Observer Design for Takagi-Sugeno Descriptor System with Lipschitz Constraints

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1 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl Obseve Desgn fo akag-sugeno Descpto System wth Lpschtz Constants Klan Ilhem,Jab Dalel, Bel Hadj Al Saloua and Abdelkm Mohamed Naceu MACS Unt - Natonal Engneeng School of Gabès unsa,unvesty of Gabès klan.lhem@gmal.com Unvesty of Rems Champagne-Adenne CReSIC EA384, Mouln de la Housse BP 39, 5687 Rems Cedex, Fance. dalel.jab@yahoo.f ABSRAC hs pape nvestgates the desgn poblem of obseves fo nonlnea descpto systems descbed by akag-sugeno (S) system; Dependng on the avalable knowledge on the pemse vaables two cases ae consdeed. Fst a S descpto system wth measuables pemses vaables ae poposed. Second, an obseve desgn whch satsfyng the Lpschtz condton s poposed when the pemses vaables ae unmeasuables. he convegence of the state estmaton eo s studed usng the Lyapunov theoy and the stablty condtons ae gven n tems of Lnea Matx Inequaltes (LMIs). Examples ae ncluded to llustate those methods. KEYWORDS Descpto S system, Lpschtz Condton, Unmeasuable Pemse Vaable & Obseve Desgn.. INRODUCION Snce moe than two decades ago, akag Sugeno ( S) fuzzy models have attacted wde attenton fom scentsts and engnees, essentally because the well-known fuzzy models can effectvely appoxmate a wde class of nonlnea systems. Relaxed suffcent condtons fo fuzzy contolles and fuzzy obseves ae poposed n va a multple Lyapunov functon appoach []. System modelng by descpto method has an mpotant use n the lteatue snce t epesents many class of non lnea system. he enhancement of the modelng ablty s due to the stuctue of the dynamc equaton whch encompasses not only dynamc equatons, but also algebac elatons [].Snce both S and descpto fomalsm ae attactve n the feld of modelng, the S epesentaton has been genealzed to descpto systems. he stablty and the desgn of state-feedback contolles fo -S descptos systems ae chaactezed va LMI n [3], [4]. System modelng by descpto method has an mpotant use n the lteatue snce t epesents many class of non lnea system. In othe hand, many poblems n decson makng, contol and montong eque a state estmaton based on a dynamc system model. A genec method fo the obseve desgn, vald fo DOI :.5/jcs.. 3

2 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl all types of nonlnea systems, has not yet been developed. hs poblem eceved consdeable attenton n the last thee decades; t s of geat mpotance n theoy and pactce, snce thee ae many stuatons whee pemse vaable ae naccessble. In ths pape, we popose to desgn a state obseve to estmate exactly the descpto s states. Fo a S fuzzy model, well-establshed methods and algothms can be used to desgn obseves that estmate measuable states. Seveal types of such obseves have been developed fo S fuzzy systems, among whch: fuzzy hau Luenbege obseves educed-ode obseves and sldngmode obseves. In geneal, the desgn methods fo obseves also lead to an LMI feasblty poblem [5], [6], [7], [8]. Fo the unmeasuables pemses vaables many seaches studed the obseve desgn fo non lnea descpto system wth lpschtz constant [9] [][][].he fuzzy S systems ae studed n [3]. he man objectve of ths pape s to develop an obseve desgn fo S descpto system wth lpschtz constant. Unde some suffcent condtons, the desgn of the obseve s educed to the detemnaton of a matx. he choce of ths paamete s pefomed by solvng stct LMIs (lnea matx nequaltes). he outlne of the pape s as follows. Fst, the class of studed systems and obseve ae gven. Second poblem fomulaton fo obseve wth measuable vaables s dealt. hd the Lpschtz obseve wll be studed. Fnally we pesent an example to llustate the effectveness of the poposed method. Concludng emaks fnsh ths pape.. SYSEM DESCRIPION AND PRELIMINARIES.. Descpto System Let us consde the class of non lnea descpto system whch s defned as: E ( x ) x& = A( x ) x + B( x ) u y = C ( x ) x n q Whee x( t ) R, u R and y m R n n the output vectos; A( x ), E ( x ) R n m, B( x ) R and C ( x ) q n R matces functons. Fo smplcty, we should always consde that ( ) n x R. hen the S descpto system s gven as: l ( ) & = ( )( + ) k k k = = y = Cx v z t E x t h z t A x t B u t Whee z epesent the pemse vaable; h ( z ) and k ( ) () epesent espectvely the state, the contol nput and ae non lnea E x t s egula fo each () v z t epesent espectvely the ght and the left actvatng functon; and l s espectvely the ght and the left numbe of fuzzy ules ; A, B and C ae constant matces; Whee w g ( z ) h ( z ) =, w ( z ) = Fj ( z j ) (3) = h z t = ( ) Ek 4

