequecy doai of of he ye igal, which i aued o e i he doai of Ω Ω. I addiio, he reaiig par Ω>Ω (e.g. eauree oie) i o ipora a copared o he high frequecy ode of he oerver ad hece, ca e igored. herefore, he high frequecy par of he ai ye igal, i.e. y () ad x (), ca e eliiaed. I oher word, i ca e aued ha x = ad y () =. Propoiio : Sigal y() i he low ad fa-ode ca e wrie a ( ) ˆ y y ( ) = ˆ y ( τ) ϕ( τ) dτ for Ω Ω for Ω>Ω (9) repecively, where ϕ() i he ivere Laplace rafor of Φ( ) proof: Uig (6) ad (7), y ( ) ca e wrie a y ( ) ( ) y preeed i (7). Moreover, σ = Cσ () ( ) () = ˆ () ( ) σ C x ϕ τ dτ () { } y ( ) = Yj ˆ( Ω) H( jω) L ()

5 L idicae he ivere Laplace rafor operaor. he low ad fa ode of hi igal are where { } { } () = L ˆ( Ω) ( Ω) Ω Ω = ˆ { Y ( jω) H ( jω) Ω Ω Ω Ω } y Y j H j L (3) ( τ) δ( τ) τ () = yˆ d = ˆ y, { } () = L ˆ( Ω) ( Ω) Ω>Ω = ˆ { Y( jω) H ( jω) Ω>Ω Ω>Ω } y Y j H j ( ) ( ) = yˆ τ ϕ τ dτ. L (4) Hece, he error ewee he eiaed ad acual oupu i he low ad fa-ode are ( ) ( ) ( ) ( ( ) ( )) ( ) σ ˆ y = y y = Cx x = Cσ, (5) ( ) ( ( ) ( ˆ x ( τ) ϕ( τ) dτ )). ( ) = ( ) ( ) = Cx ( ) ˆ ( ) ( ) σ y y y ϕ τ dτ y = Cx he, accordig o Reark, uig a low-pa filer, σ () ecoe he Laplace rafor of (7) i Σ ( ) = L (), Χˆ ( ) = { xˆ () } where y { σ y } y y ( ) () = ˆ ( ) ( ) (6) σ C x τ ϕ τ dτ. (7) ( ) ˆ ( ) ( ) ( ) ( ) Σ = CΧ Φ = CΣ Φ, (8) y L ad ( ) = { σ () } Σ L. Reark : Decopoiio of igal io he low ad fa ode i ju eeded i he wichig ae ad i he oral operaio (i.e. efore he wichig codiio occur) where igal operae i he frequecy doai Ω Ω. Hece, uig (5) ad Propoiio, efore he ye eer io he wichig ae σ y ( ) = Cσ ( ). (9) I he ex ecio, he oerver deig for hee wo ae (i.e. he raie ae ad he wichig ae) will e give. 3. Oerver Deig I hi ecio, he oerver deig will e preeed for wo differe ae. Fir, i he raie ae, codiio for forcig he eiaed ae variale o ove quickly o he wichig ae will e defied. Due o he high apliude of he relay gai d, hi ode ac a a high-gai oerver. Nex, i he wichig ae, paraeer will e deiged uch ha a fa raie i oaied for he low-ode ad a iple high-frequecy rejecio i provided for he fa-ode.

6 3.. raie ae I hi uecio, i will e how ha proper elecio of he relay paraeer (d) guaraee covergece of he error dyaic o he lidig urface i fiie ie ad ha he wichig codiio occur. Le defie σ y = a he lidig urface. I i well kow ha he lidig urface ad i derivaive u aify σ y = σ y = i he wichig ode. he codiio for exiece of a wichig ode i [9]: σ yσ y < σy ad. () he, i i aid ha he urface σ y = i a aracig urface ad he wichig codiio occur. Aupio 4: Aue ha he poiive defiie arix P = P ad arix L aify he followig equaio: where A i Hurwiz ad Q i poiive defiie. Aupio 5: Aue ha σ Nσ >, where N = PEC. ( A LC) P + P ( A LC) = Q, () Lea : Coider Aupio --5, Reark, ad he error dyaic (8) ad (9). Selecig he lidig paraeer d a where ad σ( ) i σ () a ( ( ) ) ( ) d d () d = λ Q + P γ σ + λ E P E P i =, guaraee ha σ ( ) li. I addiio, he lidig urface will reach he lidig = aifold σ y = i fiie ie ad he wichig ode will occur. Proof: Le defie he followig Lyapuov fucio: he ie derivaive of hi fucio i V = σ Pσ. (3) ( f (, ) f ( ˆ, ) λ (, )) V = σ Q σ+ σ P x u x u E x u σ PEu Cσ λi( Q) σ + P γ σ + λ E P σ dσ PE Cσ. (4) Coiderig Aupio 5 ad he fac ha Cσ i a calar, i give V σσ PE ( λi ( Q) + P γ) σ + λ E P σ d C, (5) C σ Hece, V ecoe egaive if or equivalely σσ PE 3 d C λi Q + P γ σ + λ E P σ C ( ( ) ), (6)

