An observation algorithm for nonlinear systems with unknown inputs

Transcription

1 An observaton algorthm or nonlnear systems wth unknown nputs Jean-Perre Barbot, Drss Boutat, Therry Floquet To cte ths verson: Jean-Perre Barbot, Drss Boutat, Therry Floquet An observaton algorthm or nonlnear systems wth unknown nputs Automatca, Elsever, 2009, 45 (8), pp <101016/jautomatca > <nra > HAL Id: nra Submtted on 4 Jun 2009 HAL s a mult-dscplnary open access archve or the depost and dssemnaton o scentc research documents, whether they are publshed or not The documents may come rom teachng and research nsttutons n France or abroad, or rom publc or prvate research centers L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la duson de documents scentques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche ranças ou étrangers, des laboratores publcs ou prvés

2 An observaton algorthm or nonlnear systems wth unknown nputs Jean-Perre Barbot 1,4, Drss Boutat 2 and T Floquet 3,4 1 Equpe Commande des Systèmes (ECS), ENSEA, 6 Avenue du Ponceau, Cergy, France 2 LVR/ENSIB, 10 Boulevard de Lahtolle, Bourges, France 3 LAGIS UMR CNRS 8146, Ecole Centrale de Llle, BP 48, Cté Scentque, Vlleneuve-d Ascq, France 4 Equpe Projet ALIEN, INRIA Llle - Nord Europe, France Abstract Ths paper provdes some new developments n the desgn o unknown nput observers or nonlnear systems An algorthm whch states the state and the unknown nput o the system can be recovered n nte tme s ntroduced Ths algorthm leads to the transormaton o the system nto an extended block trangular observable orm sutable or the desgn o nte tme observers The proposed method s useul to relax some restrctve condtons o exstng nonlnear unknown nput observer desgn procedures 1 Introducton It s o mportance to desgn observers or multvarable lnear or nonlnear systems partally drven by unknown nputs Such a problem arses n systems subject to dsturbances or wth naccessble nputs and n many applcatons such as parameter dentcaton, ault detecton and solaton or cryptography The desgn o observers or nonlnear systems s a challengng problem, even or accurately known systems It has receved a consderable amount o attenton n the lterature In many approaches, nonlnear coordnate transormatons are used to transorm the system nto sutable observer canoncal orms Then, observers wth lnearzable error dynamcs (see eg [9, 20, 21]), hgh gan observers [8] or backsteppng observers [15] can be desgned Few works deal wth the desgn o unknown nput observers or nonlnear systems Some o them are concerned wth applcatons n the eld o ault detecton and dentcaton, and n partcular the nonlnear Fundamental Problem o Resdual Generaton [4, 10, 19] Other ones deal wth nonlnear systems subject to exogenous perturbatons and are based on sldng mode consderatons [22] The convergence s obtaned n nte tme under the assumptons that the system can be put 1

3 nto a set o trangular observable orms, where the unknown nputs act only on the last dynamcs o each trangular orm Ths assumpton s known as the observablty matchng condton The present work ams at the development o a systematc method leadng to the nte tme observaton o a class o nonlnear systems wth unknown nputs even the observablty matchng condton s not ullled To ths end, the procedure gven n [5] s extended to the nonlnear case It also results n a constructve algorthm that transorms the system nto a smlar type o block trangular observable orms Ths transormaton reles on the ntroducton o sutable cttous outputs Sucent condtons or the exstence o such auxlary varables are gven Ater transormaton o the system, t s shown that t s possble to desgn observers that provde the nte tme estmaton o both the state varables and the unknown nputs An llustratve example hghlghts the ecency o the proposed methodology 2 Problem statement and motvatons Consder, on an open set U, the nonlnear system: ẋ = (x) + g(x)w = (x) + m g (x)w =1 (1) y = h(x) = [h 1 (x),, h p (x)] T where x U R n s the state vector, y R p s the output vector and where w = [w 1,, w m ] R m represents the unknown nputs The vector elds and g 1,,g m, and the unctons h 1,, h p, are assumed to be sucently smooth on U Wthout loss o generalty, t s assumed that p m, and that or all x U, the dstrbuton G = span {g 1,,g m } and the codstrbuton span {dh 1,,dh p } are nonsngular on U Lke n [17], let us rst dene the unknown nput characterstc ndexes {ρ 1,, ρ p } such that, or 1 p: L gj L k h (x) = 0, or k < ρ 1, and or all 1 j m, L gj L ρ 1 h (x) 0, or at least one 1 j m, or all x U The system (1) wth w = 0 s supposed to be locally weakly observable on U [11] Thus, there exsts a change o coordnates ( φ = h 1,, ) T h 1,, h p,, L νp 1 h p L ν1 1 such that, ater a sutable reorderng o the state components, the system (1) s locally transormed nto ξ = A δ ξ + H δ V δ (x,w) (2) η = a(ξ,η) + b(ξ,η)w (3) y = C δ ξ 2

