State tracking control for Takagi-Sugeno models

Transcription

1 State tacing contol fo Taagi-Sugeno models Souad Bezzaoucha, Benoît Max,3,DidieMaquin,3 and José Ragot,3 Abstact This wo addesses the model efeence tacing contol poblem It aims to highlight the encouteed difficulties and the poposed solutions to achieve the tacing objective fo nonlinea systems descibed by Taagi-Sugeno (T-S) models Diffeent contol stategies ae exposed Exact state tacing is poposed and stuctual conditions fo it ae given Appoximate state tacing is also studied The choice of the efeence model to be taced is discussed, as well as the citeion to be minimized to achieve given tacing objectives Intoduction and pape outline The main objective of this wo is to deal with efeence model tacing fo nonlinea systems descibed by Taagi-Sugeno (T-S) models The T-S models ae nown to be an efficient way to deal with the poblems of estimation and contol of nonlinea systems by witing them in a polytopic fom Oiginally intoduced by [], the T-S epesentation allows to exactly descibe nonlinea systems, povided that the nonlineaities ae bounded This is easonable since state vaiables as well as paametes of physical systems ae bounded, and so is the input of the system which may be consideed natually stable, o equipped with a stabilizing contol (see fo example [] and the efeences theein) Despite an abundant liteatue on stability conditions of T-S models, few authos have dealt with the tacing poblem fo all the state of the system; indeed most of the wos deal with the tacing of the system output, which geneally educes to some combinations of these vaiables One can efe to some wos concened with Univesity of Bodeaux, IMS lab-cnrs, Automatic contol goup, 35 cous de la libeation, 3345 Talence, Fance souadbezzaoucha@ims-bodeauxf UnivesitédeLoaine,CRAN,UMR739,avenuedelaFoêt de Haye, Vandoeuve-lès-Nancy Cedex, 5456, Fance 3 CNRS, CRAN, UMR 739, Fance s:benoitmax, didiemaquin, state o output feedbac with H pefomances [3], [4] and [5] The nonlinea tacing contol poblem is expessed in tems of Linea Matix Inequality (LMI) and is based on the T-S and Paallel Distibuted Compensation (PDC) stuctues (an L tacing pefomance elated to the tacing eo is fomulated and a PDC state feedbac contol (o output feedbac contol) is designed See [5], [4] fo examples) Howeve, in the cited efeences, a efeed suitable choice fo the efeence model is made without any explanations no details The last ema motivated the pesent study In fact, eithe fo the linea case o the nonlinea one, few wos detail the influence of the efeence model choice, which is not a tivial tas In [6] fo example, the authos efeed to the Ezbege s conditions, but with no futhe explanations Fo these easons, in the poposed wo, a focus is made not only on the design contol pocedue, but also on the tacing (matching) conditions The pape is oganized as follows In section II, the stuctual conditions to achieve state tacing ae intoduced in the T-S case These conditions ae an extension of the well nown Ezbege s conditions Two pocedues fo state tacing ae consideed: fo the fist one, the contolle stuctue is fistly fixed and then the stuctual conditions and the appopiate gains of the contolle ae deduced Fo the second one, no peequisited contol law is consideed, with the objective to achieve a null tacing eo fo both stategies In section III, the quadatic optimal contol with an extension to the Model Pedictive Contol fo T-S models ae pesented Simulation examples ae given in section IV Finally, section V summaizes the obtained esults Stuctual conditions fo exact state tacing fo T-S systems Model and objective Let us conside the following T-S model []: x + = A x + B u ()

