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

Size: px

Start display at page:

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

Berniece Lorraine Cole
6 years ago
Views:

1 Simultaneous state and unknown inputs estimation with PI and PMI obseves fo Takagi Sugeno model with unmeasuable pemise vaiables Dalil Ichalal, Benoît Max, José Ragot and Didie Maquin Abstact In this pape, a popotional integal (PI) and a popotional multiple integal obseve (PMI) ae poposed in ode to estimate the state and the unknown inputs of nonlinea systems descibed by a Takagi-Sugeno model with unmeasuable pemise vaiables This wok is an extension to nonlinea systems of the PI and PMI obseves developed fo linea systems The state estimation eo is witten as a petubed system Fist, the convegence conditions of the state estimation eos between the system and each obseve ae given in LMI (Linea Matix Inequality) fomulation Secondly, a compaison between the two obseves is made though an academic example I INTRODUCTION Model-based appoaches have been impotant and useful means to constuct a fault diagnosis module fo nonlinea systems in ode to detect, isolate and identify actuato, senso and system faults Geneally, the implementation of these functions is ealized with obseves Moeove, obseves povide an estimation of accessible and inaccessible states, outputs and faults of nonlinea systems The estimated signals ae used fo example to elaboate feedback contol laws, fault detection and isolation pocedue (FDI) and fault toleant contol (FTC) 8, The poposed wok focuses on the class of nonlinea systems descibed by Takagi-Sugeno models with unmeasuable pemise vaiables The T-S model povides a useful tool to epesent with a good pecision a lage class of nonlinea systems and can even descibe exactly cetain classes of nonlinea systems 4 by using the nonlinea secto tansfomation In the ecent yeas, consideable effots have been povided to study stability and stabilization of this class of systems 4,, 8, 5 The topic of state estimation has also been widely studied in many woks In, 9, 6,, the authos poposed diffeent methods in ode to estimate the state of T-S systems fo the pupose of diagnosis The main advantage of T-S stuctue is its simplicity because it oiginates fom the intepolation between linea systems Thus, analysis and design methods developed fo linea systems can be genealized to nonlinea systems as used in the woks cited above This eseach was suppoted by the TASSILI No 7 pogam unde MDU gant 74 The authos ae with Cente de Recheche en Automatique de Nancy (CRAN), Nancy-Univesité, CNRS, avenue de la foêt de Haye F-5456 Vandoeuve-les-Nancy {dalilichalal, benoitmax, joseagot, didiemaquin}@enseminpl-nancyf In the context of obust obseve design, one of the most successful technique is the use of PI obseve, in which the unknown inputs ae estimated simultaneously with the states of the system The PI obseve was fist poposed by Wojciechowsky in 5 fo single input-single output LTI systems A genealization scheme was pefomed by Kaczoek to multivaiable systems Theeafte, the PI obseve has been used in diffeent studies In a linea PI obseve is designed and applied to a physical system In 5 a PI obseve fo linea descipto systems is poposed Howeve, this obseve can be used 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 dynamic of the system In othe cases, this poblem can be solved by using multiple integals in the obseve in ode to estimate all of the deivatives of the unknown inputs A PMI obseve was fistly poposed by Jiang in In 7, a popotional multiple integal obseve is poposed to estimate a lage class of signals descibed in a polynomial fom fo LTI descipto systems An othe pape 9, pesented in the same confeence, deals with the state estimation using a new method consisting in the tansfomation of the TS system with unmeasuable pemise vaiables into an uncetain TS system with estimated pemise vaiables, in addition, the method is extended to estimate the unknown input using a PI obseve We popose, in this pape, a genealization of the PI and PMI obseves to nonlinea systems descibed by T-S models with unmeasuable pemise vaiables The pape is oganized as follows Section II pesents the T-S stuctue and the poblem of state estimation, and gives the motivation of this wok In section IIIA the design of PI obseve is addessed and in section IIIB the PMI obseve is studied Section IV pesents a numeical example with discussion about the pefomances of the two poposed obseves Finally, this note is ending with conclusions and pespectives The idea is based on two steps: the fist step consists to tansfom the TS system with unmeasuable pemise vaiable into a petubed TS system with estimated pemise vaiable The petubation tem is due to the unmeasuable pemise vaiable The second step is to make the system in an augmented fom by adding integatos to estimate the unknown input

