STATE OBSERVATION FOR NONLINEAR SWITCHED SYSTEMS USING NONHOMOGENEOUS HIGH-ORDER SLIDING MODE OBSERVERS

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1 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 Publshed onlne n Wley Onlne Lbay (wleyonlnelbay.com) DOI: 0.002/asc.56 STATE OBSERVATION FOR NONLINEAR SWITCHED SYSTEMS USING NONHOMOGENEOUS HIGH-ORDER SLIDING MODE OBSERVERS J. Davla H. Ríos and L. Fdman ABSTRACT Ths atcle pesents the poblem of fnte tme econstucton of the contnuous state and opeatng mode fo a class of nonlnea swtched systems. The poposed method s based on the nonhomogeneous hgh-ode sldng mode appoach. It s able to econstuct both the state and opeatng mode of a swtched system based only on ts measuable outputs and though the use of the featues of the equvalent output necton. The obsevablty s deved n tems of cetan geometc estctons on the vecto felds of the swtched system that eque the avalablty of all ts modes. The method does not eque the system to be tansfomed nto any nomal fom. Smulaton esults suppot the poposed method. Key Wods: Hgh-ode sldng modes nonlnea obseves swtched systems. I. INTRODUCTION Swtched systems whose behavo can be epesented by the nteacton of contnuous and dscete dynamcs have been wdely studed dung ecent decades snce they can be used to descbe a wde ange of physcal and engneeng systems. Most of the attenton pad to these knds of system has focused on the poblems of stablty and stablzaton wth etensve and satsfactoy esults (see e.g. [] [4]). Sldng-mode-based obust state obsevaton has been developed successfully n Vaable Stuctue Theoy n ecent yeas (see e.g. [5] [7]). The obseve desgn poblem fo swtched systems.e. the estmaton of the contnuous and dscete state s of geat nteest fo many aeas of contol. The man dffeence among the estent appoaches s elated to the knowledge of the actve dscete state o opeatng mode: some appoaches consde only contnuous state uncetanty wth a known opeatng mode whle othes assume that both the opeatng mode and the contnuous state ae unknown. In [8] a Luenbege obseve appoach fo lnea systems s poposed fo Manuscpt eceved Apl 29 20; evsed Octobe 3 20; accepted Decembe J. Davla s wth Natonal Polytechnc Insttute Secton of Gaduate Studes and Reseach ESIME-UPT C.P Meco D.F. (e-mal: adavla@pn.m). L. Fdman s wth Automatc Contol Depatment CINVESTAV-IPN on leave on Natonal Autonomous Unvesty of Meco Depatment of Contol Dvson of Electcal Engneeng Engneeng Faculty C.P Meco D.F. (e-mal: lfdman@unam.m). H. Ríos s wth Natonal Autonomous Unvesty of Meco Depatment of Contol Dvson of Electcal Engneeng Engneeng Faculty C.P Meco D.F. (e-mal: hectoos@comundad.unam.m). The authos gatefully acknowledge the fnancal suppot fom PAPIIT 72 CONACyT and 5855 FONCICyT SIP-IPN and PAEP-IPN. the known opeatng mode case. In othe wok consdeng that the contnuous state s known an algothm fo econstuctng the dscete state n nonlnea uncetan swtched systems s pesented n [9] based on sldng mode contol theoy. Fo the unknown opeatng mode case two state obseves fo some classes of swtched lnea systems wth unknown nputs ae desgned n [0] based on a popety of stong detectablty and usng a lnea mat nequalty (LMI) appoach. A nonlnea fnte tme obseve s poposed n [] to estmate the capacto voltage fo multcellula convetes whch have a swtched behavo. In [2] and [3] the obsevablty of hybd systems s studed whee the dscete state depends on the state taectoes. In ths wok a hgh-ode sldng-mode multobseve appoach s consdeed fo the state econstucton and opeatng mode dentfcaton poblems. The nonhomogeneous methods pesented n [4] and [5] ae appled to econstuct n fnte-tme both the state and the opeatng mode of a class of nonlnea swtched systems. The state obsevaton featues and the equvalent output necton ae eploted to econstuct n fnte tme the contnuous state and the opeatng mode. Nonhomogeneous hgh-ode sldng mode methods povde smalle tansent tmes than conventonal homogeneous methods. Man contbuton. A mult-obseve appoach based on nonhomogeneous hgh-ode sldng mode methods to contnuous state and opeatng mode econstucton fo nonlnea swtched systems s poposed. The method allows the fnte tme econstucton of both the contnuous state and the opeatng mode usng the equvalent output necton wthout equng any system tansfomaton and based only on the 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

2 2 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 measuable outputs. As fa as the authos know t s the fst pape n whch the nonhomogeneous hgh-ode sldngmode methods ae appled to the desgn of obseves. The pape has the followng stuctue. Secton II deals wth the poblem statement. Secton III pesents the obseve stuctue and the man assumptons undelyng the feasblty of the poposed pocedue. In secton IV the obsevaton eo dynamcs ae studed and a methodology fo the selecton of the obseve gans s suggested. Secton V pesents a method fo dentfyng the opeatng mode. An eample s gven n Secton VI. Some concludng emaks ae gven n Secton VII. II. PROBLEM STATEMENT Consde the nonlnea swtched system ẋ= fλ ( t) ( ) y= h( ) () whee X R n s the state vecto and l(t) { 2... q} s the so-called swtchng sgnal. The swtchng sgnal detemnes the cuent system dynamcs among the possble q opeatng modes f () f 2()... f q(). The output vecto s y Y R p. The vecto felds f ():X R n and the functons h() = [h h 2 h p] T : X Y epesent the known nomnal pat of the system dynamcs. The followng defntons ae taken fom [6]. Defnton. A hybd automaton H s a collecton H = (Q X f Int D E G R) whee Q = { 2... q} s the fnte set of dscete vaables; X = { 2... n} s the fnte set of contnuous vaables; {f () f 2()...f q()} ae vecto felds; Int = {Q X} s the set of ntal states; D(q ) = { X R n } s a doman; E = {( ) = = 2... q } s the set of edges; l(t) s the guad condton; R( ) = s the eset map (n ths case the dentty map fo ). Defnton 2. A hybd tme taectoy s a fnte o nfnte sequence of ntevals τ = { } N I such that: I = 0 = [ τ τ ] fo all < N; fn < then ethe I N = [ τ N τ N ]oi N = [ τ N τ N ); τ τ = τ + fo all. In othe wods a hybd tme taectoy s a sequence of ntevals of the eal lne whose end ponts ovelap. Fo a N hybd tme taectoy τ = { I } = 0 we defne t as the set { N} fn s fnte and { 2...} f N = and τ = τ ( τ τ ). Fnally let us ntoduce the defnton of eecuton. Defnton 3. An eecuton of a hybd automaton H s a collecton =(t q ) whee t s a hybd tme taectoy q: t Q s a map and = { : t } s a collecton of dffeentable maps : I X such that (q(0) 0 (0)) Int; fo all t [ τ τ ) ẋ ()= t fq ( () t ) and (t) X; fo all t \{N} e = (q() q( + )) E + ( τ ) G( e) and ( τ+ ) R( e ( τ )) The eecuton of a hybd automaton s a smla concept to the soluton of a contnuous dynamc systems. Zeno eecutons ae not allowed. The zeno phenomena can be descbed by an nfnte eecuton wth t <. The followng defnton of obsevablty fo a hybd automaton s adapted fom [7]. Defnton 4. Consde the system () and the vaable = (t ). Let (t ) be a taectoy of the automaton H wth a hybd tme taectoy T N and T N. Suppose that fo any taectoy (t 2 ) of H wth the same T N and T N the equalty y(t ) = y(t 2 ) a.e. n[t n t end] mples (t ) = (t 2 ) a.e. n[t n t end] then we say that = (t )sz(t N) obsevable along the taectoy (t ). The system () s Z(T N) obsevable along any taectoy (t ) and fo any possble hybd tme taectoy T N. The am of ths pape s to desgn a fnte-tme convegng obseve fo both the state and the opeatng mode of the system by means of the knowledge of the multple outputs. In ths pape we study the systems of the fom () whose hybd tme taectoes satsfy that τ τ Tδ fo all =... N and a constant paamete T d > 0 called the mnmal dwell tme. III. MULTI-OBSERVER DESIGN Consde the followng obseve stuctue: ˆ = f ( ˆ)+ G( ˆ ) σ = 2 q yˆ h ˆ = n wth the estmated state vecto ˆ R and estmated output p n n ŷ R. G( ˆ ) R wll be desgned futhe n the pape as a dstbuton mat ensung that the effects of the unknown nputs ae compensated by the dscontnuous T T T T n coecton tems σ = [ σ σ 2 σ p] R whch n tun wll be desgned usng hgh-ode sldng mode technques. The solutons of (2) ae undestood n the Flppov sense n ode to make t possble to use dscontnuous sgnals n the obseves and to concde wth the usual solutons when the ght-hand sdes ae contnuous. It s also assumed that all consdeed coecton tems allow the estence and etenson of solutons to the whole sem-as t 0. Fstly wth efeence to a scala functon q wth vecto agument defned n an open set W R n such that (2) 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

3 J. Davla et al.: State Obsevaton fo Nonlnea Swtched Systems 3 q q q q():r n R denote dq( )= = n. Select a set of outputs such that the followng matces: Φ ( ˆ ) ˆ dh ˆ dlf ( ) h ˆ ˆ = dhp( ˆ ) dl ( ) p ˆ f ˆ h p = 2 q (3) L k h wth k f Lf h ( ) f ( ) ()= () () satsfy p = n () " =...q and fo all X; hee denotes the elatve degee of the th ow of the output n the opeaton mode. Now the followng assumptons ae establshed. Φ ( ˆ ) Assumpton. The q matces n (3) ae ˆ nonsngula fo evey possble value of ˆ X. Assumpton 2. The mappngs F () ae dffeomophsms on X " =...q. Remak. Assumpton 2 mples the estence of local dffeomophsms n the doman X fo each opeaton mode. Consdeng the above assumptons the dstbuton matces G ˆ ae desgned as ˆ G( ˆ )= Φ q. ˆ = (4) s well defned " X. Accodng to Assumpton G ˆ The followng step s to analyze the obsevaton eo dynamcs between swtchngs. 3. Coecton tems desgn Assumpton 3. Thee ae known constants Γ > 0 Γ 2 > 0 and known Lpschtz functons ρ ( e y )> 0 such that " =... q the followng nequaltes ae satsfed: Lf h Lf h e ( ˆ ) ( ˆ ) < Γ ρ ( y )+ Γ 2 (5) whee ey = h h ae the output eos. The coecton tems could be calculated ntoducng the followng nonhomogeneous hgh-ode sldng mode dffeentato [4] as aulay dynamcs: y 2 ϑ = ϑ α N e sgn( e y ) ϑ ϑ α N ϑ ϑ 2= sgn ϑ 2 ϑ = α N sgn ϑ ϑ ( ϑ 2 ) (6) In ths way the coecton tems take the followng fom " =...p: σ α N ey sgn e ( y ) 2 = α N 2 2 sgn( 2 ) α N sgn( ) whee 2 = ϑ 2 ϑ = ϑ ϑ the constants α k ae chosen ecusvely and ae suffcently lage. In patcula accodng to [8] one possble choce s α 6 =. α 5 = 5. α 4 = 2 α 3 = 3 α 2 = 5 whch s enough fo the case when 5 " =...pand " =...q. The gans N ae uppe bounds fo each (5) and a poposton fo the desgn wll be establshed late n the pape. The contnuous tme econstucton consdeng the tme nteval between swtchngs wll be descbed net. IV. OBSERVATION ERROR DYNAMICS BETWEEN SWITCHINGS Consde l(t) l * = const. "t [0 t ) whee t s the tme n whch the fst swtchng occus. Theefoe the system dynamcs on the opeatng mode l * ae gven by (7) ẋ fλ* t 0 t (8) y= h( ) = [ ) Thus any of the q obseves (2) can be assocated wth the coespondng output eo ey = y ˆ y and the state eo e = ˆ n the tme nteval [0 t). Takng nto account the pevous eplanatons the followng theoem can be stated. Theoem. Consde that the obseves (2) wth the coecton tems desgned accodng to (7) ae appled to system (8) and let Assumptons 3 be satsfed. Then povded that a ae chosen popely and N ae uppe bounds fo each (5) the state estmaton eo e = ˆ conveges to zeo n fnte tme. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

4 4 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 Poof. System (8) unde Assumptons 2 can be epesented on new coodnates as whee ż = Az+ Bϕ λ* ( z) (9) y = Cz z A dag A A A A = ( 2 p ) n n 0 0 = B 0 0 B B = B p T B = [ 0 0 ] C 0 0 C C = C p C ϕ λ * = [ 0 0] Lf h λ* ( ) ( z)= p Lf h λ* p n p p n = Φ λ* z Now the obseves (2) can be epesented as: zˆ = Azˆ + Bϕ ( zˆ )+ σ = q y = Czˆ zˆ have the same stuctue as n (0). (0) () whee ϕ ẑ Defne the state obsevaton eos as ez = z ˆ z " =... q. The state obsevaton eo dynamcs take the followng fom: e z = Aez + BΨ z z q ( ˆ )+ σ = (2) = whee Ψ zˆ z ϕ zˆ ϕ λ * z. Now f t s possble to fnd appopate coecton tems whch can stee the vectos e z to zeo then the equalty ẑ = zwll be satsfed when = l *. Nevetheless t s not desable to desgn the coecton tems n the coodnates ẑ but athe n the coodnates ˆ. Theefoe etunng to the ognal coodnates and defnng the followng output eo vecto: ε ε e y = = (3) ε ( ) ey The state obsevaton eo dynamcs (2) tun nto output obsevaton eo dynamcs block fom as follows: ε ε σ = 2+ ε ε σ 2= 3+ 2 ε = Ψ ( Φ ( ) Φ λ * ( ) )+ σ (4) Notce that the dynamc stuctues (4) ae vey smla to (6). Thus f a vaable change s ealzed n the stuctue (6) t s possble to obtan the followng sldng mode dffeentato foms: ε ε α N ε sgn ε ε 2= ε 3 2 = 2 2N 2 sgn 2 ( ) α ε ε ε ε ε = Ψ () α N sgn ε ε (5) Now f Assumpton 3 s satsfed and f the paametes α k ae chosen ecusvely accodng to the hgh-ode sldng mode dffeentato popetes descbed n [8] the followng equalty s satsfed only when = l * n fnte tme: ε ε ε 00 0 p λ* λ* 2 λ* [ ] =. Then the condton ε λ* 0 t [ t* t ) mples that ε ε λ* λ* λ* [ ] t [ 0 t * ]. To pove ths assume that the condton ε λ* 0 s satsfed n a nonzeo tme nteval. Ths condton mples that ε λ* 0 n the same tme nteval. Thus fom the fst ow of (5) t s obtaned that ε λ* 2 0. Then snce ε λ* 2 0 and ε λ* 0 fom the second ow of (5) t s obtaned that ε λ* 3 0. If the same pocedue s teated the followng epessons ae obtaned ε λ* 0 " =...p. Gven Assumpton 3 and takng nto account that the gans N ae uppe bounds fo (5) the last ow of (5) defnes the followng dffeental ncluson ε λ λ * [ Nλ Nλ ] * * * 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

5 J. Davla et al.: State Obsevaton fo Nonlnea Swtched Systems 5 [ ] α ε ε N sgn whee Ψ λ* () N λ* N λ* " =...p. Theefoe the dynamcs (4) convege to zeo afte a fnte tme.e. ε λ* 0 t [ t* t) and accodng to Assumpton 2 t s ensued that the state estmaton eo e λ = λ * * also conveges to zeo n fnte tme. Notce that t s always possble to select the gans N to be suffcently lage such that each t * < t " =...p. Q.E.D. Theoem states that when the actve dynamc s the l * - th one accodng to (8) then the obsevaton eo e λ* of the l * - th obseve conveges to zeo n a fnte tme. Remak 2. Notce that n ode to desgn the obseves only the calculaton of the nvese matces Φ ( ) s necessay and not of the nvese tansfomaton Φ ( z). Now a futhe esult consstng of an addtonal assumpton nvolvng all the possble system dynamcs f () f 2()...f q() guaanteeng that all obseves povde the coect estmate of the contnuous state espectvely of the cuent value of the opeatng mode s gven. Assumpton 4. Let the functons f () f k() and h() be such that fo each k = 2... q k wth =...p the followng equalty s satsfed: Lf h ι k ( ) 0 = 2 Lf Lf h ι k ι 0 (6) whee fι k = fι fk. Theefoe f Assumpton 4 s satsfed then all obseves povde the coect estmate of the contnuous state espectvely of the cuent value of the opeatng mode. Notce that the poposed method does not eque that the swtchng paamete satsfes the matchng condton (see [9]). Ths mpovement s a esult of the mult-estmato appoach. To pove ths consde the dynamcs (5) n tme nstants befoe and afte the swtchng tme t.e. ε ε α N ε sgn ε ε 2 = 2 2= ε 3 α 2N ε 2 ε ε sgn 2 ε = Ψ () α N sgn ε ε whee ε ε ε l ι l t k l t Ψ l Ψ t Ψ = = + + ι k t ( ε ) wth the paametes α α α l = ι l k l and the gans N = Nι Nk fo k = 2... q and k "l =...n. In ode to peseve the estmaton n the swtchng tmes.e. so that the equalty e D 0 s kept t s suffcent that Assumpton 4 be satsfed. In ths way the swtchngs do not affect the obsevablty mappngs. It s mpotant to emak that afte each opeatng mode swtches the obseves could loose the coect estmaton (due to the dscontnutes n the hghe ode output devatves) f Assumpton 4 s not satsfed. Howeve afte a tansent whch can be made abtaly small by takng suffcently lage values of the coecton tems paametes the coect value of the state s ecoveed. The appopate selecton of the dffeentato s gans ensue the convegence of the dffeence ˆ to zeo n a tme smalle than T d. Ths means that unde the pesence of umps n the contnuous state the poposed algothm ensues the estmaton of afte a fnte-tme tansent smalle than T d. 4. Gans Adaptaton fo the coecton tems Theoem solves the contnuous state obsevablty poblem wheneve the gans N ae chosen appopately but t does not eplan how to choose the above mentoned gans. Based on [5] that show how to select adaptable gans N fo the hgh-ode sldng mode dffeentato the followng poposton s stated to choose the gans N lke tme functons and s adaptable wth espect to the output eo. Poposton. Consde the dynamcs (5) and that 0 ε ( 0) ε whee ε 0 ae known constants. To adapt the gans N (t) of evey coecton tem n (7) the followng algothm s consdeed:. Set N ()= t N 0 fo 0 t t * wth t * the tme nstants n whch evey dynamc block of (5) conveges to zeo wth N 0 a suffcently lage constant. To detect that evey dynamc block has conveged to zeo t s suffcent to vefy that the followng nequalty s satsfed [5] ey t N t () γ 0 0δ 0 γ tδ (7) whee γ 0 and γ t ae postve constants and d > 0s the sample tme. 2. Set N t λ ρ e λ 2 fo all t > t * wth ()= + y λ λ p > 0 2> 0 = (8) Then the convegence of ey ()to t zeo n fnte tme s ensued. Moeove the logathmc devatves Ṅ() t N() t ae unfomly bounded fo evey dynamc block. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

6 6 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 Poof. Fo the poof each step of the algothm s consdeed.. It s necessay to show that the gans N 0 ae suffcently lage fo the convegence of the dynamcs (5) n tme ntevals [ 0 t *]. Then takng nto account that = l * and Assumpton 3 t s obtaned that ε Γ ρ ( e )+ Γ λ* λ* λ* λ* yλ * λ* 2 ε + α λ* λ* N sg λ* 0 n ε ε λ* λ* λ* λ* + Γ ρ + Γ 2 + α N 0sgn() λ* λ* λ* λ* λ* λ* λ* λ* (9) (20) It s well known [8] that t s possble to select the gans N 0 " =... p to be suffcently lage to povde any convegence tme n ths case suffcently lage such that dynamcs (5) convege wthn [ 0 t *]. Then N 0 can be chosen n the followng fom: + N 0> Γ * * + Γ λ λ λ* 2 ρ (2) Also accodng to [5] t s possble to detect that dynamcs (5) have conveged vefyng the nequalty (7) " =...p. 2. Once dynamcs (5) have convegence " =... p the dentty ε λ* 0 " =...ps tue and accodng to Theoem t s ensued that the state estmaton eo e λ = λ * * s equal to zeo.e. ˆ λ *. In ths way t s obtaned that ε λ* α λ* N 0 sgn() (22) λ* λ* λ* whee Nλ* ()= t λ ρ e λ λ y λ * * ( λ* )+ λ* 2 wth λ > 0 λ > 0 λ* λ* " =...p; makes sue that N t 2 λ* ()s also an uppe bound of Ψ λ* ()mantanng the convegence of the dynamcs (5) fom t * fowad fo all =...p. Now t s necessay to show that the logathmc devatve of the gan Nλ* ()s t unfomly bounded. The gan devatve of Nλ* ()s t gven by N ()= t λ ρ ( e λ p λ* )+ 2 = λ* λ* λ* y λ* By Assumpton 3 the functons ρ λ* () ae known Lpschtz functons. Theefoe ρ λ* () " =... p est and ae bounded by popely constants ρ + L λ*. Then computng the logathmc devatve the followng s obtaned: Ṅ N λ* λ* L () t λ ρ λ () t + + λ ρ + λ + λ* λ* λ* 2 λ* λ* λ* 2 = p Ṅλ* () t It s easy to see that s unfomly bounded by a Nλ* () t constant fo all =...p. Q.E.D. It s natual to estmate the constants γ 0 and γ t though smulaton. Fom (7) one can conclude that fo any small N the accuacy of the eo wll be bette. Howeve f the ntal condton s vey lage N has to be lage. Then when the taectoes of the system ae close to the ogn the gan N must be small. Theefoe Poposton s a good opton to use a vaable gan N and n ths way mpove the accuacy of the eo. Now the followng step to solve the poposed obsevablty poblem s to establsh a method fo econstuctng the opeatng mode to complete the obseve desgn. V. SWITCHING SIGNAL IDENTIFICATION In ths secton the method fo econstuctng the swtchng sgnal s outlned. In steady state all entes of vectos e and e ae dentcally zeo whle the tems ė =... p ae dectly affected by the dscontnuous coecton tems.e. ae zeo n the aveage sense. Thus we ae n poston to eplot one of the man featues of sldng mode obseves the equvalent output necton pncple. The epesson fo ė =...p s ė = f ( ˆ )+ G ( ˆ ) σ f ( ) λ * (23) Statng fom the moment at whch the eact state econstucton s acheved (23) smplfes as ė = f ( ˆ )+ G( ˆ ) σ fλ* ( )= 0. Then the coecton tems wll take the value of the equvalent output necton σ eq.e. G( ˆ) σ = f f ˆ eq λ* whch deves fom mposng the zeong of ė = 0 (equvalent contol method). The above equaton mples that among the q obseves (2) thee s only one wth all the assocated equvalent output nectons beng dentcally zeo accodng to the followng condton of econstuctablty of the swtchng sgnal: Condton. The swtchng sgnal l(t) can be econstucted by means of equvalent output necton accodng to = G ˆ σ eq 0 λ* (24) G ˆ σ eq 0 λ* (25) povded that the followng set: M = { R n f λ *( )= f ( ) λ* }. s a dscete set. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

7 J. Davla et al.: State Obsevaton fo Nonlnea Swtched Systems 7 Snce has dscontnuous tems the equvalence σ = σ eq holds only n the Flppov sense so that the ecovey of the equvalent output necton σ eq fom the dscontnuous output necton eques fltaton. Let us defne the followng equvalent output necton estmatos of σ eq : τσ ˆ = σ σˆ eq (26) eq whee t ae desgned accodng to d <<t << wth τ = δ 2 whee d s the sample tme. The contnuous sgnals ˆσ eq must be analyzed n ode to etact the nfomaton about the cuent value of the swtchng sgnal. Theoetcally a smple theshold would be enough. Indeed t was shown that one and only one of the sgnals ˆσ eq becomes dentcally zeo and stays n ths value untl l(t) changes value. Howeve all sgnals ˆσ eq can occasonally coss the zeo value. Theefoe a logc should be mplemented that looks fo the sgnal beng close to zeo ove a sutable ecedng-hozon tme nteval of fnte length. Ths can be done easly va the numecal method descbed below. Let T s be a small samplng tme. The followng nonnegatve quanttes ae evaluated onlne at any samplng nstants t = kt s k = ν µ eq s σ 2 q (27) = 0 The value of fo whch m s mnmum s evaluated and ths value wll be the estmated opeatng mode ˆλ t µ VI. SIMULATION EXAMPLE ()= agmn. Consde the sth-ode nonlnea system composed by the nteconnecton of a Chua ccut and a Rössle oscllato: αλ( t) cλ( t) + αλ( t) 2 αλ( t) βλ( t) 2 ẋ = a t λ 5 bλ( t) + 6( 4 dλ( t) ) (28) whee l(t) { 2 3} and a l(t) = { } b l(t) = { } c l(t) = { } d l(t) = {6 2 4} a l(t) = {0 8 2} and b l(t) = {6 6 6} ae constant paametes. System (28) epesents a swtched veson of the chaotc Chua-Rössle dynamcs. Consde the measuable system output y = [ 3 5] T. The matces Φ ˆ defned n ˆ (3) ae Φ ( ˆ ) ˆ β β β β = a a ( a ) It s easy to calculate that Φ ˆ s nonsngula " = 23. ˆ Theefoe Assumptons and 2 ae satsfed. Notce that fo ths patcula eample all dffeomophsms ae global. In ths way matces G( ˆ )ae desgned accodng to (4). Note that fo evey pa of systems (f f k) wth k = 2 3 the equaltes Lf h ι k = 0 and Lf Lf h ι k ι = 0 hold. Then by consequence Assumpton 4 s fulflled and all the desgned obseves wll povde the coect estmaton of the contnuous state. The obseves ae desgned as follows: ẋˆ Table I. Coecton tems paametes. Paamete = = α α.5.5 α 2.. N N (t) 0 e y e y α cˆ + α ˆ α ˆ ˆ ˆ 2 + ˆ 3 β ˆ 2 = ˆ 5 ˆ 6 + ˆ ˆ 4 + aˆ 5 b + ˆ 6( ˆ 4 d) σ G ˆ σ 2 whee the coecton tems ae calculated usng the followng aulay dynamcs 2 3 N e 3 y sgn e 2 y ϑ ϑ α = 2 ϑ = ϑ α N ϑ ϑ 2 sgn ϑ ϑ ϑ = α N sgn ϑ ϑ wth ey = ˆ 3 3 and ey = ˆ " = 23. The paamete values of the coectons tems ae shown n Table I. (The same paametes ae used fo the thee obseves takng nto account Poposton ). 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

8 8 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 Fg.. Actual and econstucted opeatng mode. Top: Real dscete state and ts econstucton (Left top: Tansent eo econstucton). Bottom: A zoom acoss the fst swtchng. The unknown swtchng sgnal l(t) s selected fo smulaton pupose only as shown n Fg.. The system ntal condtons ae set as (0) = [ ] T. The obseve ntal condtons ae taken as zeo. Smulatons wee done n the MATLAB Smulnk envonment wth the Eule dscetzaton method and samplng tme d = sec. The eal and estmated contnuous taectoes of the system ae depcted n Fg. 2 only fo the obseve to llustate the behavo. Howeve all the obseves ae capable to estmatng the contnuous state coectly. The convegence to zeo of the contnuous state estmaton eos s obtaned fo all obseves. As shown n Fg. 3 because Assumpton 4 s satsfed all the obseves estmate the contnuous state n a coect way. Moeove even n the pesence of swtchngs the estmaton eo stll equal to zeo. On the othe hand n Fg. 4 t s shown the behavo of the desgned coectons tems gans usng Poposton fo the Obseve. It s easy to see that the gans N (t) and N 2(t) dmnsh when estmaton eo has conveged and they do not suffe any change because the eo emans n zeo. The estmated equvalent output nectons can be seen n Fg. 5. Notce that afte the tansent the estmated equvalent output nectons ae dentcally zeo only when the coespondng mode s actve e.g. n the tme nteval t [020] the opeatng mode l = s actve then both equvalent output nectons and 2 ae equal to zeo n ths opeatng mode what does not happen fo the othes equvalent output nectons n the same nteval of tme (see ght column n Fg. 5). Thus the opeatng mode can be econstucted by detectng whch estmated equvalent output necton s dentcally zeo. The logc suggested n (27) s appled and the coespondng esults ae shown n Fg.. The uppe plot shows the actual and econstucted opeatng mode. The zoomed plot shows that the duaton of the dentfcaton tansent followng a swtched opeatng mode s appomately 0.05 seconds. Ths length could be abtaly educed by takng dffeent values fo the paamete t n the σ eq estmato (26). Due to ths paamete modfes the estmaton velocty of the equvalent output necton t s possble to mpove the delay econstucton of the dscete state changng ths one. VII. CONCLUSIONS In ths atcle a method based on the nonhomogeneous hgh-ode sldng mode appoach fo the fnte tme state 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

9 J. Davla et al.: State Obsevaton fo Nonlnea Swtched Systems 9 Fg. 2. Contnuous state taectoes. Left column: Contnuous taectoes of the system and the taectoes estmated by the Obseve. Rght column: A zoom of the eal and estmated taectoes. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

10 0 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 Fg. 3. Estmaton eo convegence. Left column: Eo convegence fo evey obseve. Rght column: A zoom of the eo convegence. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

11 J. Davla et al.: State Obsevaton fo Nonlnea Swtched Systems Fg. 4. Coecton tems gans of the Obseve. Top left: Behavo of the coecton tems gans when the state estmaton eo has conveged. Fg. 5. Estmated equvalent output nectons. Left column: Full equvalent output nectons fo evey obseve. Rght column: A zoom of the equvalent output nectons. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

12 2 Asan Jounal of Contol Vol. 5 No. pp. 3 Januay 203 Fg. 5. Contnued. obsevaton of the contnuous state and opeatng mode fo a class of nonlnea swtched systems has been poposed. The appoach s able to econstuct both the contnuous state and opeatng mode of a swtched system based only on ts measuable outputs and though the use of the featues of the equvalent output necton. Geometc stuctual estctons on the vecto felds of the swtched system that eque the avalablty of all ts modes ae gven to guaantee the fnte tme eact state econstucton. REFERENCES. Bancky M. Multple Lyapunov functons and othe analyss tools fo swtched and hybd systems IEEE Tans. Autom. Contol Vol. 43 No. 8 pp (998). 2. Lbezon D. Swtchng n Systems and Contol. Systems and Contol: Foundatons and Applcatons Bkhause Boston (2003). 3. Lan J. and J. Zhao Output feedback vaable stuctue contol fo a class of uncetan swtched systems Asan J. Contol Vol. No. pp (2009). 4. Psano A. and E. Usa Contact foce estmaton and egulaton n actve pantogaphs: an algebac obsevablty appoach Asan J. Contol Vol. 3 No. 6 pp (20). 5. Shtessel Y. I. Shkolnkov and M. Bown An asymptotc second-ode smooth sldng mode contol Asan J. Contol Vol. 5 No. 4 pp (2003). 6. Davla J. L. Fdman and A. Levant Second-ode sldng-mode obseve fo mechancal systems IEEE Tans. Autom. Contol. Vol. 50 pp Novembe (2005). 7. Batoln G. and E. Punta Reduced-ode obseve n the sldng-mode contol of nonlnea nonaffne systems IEEE Tans. Autom. Contol Vol. 55 No. 0 pp (200). 8. Alessand A. and P. Coletta Desgn of obseves fo swtched dscete-tme lnea systems n Poc Amecan Contol Confeence (Denve CO USA) pp (2003). 9. Oan N. A. Psano M. Fanceschell A. Gua and E. Usa Robust econstucton of the dscete state fo a class of nonlnea uncetan swtched systems Nonlnea Anal.-Hybd Syst. Vol. 5 No. 2 pp (20). 0. Beaano F. and A. Psano Swtched obseves fo swtched lnea systems wth unknown nputs IEEE Tans. Autom. Contol Vol. 56 No. 3 pp (20).. Dema M. K. Busawon K.Benmansou and A. Maouf Hgh-ode sldng mode contol of a dc 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety

13 J. Davla et al.: State Obsevaton fo Nonlnea Swtched Systems 3 moto dve va a swtched contolled mult-cellula convete Int. J. Syst. Sc. Vol. 42 No. pp (20). 2. Bempoad A. G. Fea-Tecate and M. Moa Obsevablty and contollablty of pecewse affne and hybd systems IEEE Tans. Autom. Contol Vol. 45 No. 0 pp (2000). 3. Babot J. H. Saadaou M. Dema and N. Manamann Nonlnea obseve fo autonomous swtchng systems wth umps Nonlnea Anal.-Hybd Syst. Vol. No. 4 pp (2007). 4. Levant A. Eact dffeentaton of sgnals wth unbounded hghe devatves n Poc. 45th IEEE Conf.Decson Contol (San Dego CA USA) pp (2006). 5. Angulo M. L. Fdman and A. Levant Robust eact fnte-tme output based contol usng hgh-ode sldng modes Int. J. Syst. Sc. Vol. 42 No (20). 6. Lygeos J. K. Johansson S. Smc J. Zhang and S. Sasty Dynamcal popetes of hybd automata IEEE Tans. Autom. Contol Vol. 48 No. pp. 2 8 (2003). 7. Kang W. J.-P. Babot and L. Xu On the obsevablty of nonlnea and swtched systems n Emegent Poblems n Nonlnea Systems and Contol Vol. 393 (B. Ghosh C. Matn and Y. Zhou eds.) pp Spnge Beln/Hedelbeg (2009). 8. Levant A. Hgh-ode sldng modes: dffeentaton and output-feedback contol Int. J. Contol Vol. 76 No. 9 0 pp (2003). 9. Yu L. J. Babot D. Boutat and D. Benmezouk Obsevablty foms fo swtched systems wth zeno phenomenon o hgh swtchng fequency IEEE Tans. Autom. Contol Vol. 56 No. 2 pp (20). Joge Davla was bon n Meco Cty Meco n 977. He eceved B.Sc. and M.Sc. degees fom Natonal Autonomous Unvesty of Meco (UNAM) n 2000 and 2004 espectvely. In 2008 he eceved Ph.D. degee and was awaded the Alfonso Caso medal fo the best Gaduate student fom UNAM. He was apponted to Postdoctoal Reseach poston n Cento de Investgacon y Estudos Avanzados (CINVESTAV) dung He cuently holds Reseach Pofesso poston at Secton of Gaduate Studes and Reseach of Natonal Polytechnc Insttute (ESIME UPT). Hs pofessonal actvtes ae concentated n the felds of obsevaton of lnea systems wth unknown nputs obseves fo swtchng systems nonlnea obsevaton theoy obust contol desgn hgh-ode sldng-mode contol and the applcaton to the aeonautc and aeospace technologes. Hécto Ríos was bon n Meco D.F. n 984. He eceved B.Sc. and M.Sc. degees fom Natonal Autonomous Unvesty of Meco (UNAM) n 2008 and 200 espectvely. Snce 200 he has been Ph.D. student of UNAM. Hs pofessonal actvtes have been concentated n the obsevaton of lnea and nonlnea systems wth unknown nputs the obsevaton of swtched systems the fault detecton solaton and dentfcaton poblem and hgh-ode sldng-mode contol and ts applcatons. L. Fdman eceved M.S. degee n Mathematcs fom Kubyshev State Unvesty Samaa Russa n 976 Ph.D. degee n Appled Mathematcs fom Insttute of Contol Scence Moscow Russa n 988 and D.Sc. degee n contol scence fom Moscow State Unvesty of Mathematcs and Electoncs Moscow Russa n 998. Fom 976 to 999 he was wth Depatment of Mathematcs Samaa State Achtectue and Cvl Engneeng Academy. Fom 2000 to 2002 he was wth Depatment of Postgaduate Study and Investgatons at Chhuahua Insttute of Technology Chhuahua Meco. In 2002 he oned Depatment of Contol Dvson of Electcal Engneeng of Engneeng Faculty Natonal Autonomous Unvesty of Meco (UNAM) Méco. He has publshed ove 250 techncal papes. Hs man eseach nteests ae vaable stuctue systems. D. Fdman s an Assocate Edto of the Intenatonal Jounal of System Scence Jounal of Fankln Insttute and Confeence Edtoal Boad of IEEE Contol Systems Socety Membe of TC on Vaable Stuctue Systems and Sldng mode contol of IEEE Contol Systems Socety. He has woked as Invted Pofesso n 4 unvestes and scentfc centes of Fance Gemany Italy and Isael. 202 John Wley and Sons Asa Pte Ltd and Chnese Automatc Contol Socety