Observer Design for a Class of Discrete-Time Takagi-Sugeno Implicit Models Subject to Unknown Inputs

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1 Amercan Journal of Computatonal and Appled Mathematcs 7 7(4): 5- DOI:.593/j.ajcam Observer Desgn for a Class of Dscrete-me aag-sugeno Implct Models Subject to Unnown Inputs Abdellah Louzm Abdellatf El Assoud Jalal Soulam * El Hassane El Yaagoub Laboratory of Hgh Energy Physcs and Condensed Matter Faculty of Scence Hassan II Unversty of Casablanca Casablanca Morocco ECPI Department of Electrcal Engneerng ENSEM Hassan II Unversty of Casablanca Casablanca Morocco Abstract hs paper nvestgates the problem of the fuzzy unnown nputs observer (FUIO) desgn for a class of dscrete-tme aag-sugeno mplct models (DSIMs) wth unmeasurable premse varables whch satsfyng Lpschtz condtons. he unnown nputs (UIs) affect both state and output of the model. he dea of the proposed result s based on the separaton between dynamc and statc euatons n the consdered DSIM. Frst the method used to separate dynamc euatons from statc euatons s developed. Next based on the augmented system structure whch contans the dynamc euatons and the unnown nputs a new observer desgn n explct structure to estmate smultaneously the system state and the unnown nputs s establshed. he convergence of the state estmaton error of the augmented system s studed by usng the Lyapunov theory and the gan matrx of the FUIO s obtaned by solvng only one lnear matrx neualty (LMI). At last an llustratve example s gven to show the effectveness of the proposed technue. Keywords Dscrete-tme aag-sugeno mplct model Fuzzy unnown nput observer Lpschtz condton LMI. Introducton It s well-nown that to provde on-lne estmaton of unmeasurable states of varous ndustral processes the approach conssts n combnng a pror nowledge about a process wth expermental data. he algorthm permttng to realze a such estmaton s called an observer. hs last one s a dynamcal system whch combnes a nomnal model of the process wth on-lne measurements of the nput and the output of the process to estmate the unmeasurable states. he feld of the observer desgn for dynamc systems has attracted much attenton of the researchers for a long tme. hs s due to ts mportant role n the control and fault-tolerant areas. Moreover UIs can result ether from uncertanty n the model or from the presence of unnown external exctaton. hus due to the ncreasng demand for relablty and mantenablty of the automatc control process FUIO desgn s wdely used n the area of fault detecton and desgn of fault tolerant control strategy. hs s one of the most attractve research areas n both theoretcal and practcal felds durng these last two decades. Indeed many * Correspondng author: jalal.soulam@gmal.com (Jalal Soulam) Publshed onlne at Copyrght 7 Scentfc & Academc Publshng. All Rghts Reserved wors usng dfferent approaches can be found n the lterature [-3]. In ths perspectve based on the aag-sugeno (-S) fuzzy approach we study here the problem of observer desgn for a class of dscrete-tme nonlnear mplct models subject to UIs. Currently t s well-nown that ordnary -S fuzzy approach [4 5] have been wdely and successfully used n the nonlnear processes modellng to descrbe the behavour of many chemcal and physcal processes. he dea of ths approach s to apprehend the global behavour of a process by a set of local models. he success of such approach reles on the fact that once the -S fuzzy models are obtaned some analyss and desgn tools developed n the lnear case can be used whch facltates observer or/and controller synthess for complex nonlnear systems see for example [6 7] and the references theren. On the other hand n realty many ndustral processes are naturally modelled as systems of dfferental and algebrac euatons also called descrptor models or sngular models or mplct models see [8-] for some real applcatons of mplct models. he numercal smulaton of such models usually combnes an ODE numercal method together wth an optmzaton algorthm. Moreover notce that the ordnary -S fuzzy model s a specal case of the fuzzy mplct model. Indeed n [ ] a fuzzy mplct model s defned by extendng the -S fuzzy model [4].