4 wth ξ = ( ) ξ1 T,, ξp T T, ξ R δ, 1 p V δ (x, w) = L δ h (x) + m L gj L δ 1 h (x)w j R = A δ = R δ δ H δ = ( ) T R δ, C δ = ( ) R 1 δ and a, b are smooth vector elds on U The ntegers (ν 1,ν 2,,ν p ) are the so-called observablty ndces o system (1) (see [14] or a denton) and thus satsy ν ν p = n From the denton o the ρ, one has δ = mn (ν,ρ ) and g δ = 0 ν < ρ (e δ = ν ) 21 A trangular observable orm Most exstng nonlnear observers or system (1) are desgned under the assumpton that the system satses the so-called observablty matchng condton Ths condton was rst ormulated or SISO lnear systems n [18], Chapter 4 In the general case o MIMO nonlnear systems, a necessary and sucent condton s gven by: ν ρ or all 1 p (4) Then, δ = ν or all 1 p, and under the change o coordnates ξ = φ(x) the system (1) s transormed nto the orm: ξ = A ν ξ + H ν V ν (ξ,w) (5) y = C ν ξ wth m L gj L ν 1 h (x)w j 0 and only ν = ρ Each subsystem o (5) s n =1 the so-called trangular observable orm Fnte tme observers or such a orm can be ound n the lterature, based on step-by-step sldng mode technques [22], hgher order sldng modes [6, 16], numercal approaches [3, 13], or algebrac methods [2] The desgn o such observers s let to the reader, snce they are straghtorward applcatons o exstng results Nevertheless, dependng on the choce o the observer, some assumptons have to be ntroduced For nstance, n all the prevously mentoned works, the unknown nput w has, at least, to be bounded A necessary and sucent condton or the recovery o the unknown nputs s that Γ(x) = L g1 L ρ1 1 h 1 (x) L gm L ρ1 1 h 1 (x) L g1 L ρp 1 h p (x) L gm L ρp 1 h p (x) 3

5 has rank m In ths case, ρ = {ρ 1,, ρ p } s the vector relatve degree as dened n [12], p 220, when m = p 22 An extended trangular observable orm The am o ths paper s to provde an observaton algorthm that allows or the nte tme estmaton o both the state and the unknown nputs o (1) even ν j > ρ j or at least a j n {1,, p} Consder agan the general orm (2-3) Applyng any o the aorementoned nte tme observers: () or 1 p, ξ can be estmated n nte tme; () one can also recover n nte tme the last component V δ o each subsystem o (2) The problem s to recover the remanng state η Denote: where V (x) = V δ1 (x,w) V δ2 (x,w) V δp (x,w) = L δ1 h 1(x) L δ2 h 2(x) L δp h p(x) + Γ δ(x)w L g1 L δ1 1 h 1 (x) L gm L δ1 1 h 1 (x) Γ δ (x) = L g1 L δp 1 h p (x) L gm L δp 1 h p (x) Let be the commutatve algebra o the measured outputs and ther successve Le dervatves up to order δ : = span{h 1,, L δ1 1 h 1,, h p,,l δp 1 h p } and let d be the codstrbuton: d = span{dh 1,,dL δ1 1 h 1,,dh p,,dl δp 1 h p } Assume there exsts a 1 p row vector K(x) = (k 1 (x),, k p (x)) 0, k or 1 p, such that: K(x)Γ δ (x) = 0 or all x U (6) and set: ȳ = h(x) = K(x)V (x) = p k (x)l δ h (x) (7) Note that ȳ s an avalable normaton (ater a nte tme) and s not aected by the unknown nputs Thereore, d +span { d h } d, ȳ can be consdered as an addtonal cttous output Then, let ρ and ν be the unknown nput characterstc ndexes and the observablty ndces o (1) wth respect to the extended output [ y T, ȳ ] T = [ h T, h(x) ] T I ν ρ or all 1 p + 1, the =1 4

