Phi-Calculus fuzzy arithmetic in control: Application to model based control

Transcription

1 ISA-EUSLAT 9 Ph-Clculus fuzz rthmetc n control: Applcton to model bsed control Lurent Vermeren 3 Therr Mre Guerr 3 Hm Lmr 3 Unv Llle ord de rnce -59 Llle rnce UVHC LAMIH Vlencennes rnce 3 CRS UMR Vlencennes rnce Eml: {lurentvermeren guerr}@unv-vlencennesfr Abstrct Ths pper ms t pplng fuzz rthmetc of ntervls clculus nd fuzz qunttes to utomtc control Ths one clled clculus fuzz rthmetc s more prctcl thn the extenson prncple one nd -cut bsed methods It comes from dfferent representton of fuzz numbers The present pper follows up wor n ntroducng The present pper s nterested n ts use for fuzz nternl model control scheme bsed Kewords uzz rthmetc fuzz model nverson fuzz number weghted fuzz fuson model bsed control Introducton The fuzz set theor elborted b LA Zdeh [] hs been shown to be used n the chrcterzton of fuzzness nd/or uncertnt usng coherent mthemtcl model Vrous pplctons cme t hnd n prtculr n the fuzz control re [ 3] The uncertn clculus ws undergong qute boom these lst ers Severl wors exhbt nterestng results often referrng to ntervls theor [4 5] or rthmetc bsed on fuzz numbers such s Trngulr or Trpezodl uzz umbers [6 7] In the fuzz rthmetc cse the dfferent pproches re generll bsed ether on Zdeh s extenson prncple [ 6] ether on fuzz relton [8] or fnll use the -cuts [9] However no generl pproch llowng common rthmetc opertons to be used on fuzz numbers s vlble Ths wor proposes the use of soclled clculus fuzz rthmetc bsed on dfferent modelng of fuzz numbers [ ] or ths lgebr the modelng of fuzz numbers s consdered through the dstrbuton functon nsted of the clsscl membershp functon The frst prt presents some generltes on ths clculus rthmetc One of the mn nterests of ths lgebr s to provde some nce propertes for wht s clled exct clculus wth fuzz numbers These propertes cn be used n order to nvert fuzz model Therefore the second prt presents possble pplcton n control to fuzz nternl model control scheme uzz rthmetc: -clculus In the lterture mn modelng pproches of mprecson whch s nvolved n mn pplctons nd domns use fuzz numbers nd fuzz rthmetc In mn wors the methods hve led to the development of vrous membershp functons for representng fuzz numbers uzz numbers re often represented n pplctons b LR fuzz sets nd n ISB: prtculr trngulr nd trpezodl fuzz sets There exst two pproches for fuzz rthmetc on one hnd the Zdeh s extenson prncple on the other hnd the lphcuts nd ntervls rthmetc Our wors concern the frst cse usng new modelng for fuzz numbers [] nd t llows ncludng lrge prt of the results exstng n the domn of rthmetc [4 6 8 ] uzz numbers modelng Usull fuzz number s modeled b ts membershp functon µ not null on bounded set Supp( or exmple trngulr fuzz number (T cn be represented b the shorthnd smbol ( bmc wth µ ( x for x m (mode nd the ernel defned b the ntervl m b m+ c [ ] Heren nsted of the clsscl membershp functon the modelng used for the representton of fuzz numbers s bsed on the dstrbuton functon defned b the followng expresson: x ( t µ dt ( x ( + µ dt + ( t Convergence of µ ( t dt (fnte crdnlt of s ssumed b consderng onl membershp functons on compct support I Thus the mjor nterest s tht the dstrbuton functon for ll fuzz number s lws n ncresng monotone functon from ( Thus n nverse functon Supp to [ ] from [ ] to Supp( [ ] cn lws be defned nd ts defnton s ver mportnt for the opertons The set of fuzz numbers represented b dstrbuton functon s noted Φ It should be notced tht the dstrbuton functon for sngleton s lso sngleton nd the one for n ntervl s lne between ts bounds 736