2 6 Abdellah Louzm et al.: Observer Desgn for a Class of Dscrete-me aag-sugeno Implct Models Subject to Unnown Inputs he FUIO desgn problem for -S explct or mplct models has receved consderable attenton and s stll an actve area of research n both contnuous-tme and dscrete-tme cases durng the two last decades. Indeed concernng the case of contnuous-tme -S explct or mplct models subject to UIs varous developments on fuzzy observer and ts applcaton to fault detecton exst n the lterature. We may cte [3-7] for explct models and [ 8-3] for mplct models. Lewse n dscrete-tme -S models case several wors exst for explct or mplct structures see e.g. [4-8]. It should be noted that generally an nterestng way to solve the varous FUIO rased prevously s to wrte the convergence condtons on the LMI form [9]. he man contrbuton of ths paper conssts to propose a new result of FUIO desgn for a class of DSIMs satsfyng the Lpschtz condtons allowng the smultaneous estmaton of the unnown states and unnown nputs. he procedure s based on the separaton between dynamc and statc euatons n the DSIM n order to use the augmented system structure whch contan the dynamc euatons and the UIs. he global exponental stablty of the state estmaton error of the augmented system s studed by usng the Lyapunov theory and the stablty condton s gven n term of only one LMI. Besdes the proposed result s gven wthout the use of an optmzaton algorthm. he rest of the paper s structured as follows. he consdered class of DSIMs subject to UIs s presented n Secton. he man contrbuton about FUIO desgn permttng to estmate unnown states and UIs s stated n Secton 3. Fnally a numercal example to show the good performance of the proposed technue s gven n Secton 4. hroughout the paper the followng notatons are adopted. Matrx (or ) means that s symmetrc and postve defnte (or negatve defnte). denotes the transpose of. he symbol I (or ) represents the dentty matrx (or zero matrx) wth n n m approprate dmenson. R and R denote the spaces of n-dmensonal real vectors and n m real matrces respectvely.. System Descrpton and Prelmnares In ths paper the followng class of DSIMs subject to UIs s adopted: Ex h ( x )( A x B u D d ) y Cx + Fd n x R denotes the state vector n wth R s the vector of dfference varables n R () s the vector of algebrac varables wth nn n (measured) y B p u d R m r s the nput vector whch s nown R s the unnown nput vector R s the measurement vector. n m R ran( E) n matrces wth: A p n R C F p r R D n r R A E n n R n n R wth are real nown constant A A B D ; B ; D A A B D A I E ; C C () (3) are supposed nvertble. s the number of sub-models and h( x ) are the weghtng functons that verfy the so-called convex sum propertes: h ( x ) h ( x ) (4) hey ensure the transton between the contrbuton of each sub model: Ex A x Bu D d y Cx + Fd Before gvng the man result let us mae the followng assumpton [8 9]: Assumpton : Suppose that: ( EA ) s regular.e. det( ze A ) z All sub-models (5) are mpulse observable and detectable. In what follows as mentoned above we proceed to the separaton of the dynamc euatons from statc euatons of the model (). Indeed from ()-(3) sub-model (5) can be rewrtten as follows: A A B u D d A A Bu Dd y C + Fd From (6) and usng the fact that A s nvertble the algebrac euatons can be solved drectly to obtan: K (5) (6) J u L d (7) K L J A A A A B D hus substtutng (7) n (6) we obtan: (8)