6 system (1) can be transormed nto: ξ = A ν ξ + H ν V ν ( ξ, w) or 1 p y = C ν ξ ξ = A νp+1 ξ + H νp+1 V νp+1 ( ξ,w) ȳ = C νp+1 ξ where ξ = ( ξ T 1,,ξ T p, ξ T) T R n Then, t s possble to recover both the state and the unknown nputs n nte tme Let us gve sucent condtons or the exstence o a sutable cttous output ȳ For ths, the ollowng notatons are ntroduced: ) G = span{ 1,, n m }, the annhlator o G ( are 1-orms such that ι gk = 0, where ι g s the nner product o the vector eld g and the 1-orm ) ) Ω, the module spanned by d over Proposton 1 The ollowng condtons are equvalent: ) Equaton (6) has a soluton K and KV / ) Ξ = span{ G Ω such that ι / } = {0} Proo: Set = p k dl δ 1 h wth k Clearly, Ω and KΓ δ = K =1 p ι = ι k dl δ 1 h = =1 dl δ1 1 h 1 dl δp 1 h p p =1 k L δ h = KV = ȳ [ ] [ ] g1 g m = g1 g m Thus: {K s a soluton o (6) such that KV / } { Ω,ι τ = 0 or any τ G and ι / } { G Ω and ι / } Ξ {0} The dscusson above can be recursvely generalzed as ollows Assume that the condton (4) s not stll satsed wth the extended output obtaned wth the solutons o (6) On the bass o ths new output, the correspondng matrx Γ δ can be computed and another set o cttous outputs can eventually be ound One can terate ths procedure untl the condton (4) s ullled or a new extended output Then, the orgnal system can be put nto an extended 5

7 block trangular observable orm 1 : ξ 1 = A ν 1 ξ 1 + H ν 1 V ν 1 (ξ, w) y 1 = y = C ν 1 ξ 1, 1 p 1 ξ 2 = A ν 2 ξ 2 + H ν 2 V ν 2 (ξ, w) y 2 = C ν 2 ξ 2, 1 p 2 ξ k y k = A ν k ξ k + H ν k V ν k (ξ, w) = C ν k ξ k, 1 p k (8) where the ntegers ν j are the observablty ndces o the system (1) wth the new outputs y j The rst subsystem s ed by the orgnal outputs o the system A nte tme observer s desgned to estmate the state o ths subsystem and to provde n nte tme the knowledge o the cttous outputs y 2, 1 p2 Then, the state o the second trangular observable orm can be estmated as well as the cttous outputs y 3 Thus, one can recursvely obtan the whole state o the system n nte tme Remark 1 The nte tme property s requred to ensure that one obtans a ast and accurate estmaton o the cttous outputs (or nstance, va the equvalent output njecton n the case o sldng mode observers, see [6] and the reerences theren) Furthermore, ths property s oten desrable n the ramework o observaton and partcularly or the purpose o observer-based controller desgn or nonlnear systems Then, or a large class o nonlnear systems, the observer can be desgned separately rom the controller and the separaton prncple does not need to be proved It can also be o paramount mportance n applcatons that requre ast estmatons o some unknown nputs lke ault detecton and dentcaton or on-lne parameter dentcaton 3 Nonlnear unknown nput observer algorthm An algorthm that states the system can be transormed nto (8) and that provdes the ntegers p j, ν j and the auxlary outputs y j (j = 1,,k ) s now gven [ ] T Step 0: Compute G and ts annhlator G Set p 1 = p, h 1 1,, h 1 p = 1 [h 1,,h p ] T, µ 0 = 0, z 0 1 = = z 0 µ 0 = 0 Step α: [a] Consder y α = [ h α 1,,h α p α ] T R p α and reorder ts components as ollows: y α = [ h α 1,, h α l α,hα l α +1,, h α p α ] T 1 Systems that admt such a orm belong to the class o let nvertble systems wth trval zero dynamcs (see [2]) 6