2 II PRELIMINARIES AND PROBLEM STATEMENT A Multiple model appoach Conside 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 and f and h ae nonlinea functions The epesentation () is difficult to study, elsewhee in liteatue, all of the woks developed concening the nonlinea systems concen specific classes Fo example, in,, Lipschitz systems, which ae epesented by a linea pat and a nonlinea one, ae consideed The nonlinea pat is assumed to be Lipschitz with espect to the state x As mentioned in the intoduction, the T-S model appoach is a vey inteesting method to epesent nonlinea systems Diffeent methods exist to obtain a T-S model, as identification o lineaization of the system () aound diffeent opeating points o by using the nonlinea secto tansfomation The multiple model stuctue is given by: ẋ(t) = µ i (ξ(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Du(t)+Gd(t)+Wω(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 and G R q s, and W R q v The unknown inputs ae modeled by d(t) and ω(t) ae the noises affecting the state and the measuement equation In this stuctue, the output is assumed to be linea with egad to the state of the system The weighing functions µ i ae nonlinea and depend on the decision 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 model is simple and is consideed as a univesal appoximato since it can epesent any nonlinea behavio accoding to an adequate numbe of submodels The multiple model stuctue povides a mean to genealize the tools developed fo linea systems to nonlinea systems due to the popeties expessed in () B Poblem statement Diagnosis of nonlinea systems is often based on a bank of obseves to detect and isolate actuato and senso faults Fo designing obseves, it is often assumed, in the liteatue that the weighting functions µ i depend on measuable pemise vaiables u and/o y Thus, to pefom diagnosis, it is necessay to develop two diffeent multiple models The fist one whee the weighting functions depend only on the output of the system in ode to detect and isolate actuato faults The second one with weighting functions depending only on the input of the system in ode to detect and isolate senso faults To educe this difficulty, it is inteesting to develop only one multiple model using weighing functions which depend on the state of the system Thus, the same multiple model can be used to constuct obseve bank fo detecting and isolating both actuato and senso faults Howeve, the main difficulty is due to the fact that the state equation is now a nonlinea function of the state In the liteatue, only few woks ae developed fo obseve design 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 petubed o uncetain T-S system with measuable pemise vaiables III MAIN RESULT Along this pape, we assume that the following assumptions hold: A The system is stable A The signals u(t), d(t) and ω(t) ae bounded Pactically, these assumptions ae often not estictive A Extension of classical PI obseve Conside the following T-S fuzzy system with weighting functions µ i depending on the state of the system: ẋ(t) = µ i (x(t))(a i x(t)+b i u(t)+e i d(t)+w i ω(t)) y(t) = Cx(t)+Gd(t)+Wω(t) (4) In the next, fo sake of simplicity, the time vaiable t is omitted In this section, the unknown inputs d(t) ae assumed to be constant: A d = This assumption will be elaxed in the section IIIB The poposed PI obseve is given by the following equations: ˆx = µ i ( ˆx) ( A i ˆx+B i u+e i d ˆ+ K Pi (y ŷ) ) ŷ = C ˆx+G dˆ d ˆ = µ i ( ˆx)K (y ŷ) whee ˆx and dˆ ae the estimates of x and d In ode to facilitate the compaison between the system and its obseve, the system (4) can be witten as a petubed system with weighting functions µ i depending on the estimated state as follows: ν = ẋ = (5) µ i ( ˆx)(A i x+b i u+e i d +W i ω + ν) (6) (µ i (x) µ i ( ˆx))(A i x+b i u+e i d +W i ω) (7) This tem is seen as a bounded vanishing petubation to minimize Indeed, due to the assumptions A, A and the definition of the weighting functions (), ν(t) is bounded and if ˆx x then ν