2 st x R n x and u R n u with: A = µ i, (ξ )A i, B = µ i, (ξ )B i () whee B is supposed to be a full column an matix and whee the weighting functions µ i, (ξ ) depend on the so-called pemise vaiable ξ which may be a state, input, o output combination These weighting functions satisfy the following convex sum popety: µ i, (ξ ), µ i, (ξ )= (3) The consideed linea efeence model is the following: x,+ = A x, + B u, (4) st x, R n x, u, R n u and whee the desied pefomances ae defined by the choice of the matices A and B The ideal tacing objective is to adjust, at each instant, the contol u in such a way that the system state x follows the efeence model state x, with a null tacing eo Fo this pupose, two pocedues may be consideed The fist stategy consists in setting the contolle stuctue and then deduce the appopiate stuctual conditions that must satisfy the efeence model and contolle gains The second stategy is not based on a peequisited contol law stuctue The idea is to find, as fo the fist stategy, the appopiate stuctual conditions, but also an analytical expession fo the contol law If the ideal tacing is not eachable, some compomises need to be defined such as, fo example, tacing of a subset of the states, instead of all of them Peequisited contol law In ode to achieve the tacing objective, the following contol law is consideed: u = K x + K, u, (5) Substituting (5) into (), the closed-loop system is: x + =(A + B K ) x + B K, u, (6) The matching conditions fo the efeence model and the system ae then obtained by compaing the closedloop system (6) and the efeence model (4) They ae given by: { A + B K = A B K, = B (7) Fom (7), in ode to have a solution in espect to the gain K and K,,thefollowinganconditionshaveto be fullfiled: { an[b ] = an[b A A ] an[b ] = an[b B ] (8) whee B and A ae defined in () If conditions (8) ae fulfilled, then at each sampling time, the gains K and K, ae given by: K, = B + B, K = B + (A A ) (9) with B + asuitablepseudo-invesematixofthefullcolumn an B matix Note that in ode to satisfy the matching conditions (8), fom definitions (), since the system matices A and B depend on the time, one sufficient, but not unique, condition is to conside the matices A i, B i and A, B in the following canonical fom: A i = A A,A = A i A ) ) () B i = ( nx n u b i,b = ( nx n u with A amatixofdimension(n x n u ) n x, A i and A matices of dimensions n u n x b i and b ae of dimension n u n u Thestuctue()meansthat: the (n x n u ) fist ows of the matices A i ae equal to the (n x n u ) fist ows of the matix A the (n x n u ) fist ows of the matices B i ae null 3 the (n x n u ) fist ows of the matix B ae null allowing to fully satisfy the an conditions (8) It is impotant to note that the matching conditions (8) between the efeence model and the system depend on the choice of the contol law, u given by (5) It means that these conditions have to be adapted when changing the stuctue of the contol law 3 Numeical example To illustate the above conditions, let us conside the following academic example: 5 A = 99,B = 5 A = 99,B = 5 b

3 5 A = 99,B = 5 () The weighting functions ae taen as: µ, = sin(x,) tanh(x, ), µ, = µ, () 4 Applying the tacing contol law (5) with (9), the system and model efeence states ae depicted in figue (espectively noted x i and x i, i =,,3) In figue, the contol inputs u, and u ae epesented the contol law u has to veify: B u = x,+ A x (4) Note that the exact state tacing is ensued, ie (3) has asolutioninespecttou, only if the following matching condition is fulfilled at each sampling time: an[b ]=an [ ] B x,+ A x (5) The tacing contol is deduced and given by the following equation: u =(B T B ) B T (x,+ A x ) (6) x x and the eade is invited to compae the two stuctues (6) and (5) with the help of (9) Figue System and model efeence states u Figue Contol inputs u, and u Fom the depicted figues, one can see that the contol tacing is efficient fo all the thee states unde the specified stuctual conditions 4 No peequisited contol law In this subsection, no peequisited stuctue of the contol law is consideed Then, at time, in ode to achieve the tacing objective at time + with a null state tacing eo between (4) and (), meaning: x,+ = A x + B u (3) x x x 3 x 3 u Rema If the pemise vaiables ξ of the weighting functions µ i, depend on the input u,thecontollaw (6) will be implicit (ie u = F(u )) since A and B ae input depending () A solution may be given by an iteative algoithm with the following ecuence: ( ) u ( j+) = ( )T ( with = µ i, (u ( j) )B i,a ( j) = ) T (x,+ A ( j) x ) (7) µ i, (u ( j) )A i, j =,,NwithNthenumbeofiteationsandu the input initialization (may be taen as u, fo example) The convegence of this algoithm may be poved locally, but it will not be consideed in the pesent wo since it is not the pupose of the study (see [7], [8], [9] fo moe details) As mentioned in the beginning of this section, to achieve the exact tacing, the system and model efeence must be in a cetain canonical fom (stuctual conditions) Depending on the efeence and system models, these conditions may be impossible to meet and thus must be elaxed One way is to apply the tacing to some states only, but it can affect the functioning of the system since othe states ae left fee A bette way is to use an appoximate tacing of all the states 3 Appoximate state tacing fo T-S systems In this section, the optimized tacing using the nom of the tacing eo is addessed This pat concens the optimal contol with the intoduction of the MPC fo T-S models As it was mentioned above, due to the estictiveness of the exact tacing stuctual conditions, one can conside anothe appoach to elax these