7 d 3 ( ( λ ( ) + γ) + λ ) i Q P σ E P σ C 3 ( ( λi ( ) + γ) + λ ) ( ( ) ) Cσσ PE Q P σ E P σ C C σσ λ Q + P γ σ + λ E P i E P E P. (7) Now, le defie ( ( ) ) ( ) d = λ Q + P γ σ + λ E P E P. i Hece, elecig d d guaraee V <. Coequely, ( ) = σ ca e reached i fiie ie. Nex, o how ha he wichig codiio occur, le defie V = σ. Accordig o (9), he ie derivaive of V i V = σ σ y y y (( ) f (, ) f ( ˆ, ) λ(, )) C σ ( A LC ) C E CEd C σ ( A LC ) C E CEd = σyc A LC σ + x u x u + E x u Eu σ y + γ + λ σ = σ y + γ + λ Sice i wa how ha σ ad d d, i ca e readily ee ha he la racke of (8) ecoe egaive ad coequely V. herefore, codiio () eure ha he lidig urface σ y = ca e reached i y (8) fiie ie ad he wichig codiio occur. 3.. Swichig ae I i well kow ha he decriig fucio (DF) of relay elee i he wichig ae ca e oaied aed o i ipu apliude ad i d paraeer []. Nex, he DF for he relay elee will e defied. 3.. Relay odel i he wichig ae he relay elee wa defied i (4), where d i he relay gai ad σ y i he ipu defied i (5). A i will e how laer, he fa-ode dyaic of he oerver i a liear ye. he ocillaig ad high frequecy par of σ y () ca e preeed a [] where a ad y ( ) ai( ) σ = Ω, (9) Ω are he apliude ad frequecy of he lii cycle (or ocillaio), repecively. Uig he Fourier rafor, u() ca e preeed a [] where K, K, ( ) σ ( ) ( ) ( ) u = K + Kai Ω + K ai Ω +, (3) y K, ca e deeried uig he Fourier rafor. Hece, u ( ) ( ) σ ( ) σ ( ) ( ) u K y K y N ε ca e wrie a = + +, (3)

8 where N ε () i he odelig error. Sice Ω i eleced high eough i he deig procedure ad ecaue o ye i egieerig applicaio ac a low pa filer ad i view of he fac ha N ε () coai igal wih frequecie larger ha Ω, he effec of () i he eiaed ae ca e igored wihou ay lo of N ε geeraliy. Neverhele, i will e coidered i aalyi. Aupio 6: N ε () i ukow u ouded ad i or ca e preeed a N ε < β σ + βσ, (3) () where β ad β are poiive coa ad σ a σ y. Uig (), (3) ca e wrie a ( ) = ( ) + σ ( ) + ( ) u K K y N ε Cσ, (33) where K ad K ca e copued a [, ] K u d, (34) σ πa = = y u = Hece, he relay elee pae he low-ode of igal wih he gai he gai K ; i.e. 4d K =. (35) πa ( ) σ ( ) y K ad he fa-ode of igal wih u = K, (36) ( ) ( ) u = K Cσ. (37) 3.. Oerver rucure i wichig ae I order o derive he ai reul, fir y coiderig wo ode of igal i he wichig ae, le rewrie (8) a ( ) ( ˆ ˆ ) ( ) σ + σ = Aσ +Aσ + f x + x, u f x + x, u + λ x + x, u Eu, (38) y + - y H( ) Oerver Srucure ŷ C ˆx σ y L - g( ) Ed σ y + A + LPF u ( ˆ,u) fx + ˆx Figure. Block diagra of propoed oerver.