2 ISA-EUSLAT 9 uzz relzton nd extenson Smlrl to defuzzfcton nd fuzzfcton concepts t s necessr to defne relton between cr vlue nd fuzz number Therefore n pplcton from Φ to whch ssoctes cr number to fuzz number s clled fuzz relzton Conversel n pplcton from to Φ s clled fuzz extenson The choce of relzton depends obvousl on the pplcton The most frequentl encountered re: - The medn relston noted Rmed ( t ssoctes to dstrbuton functon the number such s ( 5 Through ths medn relzton the followng equvlence relton R n Φ cn be defned: the set of the functons such s ( 5 defnes the clss of equvlence of Ever element of ths clss wll be clled fuzz nd denoted R s defned s: - The modl relston noted mod ( R ( ( Mxµ ( x x - The men relzton noted R ( mod ( ( R d men men s defned s: 3 rthmetc opertons Consder two fuzz numbers nd b usng ther dstrbuton nd b or fuzz number whose support ncludes let us lso defne ts negtve nd postve prts s + f x < ( x ( x f x [ ] f x > ( f x < ( x ( x f x [ ] f x > (3 The clsscl rthmetc opertons on fuzz numbers (ddton + b pseudo-opposte subtrcton b multplcton b pseudo-nverse onl for non mxedtpe fuzz number b b nd dvson b re defned herenfter [9] ( x ( x + ( x + b b ( x ( x x b( x ( x + b( x (4 x x + x ( ( ( ( x ( x x ( x ( x b b b b b wth mn b b b nd mx b b b Let us notce tht these four clsscl opertons re comptble wth the Moore s ntervl opertons 4 Weghted fuzz fuson opertor Ths prt dels wth fuson opertor ntroduced wthn the frmewor of ths Φ -clculus lgebr Ths opertor s clled the weghted fuzz fuson (W [3] nd s useful for the next sectons Consder two fuzz numbers b Φ nd ther dstrbuton [ ] : x b The weghted fuzz fuson (W of the two fuzz numbers s defned s: p ( x + pb b ( x W ( x x (5 p + pb wth p p nd p + p b 3 Internl model control 3 Prncple of the structure As wth n open-loop control scheme the nternl model control (IMC structure pples onl on stble sstems wth mnmum phse behvor ccordng to g d u p C G b G m gure : Internl model control scheme The flter s ntroduced n order to flter out the mesurement nose nd to ntroduce some robustness n the loop Its sttc gn s Clsscll for lner models t cn be shown tht the output equton (g s gven b: q + S q d (6 wth: ( ( B d S B d ( q ( q G( q C( q ( ( ( m ( ( q C( q ( q ( ( ( m ( + C q q G q G q + C q q G q G q nd the control lw equton s gven b: m ISB:

3 ISA-EUSLAT 9 C( q ( ( ( ( m ( u q C q q G q c The mnmum purpose of the nternl model structure s to gurntee closed-loop stblt nd dsturbnce rejecton e ( nd S ( B d Recll lso tht the nternl model control performs nd of model smplfcton Therefore ts extenson to fuzz nternl control needs the defnton of n nverse model The next prt trets ths ssue 3 Model nverson Let us consder stble tme-nvrnt controllble SISO lner process model whose mthemtcl descrpton s gven n the form of recurrent equton: ( + b u (8 where nd u correond to the process output nd nput t smple b { } denote the model prmeters Suppose now tht these prmeters re fuzz numbers b ledng to: { } ( (7 + b u (9 or model nverson the followng queston rses [4]: nowng the descrpton of uncertntes for the model s t possble to snthesze controller bsed on the nverse model ble to mntn the model output wthn tolernce envelope round the exct trjector? Whch mens: Model output + ( where s the ccepted tolernce round the nomnl trjector To nswer to ths queston