8 such that or 1 j l α : [1,,m], k N L g L k h α j = 0 and or 1 j p α l α, there exsts an nteger ρ α j such that: [1,,m] L g L k h α l α +j = 0 k < ρ α j 1 [1,,m] L g L ρα j 1 h α l α +j 0 [b] Dene Φ α = {dh α 1,, dl n 1 h α 1,, dh α l I α = span α,, dln 1 {( dz α 1 1,,dz α 1 µ α 1 ) Φ α } Let dmi α = µ α 1 + ϕ α I α can be wrtten as ollows: h α lα} Compute I α = span{dz1 α 1,, dz α 1 µ,dh α α 1 1,,dL ϕα 1 1 h α 1,, dh α l α,,dlϕα l α 1 h α l α} wth lα ϕ α = ϕα I µ α 1 + ϕ α = n, set =1 {dz k 1,, dz k µ k } µ k = µ α 1 + ϕ α = {dz α 1 1,, dz α 1 µ,dh α α 1 1,, dl ϕα 1 1 h α 1,, dh α l α,, dlϕα l α 1 h α l α} and stop the algorthm [c] I µ α 1 + ϕ α < n, consder the outputs aected by the unknown nputs and dene: Υ α = {dh α l +1,, dl ρα α 1 1 h α l +1,, dh α α p α,, dlρα p α l α 1 h α p α} Compute the codstrbuton Ω α = span {I α Υ α } Let dmω α = µ α 1 + ϕ α + κ α = µ α and wrte Ω α = span{dz1 α,,dzµ α α} wth and pα l α =1 {dz1 α,,dzµ α α} = {dzα 1 1,,dz α 1 µ, dh α α 1 1,, dl ϕα 1 1 h α 1, h α l α,dhα l α +1,, dlκα 1 1 h α l α +1,, dh α l α,, dlϕα l α 1, dh α p α,,dlκα p α l α 1 h α p α} κ α = κα I µ α = n, the algorthm stops [d] Otherwse, µ α < n Dene α = span{z1 α,, zµ α { α} µ α } Ω α = span φ dz α, φ α =1 Ξ α = span { G Ω α such that ι / α} 7

9 Let p α+1 = dm Ξ α I p α+1 = 0, the state o the system (1) can not be recovered wth the method descrbed n ths paper and the algorthm stops Otherwse, there exst p α+1 one-orms such that Ξ α = span { 1,, pα+1} and one can dene the ollowng vector o cttous outputs, sutable to the problem (see Proposton 1): y α+1 = [ ] ] T T ι 1,,ι Set [h α+1 pα+1 1,, h α+1 p = [ ] α+1 T ι 1,,ι Go to [a] pα+1 I the algorthm stops or some µ k = n, the change o coordnates φ = (z k 1,,z k µ k ) T s well dened and transorms the system nto a set o block trangular observable orms smlar to (8) The unctons h j, and the ntegers p, l, ϕ j, κ j are obtaned n each -th teraton o the algorthm Remark 2 I the condton (4) s satsed or the measured outputs o the system (1), µ 1 = n ater the rst teraton o the algorthm and the system s exactly transormed nto the orm (5) that s usually consdered or the desgn o asymptotc (see eg [8, 15]) or robust nte tme ([6, 22]) nonlnear observers 4 Estmaton o the unknown nputs I the algorthm ends n a postve way, the unknown nputs can also be obtaned n a nte tme Indeed, the use o a nte tme observer provdes an estmaton o the state, say x, and the knowledge o the ollowng quanttes (rom the last lne o each block o the trangular orm): Θ( x) = m θ j = L κ j h l +j ( x) + L gs L κ j 1 h l +j ( x)w s, (9) s=1 or 1 j p l, and 1 k The relatons (9) can be rewrtten as: Λ( x)w = Θ( x) wth L g1 L κ1 1 1 h 1 l 1 +1 ( x) L g m L κ1 1 1 h 1 l 1 +1 ( x) Λ( x) = L g1 L κk p k l 1 h k k ( x) L p g m L κ k p k l 1 h k k ( x) p θ 1 1 L κ1 1 h1 l 1 +1 ( x) θ k L κ p k l k p k l h k p k ( x) 8