9 where A = A LC. Noe ha aed o Reark, he ai ye igal wih high frequecy ode (i.e. x ) ca e eliiaed fro (38). Nex, i he oerver rucure, he high frequecy ode of eiaed igal ( x ˆ ) i filered if i i ju ued a ae of a oliear fucio uch a f (). hi ak chage he high frequecy ode of he propoed oerver io a liear ye ad coequely properie of he high frequecy ode of he oerver (uch a he frequecy ad he relay equivale gai), will oey liear relay feedack ye. Figure pree he lock diagra of he propoed oerver, i which LPF deoe a liear Low Pa Filer, deiged aed o Reark. Hece, (38) ca e wrie a he, uig (33) ( ) ( ˆ ) λ( ) σ σ Aσ +Aσ f x f x E x E (39) + = +, u, u +, u u. ( ) ( ˆ ) λ( ) ( ) σ ( ) ( ) σ + σ = Aσ +Aσ + f x, u f x, u + E x, u KECσ KE y N ε E. (4) Coequely, he eiaio error dyaic of he oerver ca e decopoed io he fa ad low ode, repecively, a where σ ad σ G : ( ) K σ ( ) σ = A LC σ E, (4) FO y ( ) ( ) ( ˆ ) λ( ) ( ) G SO : σ = A KEC LC σ + f x, u f x, u + E x, u N ε E, (4) are he oerver error dyaic aociaed wih he low ad fa ode, repecively. hee wo dyaic eed o e deiged for differe ode. For he low ode, ecoe very all a. O he oher had, i he fa-ode, H( jω) proper value for K u e defied uch ha σ ad L u e deiged o provide K ad i addiio, he ocillaio frequecy ecoe large eough. Moreover, o deerie he relay equivale gai K ad K (a i will e how i heore 3) here i a radeoff ewee eer ae eiaio ad le eiiviy o he eauree oie Slow-ode dyaic of oerver i wichig ae Lea guaraee covergece of he error dyaic o he lidig urface, which i ur eure occurrece of he wichig ae. he, i he wichig ae, a i wa how i (4) ad (4), he oerver rucure coi of wo par. I hi ecio, ha all apliude. K i deiged uch ha he low-ode of he eiaio error, i.e. σ, Aupio 7: Le P = P e a poiive-defiie arix ad alo L, E ad K e uch ha ( K ) ( K ) A EC LC P + P A EC LC = Q (43) where A i he ae a efore ad Q i a poiive-defiie arix. heore : Coider he low-ode of he oervaio error (4) ad Aupio --7. he, he error dyaic for he low ode ( σ ) i uiforly uliaely ouded. Moreover, hi oud ca e ade all eough. Proof: Coider he followig Lyapuov fucio:

10 V = σ P σ 3 where P i he ae a defied i Aupio 4. he ie derivaive of (44) i ( )( (, ) ( ˆ, ) λ(, )) ε ( )( ) V = σ Q σ + σ P f x u f x u + E x u K σ PECσ N σ PE 3 ( ) K ( ) N () λ Q σ + γ P σ + λ PE σ λ N σ + σ PE, i i ε where arix N wa defied i Aupio 5. Uig Aupio 6 ad Youg iequaliy () ( ) σ N ε PE + λ PE σ λ ax P β σ E + βσ E + λ E Uig (46), (45) ecoe ( ) ( ) β E σ λax P + σ β E σλax P + λ E σ + β E λax( P) + β σλax( ) + λ. E P E (44) (45) (46) 3 σ + (47) V c c where = λ ( Q ) + λ ( N) γ P β E λ ( P ) ad ( ) herefore, if c K i i ax K deiged uch ha he followig iequaliy hold: ( Q ) ( N) P E ( P ) λ + λ γ + β λ + i K i ax ad defiig he followig copac e aroud he origi: i ca e cocluded ha V 3 { σ c σ c} Ψ= :, c = β σλax + λ E P E., (48) (49) whe he error i ouide of he copac e Ψ. Hece, accordig o he exeio of he adard Lyapuov heore [], he error rajecory σ i uliaely ouded. Moreover, hi oud ca e ade all uig large value for c. hi ca e accoplihed y deigig appropriae value for he oerver paraeer uch a K ad L. Reark 3: Accordig o (49), y elargig he equivale relay gai K ad coequely c, he copac e Ψ ecoe all. I.e., if σ. K ecoe large eough, he he radiu of he copac e ecoe egligile or A i i clear fro heore, i addiio o vecor L, wichig codiio. I he ex ecio, i will e how ha of ocillaio. Hece, K alo affec he dyaic of he low-ode i he K i a fucio of he apliude ad frequecy K ca e corolled y paraeer of he fa-ode dyaic of he oerver Fa-ode dyaic of oerver i wichig ae I heore, i wa how ha he low-ode of he oervaio error dyaic ca ecoe very all y he appropriae deig of arix L ad epecially he gai of he low ode ( K ). I he followig, he fa-ode of he oerver i wichig codiio i aalyzed.

