Simultaneous state and unknown inputs estimation with PI and PMI observers for Takagi Sugeno model with unmeasurable premise variables

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1 Simultaneous state an unknown inputs estimation with PI an PMI obseves fo Takagi Sugeno moel with unmeasuable pemise vaiables Dalil Ichalal, Benoît Max, José Ragot an Diie Maquin Abstact In this pape, a popotional integal (PI) an a popotional multiple integal obseve (PMI) ae popose in oe to estimate the state an the unknown inputs of nonlinea systems escibe by a Takagi-Sugeno moel with unmeasuable pemise vaiables This wok is an extension to nonlinea systems of the PI an PMI obseves evelope fo linea systems The state estimation eo is witten as a petube system Fist, the convegence conitions of the state estimation eos between the system an each obseve ae given in LMI (Linea Matix Inequality) fomulation Seconly, a compaison between the two obseves is mae though an acaemic example I INTRODUCTION Moel-base appoaches have been impotant an useful means to constuct a fault iagnosis moule fo nonlinea systems in oe to etect, isolate an ientify actuato, senso an system faults Geneally, the implementation of these functions is ealize with obseves Moeove, obseves povie an estimation of accessible an inaccessible states, outputs an faults of nonlinea systems The estimate signals ae use fo example to elaboate feeback contol laws, fault etection an isolation poceue (FDI) an fault toleant contol (FTC) 8, The popose wok focuses on the class of nonlinea systems escibe by Takagi-Sugeno moels with unmeasuable pemise vaiables The T-S moel povies a useful tool to epesent with a goo pecision a lage class of nonlinea systems an can even escibe exactly cetain classes of nonlinea systems 4 by using the nonlinea secto tansfomation In the ecent yeas, consieable effots have been povie to stuy stability an stabilization of this class of systems 4,, 8, 5 The topic of state estimation has also been wiely stuie in many woks In, 9, 6,, the authos popose iffeent methos in oe to estimate the state of T-S systems fo the pupose of iagnosis The avantage of T-S stuctue is its simplicity because it oiginates fom the intepolation between local linea systems Thus, analysis an esign methos evelope fo linea systems can be genealize to nonlinea systems as use in the woks cite above The authos ae with Cente e Recheche en Automatique e Nancy (CRAN), Nancy-Univesité, CNRS, avenue e la foêt e Haye F-5456 Vanoeuve-les-Nancy {alilichalal, benoitmax, joseagot, iiemaquin}@enseminpl-nancyf In the context of obust obseve esign, one of the most successful technique is the use of PI obseve, in which the unknown inputs ae estimate simultaneously with the states of the system The PI obseve was fist popose by Wojciechowsky in 5 fo single input-single output LTI systems A genealization scheme was pefome by Kaczoek to multivaiable systems Theeafte, the PI obseve has been use in iffeent stuies In a linea PI obseve is esigne an applie to a physical system In 5 a PI obseve fo linea escipto systems is popose Howeve, this obseve can be use only if the unknown inputs ae constant ove the time, nevetheless in pactical cases the appoach is effective if the vaiations of the unknown inputs ae slow in espect to the ynamic of the system In othe cases, this poblem can be solve by using multiple integals in the obseve in oe to estimate all of the eivatives of the unknown inputs A PMI obseve was fistly popose by Jiang in In 7, a popotional multiple integal obseve is popose to estimate a lage