10 Snce the dstrbuton span {g 1,,g m } s assumed to be nonsngular, the matrx dl κ1 1 1 h 1 l 1 +1 ( x) [ ] Λ( x) = g1 g m dl κk p k l 1 h k p k ( x) has rank m on every subset o U where at least m one-orms dl κ j 1 h l +j does not belong to G Thus, an estmaton o the unknown nput s gven by: w = Λ + ( x)θ( x) where Λ + s a well dened pseudo-nverse o Λ 5 Example As a way o llustraton 2, consder the ollowng nonlnear system subject to the unknown nput w = [w 1,w 2 ] T ẋ 1 = x 2 x 3 1 ẋ 2 = x 3 + x 2 2 x a(x 3,x 4 )w 1 ẋ 3 = x 5 (10) ẋ 4 = x 4 + x b(x 2,x 3 )w 1 ẋ 5 = x 3 + x 2 w 2 ẋ 6 = x 6 + w 2 wth outputs y 1 = x 1, y 2 = x 4 and y 3 = x 6 The scalar unctons a and b are such that a(x 3, x 4 ) = a 1 (x 3 )a 2 (x 4 ), b(x 2,x 3 ) = a 1 (x 3 )b 2 (x 2 ) and where a(0,0) 0 and b(0,0) 0, and b 2 (x 2 ) 0, x 2 R For ths system, one has ρ 1 = 2, ρ 2 = ρ 3 = 1 and ν 1 = 4, ν 2 = ν 3 = 1 (as a consequence δ 1 = 2, δ 2 = δ 3 = 1) Thus, the necessary and sucent condton (4) or a system to be transormed nto a orm smlar to (5) s not ullled Furthermore, the dstrbuton G s not nvolutve As a consequence, the sldng mode observer proposed n [22], where the dstrbuton spanned by the unknown nput channels has to be nvolutve, can not be desgned here However, the procedure proposed n ths paper s applcable The annhlator o G s gven by: G = span{dx 1, dx 3, b 2 dx 2 a 2 dx 4,dx 5 x 2 dx 6 } In the rst step o the algorthm, I 1 s empty and Ω 1 = span{dh 1,dL h 1,dh 2, dh 3 } = span{dx 1, 3x 2 1dx 1 + dx 2,dx 4,dx 6 } wth dm Ω 1 = 4 < n One has 1 = span{h 1,L h 1,h 2,h 3 } = span{x 1,x 2 x 3 1,x 4,x 6 } = span{x 1,x 2,x 4,x 6 } 2 A physcal applcaton o the proposed algorthm n the eld o chaotc synchronzaton or secure communcaton can be ound n [2] 9

11 and b 2 dx 2 a 2 dx 4 Ω 1 Then G Ω 1 = span{dx 1, b 2 dx 2 a 2 dx 4 } Ξ 1 = span{b 2 dx 2 a 2 dx 4 } Ξ 1 0 and one can dene the ollowng cttous output: y 2 = ι (b 2 dx 2 a 2 dx 4 ) = b 2 (x 3 + x 2 2 x 3 2) a 2 ( x 4 + x 2 2) = b 2 (x 2 )x 3 mod[ ] Snce b 2 (x 2 ) 0 or all x 2, the knowledge o y 2 s equvalent to the knowledge o x 3 The second step starts by denng the output vector: y 2 = x 3 I 2 s empty and Ω 2 = span{dx 1,dx 2,dx 4,dx 6,dx 3,dx 5 } wth dm Ω 2 = 6 Then, one can dene the change o coordnates: z = ψ(x) = [x 1,x 2,x 4,x 6,x 3,x 5 ] T In the new coordnates, the system s rewrtten as: ż 1 = z 2 z 3 1 ż 2 = z 5 + z2 2 z2 3 + a 1 (z 5 )a 2 (z 3 )w 1 ż 3 = z 3 + z2 2 + a 1 (z 5 )b 2 (z 2 )w 1 (11) ż 4 = z 4 + w 2 ż 5 = z 6 ż 6 = z 5 + z 2 w 2 By the means o a nte tme observer ed by the orgnal outputs o system (10), e y 1 = z 1, y 2 = z 3 and y 3 = z 4, one can recover the state z 2 and V 1 = z 5 + z 2 2 z a 1 (z 5 )a 2 (z 3 )w 1 V 2 = z 3 + z a 1 (z 5 )b 2 (z 2 )w 1 V 3 = z 4 + w 2 For the sake o place, the desgn o the observer s not gven here However, the reader can or nstance reer to [7] where an example o hgher order sldng mode observer, desgned or a smlar problem n the lnear case, can be extended to the problem o nonlnear systems Then, y 2 = ι (b 2 (z 2 )dz 2 a 2 (z 3 )dz 3 ) = b 2 (z 2 ) ( z 5 + z 2 2 z a 1 (z 5 )a 2 (z 3 )w 1 ) a 2 (z 3 ) ( z a 1 (z 5 )b 2 (z 2 )w 1 ) = b2 (z 2 )V 1 a 2 (z 3 )V 2 = b 2 (z 2 )(z 5 + z 2 2 z 3 2) a 2 (z 3 )( z 3 + z 2 2) s known ater a nte tme and z 5 = y2 +a 2(z 3)( z 3+z 2 2 ) b 2(z 2) + z2 3 z2 2 s an avalable normaton Agan, a nte tme observer leads to the recovery o z 6 Then, the unknown nput can also be obtaned snce: w 1 = V1 z5 z2 2 +z3 2 a 1(z 5)a 2(z 3) and w 2 = V 3 +z 4 10

