Iterative Feedback Tuning with Application to Robotics

Transcription

1 ISSN ISRN LUTFD/TFRT SE Ieave Feeback Tnng wh Applcaon o Robocs Alessano Bn Depamen of Aomac onol Ln Inse of Technolog Decembe 3

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3 Depamen of Aomac onol Ln Inse of Technolog Box 8 SE- Ln Sween Ahos Alessano Bn Docmen name MASTER THESIS Dae of sse Decembe 3 Docmen Nmbe ISRN LUTFD/TFRT SE Spevso Rolf Johansson an Anes Robesson a Depamen of Aomac onol n Ln. Eoao Mosca a Unvesá Fenze. Sponsong oganzaon Tle an sble Ieave Feeback Tnng wh Applcaon o Robocs. Ieav mnng av nfomaonsåeföng me obokllämpnng. Absac Man conol objecves can be expesse n ems of a ceon fncon. Geneall, explc solons o sch opmzaon poblem eqe fll knowlege of he plan an sbances an complee feeom n he complex of he conolle. In pacce, he plan an he sbances ae selom known, an s ofen esable o acheve he bes possble pefomance wh a conolle of pescbe complex sch as fo example a PID conolle. The opmzaon of sch conol pefomance ceon pcall eqes eave gaen-base mnmzaon pocees. The majo smblng block fo he solon of hs opmal conol poblem s he compaon of he gaen of he ceon fncon wh espec o he conolle paamees: s a fal complcae fncon of he plan an sbance namcs. When hese ae nknown, s no clea how hs gaen can be compe. Ieave Feeback Tnng IFT s a npop aa-base esgn meho fo he nng of esce complex conolles. I oes no epen on he plan moel, lzes I/O aa onl. Theefoe IFT s obs agans he plan moel ncean. A each eaon, an pae fo he paamees of he conolle s esmae fom aa obane pal fom he nomal opeaon of he close loop ssem an pal fom a specal expemen. No enfcaon pocee s nvolve. In hs hess nng of obo jon conolles sng IFT s consee. The ffeen IFT-schemes have been vefe n smlaon an n eal expemens on an nsal obo manplao ABB Ib-. Kewos lassfcaon ssem an/o nex ems f an Spplemena bblogaphcal nfomaon ISSN an ke le Langage Englsh Sec classfcaon Nmbe of pages Recpen s noes ISBN The epo ma be oee fom he Depamen of Aomac onol o boowe hogh: Ln Unves Lba, Box 3, SE- Ln, Sween. Phone 46 46, Fax

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5 3 To m faml

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7 onens Acknowlegmens.. 7. Inocon.. 9. IFT Descpon 3. The Nonlnea ase Mofcaons an Impovemens o IFT Applcaons on an Insal Robo 7 6. onclsons Bblogaph

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9 Acknowlegmens I wol lke o hank Pofesso Rolf Johansson fo he welcome, he avalabl an he me ha he gave me ng hs wok. Hs hman an pofessonal wll be wh me foeve. I wol lke o hank D. Anes Robesson fo he sppo an connos help ha he gave me fo he heoecal pa an moeove fo he as an nghs n whch we woke ogehe n he obo laboao whee we also fomlae a heo smmaze as he bes esls come beween mngh an fve n he monng. He s a peson ha caes abo eveone an I wll neve foge hm. I wol also lke o hank m Ialan spevso Pofesso Eoao Mosca fo hs avce o se o on hs expeence n Ln an also fo hs kn avalabl ng hese monhs. I am also gaefl o all he people of he epamen of Aomac onol a Ln Inse of Technolog fo he help an sppo. 7