class of signals escibe in a polynomial fom fo LTI escipto systems We popose, in this pape, a genealization of the PI an PMI obseves to nonlinea systems escibe by T-S moels with unmeasuable pemise vaiables The pape is oganize as follows Section pesents the T-S stuctue an the poblem of state estimation, an gives the motivation of this wok In section A the esign of PI obseve is aesse an in section B the PMI obseve is stuie Section 4 pesents a numeical example with iscussion about the pefomances of the two popose obseves Finally, this note is ening with conclusions an pespectives Contaily to 9, this pape iscuss the simultaneous state an unknown input estimation using a PI an PMI obseve The iea is base on two steps: the fist step consists to tansfom the TS system with unmeasuable pemise vaiable into a petube TS system with estimate pemise vaiable The petubation tem is ue to the unmeasuable pemise vaiable The secon step is to make the system in an augmente fom by aing integatos to estimate the unknown input The pape 9 eals with the state estimation using a new metho consisting on the tansfomation of the TS system with unmeasuable pemise vaiables into an uncetain TS system with estimate pemise vaiables, in aition, the metho is extene to estimate the unknown input using a PI obseve

2 II PRELIMINARIES AND PROBLEM STATEMENT A Multiple moel appoach Consie the following geneal fom of continuous-time nonlinea systems: { ẋ(t) = f(x(t),u(t)) () y(t) = h(x(t),u(t)) whee x R n, u R m, y R q an f an h ae nonlinea functions The epesentation () is ifficult to stuy, elsewhee in liteatue, all of the woks evelope concening the nonlinea systems concen specific classes Fo example, in, Lipschitz systems, which ae epesente by a linea pat an a nonlinea one, ae consiee The nonlinea pat is assume to be Lipschitz with espect to the state x As mentione in the intouction, the T-S moel appoach is a vey inteesting metho to epesent nonlinea systems Diffeent methos exist to obtain a T-S moel, as ientification o lineaization of the system () aoun iffeent opeating points o by using the nonlinea secto tansfomation The multiple moel stuctue is given by: ẋ(t) = µ i (ξ(t))(a i x(t)+b i u(t)+e i (t)+w i ω(t)) y(t) = Cx(t)+Du(t)+G(t)+W c ω(t) () whee A i R n n, B i R n m, C R q n, D R q m, E i R n s, W i R n v an G R q s, an W c R q v The unknown inputs ae moele by (t) an ω(t) ae the noises affecting the state an the measuement equation In this stuctue, the output is assume to be linea with ega to the state of the system The weighing functions µ i ae nonlinea an epen on the ecision vaiable ξ(t) which can be measuable like {u(t), y(t)} o not measuable like the state x(t) of the system The weighting functions satisfy the following popeties: µ i (ξ(t)) () µ i (ξ(t)) = Thus the stuctue of the multiple moel is simple an is consiee as a univesal appoximato since it can epesent any nonlinea behavio accoing to an aequate numbe of the local moels The multiple moel stuctue povies a mean to genealize the tools evelope fo linea systems to nonlinea systems ue to the popeties expesse in () B Poblem statement Diagnosis of nonlinea systems is often base on a bank of obseves to etect an isolate actuato an senso faults Fo esigning obseves, it is often assume, in the liteatue that the weighting functions µ i epen on measuable pemises vaiables u an/o y Thus, to pefom iagnosis, it is necessay to evelop two iffeent multiple moels The fist one whee the weighting functions epen only on the output of the system in oe to etect an isolate actuato faults The secon one with