12 6 Concluson In ths paper, the problem o the observaton o nonlnear systems subject to unknown nputs was consdered An observaton algorthm that determnes whether t s possble to recover the state and unknown nput n nte tme was ntroduced When the answer s yes, the algorthm provdes a change o coordnates that transorms the system n a new type o block trangular observable orm well suted to the desgn o nte tme observers The observablty matchng condton usually requred or the desgn o nonlnear unknown nput observers s relaxed wth the proposed method Reerences [1] J-P Barbot, I Belmouhoub and L Boutat-Baddas, Observablty Normal Forms, n: New trends n Nonlnear dynamcs and control, LNCIS 295, W Kang et al, Eds, Sprnger Verlag, 2003, pp 1 24 [2] J-P Barbot, M Fless, T Floquet, An algebrac ramework or the desgn o nonlnear observers wth unknown nputs, IEEE Con on Decson and Control, New-Orleans, USA, 2007 [3] S Dop, J W Grzzle and F Chaplas, On numercal derentaton algorthms or nonlnear estmaton, n IEEE Conerence on Decson and Control, pp , 2000 [4] T Floquet, J-P Barbot, W Perruquett and M Djemaï, On the robust ault detecton va a sldng mode dsturbance observer, Internatonal Journal o control, vol 77, 2004, pp [5] T Floquet and J-P Barbot, An observablty orm or lnear systems wth unknown nputs, Internatonal Journal o control, vol 79, 2006, pp [6] T Floquet, JP Barbot, Super twstng algorthm based step-by-step sldng mode observers or nonlnear systems wth unknown nputs, Internatonal Journal o Systems Scence, vol 38, 2007, pp [7] T Floquet and JP Barbot, A canoncal orm or the desgn o unknown nput sldng mode observers, n Advances n Varable Structure and Sldng Mode Control, Lecture Notes n Control and Inormaton Scences, Vol 334, C Edwards, E Fossas Colet, L Frdman, (Eds), Sprnger Edton, 2006 [8] J P Gauther, H Hammour and S Othman, A smple observer or nonlnear systems wth applcatons to boreactors, IEEE Transactons on Automatc Control, Vol 37, 1992, pp

13 [9] A Glumneau, C H Moog, and F Plestan, New Algebro-Geometrc Condtons or the Lnearzaton by Input-Output Injecton, IEEE Transactons on Automatc Control, vol 41, 1996, pp [10] H Hammour, M Knnaert, and E H El Yaagoub, Observer-Based Approach to Fault Detecton and Isolaton or Nonlnear Systems, IEEE Transactons on Automatc Control, vol 44, 1999, pp [11] R Hermann and A J Krener, Nonlnear controllablty and observablty, IEEE Transactons on Automatc Control, Vol 22, pp , 1977 [12] A Isdor, Nonlnear Control Systems, Communcaton and Control Engneerng Seres, Thrd edton, Sprnger-Verlag, 1995 [13] W Kang, Movng Horzon Numercal Observers o Nonlnear Control Systems, IEEE Transactons on Automatc Control, Vol 51, No 2, pp , 2006 [14] A J Krener and W Respondek, Nonlnear observers wth lnearzable error dynamcs, SIAM J Control Optm, Vol 23, 1985, pp [15] A J Krener and W Kang, Locally convergent nonlnear observers, SIAM J Control Optm, Vol 42, , 2003 [16] A Levant, Robust Exact Derentaton va sldng mode technque, Automatca, Vol 34, No 3, pp , 1998 [17] R Marno, W Respondek, A J Van der Schat, Almost dsturbance decouplng or sngle-nput sngle-output nonlnear systems, IEEE Transactons on Automatc Control, Vol 34, pp , 1989 [18] W Perruquett and J-P Barbot, Sldng Mode Control n Engneerng, Ed Marcel Dekker, 2002 [19] C de Perss and A Isdor, A geometrc approach to nonlnear ault detecton and solaton, IEEE Transactons on Automatc Control, vol 46, 2001, pp [20] W Respondek, A Pogromsky, H Njmejer, Tme scalng or observer desgn wth lnearzable error dynamcs, Automatca, Vol 40, No 2, pp , 2004 [21] X H Xa and W B Gao, Nonlnear observer desgn by observer error lnearzaton, SIAM J Control Optm, vol 27, 1989, pp [22] Y Xong and M Sa, Sldng Mode Observer or Nonlnear Uncertan Systems, IEEE Transactons on Automatc Control, vol 46, 2001, pp