3 The assumption A allows to make the system (6) in the augmented fom: ẋ a = µ i ( ˆx) ( Ã i x a + B i u+ Γ i ω ) (8) y = Cx a + D ω Ai E Ã i = i Bi, B i = I Wi, Γ i = C = C G, D = W,x a = x d ν, ω = ω A simila easoning makes it possible to tansfom the poposed PI obseve (5) in the following augmented fom: ˆx a = µ i ( ˆx) ( Ã i ˆx a + B i u+ K i (y ŷ) ) (9) ŷ = C ˆx a KPi K i = K Let us conside the augmented state estimation eo: e a = x a ˆx a () whose dynamics is given by: ė a = µ i ( ˆx) ( (Ã i K i C)e a +( Γ i K i D) ω ) () The goal is to detemine the gain matices K i of the obseve in ode to stabilize the system (), ie to guaantee the convegence of the state estimation eo towad zeo when the petubation ω is nul and to attenuate the tansfe gain fom the bounded petubation ω(t) to the state estimation eo e a (t) when ω(t) is diffeent fom zeo ( ω(t) is bounded since assumptions A and A ae satisfied) In ode to establish the existence conditions of the PI obseve in theoem, let us fist intoduce the following lemma: Lemma : 4 Conside the continuous-time TS-system defined by: ẋ(t) = µ i (x(t))(a i x(t)+b i u(t)) y(t) = Cx(t) () The system () is stable and veifies the L -gain condition: y(t) < γ u(t) if thee exists a symmetic positive definite matix P such that () is satisfied 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 detemined by minimizing γ unde the following LMI constaints in the vaiables P = P T >, M i and γ fo i =,,: ÃT i P+PÃ 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 deived fom K i = P M i and the attenuation level is calculated by γ = γ Poof: Accoding to the assumptions A and A, ω(t) is bounded Then, by applying lemma with e a (t) < γ ω(t), we obtain: ÃT i P+PÃ 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 < (5) The LMI fomulation in theoem is obtained by using the following changes of vaiables: M i = P K i, γ = γ Remak : The minimization of γ may esult in slow dynamics of the state estimation eo This poblem can be solved by pole assignment of the matices (Ã i K i C) in the left half complex plane defined by: {z Re(z) < λ}, λ > (6) Thus, the LMIs in theoem ae solved simultaneously with the following constaint (to impose Re(λ i ) < λ, whee λ i ae the eigenvalues of Ã i and λ > ): P(Ã i + λi)+(ã i + λi) T P M i C C T M T i < (7) Moe pecise pole clusteing can be obtained by adding LMI constaints 6 This appoach emains effective in pactical cases whee the assumption is not satisfied Howeve, the unknown inputs must vay slowly Othewise, bad state and unknown inputs estimation ae obtained by using this method In the next section, anothe method to estimate the state and the unknown inputs is poposed It is based on the popotional multiple integal obseve This obseve is inteesting because the assumption A is not equied in the theoetic poof, so it is possible to estimate a lage class of unknown inputs B Popotional multiple integal obseve Let us conside the multiple model with unmeasuable pemise vaiables descibed in (4) The unknown input is assumed to be a bounded time vaying signal with null q th deivative: A4 d (q) (t) =