weighting functions epening only on the input of the system in oe to etect an isolate senso faults To euce this ifficulty, it is inteesting to evelop only one multiple moel using weighing functions which epen on the state of the system Thus, the same multiple moel can be use to constuct obseve bank fo etecting an isolating actuato an senso faults Howeve, a main ifficulty appeas ue to the fact that the state equation is now a nonlinea function of the state In the liteatue, only few woks ae evelope fo obseve esign fo T-S systems with unmeasuable pemise vaiables Nevetheless, we can cite 6, 7, 6, 4, whee the authos e-wite the system eithe as a petube o uncetain T-S system with measuable pemise vaiables III MAIN RESULT Along this pape, we assume that the following assumptions hol: A The system is stable A The signals u(t), (t) an ω(t) ae boune Pactically, these assumptions ae often not estictive A Extension of classical PI obseve Consie the following T-S fuzzy system with weighting functions µ i epening on the state of the system: ẋ(t) = µ i (x(t))(a i x(t)+b i u(t)+e i (t)+w i ω(t)) y(t) = Cx(t)+G(t)+Wω(t) (4) In the next, fo sake of simplicity, the time vaiable t is omitte The popose PI obseve is given by the following equations: ˆx = µ i ( ˆx) ( A i ˆx+B i u+e i ˆ+ K Pi (y ŷ) ) ŷ = C ˆx+G ˆ ˆ = µ i ( ˆx)K (y ŷ) on which ˆx an ˆ ae the estimates of x an In oe to facilitate the compaison between the system an its obseve, the system (4) can be witten as a petube system with weighting functions µ i epening on the estimate state as follows: whee: ν = ẋ = (5) µ i ( ˆx)(A i x+b i u+e i +W i ω + ν) (6) (µ i (x) µ i ( ˆx))(A i x+b i u+e i +W i ω) (7) This tem is seen as a boune vanishing petubation to minimize Inee, ue to the assumptions A, A an the efinition of the weighting functions (), ν(t) is boune an if ˆx x then ν The unknown inputs (t) ae assume to be constant: A = The assumption allows to make the system (6) in the augmente fom: ẋ a = µ i ( ˆx) ( Ã i x a + B i u+ Γ i ω ) (8) y = Cx a + D ω

3 whee: Ai E à i = i Bi, B i = C = C G, D = W I Wi, Γ i =,x a = ν, ω = ω x A simila easoning makes it possible to tansfom the popose PI obseve (5) in the following augmente fom: ˆx a = µ i ( ˆx) ( à i ˆx a + B i u+ K i (y ŷ) ) (9) ŷ = C ˆx a whee: KPi K i = K Let us consie the augmente state estimation eo: e a = x a ˆx a () whose ynamic is given by: ė a = µ i ( ˆx) ( (à i K i C)e a +( Γ i K i D) ω ) () The goal is to etemine the gain matices K i of the obseve in oe to stabilize the system (), ie to guaantee the convegence of the state estimation eo towa zeo when the petubation ω is nul an to attenuate the tansfe gain fom the boune petubation ω(t) to the state estimation eo e a (t) when ω(t) is iffeent fom zeo ( ω(t) is boune since assumptions A an A ae satisfie) In oe to establish the existence conitions of the PI obseve in theoem, let us fist intouce the following lemma: Lemma : 4 Consie the continuous-time TS-system efine by: ẋ(t) = µ i (x(t))(a i x(t)+b i u(t)) y(t) = Cx(t) () The system () is stable an veifies the L -gain conition: y(t) < γ u(t) if thee exists a symmetic positive efinite matix P such that () is satisfie fo i =,,: A T i P+PA i +C T C PB i B T i P < () γ I Theoem : The PI obseve (9) fo the system (8) is etemine by minimizing γ une the following LMI constaints in the vaiables P = P T >, M i an γ fo i =,,: ÃT i P+Pà i M i C C T M i + I P Γ i M i D Γ T i P D T Mi T < (4) γi The gains of the obseve ae eive fom: an the attenuation level is calculate by: K i = P M i (5) γ = γ (6) Poof: Accoing to the