3 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl h ( z ) has also the same chaactestc as k ( ) h ( z ) and h ( z) fo,..., v z t and t satsfes: = > = (4) he passage fom the nonlnea system to the S descpto system s obtaned by the secto nonlneaty appoach. In ths case, the S descpto matches exactly the nonlnea model n a compact set of the state vaables... Descpto obseve In geneal a S obseve s defned by the nteconnecton between many locals Luenbege obseves [4], [5]. It s wtten as: ( ˆ) xˆ & = hj z t A xˆ + Bu L y yˆ = yˆ = Cxˆ Whee ˆx ( t ) and ẑ t epesent espectvely the estmate state and the estmate pemse vaable; L ae the obseve gan matces. In ths study, the poposed obseve s n the descpto fom. Usng the geneal S descpto fom, t s possble to suppose a nonlnea obseve based on ths fom: l ( ) & ˆ v z t E xˆ = h z t A xˆ + B u L y yˆ k k j k = j= yˆ = C xˆ (6) hee ae two cases to defne wtch elated to the accessblty of the pemses vaables. In the z t ae eal tme avalable ẑ ( t ) = z ( t ) and thus so ae fst tme we assume that the vaables the weghtng functons h ( zˆ ) = h ( z ( t )) and vk ( zˆ ) vk ( z ( t )) (5) =.But, n many pactcal stuatons, these pemse vaables depend on the state vaables that ae not always accessble. hen two cases ae consdeed. - measuables pemses vaables - unmeasuables pemses vaables. 3. OBSERVER DESIGN 3.. Measuables Pemses Vaables hs secton s devoted to the state estmaton. In fact a descpto fom of obseve wll be consdeed. he followng theoem pesents the man esult. heoem: he convegence of the eo estmaton between the system () and the obseve (6) s ensued f thee exst: P = P > and P 3, L, Y3 fo =,..., and k =,..., l such that: A P3 C Y3 + P3 A Y3 C (*) P E P P A Y C E P P E k k 3 3 k < (7) As usual, a sta (*) ndcates a tanspose quantty n a symmetc matx. 5

4 Poof: Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl Away to obtan suffcent convegence condtons fo descpto fuzzy obseve s to consde the eo estmaton as: ˆ e t = x t x t (8) Its tme devatve s gven by: ˆ e& t = x& t x & t (9) Accodng to equaton () equaton (6) becomes: l ( ) ˆ = ( )(( ˆ + ) + ( ˆ)) v z t E x & t h z t A x t B u t L Cx t Cx t () k k k = = he dffeence between ths last equatons and equaton () s gven by: l ( ) = ( )( ) v z t E e& h z t A L C e () k k k = = Let us consde the augmented eo vecto s gven by: e t = e t e& t () Equatons () can be wtten: = A e Ee& t (3) k Wth I I E = Ak and = A LC E (4) k In ode to fnd an asymptotc convegence eo, we consde the Lyapunov canddate functon ( ),, n n V e t = e t EPe t P = P P R wth: P P EP = P E >, P =, P = P > P = P3 P and (5) 4 he negatvty of the Lyapunov functon s assumed by: V& ( e ) = e& EPe + e { EP e& < (6) P E In othes wods, obvously, wth (4), the condton (6) becomes: ( + ) < e t A P P A e t k k hen fo =,..., and fo k =,..., l we obtan: (7) ( A L C ) P P P3 I * * P I E 3 P + < 4 P A k 4 LC E k (8) A P C L P P A P L C (*) P Ek P3 + P4 A P4 L C Ek P4 P4 Ek < (9) 6