ths wor wll use n pproch developed n [4] otce tht the fuzz prt of the uncertntes s not rell ten nto ccount e ths problem could be solved usng ntervl technques for exmple [5 6 7] evertheless the rthmetc provded theren gves n es w to solve non exct clculus b mens of probblt dstrbuton To compute the control lw u t smple the terms defned t prevous smples { } nd { u } re nown As b equton (6 cn be rewrtten s: u ( ( b u b Or consderng the unnown term t smple + expresson (9 cn be rewrtten s follow: ISB: ( u ( + + ψ ( z( ( b wth ψ + + ( z( b u nd T ( [ ] z u u + + or lner model wthout tme del f + s replced wth the desred reference trjector the del result s: + Therefore wth perfect compenston the model output follows the desred trjector wth pure tme del correondng to one smple Gong bc to the queston (cf ( we wnt to gurntee tht Wth equtons ( nd ( for fuzz model (9 we hve strghtforwrdl: u ( + + ψ ( z ( (3 b ψ + + ( z( b u nd T ( [ ] z + u u + Equton (3 gves model nverson descrpton usng fuzz numbers The term + s replced wth desred trjector defned b fuzz number to obtn cusl control lw or exmple f s trngulr fuzz number (T wrtten s: trple( + (4 the control lw s: u ( + ψ ( z ( (5 b ψ + + ( z( b u nd T z ( + u u + Of course the vector z ( depends on the control prt menng tht ll re replced wth desred set pont whch correonds to the schemtc dgrm of g M u M gure : uzz model nverson turll perfect cncellton e + s not possble Moreover usng rthmetc of fuzz numbers or ntervls or clculus wll ntroduce over-estmtons To llustrte ths mportnt pont n exmple ssued from [4] s dscussed herenfter Let us consder the second-order: + b u + b u (6 where b nd b re fuzz numbers defned s: 738

4 ISA-EUSLAT 9 ( ( 55 ( ( 55 trple trple b trple b trple The desred trjector s: π π 5 sn + sn 5 75 nd s chosen s ( (7 (8 trple + (9 Usng (5 the evolutons of the set pont (mnmum nd mxmum envelopes Supp nd the of ( output model envelopes re llustrted n g 3 Obvousl le ntervl clculus or α-cut fuzz rthmetc t clerl ppers tht the clculus rthmetc genertes overestmted results gure 3: Set pont nd output model envelopes We clm tht ths overestmton s not justfed n such cses Therefore we need to ntroduce some extr constrnts n order to te nto ccount some nowledge nd to reduce the pessmsm of the results 33 Exct nverse computton Equton (5 correonds to resoluton of fuzz ffne equton: b x+ ( The de developed n [5] s to exctl solve ths equton ccordng to the two followng steps Step: Solve ( wth reect to d b x The objectve s to compute n cceptble soluton d of the equton d + ; e the soluton beng exct when possble nd pproxmted otherwse otce tht the exct soluton cn lws be computed ccordng to the rthmetc used: d ( u ( u ( u ( ISB: c c smples The problem rses when the result ( u s not n ncresng functon resultng to d Φ evertheless we cn provde n pproxmte soluton to the problem d pp Φ The fct of proposng n lterntve to exct clculus s the e de Contrr to wht generll uthors ccept e no soluton for ntervl rthmetc or α-cut fuzz rthmetc cses [4] we provde n pproxmte soluton bsed on the ntl d ( u or tht purpose we propose to use the weghted fuzz fuson (W opertor on the set of prs ( ( u u d { } It results n re-orderng of the prs tht constructs n ncresng functon d pp thus d pp Φ s fesble soluton to the problem Step: ndng soluton to equton d b x wth Supp( b If b > nd d > the exct clculus uses pont b pont dvson: ( ( x u u d b ( u ( In the generl cse we use postve-negtve decomposton (see secton 3 for d nd x We