4 stuctual conditions This appoach is the so-called quadatic optimal contol fo T-S models and aims at minimizing the tacing eo This is a natual way to deal with the poblem by optimizing the tacing eo Indeed, it is less difficult and consevative to deal with a minimization poblem than an equality constaint such as expessed by the stuctual conditions of the pevious section 3 Contol law design At each time instant, theobjectiveistominimize the following citeion which is the nom of the tacing eo: Φ (u )= B u x,+ + A x W (8) whee W is a positive definite weighting matix chosen accodingly to the state components fo which some specific tacing is desied The contol tacing law is then given by: u =(B T WB ) B T W(x,+ A x ) (9) whee the matices A and B have been aleady defined in () Rema When the pemise vaiables ξ depend on the contol as explained peviously (ema ), the same pocedue goes fo the contol law (9) with the iteative esolution, it becomes fo j =,,N: u ( j+) =(( )T W ) ( )T W (x,+ A ( j) x ) () with = µ i, (u ( j) )B i,a ( j) = µ i, (u ( j) )A i The poposed contol tacing law u (9) (ie () fo input dependent pemise vaiables) aims to ensue the tacing of x,+ by x + at each time instant Since the contolle has no ability to anticipate futue events, it can not tae contol actions accodingly and the tacing pefomances may degade Based on this statement, the MPC tacing is intoduced fo the T-S models In fact, the main advantage of MPC is to allow the cuent time slot to be optimized, while eeping futue time slot in account This is achieved by optimizing a finite time hoizon, but only implementing the cuent time slot As fo linea and nonlinea model efeence tacing contol, the MPC equies iteative solution and aims to ensue that the tacing eo is minimized on a finite sliding hoizon Roughly speaing, the pocedue fo T-S models is the same as fo the conventional MPC Howeve, some difficulties occu when the pemise vaiables depend on the contol 3 Pemise vaiables independent of the input Consideing a finite hoizon of p + steps,using the state equation x + = A x + B u () it follows the state expession at time + p + whee x +p+ = A +p x +p + B +p u +p Φ,l = +p = Φ +p, x + { l= Φ +p,l+ B l u l () A A l if > l I if = l (3) with A and B defined by () Gatheing the states on the time hoizon [ + : + p + ], letusnote: A x + u x +,p = B,p = x + x +p+ p i=, u,p = u + u +p A + A,A,p = p A +p i i= (4) B A + B B + p A +p i B A +p i B + B +p i= Using () and (4), the state x is witten as: x +,p = A,p x + B,p u,p, x,p R n(p+) (5) To ensue the efeence model tacing on the time hoizon [ +: + p+],thecontolu,p is ajusted in ode to minimize the citeion: Φ,p (ū,p )= x,,p A,p x B,p u,p W (6) with x,,p = [ x,+ ] T x,+p+ R n(p+) This leads to: u,p =(B T,p WB,p) B T,p W(x,,p A,p x ) (7) whee the input at the step defined by: u = [ I nu ] ū,p (8) is then applied to the system Fo the next step, the hoizon is moved and the citeion Φ +,p is optimized in ode to obtain and apply the contol u + As explained befoe, the contol u,p (in paticula u ) is calculated while eeping futue time slot in account, which explains the anticipative chaacte of the MPC contol

5 33 Extension to pemise vaiables dependent of the input Since the weighting functions of the matices A,p and B,p (4) may depend on the contol ū,p,instead of the analytical solution (7), the following iteative algoithm is poposed: define a theshold δ fo j =, define u ( j) j ),p and u(,p 3 compute A ( j),p and B( j),p 4 while u ( j) j ),p u(,p > δ ( ) u ( j+),p = (,p )T W ( j),p (B,p )T W(x A ( j),p x ) (9) j j +, compute A ( j+),p and B ( j+),p with: A ( j),p =,p = = A ( j) +A( p i= p i= A ( j) +p i,a ( j) = µ i, (u ( j) ())A i +B( +p ib( p i= + +p ib( + B( j) +p µ i, (u ( j) ())B i (3) Afte the algoithm convegence, the contol input at the step is defined by: u = [ I nu ] u,p (3) and is applied to the system Fo the next step, the hoizon is moved and the citeion Φ +,p is optimized in ode to obtain u + 4 Numeical example In ode to illustate the poposed appoach, let us conside the following numeical example: 5 9 A = A = A = B = 3 3, B = B = The weighting functions ae input dependent and given by: µ, = + tanh(u,) (3) µ, = µ, One can veify that the exact tacing conditions (8) ae not fulfilled Fo this eason, the MPC is pefomed fo thee steps fowad (p = ) with iteative esolution (3) Fist, the objective is to ensue a good tacing of the fouth and fifth state components Consequently, the weighting matix is chosen as W = diag(,,,,), implying a elaxation of the tacing of the fist thee state components The system and efeence model states ae depicted in figue 3 (espectively noted x i