4 Geneally, the use of a PI obseve equies the condition that the unknown input is constant (ie: d = ), thus, the unknown inputs which satisfies A4 cannot be estimated with a good pecision Then, PMI obseve is moe adequate fo this poblem, because the obseve estimates the (q ) th deivatives of the unknown input and gives a good pecision of the estimated unknown inputs Conside the genealization of the popotional multipleintegal obseve to T-S systems of the PMI obseve poposed in 7 fo linea descipto systems: ˆx = µ i ( ˆx)(A i ˆx+B i u+e i dˆ + K Pi (y ŷ)) ŷ = C ˆx+G dˆ d ˆ = µ i ( ˆx)K (y ŷ)+ dˆ ˆ d = ˆ d q = µ i ( ˆx)K (y ŷ)+ ˆ d µ i ( ˆx)K q (y ŷ)+ dˆ q (8) d ˆ q = µ i ( ˆx)K q (y ŷ) whee dˆ i, i =,,,(q ) ae the estimation of the (q ) fist deivatives of the unknown input d(t) The state and unknown inputs estimation eos ae: e = x ˆx, e = d d ˆ,, e q = d q d ˆ q Thei dynamics 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 = µ i ( ˆx)( K q Ce K W ω K q Ge ) Γ i = I n W i, W = W (9) The equations (9) can be ewitten in the following augmented fom: e e ẽ = µ i ( ˆx)((Ã i K i C)ẽ+( Γ i K i W) ω) () = Cẽ () 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 inteested with paticula component e and e of ẽ: e = Cẽ () e C = In I s epesents null matix with appopiate dimensions Theoem : The PMI obseve (8) fo the system (8) that minimizes the tansfe fom ω(t) to e(t) T e (t) T is obtained by finding the matices P = P T >, M i and γ that minimize γ unde the following LMI constaints fo i =,,: ÃT i P+PÃ 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 deived fom: and the attenuation level is calculated by: < () K i = P M i (4) γ = γ (5) Poof: The poof of theoem is simila to the poof of theoem by using the lemma with the system () Remak : When the condition A is not satisfied ie d (q) but d (q) is bounded then, we can conside the q th deivative of d(t) as a petubation The new petubation vecto is then given by: ω(t) = ν(t) T ω(t) T d (q) (t) T T The additional component d q is added in the state vecto The matices Ã i, Γ i, W, C ae augmented Then, the Theoem can be applied in ode to design the Popotional Multiple Integal Obseve with minimization of the new bounded petubation ω(t)

5 IV NUMERICAL EXAMPLE AND SIMULATIONS In this section, the poposed method is illustated though an academic example Conside a continuous-time T-S system (4) defined by: and 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 input vecto d(t) is made up of d (t) which affects the outputs of the system and d (t) affecting the dynamics of the system (see the matices E, E and G) Fo example, we can conside d as a senso fault and d as an actuato one The weighting functions depend on the fist component x of the state vecto x and ae defined as follows: { µ (x) = tanh(x ) (6) µ (x) = µ (x) The weighting functions obtained without petubations and unknown inputs ae shown in figue This figue shows that the system is clealy nonlinea since µ and µ ae not constant functions,,, oiginal d estimated d Fig Unknown input estimation with a PI obseve oiginal d estimated d oiginal d estimated d oiginal d estimated d 8 µ (t) µ (t) Fig Unknown input estimation with a PMI obseve Fig Weighting functions µ and µ The petubation ω is chosen as andom signal unifomly distibuted in 5 5 The consideed unknown inputs d (t) and d (t) ae time vaying signals with neglected fouth deivatives Afte synthesizing a PI obseve accoding to the theoem and a PMI obseve with q = 4 accoding to the theoem, we obtain the simulation esults depicted in the figues,, 4 and 5 Figues and show the unknown input and thei estimations with PI and PMI obseves It is known that the PI obseve gives an acceptable state and unknown inputs estimation even if the assumption A is not satisfied Howeve, in this example, the unknown inputs have fast vaiations esulting on bad state and unknown inputs estimation (figues and 4) compaed to the esults given by the PMI obseve (figues and 5) V CONCLUSIONS AND FUTURE WORKS The design of popotional integal (PI) and popotional multiple integals (PMI) ae studied in this pape This wok is an extension of the PI and PMI obseves developed fo linea systems to nonlinea T-S systems with unmeasuable pemise vaiables The convegence conditions of the state