11 Fro (4) i ca e how ha σ y K u GFO ( ) σ FO H ( ) y Figure. Fa-ode dyaic of oerver G FO ( ) ( u ) σ = A LC σ + E σ FO = Cσ (5) where σ FO i he oupu of he fa-ode dyaic of he oerver ad u, a how i (36), i he ipu o he fa-ode oerver a well a he oupu of he relay elee. Hece, G FO G FO ( ) where U ( ) = L { u () } ad Σ ( ) = L { () } L { Cσ() } L { ()} CΣ ( ) ( ) = = u U σ. ca e repreeed a, (5) he ex propoiio deorae he relaiohip ewee he ipu ad oupu igal of he relay elee i he fa-ode i he frequecy doai. Propoiio : Coider (8), (36) ad (5). he relaiohip ewee he ipu ad oupu of he relay elee i he fa-ode i L L { σy () } { ()} ( ) ( ) Σy = = G u U FO ( ) H( ) where u ad σ y are he ipu ad oupu of he relay elee i he fa-ode operaio, repecively. { y Proof: Fro (36) we have U ( ) K σ ()} = L. Coiderig (8) ad (5), i give (5) Hece, coiderig (7) Σ ( ) = CΣ ( ) Φ ( ) = U ( ) G ( ) Φ( ). (53) y FO Σ U y ( ) ( ) FO ( ) ( ) = G H, (54) where Σ y ( ) U ( ) i equal o he ivere of he rafer fucio of he relay elee i he fa-ode operaio, i.e. K. herefore, GFO ( j ) H ( j ) K = Ω Ω. (55) hi coplee he proof. Figure pree he fa-ode dyaic of he oerver. O he oher had, uig (34), (35) ad (55), i give K = G j H j ( Ω ) ( Ω ) FO (56)

12 A i wa how i hi ecio, he relay equivale gai i he fa ad low ode ca e regulaed y he appropriae rucure deig of liear ye G FO ( ) ad H( ) ad alo y he frequecy of ocillaio Ω. I oher word, he larger G ( jω FO ) H ( jω ), he aller K ad K. Baed o heore, hee paraeer direcly affec he ailiy ad he eiaio error of he oerver. Equaio (34), (35), (55) ad (56) help he deiger o deerie K, ocillaio, which will e coidered i he followig ecio. K ad he frequecy ad apliude of Paraeer Deig Accordig o Propoiio ad fro he heory of elf-ocillaig adapive ye, aed o relay feedack ye, H( ) ca e deiged uch ha he frequecy of he ocillaio (or he lii cycle) Ω, wih he followig propery []: ( ) ( ) 8 G jω + H jω =, (57) FO ca e corolled a deired. Moreover, he apliude of ocillaio i he fa-ode operaio ca e derived uig (35) ad (55) a 4d a = GFO ( jω ) H ( jω ). (58) π Sice he apliude of GFO ( jω ) i a decreaig fucio wih repec o he frequecy, larger elecio of Ω lead o aller value for FO ( ) G jω ad larger value for K ad K. herefore, akig he wichig frequecy large eough ad far eough fro he adwidh of he low-ode of he oerver ha wo eefi: ) he ocillaig ode of igal ca e eaily reoved uig a low-pa filer ad ) Accordig o heore ad (56), i he wichig ae, he relay equivale gai ecoe large. Hece, he oerver ac a a high gai oerver, which provide faer raie ie ad eer diurace rejecio. hi propery ove he oerver cloer o he ideal lidig-ode codiio, where he wichig frequecy grow o ifiie. he followig heore how codiio for he ideal lidig ode. heore. If he rafer fucio W l () i a quoie of wo polyoial B () ad A () of degree ad, repecively, wih o-egaive coefficie, he for he exiece of he ideal lidig ode, i i eceary ha he relaive degree ( ) of W l () e oe or wo. Proof: See [6]. Reark 4: Accordig o heore, whe codiio (57) i o aified (i.e. whe he relaive degree of GFO( ) H ( ) i le ha wo, ad for oe ye wih he relaive degree equal o wo), he he ideal lidig ode occur. I he ideally lidig-ode codiio, he wichig frequecy grow wih o oud. However, i pracice, ice he aplig ie τ i o zero, he wichig frequecy cao e ifiie. I hi codiio, he ocillaio frequecy oey he followig equaio [4, 6]:

13 ( ) ( ) exp(.) 8 G jω + H jω + jω τ = (59) FO herefore, i ca e cocluded ha for ye i which G ( ) H ( ) aifie he ideal lidig-ode codiio, he wichig frequecy grow o he large poile wichig frequecy ha i ca e copued fro (59). FO Reark 5: Accordig o he fa-ode dyaic of he oerver G FO, preeed i (5), E ad L hould e uch ha codiio give i heore (i.e. codiio for he ideal lidig ode) are aified. A i will e how i heore 3, he ocillaio frequecy or relay equivale gai hould o e very large o ake he oerver ucepile o eauree oie. 4. Noie Effec Reducio ad Deig of H() I he la ecio, i wa howed ha G ( ) ca e deeried y vecor E ad L. I hi ecio, he effec FO of he eauree oie o he eiaed ae are aalyzed. he, i order o uppre he eauree oie, a procedure for he deig of H ( ), GFO ( ) ad d will e give. 4.. Noie effec aalyi Aue he ye oupu i () i corruped wih eauree oie a y ω = Cx + ω, (6) where y ω R idicae he oiy oupu igal of he ye ad ω R i a addiive ouded oie wih ω χ. heore 3: Coider he ye i (), he ipu igal of he relay i () ad () ad he oiy ye oupu i (6). he, icreaig he apliude of GFO ( j ) H ( j ) Ω Ω decreae he oie effec i he fa ad lowode of he ipu igal of he relay of he relay elee d ha he ae effec o he fa-ode dyaic. Proof: Fir, he oie effec o he fa-ode of σ y, ad hece, eer eiaio of ae. Moreover, icreaig he gai σ y i aalyzed. Le y ω e i he frequecy doai of he faode igal. he, he fa-ode apliude of (5) ca e wrie i he frequecy doai a ( ) ( ) ( ) Σ jω = Y jω Y jω = a, (6) y ω where a i he ae a efore (i.e. he apliude of he ipu igal of he relay elee), Y ( ) { y ()} ad Y ( ) { y ()} = L. Uig (55), (5) ca e wrie a he, uig (35) ad (6), i give ( Ω ) = ( Ω ) ( Ω ) ( Ω ) ( Ω ) Y j Y j KG j H j Σ j. (6) ω FO y ω = L ω

