TRACKING CONTROL DESIGN FOR A CLASS OF AFFINE MIMO TAKAGI-SUGENO MODELS

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1 TRACKING CONTROL DESIGN FOR A CLASS OF AFFINE MIMO TAKAGI-SUGENO MODELS Carlos Arño Deparmen of Sysems Engneerng and Desgn, Jaume I Unversy, Sos Bayna S/N, Caselló de la Plana, Span arno@esdujes Anono Sala, Jose Lus Navarro Deparmen of Sysems Engneerng and Conrol, Unversdad Polcnca de Valenca, Camno de Vera, 4, Valenca, Span asala@saupves, jlnavarro@saupves Keywords: Absrac: Fuzzy conrol, affne Takag-Sugeno models, local models, LMI, sepon change When conrollng Takag-Sugeno fuzzy sysems, verfcaon of some secor condons s usually assumed However, sepon changes may aler he secor bounds Alernavely, sepon changes may be consdered as an offse addon n many cases, and hence affne Takag-Sugeno models may be beer sued o hs problem Ths work dscusses a nonconsan change of varable n order o carry ou offse-ellmnaon n a class of MIMO canoncal affne Takag-Sugeno models Once he offse s cancelled, sandard fuzzy conrol desgn echnques can be appled for arbrary sepons The canoncal models suded use as sae represenaon a se of basc varables and her dervaves Some examples are ncluded o llusrae he procedure INTRODUCTION In he las decade, desgn of fuzzy conrollers based on he so-called Takag-Sugeno TS models (Takag and Sugeno, 985) has reached maury (Sala e al, 5) TS models express he behavour of a sysem va a convex nerpolaon of local (homogeneous) lnear models, where he nerpolaon funcons are fuzzy membershp funcons wh and add- condons In parcular, desgns usng he Lnear Marx Inequaly framework (Tanaka and Wang, ; Guerra and Vermeren, 4) have become wdespread Par of he success of such echnques s due o he exsence of sysemac mehodologes for TS fuzzy denfcaon (Takag and Sugeno, 985; Tanaka and Wang, ; Babuska, 998; Nelles e al, ) One parcular characersc of he mansream TS conrol desgn framework s ha all of he local models mus share he same equlbrum pon, usually se o x = for convenence Ths s no a severe problem, as he denfcaon procedures above referred need only be appled wh a consan change of varable, used n he conex of Taylor lnearsaon n conrol desgn for decades Once ha change of varable s carred ou, global sably and performance regardng reachng x = from any nal condons can be proved The reader s referred o (Tanaka and Wang, ; Guerra and Vermeren, 4) for deals on he mehodology An affne srucure for TS models was also orgnally addressed n (Takag and Sugeno, 985), whch consders local models whou he shared equlbrum pon Ths srucure, o be denoed as Takag- Sugeno-Offse (TSO) may orgnae eher drecly from he denfcaon process or when consderng rackng asks wh varyng sepons n ordnary TS models Indeed, n he laer case, he change of varable needed o ransform he new operang pon no x = should nvolve changng he shape of he membershp funcons and he parameers of he local models Oherwse, he resulng models lose he shared equlbrum pon The above menoned conrol mehodologes mus be adaped o TSO models Some deas appear n (Km and Km, ; Johansson, 999), where quadrac Lyapunov funcons and S-procedure LMIs (Boyd e al, 994) are used o prove sably of he orgn However, sepon changes are no consdered, and he dvson no ellpsodal zones of he operang regme resuls n a cumbersome procedure whch nvolves consderng he dfferen regons of overlap of he aneceden membershp funcons As an alernave, hs paper presens a parcular 48

2 TRACKING CONTROL DESIGN FOR A CLASS OF AFFINE MIMO TAKAGI-SUGENO MODELS class of TSO fuzzy models semmng from a canoncal represenaon whch, bascally, akes he conrolled oupus and s dervaves as he chosen se of sae varables Ths canoncal represenaon has a clear physcal nsgh and sems from usual canoncal forms n lnear and nonlnear sysems (Ansakls Panos and Mchel Anhony, 997; Slone and L, 99) Once he canoncal TSO models are nroduced, an offse-removng ransformaon for sae feedback s dscussed, whch s he man resul of hs work The offse-removng ransformaon wll allow o express he orgnal TSO model as an ordnary TS form on he ransformed varables, enablng sandard fuzzy TS conrol desgn echnques o be used n such a sysem The srucure of he paper s as follows Secon wll presen he defnons for he canoncal TSO framework Secon 3 wll presen resuls on he equlbrum pon of he canoncal represenaon and he man offse-removng ransformaon Some examples llusrang he approach wll be gven n Secon 4, and a concluson secon wll close he paper CANONICAL TSO MODELS In hs secon, basc defnons of he fuzzy sysems under sudy wll be presened, whch generalse he classcal Takag-Sugeno fuzzy sysem used n mos curren desgn echnques (Tanaka and Wang, ; Guerra and Vermeren, 4; Sala e al, 5) descrbed by ẋ = n = µ (z)(a x+b u) µ (z) = () where z s assumed o be a se of accessble varables whch may nclude some or all of hose ones comprsng he sae vecor x plus exernal schedulng ones, and µ (z) are denoed as aneceden membershp funcons Defnon Canoncal local model wh offse Le us have a sysem wh p npus, p oupus and n saes defned by: ẋ = A x+b u+r y = C x () where he sae vecor s assumed o be paroned accordng o he followng srucure: x = [x x x r x x x r x p x p x prp ] T (3) e, p blocks of sze r,, r p respecvely, r + + r p = n, compable wh he block srucure n marces A, B, C, R o be descrbed below Le us defne an auxlary marx wh dmenson q (q ) as: T q = [ (q ) I q ] (4) where I q denoes he deny marx wh sze (q ) (q ) and (q ) he zero marx wh sze (q ) Also, he noaon [ ] s wll denoe a marx wh dmenson s wh arbrary elemens Then, marces n () have he srucure: A = T r (r ) r (r ) r p [ ] n (r ) r T r (r ) r p [ ] n (rp ) r (rp ) r p T r [ ] n [ ] p (r ) B = R = (r ) p [ ] p (r ) p [ ] p (rp ) p [ ] p C = [ ] p (r ) (r ) p [ ] p (r ) p [ ] p (rp ) p [ ] p (5) (6) [ ] p (rp ) (7) (8) Noe The above sysem srucure s smlar o he well-known reachable canoncal form (Ansakls Panos and Mchel Anhony, 997) For nsance, a canoncal SISO sysem: A = a a a 3 a q B = [ b ] T (9) () 49

3 ICINCO 7 - Inernaonal Conference on Informacs n Conrol, Auomaon and Robocs C = [ c c 3 c q ] R = [ r ] T conforms o he above srucure () () Defnon Canoncal Fuzzy Takag-Sugeno-Offse model A Fuzzy canoncal Takag-Sugeno-Offse model wll be defned accordng o he followng srucure: ẋ = y = m m µ (z)(a x+b u+r ) µ (z)c x (3) where each of he componen models has marces A, B, R y C whch follow he srucure n Defnon and µ (z) are he membershp funcons, whch are assumed o verfy µ (z) = The noaon below wll be used as shorhand for fuzzy summaons Ω(z) = µ (z)ω (4) Then, he fuzzy sysem n Defnon may be wren as: ẋ = Ã(z) x+ B(z) u+ R(z) y = C(z)x (5) by usng Ã(z) = n µ (z) A, B(z) = n µ (z) B, ec 3 OFFSET-ELLIMINATION PROCEDURE Proposon Gven a sysem wh he srucure n Defnon (), for consan npus u = u eq, he equlbrum values of he sae varables verfy j = =,, p j =,,r (6) Proof: Wh u = u eq, he equlbrum equaon s: = A + Bu eq + R (7) Usng he canoncal marx srucure, x x r verfy: (r ) = I (r ) (r ) r + (r ) p u eq u eq p he saes + (r ) (8) Hence, = I (r ) (r ) fnally obanng (6) r =,, p (9) Proposon Gven a sysem wh he srucure n Defnon, wh consan npu u = u eq, he equlbrum values for he sae vecor and he oupu are relaed, n he form: = [y eq (r ) y eq (r ) y eq p (r )] T y eq = [y eq y eq p ] T Proof: Replacng n he oupu equaon, y eq = C () gven he srucure of C (7) and he resuls from he prevous proposon, sang ha only he saes correspondng o he columns of he deny may be nonzero, we have: and fnally, y eq = p () = [y eq (r ) y eq (r ) y eq p (rp )] T Lemma Gven a canoncal fuzzy sysem wh he srucure n Defnon, e, ẋ = Ã(z) x+ B(z) u+ R(z) y = C(z)x () defnng an auxlary npu u es (z,y re f ) = ( C(z)Ã(z) B(z)) ( y re f C(z)Ã(z) R(z)) (3) under suable nverbly assumpons, and carryng ou he change of varable where and ˆx = x x re f (4) û = u u es (z,y re f ) (5) x re f = [y re f, (r ) y re f, (r ) y re f,p (r )] T (6) y re f = [y re f, y re f,p ] (7) 5

4 TRACKING CONTROL DESIGN FOR A CLASS OF AFFINE MIMO TAKAGI-SUGENO MODELS s a user-defned vecor, hen ˆx =, û = s he new equlbrum pon for he sysem n he varables ˆx, û and, moreover, he ransformed sysem has he equaons: ˆx = µ (z)(a ˆx+B ˆx) (8) e, s a sandard Takag-Sugeno fuzzy sysem (), where he offse erms have dssapeared The varables ˆx, û wll be denoed as ncremenal Proof: Le us denoe by S z he local affne sysem formed by he resul of evaluang Ã, B, C and R a a parcular (arbrary) pon z : ξ = Ã(z ) ξ+ B(z ) u+ R(z ) y = C(z )ξ (9) whch has he canoncal srucure of defnon () Le us compue he npu u es = u es (z,y re f ) so ha oupu y re f s an equlbrum pon for he above sysem S z, by usng he equlbrum equaon: = Ã(z ) ξ re f + B(z ) u es + R(z ) (3) y re f = C(z ) ξ re f (3) On he followng he dependence on z wll be omed for noaonal smplcy Carryng ou some operaons, B u es = Ã ξ re f R CÃ B u es = Cξ re f CÃ R = y re f CÃ R u es = ( CÃ B) ( y re f CÃ R) (3) Le s now oban he equlbrum sae ξ re f Indeed, as u es (z,y re f ) ensures ha he oupu y re f s an equlbrum pon, and as S z has he srucure, hen Proposon ensures ha ξ re f s: ξ re f = [y re f, r y re f, r y re f,p r ] T dencal o he sae x re f defned n (6) As z n (3) s an arbrary pon, hen = Ã(z) x re f + B(z) u es (z,y re f )+ R(z) z (33) Carryng ou he change of varable ˆx = x x re f û = u u es (z,y re f ) and usng (33), he sysem equaons may be wren as ẋ = Ãx+ Bu+ R = Ãx+ Bû+ Bu es + R ẋ = Ãx+ Bû Ãx re f R+ R = Ã(x x re f )+ Bû ẋ = Ã ˆx+ Bû If y re f s consdered as a consan sepon (ẏ re f = ), hen ẋ re f = and, hence ˆx = ẋ So he fuzzy sysem n he new varables resuls n: ˆx = Ã ˆx+ Bû = µ (z)(a ˆx+B ˆx) whose equlbrum pon s ˆx =, correspondng o x re f (and oupu y re f ) n he orgnal non-ncremenal varables The sysem n he new varables has s offse erm removed and a sandard fuzzy conroller may be desgned on, such as he ones n (Tanaka and Wang, ) usng LMI echnques, whch wll be used n he examples below The conrol acon for he orgnal sysem wll be compued by addng o he resulng conrol acon he erm u es (z,y re f ) Noe also ha, wh an ordnary TS sysem and a sepon y =, he resul s u es =, hence he proposed framework encompasses he sandard one For he canoncal sysems, s more powerful, however, as sepon changes can be mmedaely accommodaed as he examples below wll llusrae 4 EXAMPLES In hs secon, a se of examples showng he possbles of he proposed approach wll be presened Frs, he conrol of a sandard TS fuzzy sysem wh no offse wll be exended o varyng operaon pons, n order o compare wh he resuls applyng usual mehodologes nvolvng a consan change of varable Then, a second example wll llusrae he proposed mehodology n a MIMO case Example Le us have a sandard, offse-free sysem defned by: ẋ = µ (z)(a x+b u) = y = Cx (34) wh he wo models gven by: A = (35) A = (36) B = B = [ ] T (37) C = [ ] (38) and membershp funcons µ (z), defned on z = x + x + x 3 as he rapezodal paron depced n Fgure Fgure shows he nonlneary n he sysem as a funcon of z Noe ha condons o desgn a sandard PDC conroller (Tanaka and Wang, ) for he TS sysem o reach he orgn are fulflled For nsance, an LMI mehodology 5

5 ICINCO 7 - Inernaonal Conference on Informacs n Conrol, Auomaon and Robocs µ µ Y re f Y Fgure : Membershp funcons z Fgure 3: Sysem oupu Y ẋ3 u A A Fgure : Nonlneary n ẋ 3 u (Tanaka and Wang, ) may be appled To acheve a decay rae α, he followng LMIs mus be verfed: where z XA T A X + M T BT + B M αx > (39) XA T A X + M T BT + B M αx > (4) XA T A X XA T A X + M T BT + (4) +B M + M T BT + B M 4αX > (4) X = P, M = F X, M = F X (43) beng P a quadrac marx defnng a Lyapunov funcon and F and F he sae feedback gans o be mplemened, e, he conrol acon: û = (µ (z)f + µ (z)f ) ˆx (44) Ã A se of LMI condons (decay α = ) for he above sysem yelds he conroller: F = [ ] (45) F = [ ] (46) F(z) = µ (z)f (47) = whch, as expeced, behaves correcly when reachng he orgn (fgure no shown for brevy) However, when ryng o sablse he sysem around a new operang pon (u re f = 35, y re f = 55, x re f = (55,,) T ), Lyapunov condons no longer hold as he lnearsed model a ha pon has slopes ou of hose gven by he verces of he model above: n order o use a sandard mehodology n ha case, redefnng he local models would be needed Indeed, wh he consan change of varable û = u+35, ˆx = x x re f (he usual one o acheve x = as he operaon pon n many lnear and fuzzy echnques), he resulng conroller u = 35+ F(z)(x (55,,) T ) (48) yelds an unsable equlbrum pon: Fgure 3 shows how nal condons n he vcny of he desred arge drf away o anoher regon of he sae space On he conrary, he proposed mehodology n hs work provdes a conroller vald for all operang pons wh no modfcaons of he LMI condons Fgure 4 shows he non-consan u es calculaed wh lemma, whch replaces he consan value 35 n (48) above In ha way, he resulng loop has he desred operang pon as a sable equlbrum wh he desred decay rae, as shown n Fgure 5 Example Ths example wll demonsrae he mehodology on a 5h order MIMO Takag-Sugeno-Offse sysem wh wo unsable local models gven by: 5

6 TRACKING CONTROL DESIGN FOR A CLASS OF AFFINE MIMO TAKAGI-SUGENO MODELS U es [ B = [ 5 C = 5 [ C = 5 ] T (53) ] ] (54) (55) 36 x = [ ] T (56) 38 u = [ ] T (57) Fgure 4: Non-consan u es conrol acon x = [ ] T u = [ 3 ] T (58) (59) Y re f Y and he membershp funcons µ (z), defned on z = x + x 4 as: z < µ (z) = 5z z 4 z > 4 (6) Fgure 5: Sysem oupu Y wh non-consan u es beng µ (z) = µ (z) Le us group he dfferen equlbrum pons of each local model no an offse erm: R = A x B u (6) Hence, he sysem follows he srucure n Defnon So he conrol acon u es (z,y re f ), compued va (3), and he equlbrum sae are x re f = [y re f y re f ] T (6) where ẋ = y = A = µ (z)(a (x x )+B (u u )) = µ (z)c x (49) = B = A = [ 5 (5) ] T (5) (5) u es = ( CÃ B) ( y re f CÃ R) (63) where y re f and y re f are arbrary user-defned sepons for he wo plan oupus As usual, he change of varable removes he offse erms so he sysem may be expressed as ˆx = = µ (z)(a ˆx+B û) For a decay of α = 5, he LMI Conrol Toolbox n Malab obans: [ ] F = [ ] F = Hence he acual conrol acon o be appled o he plan, afer nverng he change of varable s: u = u es (z,y re f ) (µ (z)f +µ (z)f )(x x re f ) (64) Fgure 6 shows he sysem response approachng a sepon y re f = [ ] T The usefulness of he obaned conroller for sepon changes s shown n Fgure 53

7 ICINCO 7 - Inernaonal Conference on Informacs n Conrol, Auomaon and Robocs x U x5 x4 x3 x Fgure 6: Tme response of he sae varables 5 U Fgure 9: Acual overall conrol acon U 4 4 U es 5 Y Y 6 8 U es Fgure 7: Offse removng erm U es Fgure : Sysem oupu Y wh a sepon change o Y re f = (,) T Y Y Fgure 8: Sysem oupu Y 5 CONCLUSIONS Ths paper presens an offse-ellmnaon change of varable whch apples o fuzzy Takag-Sugeno-offse models wh local lnear models wh a parcular canoncal srucure The canoncal srucure may be obaned, for nsance, by akng as sae varables he oupus and s dervaves As a resul, a ransformed sysem wh equlbrum a ˆx = s obaned The dfference wh sandard changes of varable s ha s non-consan n me As a resul, he offse s nealy removed and he resulng ransformed sysem has he same represenaon for any desred sepon, and well-known conrol desgn echnques for fuzzy non-offse Takag-Sugeno sysems may be drecly appled ndependenly of he 54

8 TRACKING CONTROL DESIGN FOR A CLASS OF AFFINE MIMO TAKAGI-SUGENO MODELS chosen sepon Ineresngly, as a parcular case, he procedure apples o sepon changes n sandard offse-free Takag-Sugeno models The presened resuls apply o a sae feedback seng Furher research should be devoed o generalsng he procedure o suaons wh oupu feedback and nosy measuremens REFERENCES Ansakls Panos, J and Mchel Anhony, N (997) Lnear Sysems Ed McGraw-Hll, New York, USA Babuska, R (998) Fuzzy Modelng for Conrol Ed Kluwer Academc, Boson, USA Boyd, S, ElGhaou, L, Feron, E, and Balakrshnan, V (994) Lnear marx nequales n sysem and conrol heory Ed SIAM, Phladelpha, USA Guerra, T and Vermeren, L (4) LMI-based relaxed nonquadrac sablzaon condons for nonlnear sysems n he Takag-Sugeno s form Auomaca, 4:83 89 Johansson, M (999) Pecewse quadrac sably of fuzzy sysems IEEE Trans Fuzzy Sys, 7:73 7 Km, E and Km, S () Sably analyss and synhess for affne fuzzy conrol sysem va LMI and ILMI: Connous case IEEE Transacons on Fuzzy Sysems, :39 4 Nelles, O, Fnk, A, and Isermann, R () Local lnear model rees (lolmo) oolbox for nonlnear sysem denfcaon In Proc h IFAC Symposum on Sysem Idenfcaon Elsever Sala, A, Guerra, T, and Babuska, R (5) Perspecves of fuzzy sysems and conrol Fuzzy Ses and Sysems, page In Prn Slone, J-J E and L, W (99) Appled Nonlnear Conrol Ed Prence Hall, Englewood Clffs,New Jersey Takag, T and Sugeno, M (985) Fuzzy denfcaon of sysems and s applcaons o modelng and conrol IEEE Transacons on Sysem, Man and Cybernecs, 5:6 3 Tanaka, K and Wang, H O () Fuzzy conrol sysems desgn and analyss Ed John Wley & Sons, New York, USA 55