6 and x i,i =,,5) Infigue4,thecontolinputsu, and u ae epesented Fom the depicted figues, one can see that the contol tacing is efficient (especially fo the fouth and fifth states) although the stuctual conditions ae not fulfilled Secondly, if the main objective is an accuate tacing x,x, x,x, x 3,x, x 4,x, x 5,x, x,x, x,x, x 3,x, x 4,x, x 5,x, Figue 5 System and efeence model states fo W = diag(,,,,) Contol u Figue 3 System and efeence model states fo W = diag(,,,,) Contol u Contol u Contol u Figue 4 Contol inputs u () and u() of x 3 by x 3,oneshouldsettheweightingmatixas W = diag(,,,,) The obtained esults ae displayed in figue 5 fo the system and model efeence states and in figue 6 fo the contol inputs u, and u One can obseve that fo this case, the thid state tacing has been impoved when the fouth and fifth states tacing have been slightly deteioated In ode to quantify the impovement due to the pedictive contol, let us conside the following citeion: φ i = N = x,,i x,i Figue 6 Contol inputs u () and u() whee i the component numbe of a vecto (i =,,5), N is the simulation hoizon and x,i is obtained fom non pedictive contol (6) φ ip is analogously defined with x,i obtained with MPC: φ ip = N = x,,i x p,i Finally, the pefomance gain τ i due to MPC is obtained fom τ i = φ i φ ip φ i Fo the consideed example (W = diag(,,,,)), we obtain the following impovement (fo each state): τ = 54%, τ = 99%, τ 3 = 5%, τ 4 = 647%, and τ 5 = 456% In ode to highlight the influence of the time hoizon length p+onthetacingpefomances,theimpovement of φ i,namelyτ i, is computed fo time hoizon chaacteized by p {,,3,4} The esults ae gatheed in table I One can conclude, fo the pesented example, that a

7 p = p = p = 3 p = 4 τ 76% 54% 899% 4% τ 48% 99% 8% 885% τ 3 38% 5% 6% 767% τ 4 469% 647% 775% 8% τ 5 6% 456% 364% 48% Table hoizon of length p = 4 gives the best esults It is also impotant to highlight that depending on the dynamic chaacteistic of the efeence on a time hoizon, a too shot, as well as a too long hoizon may not give the best expected esults A compomise is then needed To quantify the best hoizon length, a compaative study as the one pesented may be a good solution [4] C-S Tseng, B-S Chen, and H Uang, Fuzzy tacing contol design fo nonlinea dynamic systems via T- S fuzzy model, IEEE Tansactions on Fuzzy Systems, vol 9, no 3, pp 38 39, [5] B Mansoui, N Manamanni, K Guelton, A Kuszewsi, and T Guea, Output feedbac LMI tacing contol conditions with H citeion fo uncetain and distubed T-S models, Infomation Sciences, vol 79, no 4, pp , 9 [6] C Nie and R Patton, Fault estimation and MRC-based active FTC, in 8 th IFAC Wold Congess, Milano, Italy, [7] E L Allgöwe and K Geog, Simplicial and continuation methods fo appoximating fixed points and solutions to systems of equations, Society fo Industial and Applied Mathematics Revue, vol, pp 8 85, 98 [8] O Axelsson, Iteative Solution Methods, C U Pess, Ed Cambidge, Austalia: Cambidge Univesity Pess, 994 [9] C T Kelley, Iteative Methods fo Linea and Nonlinea Equations Philadelphia, Pennsylvanie, USA: Society fo Industial and Applied Mathematics, Conclusion In this pape, the tacing objective fo nonlinea T- SmodelwasconsideedStuctualconditionsfopefect tacing wee established, as well as the quadatic optimal contol A Model Pedictive Contol fo the T- Scasewithfinitetimehoizonwasalsodevelopedand the pediction influence was highlighted via a numeical example Afistpespectivefothepesentwoistogenealize the matching conditions and the stuctue poposed in () fo a geneal contol stuctue law and establish the elation between the efeence model (A,B ) and the system matices (A i,b i ) Duing the study, a stong coelation between the time hoizon length and the model efeence dynamics was pointed, in fact, a second inteesting pespective will be to pesent a choice citeion that optimizes the time hoizon length accoding to the model efeence dynamics Refeences [] T Taagi and M Sugeno, Fuzzy identification of systems and its applications to modeling and contol, IEEE Tansactions on Systems, Man and Cybenetics, vol 5, no, pp 6 3, 985 [] K Tanaa and H Wang, Fuzzy Contol Systems Design and Analysis: A Linea Matix Inequality Appoach John Wiley & Sons, Inc, [3] T Taniguchi, K Tanaa, K Yumafuji, and H Wang, A new PDC fuzzy efeence models, in IEEE Intenational Fuzzy Systems Confeence, Seoul, South Koea, 999