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11 . Inocon Man conol objecves can be expesse n ems of a ceon fncon. Geneall, explc solons o sch opmzaon poblem eqe fll knowlege of he plan an sbances an complee feeom n he complex of he conolle. In pacce, he plan an he sbances ae selom known, an s ofen esable o acheve he bes possble pefomance wh a conolle of pescbe complex. Fo example, one ma wan o ne he paamees of a PID conolle n oe o exac he bes possble pefomance fom sch smple conolle. The opmzaon of sch conol pefomance ceon pcall eqes eave gaen-base mnmzaon pocees. The majo smblng block fo he solon of hs opmal conol poblem s he compaon of he gaen of he ceon fncon wh espec o he conolle paamees: s a fal complcae fncon of he plan an sbance namcs. When hese ae nknown, s no clea how hs gaen can be compe. The conbon of [Hjalmasson, Gnnasson, Geves, 994] was o show ha an nbase esmae of he gaen can be compe fom sgnals obane fom close loop expemens wh he pesen conolle opeang on he acal ssem. Fo a conolle of gven pcall low-oe sce, he mnmzaon of he ceon s hen pefome eavel b a Gass-Newon base scheme. Fo a wo-egee-of-feeom conolle, hee bach expemens ae o be pefome a each sep of he eave esgn. The fs an h smpl conss of collecng aa ne nomal opeang conons; he onl eal expemen s he secon bach whch eqes feeng back, a he efeence np, he op mease ng nomal opeaon. Hence he aconm Ieave Feeback Tnng IFT gven o hs scheme. Fo a one-egee-of-feeom conolle, onl he fs an secon expemens ae eqe. No enfcaon pocee s nvolve. As n an nmecal opmzaon one, a vaable sep sze can be se. Ths allows one o conol he ae of change beween he new conolle an he pevos one an s an mpoan aspec fom an engneeng pespecve. Fhemoe a vaable sep sze s he ke o esablshng convegence of he algohm ne nos conons. Wh a sep sze enng o zeo appopael fas, eas fom sochasc aveagng can be se o show ha, ne he conon ha he sgnal eman bone, he acheve pefomance wll convege o a local mnmm of he ceon fncon as he nmbe of aa ens owa nfn. The opmal IFT scheme of [Hjalmasson e al, 994] was nall eve n 994 an pesene a he IEEE D 994. The IFT meho s appealng o pocess conol engnees becase, ne hs scheme, he conolle paamees can be sccessvel mpove who eve openng he loop. In aon, he ea of mpovng he pefomance of an alea opeang conolle, on he bass of close-loop aa coespons o a naal wa of hnkng. 9

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13 . IFT Descpon. Two-egee-of-feeom conolles an IFT The heo n hs secon s base on [Hjalmasson, Geves, Gnnasson, Leqn, 998]. We conse an nknow e ssem escbe b he scee moel G v whee G s he lnea me-nvaan opeao, s he mease op, s he conol np, {v } s an nmeasable pocess sbance an epesens he scee me nsans emak: o eas he noaon he me nex s somemes lef o. We conse ha hs ssem s o be conolle b a wo-egee-of-feeom DOF conolle: whee an ae lnea me-nvaan ansfe fncons paameze b n some veco R, an { } s an exenal eemnsc efeence sgnal, nepenen of {v }. Noce ha s possble fo an o have common paamees. v G - Fg.. A wo-egee-of-feeom conolle sce We wll se he noaon an fo he op an he conol np of he ssem n feeback wh he conolle, n oe o make he epenence of hese sgnals on he conolle paamee veco explc.

14 Le be a ese op esponse o a efeence sgnal fo he close loop ssem. Ths esponse ma be efne as he op of a efeence moel T sch ha T 3 b fo he IFT meho knowlege of he sgnal s sffcen. The eo beween he acheve an he ese esponse s v G G G ˆ ~ So, sng 3: v G T G G ˆ ~ Ths eo consss of a conbon e o ncoec ackng of he efeence sgnal an an eo e o he sbance v. Fo a conolle of some fxe sce paameze b, s naal o fomlae he conol esgn objecve as a mnmzaon of some nom of ~ ove he conolle paamee veco. We wll conse he followng qaac conol pefomance ceon: 4 whee N s he nmbe of samples consee an E[ ] enoes expecaon wh espec o he weakl saona sbance v. The fle L s a feqenc weghng of he eo beween he ese esponse an he acheve esponse. The fle L weghs he penal of he conol effo. The can of cose be se o, b he gve ae flexbl o he esgn. The opmal conolle paamee * s efne b: mn ag * J 5 The objecve of he ceon 4 s o ne he pocess esponse o a ese eemnsc esponse of fne lengh N n a mean sqae sense. As fomlae, hs s a moel efeence poblem wh an aonal penal on he conol effo. N N L L E N J ~ λ