assumptions A an A, ω(t) is boune Then, by applying lemma with e a (t) < γ ω(t), we obtain: ÃT i P+Pà i P K i C C T K i T P+I P Γ i P K i D Γ T i P D T K i T P γ I < (7) The LMI fomulation in theoem is obtaine by using the following changes of vaiables: M i = P K i, γ = γ Remak : The minimization of γ may esult in slow ynamics of the state estimation eo This poblem can be solve by pole assignment of the matices (à i K i C) in the left half complex plane efine by: {z Re(z) < λ}, λ > (8) Thus, the LMIs in theoem ae solve simultaneously with the following constaint (to impose Re(λ i ) < λ, whee λ i ae the eigenvalues of à i an λ > ): P(à i + λi)+(ã i + λi) T P M i C C T M T i < (9) Moe pecise pole clusteing can be obtaine by aing LMI constaints 6 This appoach emains effective in pactical cases whee the assumption is not satisfie Howeve, the unknown inputs must vay slowly Othewise, ba state an unknown inputs estimation ae obtaine by using this metho In the next section, anothe metho to estimate the state an the unknown inputs is popose It is base on the popotional multiple integal obseve This obseve is inteesting because the assumption is not equie in the theoetic poof, so it is possible to estimate a lage class of unknown inputs B Popotional multiple integals obseve Let us consie the multiple moel with unmeasuable pemise vaiables escibe in (4) The unknown input is assume to be a boune time vaying signal with null q th eivative: A4 (q) (t) = Geneally, the use of a PI obseve equies the conition that the unknown input is constant (ie: = ), thus, the unknown inputs which satisfies A4 cannot be estimate with

4 a goo pecision Then, PMI obseve is moe aequate fo this poblem, because the obseve estimates the (q ) th eivatives of the unknown input an gives a goo pecision of the estimate unknown inputs Consie the genealization of the popotional multipleintegals obseve to T-S systems of the PMI obseve popose in 7 fo linea escipto systems: ˆx = µ i ( ˆx)(A i ˆx+B i u+e i ˆ + K Pi (y ŷ)) ŷ = C ˆx+G ˆ ˆ = ˆ = ˆ q = µ i ( ˆx)K (y ŷ)+ ˆ µ i ( ˆx)K (y ŷ)+ ˆ µ i ( ˆx)K q (y ŷ)+ ˆ q ˆ q = µ i ( ˆx)K q (y ŷ) () whee ˆ i, i =,,,(q) ae the estimation of the (q) fist eivatives of the unknown input (t) The state an unknown inputs estimation eos ae: e = x ˆx, e = ˆ,, e q = q ˆ q Thei ynamics ae given in the following fom: ė = µ i ( ˆx)((A i K Pi C)e+(Γ i K Pi W) ω +(E i K Pi G)e ) ė = µ i ( ˆx)( K Ce+e K W ω K Ge ) ė = µ i ( ˆx)( K Ce+e K W ω K Ge ) ė q = µ i ( ˆx)( K Ce+e q K q W ω K q Ge ) ė q = whee: µ i ( ˆx)( K q Ce K W ω K q Ge ) Γ i = I n W i, W = W c () The equations () can be ewitten in the following augmente fom: e e ẽ = µ i ( ˆx)((à i K i C)ẽ+( Γ i K i W) ω) () = Cẽ () whee: ẽ= e e e e q e q A i E i I s,ã i = I s C = C G Γ i = Γ T i T, K i = K Pi K K K q K q In the following, we ae only inteeste with paticula component e an e of ẽ: e = Cẽ (4) e whee: C = In I s epesents null matix with appopiate imensions Theoem : The PMI obseve () fo the system (8) that minimizes the tansfe fom ω(t) to e(t) T e (t) T is obtaine by fining the matices P = P T >, M i an γ that minimize γ une the following LMI constaints fo i =,,: ÃT i P+Pà i M i C C T Mi T + C T C P Γ i M i W Γ T i P W T Mi T γi The gains of the obseve ae eive fom: an the attenuation level is calculate by: < (5) K i = P M i (6) γ = γ (7) Poof: The poof of theoem is simila to the poof of theoem by using the lemma with the system () Remak : When the conition Ais not satisfie ie (q) but (q) is boune then, we can consie the q th eivative of (t) as a petubation The new petubation vecto is then given by: ω(t) = ν(t) T ω(t) T (q) (t) T T The aitional component q is ae in the state vecto The matices à i, Γ i, W, C ae augmente Then, the Theoem can be applie in oe to esign the Popotional Multiple Integals Obseve with minimization of the new boune petubation ω(t)