3 Amercan Journal of Computatonal and Appled Mathematcs 7 7(4): 5-7 M N u Pd J K u L d y C + Fd M N P A B D A J A K A L (9) () he weghtng functons h( x) can be rewrtten as: h ( x ) h ( J K u L d ) h ( ) () wth d u. So by aggregaton of the resultng sub-models (9) the followng global fuzzy model s obtaned: h ( ) M Nu Pd ( ) h J K u L d y C Fd () Assumpton : Suppose that d s consdered as a constant unnown control nput per tme nterval.e.: d d R (3) In order to tae the man contrbuton we rewrte the system () under the euvalent augmented state representaton gven by: h ( ) M Nu y ( ) h J K u d ; M P u ; M I N N ; J J L C F whch s euvalent to the followng state representaton: (4) (5) M Nu h ( ) M Nu y ( ) h J K u M N M M N N N M (6) N (7) M 3. Synthess of the Observer Based on the transformaton of the DSIM () nto the euvalent form (6) the proposed FUIO for () s taen n the followng form: ˆ N G y y ˆ yˆ M ˆ h ˆ M ˆ N u ( ) h ˆ J ˆ K u ˆ ( ) u ( ˆ ) (8) ( ˆ ˆ ) yˆ and ˆ represent the estmate of ( ) y and respectvely. Matrx G s to be determned such that ( ˆ ˆ ) converges toward ( ) exponentally. o gve the condton for the exponental convergence of the observer (8) we defne the state estmaton error: ˆ ˆ (9) From (6) and (8) the dynamc of the estmaton error can be descrbed by: M u h ( ˆ ) J h ( ) h ( ) J Ku ˆ ˆ ()

4 8 Abdellah Louzm et al.: Observer Desgn for a Class of Dscrete-me aag-sugeno Implct Models Subject to Unnown Inputs and M G () ( ˆ ) ˆ h h ( ) N h ( ˆ ) h ( ) So to prove the convergence of () toward zero t suffces to prove that converges toward zero. Assumpton 3: Assume that the followng condtons hold: u (3) are postve scalars Lpschtz constants and. Usng the Assumpton 3 the term M u can be bounded as follows: M u (4) ( M ) (5) wth ( M ) denotes the maxmum sngular value of the matrx M. he followng heorem provdes the man result of ths paper. heorem : Under above Assumpton 3 the DSIM () s globally exponentally stable f gven there exst matrces P Q and U verfyng the followng LMI: * * U P * PM U Q (6) M PM M U U M P Q P (7) he gan stablzng the estmaton error s gven by: G P U (8) Proof of heorem : Consder the followng standard Lyapunov functon: he varaton of by: From () we have: V V P P (9) V along the trajectory of () s gven V V V (3) ( P P) M u PM u M u P PM u (3) Lemma : For any matrces and Y wth approprate dmensons the followng property holds for any nvertble matrx Z : - Y Y Z Y Z Y (3) For Q by applyng Lemma (3) becomes: V ( P P) M u PM u PQ P M u QM u ang nto account (4) (33) becomes: V (33) ( P P PQ P P Q) (34) hus estmaton error convergence s exponentally ensured f the followng condton s guaranteed (see [3] as cted n [6]): V ( P P PQ P P Q) ( ) V whch s euvalent to the followng condton: (35) P PQ P P Q P (36) hen replacng from () nto (36) we can establsh the LMI condton (6) of heorem by usng the Schur complement [9] and the followng change of varable: U PG (37)

5 Amercan Journal of Computatonal and Appled Mathematcs 7 7(4): 5-9 hus from the Lypunov stablty theory f the LMI condton (6) s satsfed the system () s globally exponentally stable. hs ends the proof of heorem. 4. Illustratve Example In ths secton to llustrate the effectveness of the proposed desgn method let us consder the one-ln flexble jont robot descrbed by the dynamcal model gven n []. he dscrete-tme system () was obtaned by usng the Euler dscretsaton of a step sze e =. whch lead to the followng DSIM wth unmeasurable premse varable and subject to UI: Ex h ( x )( A x Bu Dd ) y Cx (38) 6 x R u R d R and y R are the state vector nown nput unnown nput and measured output vector respectvely. he matrces numercal values are:... A B=.... A D= E C he weghtng functons are: sn( x ) h ( x ) x sn( x ) h ( x ) x herefore to apply the proposed FUIO (8) for the model (38) as stated n heorem t suffces to rewrte the model (38) nto ts euvalent form (6) as mentoned n Secton. hus by heorem wth.95 the followng observer gan G s obtaned: G Smulaton results wth ntal condtons: ˆ ˆ are gven n Fgures and 3 the nput u sn( ) and the expresson of unnown nput sgnal d s defned as n Fgure 3. hese smulaton results show the performances of the proposed FUIO (8) wth the gan G the dashed lnes denote the state varables and unnown nput estmated by the FUIO. hey show that the FUIO gves a good estmaton of unnown states and unnown nput of the consdered DSIM.