15 3 Le T ans S enoe he acheve close loop esponse an sensv fncon wh he conolle {, }: G G T G S. Gven he nepenence of an v, J can be wen as: The fs em s he ackng eo, he secon em s he sbance conbon, an he las em s he penal on he conol effo. RITERION MINIMIZATION We examne now he mnmzaon of he ceon 4 wh espec o he conol paamees veco fo a conolle of specfe sce. To oban he mnmm a necessa conon s ha he fs evave of J w... he conolle paamee s zeo: 6 whee fo smplc we assme L L. If he gaen J col be compe, hen he solon of he pevos eqaon wol be obane b he followng eave algohm: J R γ 7 Hee γ s a posve eal scala ha eemnes he sep sze an R s some appopae posve efne max noce also ha, as we wll show lae, γ { } { } [ ] N N L E N v S L E N T L N J λ N N N E N J ~ ~ λ

16 ms obe some consans fo he algohm o convege o a local mnmm of he cos fncon J. R eemnes he pae econ an s heefoe ccal fo he pefomance of he algohm. Tpcall we choose a Gass-Newon appoxmaon of he Hessan of J, so: R N N T λ T 8 We wll see ha all he sgnals esmaes neee n hs expesson of R wll be avalable fom he IFT algohm. As sae he poblem s nacable snce nvolves expecaons ha ae nknown. J Howeve, sch a poblem can be solve b eplacng he gaen b an nbase esmae. In oe o solve hs poblem, one nees o geneae he followng qanes: he sgnals ~ an ; ~ he gaens an ; 3 nbase esmaes of he pocs ~ ~ an. These qanes can be obane b pefomng expemens on he close loop ssem fome b he acal ssem n feeback wh he conolle,. { } We wll see ha 8 can be expesse s follow: R N es es N T λ es es T. hee an n he seqel, es an an es enoe he esmaes of 4

17 5 OMPUTATION OF THE GRADIENT Nong ha ~, we can we: [ ]. ~ v S T T T v G G G G G G Fom he expesson above we see ha he gaen epens also on he no compable qanes T an S snce he epen on he nknown ssem. Theefoe, nless an accae moel of he ssem s assme o be avalable, he sgnal can onl be obane b nnng expemens on he acal close loop ssem. Noce ha he las wo ems n he expesson above nvolve a oble fleng of he sgnal an v hogh he close loop ssem: [ ]. T v S T T So we can we: T T T T The em T - can be obane b sbsacng he op sgnal fom one expemen of he close loop ssem fom he efeence an b sng hs eo sgnal as efeence sgnal n a new expemen. In each eaon of he conolle nng algohm, we wll se hee expemens, each of aon N, wh he fxe conolle { }, ˆ opeang on he acal plan. We wll see ha onl he secon expemen s a specal expemen, he fs an he h js conss of collecng aa ne nomal opeang conons. We enoe he N-lengh efeence sgnals b j, j,,3, an he coesponng op sgnals b j, j,,3.

18 6 We have: 3 So he expessons fo he op sgnal ae: 3 3 v S T v S T v S T Hee j v enoes he sbance acng on he ssem ng expemen j a eaon hese sbance can be assme o be mall nepenen snce he come fom ffeen expemens. Wh hese expemens ~ s a pefec ealzaon of ~ whee he sbscp enoes he eaon nmbe, whle ˆ 3 es s a pebe veson b he sbance v an 3 v of. So now we have an esmae of he gaen ha can be se n 6 an so n 7 fo he pang conol paamee law. In an analogos wa, we can oban an esmae of he gaen. Fom [ ] v S v G G o we can see ha he hee expemens escbe above geneae he coesponng conol sgnals: [ ] [ ] [ ] 3 3 v S v S v S

19 7 These sgnals can smlal be se o geneae he esmaes of he np elae sgnals eqe fo he esmaon of he gaen 6. Inee s a pefec ealzaon of, whle ˆ 3 es Resmng, wh hese wo gaen esmaes an expemenall base esmae of he gaen of J can be fome b akng: N N es N es N J es ~ λ

20 8 ONE-DEGREE-OF-FREEDOM ONTROLLERS In he case whee he smplfe conolle sce ˆ s se,.e.,, he algohm smplfes becase he h expemen becomes nnecessa. Theefoe, n he case of a one-egee-of-feeom conolle, he fs wo expemens ae n wh he same efeence sgnals as he case of a wo-egeeof-feeom conolle,.e., an he gaen esmaes ae obane b: es es G - v Fg.. A one-egee-of-feeom conolle sce