defne: d ( u d ( u + d ( u + (3 x ( u x ( u + x ( u + wth d ( u ( d u x ( u + nd x + ( u defned equtons (3 nd (4 At lst the exct soluton s gven b: d (( x+ d b x + d b f b > x x+ + x x+ d + ( b x ( ( d ( b f b < x ( x+ + x Once gn the problem rses when the result ( u (4 x s not n ncresng functon resultng to x Φ Therefore pproxmte soluton to the problem x pp Φ s generted n the sme w usng the W opertor on the set of prs ( x ( u u { } Consder n exmple to llustrte ths second step wth the fuzz numbers b trple( nd d trple ( As shown g 4( the exct soluton x Φ e x s not monotone ncresng functon Usng the weghted fuzz fuson (W opertor n pproxmte soluton x pp s generted b re-orderng x g 4(b shows the comprson between d nd b x In sense the pp dfference between these two fuzz numbers exhbts the pessmsm nduced b the method when no exct soluton n Φ s vlble Ths pessmsm cn be shown comprng the supports of the numbers: Supp d 3 4 Supp b x pp 87 6 ( [ ] ( [ ] 739

5 ISA-EUSLAT 9 The next exmple demonstrtes ver clerl the nterest of the complete lgorthm used to fnd solutons to the problem b x+ Consder gn the nverse model of secton 3 wth the fuzz numbers (7 nd the desred trjector (8 (9 g 5 shows the sme trl s for g 3 or ths test we obtn the result consderng e + Ths result ndctes tht n exct soluton hs been found t ech tme «Exct» result b clculus result gure 4: Results wth the numbers b nd d gure 5: Set pont nd output model envelopes In order to show the nterest of the method we wll chnge the constrnts level To do so the vlues of model uncertntes (7 re ncresed n order to del wth cses where n exct nverson s not lws possble ew bounds for the fuzz numbers (7 re defned The correond to the gretest possble vlues wth pole-zero ssgnment locted nsde the unt crcle for evdent stblt purpose Moreover the fuzz numbers re not more smmetrc nd correond to: trple( trple( (5 b trple( b trple 4 b ( g 6 repets the sme trl wth the desred trjector (8 (9 The fgure exhbts tht n exct smplfcton e + s not lws possble Contrr to hve no soluton n such cses the proposed pproxmte solutons seem perfectl dpted ISB: x x pp Rmed ( c c c smples x pp ( R med x pp b d Wth the help of ths nverse model control strteg bsed on constrnts justfed n the control frmewor t s now possble to consder fuzz nternl model control gure 6: Set pont nd output model envelopes 34 uzz nternl model control rom g 6 showng IMC scheme nd dfferent equtons (6 nd (7 t s possble to defne severl solutons rst soluton: Let ( r + + q q ( q ( q correonds to functon wth n nverse (exctl proper nd stble We chose C( q nd propose the + ( q equvlent model gven g 7 M ref q r M u gure 7: IMC equvlent scheme Indeed equton (7 cn be rewrtten s follow: ( ( ( ( ( r u q + c q q q + B nlog we hve the fuzz nverse model G m ( q ( q multpled b lner trnsfer functon r q ( q Σ d (6 or ever cse t s possble to dd model reference control M n order to ttenute the control sgnl ref c Rmed ( c c med ( R smples 74

6 ISA-EUSLAT 9 35 Exmple Consder nomnl second-order trnsfer functon: ( q B ( q 33 q + 97 q (7 u( q A( q 579 q q nd the followng defnton for the fuzz numbers: trple( trple( (8 b trple( b trple 3 ( The control structure used s the IMC g 7 wth reference model defned s: B m ( q 784 q + 7 q (9 A q 935 q + 69 q m ( g 8 shows the results for dfferent set ponts The frst setpont leds to results round the nomnl model (7 (represented b the vlue Secondl two successve vrtons re mde (set-pont vlue of 3 nd set-pont vlue of 4 lstl return to nomnl