5 IV NUMERICAL EXAMPLE AND SIMULATIONS In this section, the popose metho is illustate though an acaemic example Consie a continuous-time T-S system (4) efine by: an A = 8,A = B = 5,B =,E = 5 7 E = 6,W = W = C = 5 4 5,G = 7 5,W = 5 5 The unknown inputs vecto (t) is mae up of (t) which affects only the outputs of the system an (t) affecting only the ynamic of the system (see the matices E, E an G) Fo example, we can consie as a senso fault an as an actuato one The weighting functions epen on the fist component x of the state vecto x an ae efine as follows: { µ (x) = tanh(x ) (8) µ (x) = µ (x) The weighting functions obtaine without petubations an unknown inputs ae shown in figue This figue shows that the system is clealy nonlinea since µ an µ ae not constant functions,,, oiginal estimate Fig Unknown input estimation with PI obseve oiginal estimate oiginal estimate oiginal estimate 8 µ (t) µ (t) Fig Unknown input estimation with PMI obseve Fig Weighting functions µ an µ The petubations ω ae chosen as anom signals unifomly istibute in 5 5 The consiee unknown inputs ae given by: (t) an (t) ae time vaying signals with neglecte fouth eivatives Afte synthesizing a PI obseve accoing to the theoem an a PMI obseve with q = 4 accoing to the theoem, we obtain the simulation esults epicte in the figues,, 4 an 5 Figues an show the unknown inputs an thei estimations with PI an PMI obseves It is known that the PI obseve gives an acceptable state an unknown inputs estimation even if the assumption A is not satisfie Howeve, in this example, the unknown inputs have fast vaiations esulting on ba state an unknown inputs estimation (figues an 4) compae to the esults given by the PMI obseve (figues an 5) V CONCLUSIONS AND FUTURE WORKS The esign of popotional integal (PI) an popotional multiple integals (PMI) ae stuie in this pape This wok is an extension of the PI an PMI obseves evelope fo linea systems to nonlinea T-S systems with unmeasuable pemise vaiables The convegence conitions of the state

6 Fig 4 State estimation eo with PI obseve Fig 5 State estimation eo with PMI obseve estimation eo ae given in the LMI fomulation The obseves ae obust since they ae synthesize in oe to minimize the effect of noises on the state estimation eo by using an L appoach The PI obseve is inteesting fo the estimation of constant o slowly vaying unknown inputs an it is less sensitive to noises compae to the PMI obseve 7 In the othe han, PMI obseve is a goo way to obtain a moe pecise estimation of states an unknown inputs The futue woks will concen, fistly, the impovement of the PMI obseve by intoucing a stable weighting functions on the petubations ω(t) which allows to eflect the expecte fequency content of ω(t), seconly, the use of these obseves in nonlinea system iagnosis REFERENCES M Abbaszaeh an H Maquez, Robust H obseve esign fo a class of nonlinea uncetain systems via convex optimization, in Ameican Contol Confeence, ACC 7, 7 A Akhenak, M Chali, J Ragot, an D Maquin, Design of sliing moe unknown input obseve fo uncetain Takagi-Sugeno moel, in 5th Meiteanean Confeence on Contol an Automation, MED 7, Athens, Geece, 7 P Begsten, R Palm, an D Diankov, Fuzzy obseves, in IEEE Intenational Fuzzy Systems Confeence, Melboune Austalia, 4, Obseves fo Takagi-Sugeno