21 .. Two-egee-of-feeom conolles EXAMPLE onse he plan an he conolles G s s s.6535 q q.8469 q.6854q.39 q.8469 whee he opeao q s he fowa shf opeao, ha s qf f. We a a sbance v as n gven b a nose n whch s he op seqence of whe Gassan nose wh zeo mean an vaance. flee b he followng feqenc weghng fncon W s s s The ppose s o mpove he sep esponse ackng poblem. We ake he followng paamezaon of he wo conolles: q.8469 q 3 q.8469 so ha we have hee paamees,,, 3 o ne, sang fom: We se he followng qaac ceon:.39 3 J N 9

22 an he same fxe sep sze fo all he paamees, γ., an a Gass-Newon appoxmaon of he Hessan of J fo he max R. Pefomng he IFT algohm escbe above we ge, afe 5 eaons, he fnal paamees an he followng esls: Nmbe of Vale of he fnal eaons ceon J nal vale of he ceon.8443 Fg..3 Fg..4 Sep Response sep a, ample Doe lne: nal - Sol lne: fnal

23 .. Two-egee-of-feeom conolles EXAMPLE We now appl he IFT scheme o he nng of he conolle fo a flexble sevo, a wo-mass-ssem wo masses connece wh a spng. The followng fge show he skech of he pocess. Fg..5 We have he wo masses m an m. The spng beween he masses has he spng consan k. The vscos ampng coeffcens ae an, especvel. One of he masses s ven b a D-moo. Hee we neglec he nenal namcs of he moo. The foce fom he moo s popoonal o he volage, ha s: k F m Foce balance eqaons gve he followng namcal moel: p p k p p m F p p k p p m Inocng he sae veco T p p p p x ] [ an p, he ssem can be wen on sae-space fom, x B Ax x whee m m k m k m k m m k A, m k B m, k k

24 We se he followng consans an coeffcens: m.9 kg m.44 kg 3. N/m/s 3.73 N/m/s k 4 N/m k m.96 N/V k 8 V/m These vales coespon o he eal pocess whch wll be consee n nex secon. The ppose s o conol he poson p. Assmng ha we col mease all he saes, sng a sae feeback conol saeg, we col place he poles fo he close loop ssem feel conollabl max has fll ank b sng he conol law: Lx l b popel choosng he paamees L [l l l 3 l 4 ]. Hee s he efeence vale an l s a scala gan affecng he oveall gan an s chosen o ge he coec saona gan. We choose he followng ese poles fo he close loop ssem c.l.s.: b b, 3,4.4 ± 7.6 ± j5.7 j4.9 Fg..6 Noe he pool ampe fs pole-pa, whch we have chosen on ppose.

25 The gve an oscllao esponse fo he poson of he secon mass an o am wll be o mpove he coesponng conolle o ge a bee behavo. Assme now ha onl he poson p s measable. Ths we ae face wh a conol poblem whee we wan o feeback sgnals whch we can no mease. We nee o esmae he pocess sae veco sng an obseve Kalman fle escbe b: xˆ Axˆ B K xˆ whee xˆ enoes he saes of he obseve. K can be chosen so ha he obseve saes ae appoachng he eal saes wh an abal fas ae of convegence sall sch ha he obseve namcs wll be.5 o mes fase han he close loop ssem. We ge: L [ ] K We wll se he obseve saes n he feeback law nsea of he eal saes whch we can no mease. The Smlnk scheme of he conolle ssem s shown below. Fg..7 B ecomposng he sae-space ealzaon we can easl fn he ansfe fncons fom o an fom o, so ha we col have a wo-egee-offeeom conolle sce as below. Fg..8 3

26 The nal feefowa conolle s: s.36 s 47.7 s 99s s 5.8 s 977 s 548 s 6944 The nal feeback conolle s: 3 33 s. s 835 s s 5.8 s 977 s 548 s 6944 These nal conolles gve he followng esponse poson secon mass: Fg..9 Inal sep esponse The paamezaon chosen o be se fo he IFT algohm s: 4 s s s 4 s 5 4 q s 3 3 s 3 q 3 q 4 s 6 3 s s 3 7 q s 8 9 s 3 Noce he chose o keep he same enomnao n he wo conolles an. Ths o keep he sce e b he obseve an he sae feeback conol esgn. 4