5 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl In ode to solve stct LMIs, we suppose that P4 = P3, then A P3 C L P3 + P3 A P3 LC (*) < P Ek P3 + P3 A P3 L C Ek P3 P3 Ek () o have tems of Lnea Matx Inequaltes we consde a bjectf changement Y3 = P3 L : A P3 C Y3 + P3 A Y3 C (*) P E P P A Y C E P P E k k 3 3 k < () 3. Unmeasuables Pemses Vaables hs pat addesses the ssues of obseve desgn fo a class of descpto systems wth Lpschtz constant.he geneal fom fo the S descpto system s n the fom of equaton ().o avod to have a complex equaton, we wll suppose that all the Ek = E fo k =,..., l.hen we have a new fom of S descpto system gven as below: = ( ) Ex& = h z t A x + B u y = Cx hen the obseve coesponds to ths extenson of S descpto system wll be defned as: j= ( ˆ) Exˆ & = h ˆ ˆ j z t A x t + Bu t L y t y yˆ = C xˆ In ths secton, the pupose s to suggest a method fo the desgn of obseve fo S descpto system. he followng hypothecs and lemma wll be used n the development. Hypothec : he actvatng functon s lpschtz: () (3) ( ) ( ˆ ) γ ˆ h x t h x t x t x t ( ) ( ˆ ) ˆ ˆ h x t x t h x t x t m x t x t Whee γ and m postves scalas Lpschtz constants. Hypothec : he contol nput u (t) s bounded: β β u t, > Lemma Fo all matces X and Y, λ a postve scala, the followng popety s gven as: X Y + Y X λ X X + λ Y Y, λ> he followng theoem shows the most mpotant esult. (4) (5) (6) (7) 7

6 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl heoem: he convegence of the eo estmaton between the system () and the obseve n n (3) tend asymptotc to zeo s ensued f thee exst: P R n n and Q R symmetc postve n q defned, the matx K R and postves scalas λ, λ such that: A P + PA C K K C < Q (8) Q + λm I PA PB nγ I A P λi < B P λi n γ I λ I (9) γ β λ (3) EP = P E > he gan of the obseve s gven by: L = P K Poof: he basc dea s to concept a descpto obseve whch vefes Lpschtz condton by ntoducng some change n the system. Hence,the matces A B and C wll be wtten as :, (3) (3) A = A, A = A A = B = B, B = B B = C = C, C = C C = he system (3) wll be wtten as: y = C x = ( ) Ex& t = A x t + B u t + h x t A x t + B u t hen the obseve equaton wll be changed as: = ( ) ( ) Exˆ & t = A xˆ t + B u t h xˆ t A xˆ t + B u t + L y t yˆ t yˆ = C xˆ he estmaton eo s gven as: (33) (34) (35) (36) (37) = ˆ e t x t x t Consequently, the augmented state estmaton eo obeys to the followng nonlnea system: 8 (38)

7 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl ( )( ) ( ) ( ) ( ( ) ( )) Ee& t = h xˆ t A L C e t + A h x t x h xˆ t xˆ + B h x t h xˆ t u = = (39) o show the effeteness of hypothecs we suppose as below: ( ) ( ˆ ) ˆ t h x t x t h x t x( t ) ( ) ( ˆ t h x t h x( t )) A LC δ = = Φ = hen the dynamc eo wll be defned as: (4) = ( ) Φ + δ + Ee& t h xˆ t t e t A t B t u = = (4) In ode to fnd an asymptotc convegence eo, we consde the quadatc Lyapunov functon ( ),, n n V e t = e t EPe t P = P P R wth: EP = P E > hen the tme devatve of the Lyapunov functon wll be wtten as: ( ) = + { V& e t e& t EPe t e t EPe& t hen P E ( ) = δ + δ + + = + h ( ˆ x ) e Φ Pe + e PΦe V& e t t A Pe t e t PA t t B Pe t e t PB t he devatve of the Lyapunov functon s composed of quadatc tems n e and of tems cossed n e( t ), γ and.in ode to expess V e( t ) n a quadatc fom n poceed as follows: δ ψ ( m e ( n β e γ e& k (4) (43) e t.we (44) By applyng the lemma we lead to the followng nequaltes: t A Pe t + e t PA t t t + e t PA A Pe λ m e e + λ e PA A Pe δ δ λ δ δ λ ( t ) B Pe e PB λ λ e( t ) PB B Pe λ n β e e( t ) + λ e PB B Pe + + he devatve of the Lyapunov functon became: (45) (46) 9

8 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl ( ) = δ + δ + + = + h ( ˆ x ) e Φ Pe + e PΦe V& e t t A Pe t e t PA t t B Pe t e t PB t ( ˆ )( Φ + Φ ) + ( λ + λ β ) h x t P P m n I V& ( e ) e e t = + λ PA A P + λ PB B P (47) (48) he stablty of ths last equaton s assued, fo =..., : ( ( ) λ ) λ β λ λ e t h xˆ t Φ P + PΦ + m + n I + PA A P + PB B P e t < hs leads to the followng condtons: A L C P + P A L C < Q Q + λ m + λ n β I + λ PA A P + λ PB B P < hen we suppose vaable changement K = PL and the complement Schu. hen t wll be defned as below: A P + PA C K KC < Q (5) ( λ λ β ) Q + m + n I PA PB A P λ I <, λ > et λ > B P λi (53) o have an effectve choce we can take the nput as a vaable to be detemned whch one wll call ρ. By usng the complement Schu, nequalty (5) wll be wtten: Q + λm I PA PB nλ ρi A P λi < λ > et λ > B P λ n λ ρi λ I,, (54) (49) (5) (5) he pesence of the poduct λρ leads to a nonlnea nequalty. o ewte t n fom LMI, we pose γ = λρ : A P + PA C K K C < Q Q + λm I PA PB nγ I A P λ I < B P λ n γ I λ I (56) (55)

9 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl γ Knowng γ and λ we can deduce the value ρ = λ 4. DESIGN EXAMPLES 4.. Example hs secton s dedcated to llustate the effcency of the poposed appoach. We consde the followng example wtch llustate the fst theoem: ( ) & = ( )( + ) v z t E x t h z t A x t B u t k k k = = y = Cx Wth x =, A 3 = -, A E = and E =.. And the actvatng functons: ( ( x )) h = tanh /, h h - =, B =, B = and v cos( x ) cos( x ) =, C = [ ], =, v = v. Accodng to the gven pocedue, we desgn the descpto obseve based on theoem va the Matlab LMI toolbox. hen we obtan P : and P 3 : P = P = , 3 he coespondng obseve gan matces ae: L =.433, L.555 = Eo of x me Fgue. Eo estmaton fo x

10 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl We conclude fom the smulaton esults that the obseve desgn shows the essental ams of ths method. he estmaton eo of x xˆ s shown n Fgue and the eo of x ˆ x n Fgue ae close to zeo quckly...5. Eo of x Example he theoem s llustated by ths example: 3 = ( ) Ex& = h z t A x + B u y = Cx me Fgue. Eo estmaton fo x Wth x =, A = 3 6 E =. And the actvatng functons: ( ( x )) h = tanh /, h = h., A -3 - = 5-3, B = , B.5 =, C =.5 and Accodng to the gven pocedue, we desgn the descpto obseve based on theoem va the Matlab LMI toolbox. hen we obtan : P = and Q =

11 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl.5 Eo of x me he coespondng obseve gan matces ae: Fgue3. Eo estmaton fo x L = and L = Eo of x me Fgue4. Eo estmaton fo x he scalas ae of values: λ =.5863, λ =.94 and γ =

12 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl.5 Eo of x me Fgue5. Eo estmaton fo x3 he esults of smulaton coespondng to the evoluton of the state estmaton eo ae pesented n Fgue 3, Fgue 4 and Fgue CONCLUSIONS he desgn descpto obseve was studed n ths pape. he consdeed systems ae modeled n the akag-sugeno descpto stuctue wth measuable and unmeasuable pemse vaables. he stategy s based on the use of the Lpschtz condton. he stablty s studed wth the Lyapunov theoy and a quadatc functon that allows to deve condtons ensung the convegence of the state estmaton eo. he exstence condtons ae expessed n tems of LMIs that can be solved wth LMIS oolbox n Matlab. REFERENCES [] K.anaka,. Ikeda and H.O Wang, Fuzzy Regulatos and Fuzzy Obseves: Relaxed Stablty Condtons and LMI-Based Desgns, IEEE ans. Fuzzy Syst, Vol. 6,pp. 5-65, 998. [] L. Da, Sngula Contol Systems, Spnge, Gemany, 989. [3]. anguch, K. anaka, K. Yamafuj and H.O Wang, Fuzzy Descpto Systems: Stablty Analyss and Desgn va LMIs, Poc. of the Ame. Cont. Conf, Vol. 3, pp , 999. [4] C.S. seng, B.S. Chen and H.J. Uang, Fuzzy ackng Contol Desgn fo Nonlnea Dynamc Systems va S Fuzzy Model, IEEE ansactons On Fuzzy Systems, Vol. 9, pp ,. [5] K. anaka and H.Wang, Fuzzy Regulatos and Fuzzy Obseves: A Lnea Matx Inequalty Appoach,Poc. 36th IEEE Conf. on Decson and Contol, Vol., pp. 35-3, 997. [6] P. Begsten, R. Palm and D. Dankov, Fuzzy Obseves, Poc.th IEEE Intenat. Conf. on Fuzzy Systems,Vol., pp.7-73,. [7] P. Begsten, R. Palm and D. Dankov, Obseves fo akag Sugeno Fuzzy Systems, IEEE ansactons on Systems Man and Cybenetcs, pp.4-,. [8] R. Palm and P. Begsten, Sldng Mode Obseve fo a akag-sugeno Fuzzy System, Poc. 9th IEEE Intenat. Conf. on Fuzzy Systems,Vol., pp ,. [9] G. Lu, D. Wang and Y.Sun, Obseve Desgn fo a Class of Nonlnea Descpto Systems, Intenatonal Confeence on Contol and Automaton, Budapest, Hungay, 5. 4

13 Intenatonal Jounal of Instumentaton and Contol Systems (IJICS) Vol., No., Apl [] K.Guelton, S.Delpat and.m.guea, An Altenatve to Invese Dynamcs Jont toques Estmaton Human Stance Based on a akag Sugeno Unknown-Inputs Obseve n he Descpto Fom, Contol Engneeng Pactce,pp.44-46, 8. [] A. M. Nagy-Kss, B. Max, G. Mouot, G. Schutz and J. Ragot, State Estmaton of wo-me Scale Multple Models wth Unmeasuable Pemse Vaables. Applcaton to bologcal eactos, 49th IEEE Confeence on Decson and Contol, pp ,. [] G. Lu, D.Wang and Y.Sun, Obseve Desgn fo Descpto Systems wth Lpschtz Constant, Intenatonal Confeence on Contol and Automaton, pp. -3, 5. [3] R. Palm and P. Begsten, Sldng Mode Obseve fo a akag-sugeno Fuzzy System, Poc. 9th IEEE Intenat. Conf. on Fuzzy Systems,Vol., pp ,. [4].Bouaa, Contbuton à la Synthèse de Los de Commande pou les Descpteus de ype akag- Sugeno Incetans et Petubés, hèse de doctoat, 9. [5].M Guea and L. Vemeen, (3) Stablté et Stablsaton à Pat de Modèles Flous, Commande Flous : De la Stablsaton à La Supevson, Hemes, pp