6 Abdellah Louzm et al.: Observer Desgn for a Class of Dscrete-me aag-sugeno Implct Models Subject to Unnown Inputs Fgure. State varables x x x 3 and ther estmates Fgure. State varables x 4 x 5 x 6 and ther estmates

7 Amercan Journal of Computatonal and Appled Mathematcs 7 7(4): 5- generalzed form of aag-sugeno fuzzy systems. IEEE ransactons on Fuzzy Systems vol. 9 No.4 pp August. [6] K. anaa and H. O. Wang Fuzzy control systems desgn and analyss: A Lnear Matrx Ineualty Approach. John Wley & Sons;. [7] Zs. Lende. M. Guerra R. Babuša and B. De Schutter. Stablty analyss and nonlnear observer desgn usng aag-sugeno fuzzy models Sprnger Berln Hedelberg. [8] L. Da. Sngular Control Systems. Lecture Notes n Control and Informaton Scences. Sprnger-Verlag. Berln 989. [9] A. Kumar and P. Daoutds Control of nonlnear dfferental algebrac euaton systems. Chapman & Hall CRC 999. Fgure Conclusons Unnown nput d and ts estmate In ths paper we have presented a novel method of FUIO desgn for a class of DSIMs wth unmeasurable premse varables whch satsfyng Lpschtz condtons. he UIs affect both state and output of the model. he approach s based on the separaton between dynamc and statc euatons n the consdered fuzzy mplct model and on the use of an augmented system descrpton formed by the dynamc euatons and the UIs. Besdes the proposed result permttng to estmate smultaneously the system state and the UIs s gven wthout the use of an optmzaton algorthm. he exponental convergence of the state estmaton error s studed usng the Lyapunov theory and the exstence of the condton ensurng ths convergence s expressed n term of only one LMI. In order to valdate the proposed approach a numercal example s gven. he proposed example shows the good performance of the proposed FUIO desgn snce both state and UIs are well estmated. REFERENCES [] R. Isermann Fault-Dagnoss Systems An Introducton from Fault Detecton to Fault olerance. Sprnger-Verlag Berln Hedelberg 6. [] Z. Lende. M. Guerra R. Babuša and B. De Schutter Stablty Analyss and Nonlnear Observer Desgn Usng aag-sugeno Fuzzy Models. Sprnger-Verlag Berln Hedelberg. [3] L. L Fault Detecton and Fault-olerant Control for Nonlnear Systems. Sprnger Fachmeden Wesbaden 6. [4]. aag M. Sugeno Fuzzy dentfcaton of systems and ts applcaton to modelng and control. IEEE rans. Syst. Man and Cyber Vol.5 pp [5]. anguch K. anaa H. Ohtae and H. Wang Model constructon rule reducton and robust compensaton for [] G. R. Duan. Analyss and Desgn of Descrptor Lnear Systems. Sprnger. []. anguch K. anaa K. Yamafuj and H. O. Wang Fuzzy Descrptor Systems: Stablty Analyss and Desgn va LMIs. Proceedngs of the Amercan Control Conference. San Dego Calforna June 999 pp []. anguch K. anaa and H. O. Wang Fuzzy Descrptor Systems and Nonlnear Model Followng Control IEEE ransactons on Fuzzy Systems vol. 8 No.4 pp August. [3] D. Ichalal B. Marx J. Ragot and D. Maun State and unnown nput estmaton for nonlnear systems descrbed by aag-sugeno models wth unmeasurable premse varables. 