27 We can now appl he IFT algohm o pae he conol paamees of he feeback an he feefowa conolle. We wll se he followng cos fncon: J N N As wen above we nee hee expemens an we wll mplemen hem sng Smlnk. Acall n hs example we o no se he sbance v so we o no nee he h expemen. The fs an he secon expemens have he followng Smlnk schemes: Fg.. Fs IFT expemen Fg.. Secon IFT expemen: - The ese op s chosen as: a.5 T s a.5 whee a3.5 an as a n sep. 4 5

28 Fg.. Dese op Sang fom an nal ceon J.497 an sng a small sep sze of γ.5,, so ha we can speak n ems of fne nng, an a Gass-Newon appoxmaon of he Hessan of J fo he max R, afe n 6 eaons, we ge he followng fnal paamees: an he fnal ceon vale J 5 n

29 Fg. -3 Sep Response sep a, ample Sol lne: fnal afe 6 eaons Dashe lne: nal Doe: ese Ieaon nmbe Fnal os Fg..4 Fnal vale of he cos fncon J vess he IFT eaon nmbe Inal cos:

30 Fg..5 Lef: afe eaons Rgh: afe eaons Fg..6 Lef: afe 3 eaons Rgh: afe 4 eaons Fg..7 Lef: afe 5 eaons Rgh: afe 6 eaons 8

31 The paamees pang s shown n he followng fges. Fg Ieaon nmbe Fg..9 Fg.. 9

32 Fg.. Fg.. Fg..3 3

33 Fg..4 3

34 THE REAL PROESS AND THE PROBLEM OF FRITION An mplemenaon on he eal pocess has been e b a elevan poblem has been enconee: fcon. In hs case, a nonlnea conbon fcon oes no allow he IFT meho o wok popel an n o specfc case, he fcon was oo mpoan an he calclaons of - ng he eaons of he algohm wee soe. Fg..5 In he followng pce, we show he behavo of he ssem an we can nesan how n hs case he mplemenaon of IFT was almos mpossble becase we col nehe check he esl of he fnal conolle no keep sabl popees. Fg..6 Poson esponse o a sqae wave efeence sgnal A poblem wh fcon an a pool ne nal conolle s ha he pocess wll sck a posons whch ae fa fom he efeence vale an whch ma change a lo fom one eaon o anohe. The expemens on he eal pocess, even we ae n pesence of a lm case of hge fcon, showe he lm of IFT ha sall s base on he measemens of he eo beween he op of he ssem an he ese esponse. Howeve we wll show n he nex chape how s sll possble o appl he IFT meho o he case of eemnsc nonlnea ssems. 3

35 . onvegence of IFT onse a scee me lnea me-nvaan LTI moel v sbance G v an le be he ssem be conolle b he conolle wh R. n In hs secon we sae exac conons fo whch he conolle paamees pae wh he IFT algohm convege o he se of saona pons of he ceon Le D be a convex compac sbse of N J E N. n R. We noce he followng conons on he nose, he conolle, he close loop ssem an he sep szes of he algohm, especvel. V In an expemen, he sgnal seqence v,,, N consss of zeo mean anom vaables whch ae bone: v fo all. The consan an he secon oe sascs of v ae he same fo all expemens, whle seqences fom ffeen expemens ae mall nepenen. Thee exss a neghbohoo Θ o D sch ha s wo mes connosl ffeenable w... n Θ. All elemens of he ansfe fncons have he poles an zeos nfoml bone awa fom he n ccle on D. 33

36 S The lnea me-nvaan close loop ssem s sable an has all s poles nfoml bone awa fom he n ccle on D. A The elemens of he seqence { } A The elemens of he seqence { } γ sasfes γ an γ. γ sasfes γ <. Theoem Hjalmasson, 998 J onse he IFT algohm gven b γ R. Assme ha V,,, S, A an A hol. Assme ha R s a smmec max whch s geneae b he expemens a eaon an sasfes I R δi δ fo some δ >. Then lm ˆ : J ' D c { } on a se A { D }. The basc eqemen fo convegence s ha he sgnals eman bone hogho he eaons snce he esl onl apples o he se A noce n he heoem. The powe of he heoem s ha apa fom he assmpon of lnea an menvaance hee ae no ohe assmpons on he popees of he ssem. The same hols fo he conolle: he complex of he conolle s aba an he esl hs apples o smple PID conolles as well as o moe complex ones. I s also mpoan o noce ha even hogh he sbances have o have he same secon oe sascs fom expemen o expemen, s no necessa ha he sbances ae saona ng one expemen. 34