14 Σy( jω ) Y ( ) Y ( j ω jω ω Ω ) = = Y ( jω ) Y( jω ) 4d 8d Yω( jω ) + GFO( jω ) H( jω ) GFO( jω ) H( jω ) Yω( jω ) π π Clearly, y elecig d ad GFO ( jω ) H ( jω ) large eough, i reul ha Σ y ( jω ) Y ( jω ) ea ha he effec of he oiy oupu igal o he relay ipu igal (i.e. (63). hi σ y ) will e reduced. Siilarly, for he low-ode doai, le aue y ω e i he frequecy doai of he low-ode igal. he, accordig o (37--4) ad (6), G SO i preece of eauree oie ecoe ( ) ( ) ( ˆ ) λ( ) ( ) G : σ = A ECK LC σ + f x, u f x, u + E x, u EN K Eω SO ε (64) Hece, elecig oerver. K all eough decreae he ifluece of he eauree oie i he low-ode of he A a coequece, larger value for GFO ( j ) H ( j ) Ω Ω (or equivalely, aller value for K ) ca uppre he effec of eauree oie i he ae eiaio. hi ca e achieved y he appropriae deig of H ( jω ). Reark 6: Sice heore 3 ad Lea provide wo differe value for he apliude of he relay gai (d), he larger d hould e eleced, accordig o (63). Reark 7: Baed o heore, larger value for GFO ( j ) H ( j ) Ω Ω are deired i order o reduce he effec of he eauree oie. However, accordig o (56) ad Reark 3, hi ay yield loger raie ie ad le roue o diurace rejecio due o all value of K. herefore, he elecio of he fa ad low-ode gai ( K ad K ) i aed o he radeoff ewee he oie reducio ad he roue of he oerver. hi fac ugge he deig of a variale relay equivale gai echique, where he gai hould e large i order o oai fa raie repoe ad ore roue agai diurace. O he oher had, hi gai hould ecoe aller for eer oie reducio. he procedure for deigig hi variale gai will e give i he followig ecio. 4.. Noie effec reducio uig variale gai echique hi ecio provide a ehod ha ca help he lidig-ode oerver o e le eiive o he eauree oie. he deig i aed o he idea ha addig a zero o he G ( jω ) ake i apliude larger i he wichig ode, which i ur ake he apliude of he gai K aller (accordig o (56)). I i ipora o oe ha accordig o heore, addig a zero o he H( ) rig he oerver cloer o he ideal lidig-ode codiio. I order o deig a variale equivale gai for he relay, coider he followig rucure for H( jω ) FO

15 ( ) ( αγ ) H jω = + jω, (65) whereγ i a poiive coa ad α i a variale paraeer, which will e defied laer o. he apliude ad he phae agle of H ( jω ) are ( ) ( ) H jω = + αγ Ω (66) ( ) a ( ) H jω = αγ Ω. (67) Accordig o (66), he apliude of H ( jω ) varie proporioally wih α. herefore, accordig o (55) ad (56), icreaig α decreae K ad K ad vice vera. A i wa eioed efore ad accordig o Reark 7, larger value of H ( jω ) ake he oerver le eiive o he eauree oie u icur ore ae eiaio error, ad vie vera. Hece, γ ad α u e defied uch ha K i o oo large ad o oo all. Deig of γ O oe had, γ u e eleced uch ha K i large eough o have fa repoe ad eer diurace cacellaio; hi cae correpod o he iiu value of α (i.e. α i = ). O he oher had, γ u e eleced uch ha K i all eough o have le eiiviy o he eauree oie, aed o heore 3; hi cae correpod o he axiu value of α. Deig of α Accordig o heore ad aed o he ivere relaiohip ewee α ad he relay equivale gai i he fa ad low-ode (i.e. K ad K ), α u e equal o oe o provide a high equivale gai for he relay whe he eiaio error i igifica. hi will provide eer perforace for he ae eiaio ad he diurace rejecio. herefore, he iiu value of α i equal o oe. O he oher had, whe he eiaio error i all eough, α u ecoe larger o reduce he relay equivale gai ad coequely akig he oerver le eiive o he eauree oie. I hould e oed ha α hould o e very large ice i ake K very all, which lead o lower rackig of he ye ae. Baed o hee poi, a proper cadidae for deeriig α i propoed a follow:.5 α( σ ), y = + (68) ( χ ) σ y e where χ i he upper oud of he or of eauree oie. I (68), whe he ipu igal of he relay elee ( σ y ) chage fro χ o.χ, α icreae fro.5 o.33, repecively. Coequely, K decreae fro K + ( γω ) + ( γω) o K ( γ ) ( γ ).5 + Ω +.35 Ω, repecively, where K = K for α =. Figure 3 how (68) for χ =.. he apliude of K decreae whe he eiaio error ( σ y ) i igifica wih repec o he eauree oie (i.e. whe σ y χ ). Hece, whe σ y (i.e. he ipu igal o