7th Medterranean Conference on Control and Automaton MED'9 June Maedona Palace hessalon Greece. [4] D. Ichalal; B. Marx; J. Ragot; D. Maun New fault tolerant control strateges for nonlnear aag-sugeno systems. Internatonal Journal of Appled Mathematcs and Computer Scence (): 97-. [5] M. Kamel M. Chadl M. Chaabane Unnown nputs observer for a class of nonlnear uncertan systems; An LMI approach. Internatonal Journal of Automaton and Computng. 9(3) June [6] M. Chadl and H.R. Karm Robust observer desgn for unnown nputs aag-sugeno Models. IEEE ransactons on Fuzzy Systems (): [7]. Youssef M. Chadl and M. Zelmat Synthess of a unnown nputs proportonal ntegral observer for S fuzzy models. European Control Conference (ECC) July Zurch Swtzerland. [8] B. Marx D. Koeng and J. Ragot Desgn of observers for aag-sugeno descrptor systems wth unnown nputs and applcaton to fault dagnoss. IE Control heory and Applcatons October 7. [9] C. Mechmech H. Hamd M. Rodrgues and N. Benhadj Brae State and unnown nputs estmatons for mult-models descrptor systems. Amercan Journal of Computatonal and Appled Mathematcs (3): [] F. R. Lopez-Estrada J.C. Ponsart Dder hellol C. M. Astorga-Zaragoza S. Aberane Fault Dagnoss Based on Robust Observer for Descrptor-LPV Systems wth

8 Abdellah Louzm et al.: Observer Desgn for a Class of Dscrete-me aag-sugeno Implct Models Subject to Unnown Inputs Unmeasurable Schedulng Functons. 9th IFAC World Congress. Cape own South Afrca. August [] K. Bouassem J. Soulam A. El Assoud and E. El Yaagoub Unnown Input Observer Desgn for a Class of aag-sugeno Descrptor Systems. Nonlnear Analyss and Dfferental Euatons Vol. 4 6 no [] A. Louzm A. El Assoud J. Soulam and E. El Yaagoub Unnown Input Observer Desgn for a Class of Nonlnear Descrptor Systems: A aag-sugeno Approach wth Lpschtz Constrants. Nonlnear Analyss and Dfferental Euatons Vol. 5 7 no [3] K. Bouassem J. Soulam A. El Assoud and E. El Yaagoub Fuzzy Observer Desgn for a Class of aag-sugeno Descrptor Systems Subject to Unnown Inputs. Nonlnear Analyss and Dfferental Euatons Vol. 5 7 no [4] M. Chadl A. Ahena J. Ragot D. Maun State and unnown nput estmaton for dscrete-tme multple model. Journal of the Franln Insttute [5] E. Maherz M. Besbes S. Zemmel A. Mam Estmaton of the State and the Unnown Inputs of a Multmodel wth non Measurable Decson Varables. Journal of Appled Research and echnology (JAR) vol. (3) pp June 4. [6] V. Estrada-Manzo Z. Lende. M. Guerra Dscrete-tme aag-sugeno descrptor models: observer desgn. IFAC Cape own South Afrca. August [7] D. Rotondo M. Wtcza V. Pug F. Nejjar & M. Pazera Robust unnown nput observer for state and fault estmaton n dscrete-tme aag-sugeno systems Internatonal Journal of Systems Scence 6. [8] K. Zhang B. Jang V. Cocuempot Fuzzy unnown nput observer-based robust fault estmaton desgn for dscrete-tme fuzzy systems. Sgnal Processng Volume 8 Pages 4-47 November 6. [9] S. Boyd et al. Lnear Matrx Ineualtes n Systems and Control heory Phladelpha PA: SIAM 994. [3] A. Ichawa et al. Control Hand Boo Ohmu Publsher 993 oyo n Japanese.