37 .3 The sep sze: a ccal choce In hs secon we scss an show wh some expamples, how ccal s he choce of he sep sze γ n he paamee pae law Le s ake he followng obo moel: J γ R. G s s s s 5s conolle b a PD conolle: wh N. G s k p st T s N Fg..7 To mplemen IFT on hs ssem we se he ceon J N N an a Gass-Newon appoxmaon of he Hessan fo he max R. We se as ese op wh as he n sep an a4. T a s a Sang wh he nal paamees k p an T we ge he an nal cos J.77 an he followng esponse: 35

38 Fg..8 Response wh he nal PD paameesfll an he ese op oe Pefomng 5 eaons of IFT sng a sep sze γ. we ge a fnal cos J 5.43 an he followng esponse: Fg..9 Fnal esponse afe 5 eaons of IFT sng a sep sze of.fll, nal esponseashe an ese op oe 36

39 We can look a he cos fncon vales as fncon of he eaons nmbe: Fg..3 sep sze γ., fnal cos J 5.43 We pefom now he same nmbe of eaons, sang fom he same nal paamees b sng a bgge sep sze, γ.5. The esponse s: Fg..3 37

40 We ge he same fnal cos vale J 5.43 b f we look a he gaph of J as fncon of he eaon nmbes we can obseve an neesng behavo: Fg..3 sep sze γ.5, fnal cos J 5.43 We can see ha afe 3 eaon we ge a lowe, even f no conseabl, cos fncon vale J 5.4. So s no sefl o conne n he eaons an s enogh o sop he IFT algohm js afe 3 eaons. Ths s an mpoan aspec of he eave nng mehos snce can happen ha we fn a local mnmm js afe few eaon an f we on look a he cos vale ng he eaons can happen ha we can ge lowe pefomance fom he paamees pang. Ths s e n man pa o he egla sface of he cos fncon J an so s ccal o choose a gh sep sze o a leas a goo sop le fo he algohm. Anohe mpoan sse s he appoxmaon of he gaens an he fom of he max R. In fac, n case of ve egla sface of J, we col nee o se a ve small sep sze f he appoxmaon s no accae. Ths hols o pefom a hgh nmbe o eaons fo he IFT n oe o ge mpovemens of he conolle. 38

41 Ineesng s also he behavo of he IFT mplemenaon afe 5 eaon an wh a sep sze eqal o.9: Fg..33 sep sze γ.9, fnal cos J 5.43 In hs case we can see ha we fn moe han one local mnmm ng he IFT mplemenaon o a leas fo ffeen eaon nmbes n whch we col sop nsea of avng a 5 eaons. 39

42 4

43 3. The Nonlnea ase In hs chape we scss an analss of he popees of IFT when apple o nonlnea ssems conolle b a ne a lnea conolle. We wll assme ha he ssem o be conolle s gven b he followng nonlnea sae-space moel: x f x,, w h x, v * whee f f x,, w an h h x, v ae smooh fncons, whee x epesen he sae veco a me, whee an ae he scala nps an ops an whee w an v ae exenal sbances. We wll also assme ha he ssem s conolle b he followng lnea menvaan conolle q,: q, whee s he exenal efeence sgnal an s he paamees veco. Poceng n an analogos wa of he pevos secons, we ge, ffeenang he ssem eqaons w : whee x' f ' h x x' f x x' ' ' ' ' ' ' x' ' x ' f f x,, w f x f x,, w x f f x,, w 4

44 4 As he lnea case, f follows ha he gaens an can be obane b fs pefomng a smlaon sng he ssem * wh as efeence sgnal an collecng he sgnals,, x an w,,,n whee he sbscp enoe ha he sgnals sem fom he fs smlaon sng ssem *. Wh hese sgnals a han, n a secon smlaon se he lnea me-vang feeback ssem s x H B x A x wh efeence sgnal,, q q s an wh he specal choce of me-vang maces,,,,, v x h x H w x f B w x f x A whch ae fncon of he sgnals x,, w, v n he fs expemen. I hen hols ha an x x.