6 Fig 4 State estimation eo with a PI obseve Fig 5 State estimation eo with a PMI obseve estimation eo ae given in the LMI fomulation The obseves ae obust since they ae synthesized in ode 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 and it is less sensitive to noises compaed to the PMI obseve 7 In the othe hand, PMI obseve is a good way to obtain a moe pecise estimation of states and unknown inputs The futue woks will concen, fistly, the impovement of the PMI obseve by intoducing a stable fequency weighting functions on the petubations ω(t) which allows to eflect the expected fequency content of ω(t), secondly, the use of these obseves in nonlinea system diagnosis REFERENCES M Abbaszadeh and H Maquez, Robust H obseve design fo a class of nonlinea uncetain systems via convex optimization, in Ameican Contol Confeence, ACC 7, 7 A Akhenak, M Chadli, J Ragot, and D Maquin, Design of sliding mode unknown input obseve fo uncetain Takagi-Sugeno model, in 5th Mediteanean Confeence on Contol and Automation, MED 7, Athens, Geece, 7 P Begsten, R Palm, and D Diankov, Fuzzy obseves, in IEEE Intenational Fuzzy Systems Confeence, Melboune Austalia, 4, Obseves fo Takagi-Sugeno fuzzy systems, IEEE Tansactions on Systems, Man, and Cybenetics - Pat B: Cybenetics, vol, no, pp 4, 5 M Chadli, D Maquin, and J Ragot, Non quadatic stability analysis of Takagi-Sugeno systems, in IEEE Confeence on Decision and Contol, CDC, Las Vegas, Nevada, USA, 6 M Chilali and P Gahinet, H-infinity design 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 and D Ho, Popotional multiple-integal obseve design fo descipto systems with measuement output distubances, IEE poceeding Contol theoy and application, vol 5, no, pp 79 88, 4 8 T Guea, A Kuszewski, L Vemeien, and H Timant, Conditions of output stabilization fo nonlinea models in the Takagi-Sugeno s fom, Fuzzy Sets and Systems, vol 57, no 9, pp 48 59, May 6 9 D Ichalal, B Max, J Ragot, and D Maquin, State and unknown input estimation fo nonlinea systems descibed by Takagi-Sugeno models with unmeasuable pemise vaiables in 7th Mediteanean Confeence on Contol and Automation, MED 9, Thessaloniki, Geece, June R Isemann, Fault-diagnosis systems: An intoduction fom fault detection to fault toleance, Spinge, Ed, 7 G Jiang, S Wang, and W Song, Design of obseve with integatos fo linea systems with unknown input distubances, 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 design fo linea descipto systems: application to state and fault estimation, IEEE Tansactions on Automatic Contol, vol 5, no, pp 7, 5 4 A Kuszewski, Lois de commande pou une classe de modèles non linéaies sous la fome Takagi-Sugeno : Mise sous fome LMI, PhD dissetation, Univesité de Valenciennes et du Hainaut-Cambesis, 6, (In fench) 5 B Max, D Koenig, and D Geoges, Robust fault diagnosis fo linea descipto systems using popotional integal obseves, in 4nd IEEE Confeence on Decision and Contol, 6 R Palm and P Begsten, Sliding mode obseves fo Takagi-Sugeno fuzzy systems 9th IEEE Intenational Confeence on Fuzzy Systems, FUZZ IEEE, San Antonio, TX, USA, 7 R Palm and D Diankov, Towads a systematic analysis of fuzzy obseves, in 8th NAFIPS Confeence, New Yok, NY, USA, R Patton, P Fank, and R Clak, Fault diagnosis in dynamic systems: Theoy and application Pentice Hall intenational, R Patton, J Chen, and C Lopez-Toibio, Fuzzy obseves fo nonlinea dynamic systems fault diagnosis, in 7th IEEE Confeence on Decision and Contol, Tampa, Floida USA, 998 G Pechoto de Melo and T Souza Moais, Fault detection using state obseves with unknown input, identified by othogonal functions and PI obseves, Bazilian Society of mechanical sciences and Engineeing, 7 R Rajamani, Obseves fo Lipschitz nonlinea systems, IEEE Tansactions on Automatic Contol, vol 4, pp 97 4, Mach 998 T Takagi and M Sugeno, Fuzzy identification of systems and its applications to modeling and contol, IEEE Tansactions on Systems, Man, and Cybenetics, vol 5, pp 6, 985 K Tanaka, T Ikeda, and H Wang, Fuzzy egulatos and fuzzy obseves: Relaxed stability conditions and LMI-based designs, IEEE Tansactions on Fuzzy Systems, vol 6, no, pp 5 65, K Tanaka and H Wang, Fuzzy Contol Systems Design and Analysis: A Linea Matix Inequality Appoach, J Wiley and Sons, Eds, 5 B Wojciechowski, Analysis and synthesis of popotional-integal obseves fo single-input-single-output time-invaiant continuous systems, PhD dissetation, Gliwice, Poland, J Yoneyama, H output feedback contol fo fuzzy systems with immeasuable pemise vaiables: Discete-time case, Applied Soft Computing, vol 8, no, pp , Ma 8

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

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