16 relay elee or he eiaio error of ye oupu i he low ode) ecoe all, i order o have le eiiviy o he ipu eauree oie. K ad K decreae Baed o he aove dicuio, he followig crieria are propoed for he Variale Relay-Equivale Gai (VREG) echique: γ i eleced uch ha he apliude of ocillaio a (correpodig o α i = ) i approxiaely equal o.75 χ. α ax i eleced uch ha he apliude of ocillaio i approxiaely equal o.5 χ. Reark 8: I (65), ad aed o he aove procedure, γ i deeried for Ω=Ω, where Ω i deiged high eough ad far fro adwidh of he ye ( Ω ). Hece, γ will e a all coa, which i ur give all value for αγ j Ω i (69) for Ω<Ω. Hece, for Ω<Ω, we have H ( jω), which aifie (7). 5. Applicaio o Bioreacor I hi ecio, he perforace of he propoed oerver i illuraed uig a ypical ioreacor wih ioa producio ad urae coceraio which elog o he cla of (). he ae equaio of hi ioreacor are [, 3] ( ) ( ) ( ) ( ) ( ) + ( ) µ x( ) x( ) ( ) + ( ) µ x x x = Dx Kx x ( ) x = + x D ( f ( )) Y Kx x where he pecific growh rae i aued o follow he Cooi odel, x i he ioa coceraio, x i he urae coceraio, f coa, Y i he yield of cell a, ad x () i eaurale o lie y a ioeor [4]. (69) i he ile urae coceraio, D i he diluio rae, K i he reacio Farza e al. have how ha hi ye i oervale [5]. I pracice, ie-varyig. Hece, le () µ = µ + d ad K () = K + d, where µ i he axial pecific growh rae. I i aued ha he ioa µ ad K ay e ucerai ad µ ad K are he kow oial paraeer, repecively, ad d () ad d () are odel of he ouded addiive ie-varyig paraeric uceraiie, repecively. Hece, aed o (), he ucerai ye i defied a where M (, ) ( ) ( ) ( ) + ( ) µ x( ) x( ) ( ) + ( ) µ x x x = Dx + M,, ( ) Kx x ( ) ( x ) x ( ) = + x D M Y Y Kx x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + ( f ( )) ( x, ) µ x x µ x x x = i he ukow fucio. K x x K x x he goal i o deig a lidig ode oerver aed o (3) o eiae he proce ae i preece of he eauree oie ad odel uceraiie uig he ehod developed i hi paper.

17 I addiio, he odel ad oerver iulaio have ee carried ou uig a coa ipu urae coceraio ad diluio rae. he paraeer value ued i iulaio are a follow: µ h =, K =, Y =, D =.5h, f = 5g.l, d.i(.5π ) =, d ( π) =.5i. Wih hee paraeer, he uceraiy fucio M( x,) i how i Figure 4. he feedack gai (uig he pole placee echique) are L = [ ], which aifie codiio () ad (48). he, accordig o Lea ad heore 3 he apliude of he relay gai i d = 5. he iiial ae of he ye ad he oerver are eleced a x( ) = [ ] ad x ˆ ( ) = [.5], repecively. I order o how he fa wichig propery of he propoed ehod (Reark 5) E ad L hould e uch ha GFO( ) i ale ad i relaive degree i le ha wo. hi provide he highe poile wichig codiio or he ideal lidig-ode codiio. Sice E = [ ] he i give G ( ) = ( +.5), which ha he relaive degree of oe wih ale eigevalue. Hece, he ideal lidig-ode occur. 4 By elecig he aplig ie equal o τ = ad coiderig (59), he wichig frequecy ad he FO period of ocillaio will e equal o Ω = 57 rad/ ad Ω =, repecively. I hi codiio 4 π 4 accordig o (56), he equivale gai of he relay i he low-ode of he propoed oerver ecoe K = = = G j H j.638 ( Ω ) ( Ω ) ( ) FO Baed o (58), he apliude of ocillaio i Sice he frequecy of ocillaio frequecy igal. he eleced low-pa filer i Figure 5 how iulaio reul for 4 4 4d a = GFO ( jω ) H ( jω ) = =.4. π π Ω i eleced high eough, a iple low-pa filer ca eaily rejec high ( ) (.73 )(.73 ) G = + +. σ y. A hi Figure idicae, he period of ocillaio ad a are equal o ad 45-4, repecively, which cofir he value oaied fro he equaio. Nex, he effec of eauree oie o he ae eiaio i aalyzed. he eauree oie i a whie oie wih uifor diriuio i [.,.]. Hece, he oud of he eauree oie i equal o χ =.. o iprove he eiaio accuracy i preece of he eauree oie, he propoed VREG echique, dicued i ecio 4., i applied here. Figure 6 how perforace of he propoed ehod i oie reducio for ae eiaio for wo cae: ) Eiaio uig he coveioal lidig ode oerver wih fixed relay equivale gai, which i equal o K = 7834 (i.e. he ae value ha wa ued i he fir par of iulaio). hi propery i coo wih all coveioal lidig-ode oerver, where reearcher ue a relay elee, ) Slidig ode oerver uig he 3 VREG echique wih γ = ad ( e σ y α = + ). A i wa dicued efore, i VREG echique, K i large whe he eiaio error i coiderale ad all whe he eiaio error i all. hi propery provide faer covergece of he eiaio error ad le eiiviy o he eauree oie a copared o he coveioal lidig-ode oerver.

18 Figure 7 how he ipu igal of he relay elee ( σ y ) uig he coveioal lidig-ode oerver (i.e. wih fixed gai) ad he VREG echique i preece of he eauree oie. hi figure how ha VREG echique provide le eiiviy o he eauree oie i he ipu igal of he relay elee. ha i, he ocillaio frequecy i fixed ad i o affeced y he oie. Figure 8 how variaio of α i VREG echique α σ y Figure 3. Variaio of paraeerα wih repec o he ipu igal of he relay elee σy whe χ = ie() Figure 4. Sigal M( x, ) 3 σ y.5.5 x ie() Figure 5. Sigalσ uig coveioal lidig ode oerver y

19 x () 3 Eiaed x Uig Coveioal Slidg Mode Oerver Real x.5 Eiaed x Uig Slidg Mode Oerver wih VREG echique ie() (a) x () Eiaed x Uig Coveioal Slidig Mode Oerver Real x Eiaed x Uig Slidig Mode Oerver wih VREG echique ie() () Figure 6. x( ) ad x ( ) i preece of eauree oie.4.3 σ y σ y Uig Coveioal Slidig Mode Oever σ y Uig Slidig Mode Oever wih VREG echique ie() Figure 7. Ipu igal of he relay σ y i preece of eauree oie

20 .5 α α ie() ie() Figure 8. Variaio of α uig VREG echique 6. CONCLUSION Deigig a relay aed lidig-ode oerver wa iroduced i hi paper. hi oerver ue he high gai propery of he lidig-ode oerver efore he wichig occur i order o have fa repoe ad alo providig codiio for reachig he lidig urface. he, i he wichig ae, igal are decopoed io he low- ad fa-ode. he ai coriuio of hi paper i ha i provide a ehod for aalyi of he fa ad low-ode of igal i he wichig codiio for oliear lidig-ode oerver. Moreover, i give he relaiohip ewee hee ode. I he low ode, a ehod i propoed uch ha he high-gai oerver properie are oaied i order o rejec hi ode. I addiio, he fa ode i deiged uch ha ) he relay equivale gai provide codiio for good rackig ad ailiy i he low ode ad ) he frequecy of ocillaio ecoe high eough uch ha i ca e rejeced y a iple low pa filer. Furherore, eiiviy of he oerver agai he eauree oie wa aalyzed ad i wa howed how i ca affec he frequecy ad apliude of ocillaio. A he ed of hi paper, a variale relay equivale gai (VREG) echique wa propoed o give he oerver faer repoe ad le eiiviy o he eauree oie. Siulaio reul o a ioreacor proce howed ha he propoed echique provide good rackig ad eer oie rejecio a copared o he fixed equivale gai echique, which i ued i coveioal lidig-ode oerver. Referece [] J. Picó, H. D. Baia, F. Garelli, Sooh lidig-ode oerver for pecific growh rae ad urae fro ioa eauree, Joural of Proce Corol. 9 (9) [] S. K. Spurgeo, Slidig ode oerver: a urvey, Ieraioal Joural of Sye Sciece. 39 (8) [3] K. C. Veluvolu, Y.C. Soh, ad W. Cao, Rou oerver wih lidig ode eiaio for oliear ucerai ye, IE Corol heory Appl. (7) [4] R. A. Lόpez ad R. M. Yeca, Sae eiaio for oliear ye uder odel uceraiie: a cla of lidig-ode oerver, Joural of Proce Corol. 5 (5) [5] R. Aguilar, R. Mariez-Guerra, ad R. Maya-Yeca, Sae eiaio for parially ukow oliear ye: a cla of iegral high gai oerver, IEE Proc. Corol heory Appl. 5 (3) 4 44.

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