poston Durng both successve chnges the model s chnged n order to rech the bounds on nd b Two dsturbnces hve been lso dded t smples 9 nd 3 The reonses show good robustness ccordng to the prmetrc vrtons of the model Ths wor presents frst possble trc to use fuzz rthmetc for control purpose ext steps would be to prove stblt nd evlute the robustness of these pproches We cn thn to severl possbltes ncludng Lpunov pproch tht re nsghtfull used for Tg-Sugeno models [8] or Khrtonov polnomls [9] (t gure 8: IMC results 4 Conclusons Ths pper ttempts to show possble pplctons of the clculus rthmetc for utomtc control As frst step new chrcterzton of fuzz numbers b the dstrbuton functon nsted of the clsscl membershp functon hs been presented or ths lgebr t s possble to use set of ISB: smples rthmetc opertors (ddton opposte nd subtrcton multplcton nverse nd quotent comptble wth the clsscl lgebr b usng the medn relzton or the Moore s clculus b usng the fuzz number support A procedure llowng n exct clculus of n nverse model ws proposed on the bss of clculus Ths pproch ws successfull mplemented on nternl model control structures Thus the pplcton of clculus rthmetc to the stblt of sstem seems to open some new proects for other bsc concepts (observton stblzton Acnowledgment Ths wor ws supported n prt b Interntonl Cmpus on Sfet nd Intermodlt n Trnortton the Europen Communt (through the EDER Europen unds for Regonl Development the Délégton Régonle à l Recherche et à l Technologe the Mnstère de l Ensegnement supéreur et de l Recherche the régon ord Ps-de-Cls nd the Centre tonl de l Recherche Scentfque (CRS: the uthors grtefull cnowledge the support of these nsttutons References [] L A Zdeh uzz sets Informton & Control 8: [] A Sl T M Guerr R Bbuš Perectves of fuzz sstems nd control uzz Sets nd Sstems 56: [3] L Juln M Keffer O Ddrt E Wlter Appled Intervl Anlss Sprnger Edton [4] RE Moore Intervl nlss Prentce-Hll-Englewood Clffs ew Jerse 966 [5] UW Kulsh WL Mrner Computer rthmetc n theor nd prctce Acdemc Press 98 [6] D Dubos H rger J ortn A generlzed vertex method for computng wth fuzz ntervls Proc of the Interntonl Conference on uzz Sstems (IEEE ed [7] D Dubos & H Prde uzz rel lgebr: some results uzz Sets nd Sstems : [8] E Snchez Solutons of fuzz equtons wth extended opertors uzz Sets nd Sstems : [9] Z Zho R Govnd Solutons of lgebrc equtons nvolvng generlsed fuzz numbers Informton scences 56: [] D Roger J-M Lecomte -clcul : une rthmétque prtque pour les nombres flous LA 98 Rennes 998 [] H Lmr L Vermeren & D Roger «-clculus: A new fuzz rhtmetc» th IEEE ETA 6 Prgue Czech Republc 6 [] A Kufmnn MM Gupt Introducton to fuzz rthmetc theor nd pplcton Vn ostrnd Renhold ew Yor 99 [3] H Lmr L Vermeren D Roger Aggregton nd fuzzfcton b weghted fuzz fuson opertor IEEE-UZZ 7 London Unted Kngdom 7 [4] R Bouezzoul S Glchet L oullo onlner Internl Model Control: Applcton of nverse Model Bsed uzz Control IEEE T uzz Sstems (6: [5] R Bouezzoul L oullo S Glchet Inverse Controller Desgn for uzz Intervl Sstems IEEE T uzz Sstems 4(:-4 6 [6] E Grdeñes MA Snz L Jorb R Clm R Estel H Melgo AA Trept Modl ntervls Relble Computng 7:77- Kluwer Acdemc Publshers [7] JE Dz Improvements n the R Trcng of Implct Surfces bsed on Intervl Arthmetc Doctorl Thess Gron Spn 8 [8] TM Guerr L Vermeren LMI-bsed relxed non qudrtc stblzton condtons for non-lner sstems n the Tg-Sugeno s form Automtc 4(5 :83-89 [9] CW To nd JS Tur Robust uzz Control for plnt wth fuzz lner model IEEE T uzz Sstems 3(: