Piecewise Quadratic Stability of Closed-loop Takagi-Sugeno Fuzzy Systems

Transcription

1 7 Poceedgs of the Iteatoal Cofeece o Ifoato Autoato Decebe -8 Colobo S Laa Pecewse Quadatc Stablty of Closed-loop aag-sugeo Fuzzy Systes Pet Huše Mguel Beal Depatet of Cotol geeeg Faculty of lectcal geeg Czech echcal Uvesty Pague echcá 66 7 Pague 6 Czech Republc al: huse@felcvutcz Phoe: Fa: Abstact I ths pape pecewse quadatc stablty of closed-loop affe aag-sugeo (AS) fuzzy systes wth lea state-space subodels the cosequet of ules s addessed he cotol law s assued the fo of Paallel Dstbuted Copesato (PDC) Stablty aalyss of the closed-loop syste s based o pecewse quadatc Lyapuov fuctos hs techque educes cosevats of classcal stablty ethods seachg a coo Lyapuov fucto eve oe vefes pecewse quadatc stablty va lea at equaltes (LMIs) Sulato esults ae povded D I INRODUCION U to the satsfactoy esults dealg wth hghly olea systes wth a good copose betwee splcty accuacy fuzzy-logc-based cotol systes have had a apd gowth of dustal applcatos the ecet yeas Howeve ceta ey questos ea usolved o usatsfactoy eploed aog whch stablty aalyss s oe of the ost potat he aalyss of fuzzy systes s geeally had because dffeet ds of oleaty ae volved I the last yeas deep study has bee devoted to aag-sugeo (S) fuzzy systes wth lea subodels the ule s cosequets whch ae usually obtaed by leazato of the ogal olea syste soe pots Such odels ca appoate the behavou of a wde vaety of olea systes wth ay desed degee of accuacy ([][]) Moeove sce S fuzzy odels ae lea te-vayg systes tools owledge fo lea systes theoy ca be used to desg a cotolle Cotolles based o paallel dstbuted copesato (PDC) ae used vey ofte due to the splcty Results stablty aalyss of S fuzzy systes wth PDC cotolle ae based o the estece of a coo quadatc Lyapuov fucto ([]-[7]) he seach of ths fucto ca be stated as a cove optzato poble ca be foulated tes of lea at equaltes Howeve oe ca fd a epoetally stable syste whch adts a globally quadatc Lyapuov fucto that caot be foud usg LMI techques Moeove thee est stable systes that do ot adt a global quadatc Lyapuov fucto at all I addto stuctual foato stoed fuzzy ule atecedets s dsegaded by ths appoach I ths cotbuto ew stablty codtos fo AS fuzzy systes cotolled by PDC laws ae peseted he affe tes the state-space subodels eflect oe ealstc stuatos sce they usually appea pecewse lea descpto of olea systes he peseted ethod s based o seachg pecewse quadatc Lyapuov fuctos va LMIs he obtaed codtos ae less cosevatve by vtue of the che class of Lyapuov fucto cdates II AAI-SUNO FUZZY SYSMS AS fuzzy syste s defed by ules the followg fo: Rule : If ( ) s M (t) s z t the () t = A () t + Bu () t + a z M he dyacs of the oveall ope-loop syste ca be epessed as: = { + + } w A Bu a = & = w = { } = h A + Bu + a w ( z ) = M ( z ) h ( z ) = w w = = whee R s the state vecto z R s the pese vecto supposed as a lea cobato of the states e z( ) = C L C R R R A B a ()

2 8 Poceedgs of the Iteatoal Cofeece o Ifoato Autoato Decebe -8 Colobo S Laa M s the -th ebeshp fucto coespodg to the -th ule PDC cossts desgg a lea state-feedbac cotolle fo each pa { A B } cobg all of the shag the sae ule atecedets of the ogal S fuzzy syste ([][]) Patculaly PDC ceates a fuzzy cotolle descbed by ules Rule : If ( ) s M (t) s z t the ut () = () t z M whee the oveall output of the cotolle esults whee h u = h () = s defed as () s the feedbac ga vecto fo the -th pa { A B } Substtutg equato () () the dyacs of the closed-loop syste ca be descbed by & () t = h h { () t + } = = whee = A B a () III PICWIS QUADRAIC SABILIY ANALYSIS A suffcet codto fo asyptotcal stablty of closedloop syste () wth a = (wthout affe tes) s the estece of a coo postve defte at P such that P P < (4) + fo h h t = ([][]) he equeet of a coo Lyapuov fucto fo all subsystes s ofte too estctve Pecewse quadatc Lyapuov fuctos ela ths estcto allow us to hle a che class of Lyapuov fucto cdates I [8] [] pecewse quadatc stablty of ope-loop AS fuzzy systes s establshed a LMI-based ethod to test t s poved useful hs ethod s based o the patto of the state space duced by the ebeshp fucto suppots whch esults a polyhedal collecto { X } R whee I s the set of cell dces Fo each cell the set J ( ) cotas the dces fo the syste atces used the tepolato wth the -th cell Opeatg eges ae defed as those cells whee J ( ) cotas a sgle eleet the set of I dces whch goup these cells s de oted by I Itepolato eges ae those cells whee J ( ) cotas oe tha oe eleet the set of dces whch goup these cells s deoted by I As Lyapuov fuct o cdates we wll cosde fuctos of the fo: P X I V( ) = P X I paaetzed to be cotuous acoss cell boudaes hs codto s fulflled by eas of costat atces F = F f wth f = fo I satsfyg: F = F X X so we ca paaetze Lyapuov fuctos as: P = F F I P = F F I whee fee paaetes ae collected syetc at allowg a LMI foulato Sce the at P s oly used to descbe the Lyapuov fucto cell X we ca estct the Lyapuov fuct that cell by eas o o to f atces [ e] wth e = fo I satsfyg: X f I () = whee the vecto equalty f eas that each ety of the vecto s oegatve I ode to eted esults [ 8] to closed-loop AS fuzzy systes let us defe atces as: a heoe : If thee est syetc atces U W s uch that U W have oegatve etes such that:

3 Poceedgs of the Iteatoal Cofeece o Ifoato Autoato Decebe -8 Colobo S Laa satsfy P = F F I P = F F I > < P P + P U + W (6) fo { } α () t () t P U () t () t P () t () t hus X α () t V() t β () t codto (8) of Lea s accoplshed Alog taectoes of the syste we have: fo I J( ) dt J( ) J( ) { } d Vt () = h h P + P < P > U P + P + fo I J( ) the (t) teds to zeo epoetally fo evey cotuous pecewse C taectoy X satsfyg () ude cotol law () I Poof: Fst of all let us state the followg Lea whch wll be futhe used whch ca be foud e g [8] Lea : Let Vt () be deceasg pecewse thee ests postve scalas α β γ > such that: W (7) C If α () t V(t) β () t (8) d Vt () γ () t () dt th e () t βα ep ( γ t β ) () Cosde the Lyapuov fucto () copactly wtte as: Vt () = P X Sce costat atces F satsfy the codtos descbed above (see fo detals about the costucto) the latte Lyapuov fucto s cotuous pecewse C () t (ote that the taectoes () t ae assue d to be pecewse C ) Because of the absece of affe tes the Lyapuov fucto a ope eghbouhood of the og t s clea that Vt () β () t he costucto of costat atces ay soluto to the equaltes (6) (7) wth U f ples that thee ests a α > such that Sce h h W > whe h h ( z ) t follows that: d Vt ( ) h ( ) ( ) z h z { R } dt J( ) J( ) R = P + P + W Fo ay soluto of (6) (7) wth W f follows edately that thee ests a γ > such that P + P + W + γ I < J( ) Usg the fact that that: h h( z ) = fo all t s clea J( ) J( ) d Vt ( ) h ( ) h ( ) I dt J( ) J( ) e codto () of copleted z z ( γ ) ( γi ) = = γ X Lea holds the poof s IV XAMPL Let us cosde a two-desoal -ule AS fuzzy syste as follows: whee vecto & = h A+ Bu + a () = w = M h ( z ) = w = { } w = R s the state vecto z R s the pese [ z z] [ ] z ( ) = = ae gve as follows: 7 77 A he syste atces A = = A = A = A8

4 6 Poceedgs of the Iteatoal Cofeece o Ifoato Autoato Decebe -8 Colobo S Laa 4 A4 = A7 A6 4 4 A 4 7 B = B = = B = [ ] a a7 a = = = [ ] a = a = a4 = a = a6 = a8 = a d M ae the ebeshp fuctos descbed Fg (whte gay aeas espectvely Fgue ) Notce that thee ae egos total as the esult of cobato of ebeshp fucto s suppots Sce we ae teested the closed-loop syste pevously calculated gas ae cluded A A A A 4 4 A A We the calculate gas all poles of each subsyste followg cotol law: Fg Mebeshp fuctos u = = ode to place = [ ] whch leads to the h ( z ) () Sulato esults of the closed-loop syste statg fo seveal tal codtos dcate that the costucted AS fuzzy cotol syste s stable (see Fgue ) eve f codto (4) fals to pove quadatc stablty sce o coo at P was foud States e Fg Cotolled syste espose Fg Patto of the fuzzy cotol syste Fo each cell = those tces ( ) that a have to be tae to accout fo LMI (6) (o (7) espectvely) ae those that ae actve ths cell: fo opeatg eges ust oe ust be cosdeed whle fo tepolatg eges evey wth suoudg the cell ust be cluded o llustate ths pocedue let us cosde the tepolatg cell delted by the tevals [ ] A B a A [ ] I ths cell subsystes { } { A B a } { A B a } { } - A A A B a cotolles 6 ae actve (they touch ths cell - see Fgue 4) Accodg to heoe atces (6 atces sce ) have to be cosdeed fo LMI (7) volvg ths cell By eas of PWL oolbo [] lea at equaltes (6) (7) of heoe have a feasble soluto povg epoetal stablty of the og of syste () ude cotol law () he level cuves of the coputed Lyapuov fucto ae dcated by blac les Fgue 4 whle the state taectoy fo a ccle-aed tal codto s showed by a blue le A I ode to apply the pocedue deved above to test stablty of the closed-loop syste fstly we have to dvde the state space to opeatg tepolatg eges

5 6 Poceedgs of the Iteatoal Cofeece o Ifoato Autoato Decebe -8 Colobo S Laa State State Fg 4 Level sufaces of Lyapuov fucto Cotol apa 8 pp 4- [7] aaa Ieda HO Wag A Ufed Appoach to Cotollg Chaos va a LMI-Based Fuzzy Cotol Syste Desg I as Ccuts Syst vol 4 No 8 pp -4 [8] M Johasso A Ratze Aze Pecewse quadatc stablty of fuzzy systes I as Fuzzy Systes vol 7 pp 7-7 [] M Johasso Pecewse lea cotol systes PhD dssetato Dept Autoat Cot Lud Ist echol Lud Swede [] M Johasso A Ratze Coputato of pecewse quadatc Lyapuov fuctos fo hybd systes I asactos o Autoatc Cotol Specal Issue Hybd Syst vol 4 8 pp [] S Hedlud M Johasso A toolbo fo coputatoal aalyss of pecewse quadatc lea systes Dept Autoat Cot Lud Ist echol Lud Swede V CONCLUSION I ths pape ew stablty codtos fo closed-loop AS fuzzy systes wee peseted hey ae based o pecewse quadatc Lyapuov fucto cdates a che class of quadatc fuctos tha the globally oes Sce the pecewse quadatc Lyapuov fuctos ft bette the pecewse lea atue of aag-sugeo fuzzy systes the esultg stablty codto s less cosevatve he deved codto ca be vefed by solvg a cove optzato poble epeseted by lea at equaltes that ca be effcetly solved by coecal avalable algoths he effcecy of poposed pocedue was llustated wth a eaple ACNOWLDMN hs wo has bee suppoted by the poect INO P4LA (sposoed by the Msty of ducato of the Czech Republc) the poect ACR /4/P the poect ACR // (both sposoed by the at Agecy of the Czech Republc) RFRNCS [] aaa HO Wag Fuzzy cotol systes desg aalyss: a lea at equalty appoach Joh Wley & Sos [] HO Wag aaa MF ff A Aalytcal Faewo of Fuzzy Modelg Cotol of Nolea Systes: Stablty Desg Issues Poc Aeca Cotol Cofeece Seattle pp 7-76 [] aaa M Sugeo Stablty Aalyss of Fuzzy Systes Usg Lyapuov's Dect Method Poc of NAFIPS ' pp -6 [4] aaa aguch HO Wag Model-Based Fuzzy Cotol of ORA Syste: Fuzzy Regulato Fuzzy Obseve Desg va LMIs that Repeset Decay Rate Dstubace Reecto Robustess Optalty Seveth I Iteatoal Cofeece o Fuzzy Systes Alasa 8 pp -8 [] aaa aguch HO Wag Robust Optal Fuzzy Cotol: A Lea Mat Iequalty Appoach Iteatoal Fedeato of Autoatc Cotol (IFAC) Wold Cogess Beg July pp -8 [6] aaa aguch HO Wag Fuzzy Cotol Based o Quadatc Pefoace Fucto 7th I Cofeece o Decso