fuzzy systems, IEEE Tansactions on Systems, Man, an Cybenetics - Pat B: Cybenetics, vol, no, pp 4, 5 M Chali, D Maquin, an J Ragot, Non quaatic stability analysis of Takagi-Sugeno systems, in IEEE Confeence on Decision an Contol, CDC, Las Vegas, Nevaa, USA, 6 M Chilali an P Gahinet, H-infinity esign with pole placement constaints : an LMI appoach, IEEE Tansactions on Automatic Contol, vol 4, no, pp 58 67, 996 e e e e e e 7 Z Gao an D Ho, Popotional multiple-integal obseve esign fo escipto systems with measuement output istubances, IEE poceeing Contol theoy an application, vol 5, no, pp 79 88, 4 8 T Guea, A Kuszewski, L Vemeien, an H Timant, Conitions of output stabilization fo nonlinea moels in the Takagi-Sugeno s fom, Fuzzy Sets an Systems, vol 57, no 9, pp 48 59, May 6 9 D Ichalal, B Max, J Ragot, an D Maquin, State an unknown input estimation fo nonlinea systems escibe by takagi-sugeno moels with unmeasuable pemise vaiables in 7th Meiteanean Confeence on Contol an Automation, MED 9, Thessaloniki, Geece, June R Isemann, Fault-iagnosis systems: An intouction fom fault etection to fault toleance, Spinge, E, 7 G Jiang, S Wang, an W Song, Design of obseve with integatos fo linea systems with unknown input istubances, Electonics Lettes, vol 6, no, pp 68 69, T Kaczoek, Popotional-integal obseves fo linea multivaiable time-vaying systems, Regelungstechnik, vol 7, pp 59 6, 979 D Koenig, Unknown input popotional multiple-integal obseve esign fo linea escipto systems: application to state an fault estimation, IEEE Tansactions on Automatic Contol, vol 5, no, pp 7, 5 4 A Kuszewski, Lois e commane pou une classe e moèles non linéaies sous la fome Takagi-Sugeno : Mise sous fome LMI, PhD issetation, Univesité e Valenciennes et u Hainaut-Cambesis, 6, (In fench) 5 B Max, D Koenig, an D Geoges, Robust fault iagnosis fo linea escipto systems using popotional integal obseves, in 4n IEEE Confeence on Decision an Contol, 6 R Palm an P Begsten, Sliing moe obseves fo Takagi-Sugeno fuzzy systems 9th IEEE Intenational Confeence on Fuzzy Systems, FUZZ IEEE, San Antonio, TX, USA, 7 R Palm an D Diankov, Towas a systematic analysis of fuzzy obseves, in 8th NAFIPS Confeence, New Yok, NY, USA, R Patton, P Fank, an R Clak, Fault iagnosis in ynamic systems: Theoy an application Pentice Hall intenational, R Patton, J Chen, an C Lopez-Toibio, Fuzzy obseves fo nonlinea ynamic systems fault iagnosis, in 7th IEEE Confeence on Decision an Contol, Tampa, Floia USA, 998 G Pechoto e Melo an T Souza Moais, Fault etection using state obseves with unknown input, ientifie by othogonal functions an PI obseves, Bazilian Society of mechanical sciences an Engineeing, 7 R Rajamani, Obseves fo Lipschitz nonlinea systems, IEEE Tansactions on Automatic Contol, vol 4, pp 97 4, Mach 998 T Takagi an M Sugeno, Fuzzy ientification of systems an its applications to moeling an contol, IEEE Tansactions on Systems, Man, an Cybenetics, vol 5, pp 6, 985 K Tanaka, T Ikea, an H Wang, Fuzzy egulatos an fuzzy obseves: Relaxe stability conitions an LMI-base esigns, IEEE Tansactions on Fuzzy Systems, vol 6, no, pp 5 65, K Tanaka an H Wang, Fuzzy Contol Systems Design an Analysis: A Linea Matix Inequality Appoach, J Wiley an i Sons, Es John Wiley an Sons, inc, 5 B Wojciechowski, Analysis an synthesis of popotional-integal obseves fo single-input-single-output time-invaiant continuous systems, PhD issetation, Gliwice, Polan, J Yoneyama, H output feeback contol fo fuzzy systems with immeasuable pemise vaiables: Discete-time case, Applie Soft Computing, vol 8, no, pp , Ma 8