45 43 B epeang he secon smlaon wh a new efeence sgnal,, q q s j he e gaens wh espec o j can be obane. To oban he gaen wh espec o he complee paamee veco n R, he secon smlaon has o be pefome n mes. A awback wh hs meho s ha he nmbe of expemens s popoonal o he nmbe of paamees ha ae o be ne. An alenave appoach o avo hs s sggese b Sjöbeg an Agaval n [Sjöbeg e al, 997] an b De Bne, Aneson an Geves n [De Bne e al, 996] whee a lnea-vang moel s enfe. Anohe appoach fo he conol of non lnea ssems s he evelopmen of IFT e o Hjalmasson e al 998, ha we now escbe. Fo each eaon n J R γ he IFT meho ses wo expemens, each of aon N sa, wh he fxe conolle opeang on he acal plan. Noce ha cona o he meho olne above he nmbe of expemens s fxe o wo egaless of he menson of he paamee veco. When - s se as efeence n he secon expemen n IFT, he ssem eqaons n he secon expemen can be wen:,,, v x h w x f x Appoxmang he fs wo eqaons b a fs oe Talo expanson aon x,, w,, v, he close loop ajecoes n he fs expemen wh as efeence sgnal, gves x h f x f x x x δ ** whee w f f x f f w f w x w v h x h h v h v x v δ

46 Above he sppesse agmens ae sgnals fom he fs expemen: f f x,, w. Noce ha an δ can be egae as exenal sgnals snce he ae fncons of vaables n he fs expemen an w an v onl. Dsegang hese sgnals we see ha he fs oe appoxmaon s encal o he lnea me-vang eqaons seen pevosl ha geneae he e gaen excep fo he fac ha he efeence sgnal - n he expesson of s flee b ' wheeas n ** s no. The eason of hs las ffeence s ha n he IFT algohm hs fleng s one afe an no befoe he secon expemen. These smlaes sgges ha shol be possble o mpove he pefomance fo nonlnea ssems sng he sana IFT pocee ne he followng conons: - he fs oe Talo appoxmaon s easonabl accae; - he sgnals an δ ae small compae o f x x f an x, especvel; h x - he eo e o commng he' opeao an he close loop ssem s small. I ma seem as f hese sae conons ae qe escve. Howeve, pacce, as wll be evence below, has shown ha hs oes no seem o be he case fo man ssems. One eason fo hs s ha sffces o be able o compe a escen econ, he exac gaen s no necessa. Howeve, mgh be necessa o ece he sep-sze γ when a pebe gaen esmae s se. Fhemoe, cae has o be execse f e.g. a Gass- Newon pae s se snce he jon effec of he gaen pebaon an he mofcaon of he seach econ case b R s ave an one ma en p n an ascen econ. We concle ha fo nonlnea ssems, mgh be wse o se a small sep-sze. A SIMULATION EXAMPLE We wll conse now he followng nose-fee nonlnea ssem whch has x an z as saes: x x.x z z.x z 3 3..z 3 44

47 45 Remembeng ha he sbscps enoe expemen nmbe n he IFT pocee, fo hs ssem he gh-han se of he fs eqaon of ** becomes: ˆ z z x x x x f x f x γ ξ an.4.4. ˆ z x x ψ λ We ll conse he ssem conolle wh he PI conolle. q. We can pove ha ξ omnaes λ an also γ ψ <<. Ths ncaes ha he pebaon ψ λ s no ve sgnfcan. So we can sa ha f x f x x s a easonable appoxmaon of he secon expemen. Ths ncaes also ha, pove he commaon ' an he close loop ssem oes no nflence he sgnals oo mch, shol be possble o se he IFT on hs ssem. We choose a efeence sgnal as a peo of a sqae-wave wh peo me 5. Fg. 3. Refeence sgnal fo he fs IFT expemen The ese op s aken o be he efeence sgnal of he fs IFT expemen an flee hogh he followng low-pass fle.95.5 q q T

48 46 Fg. 3. Dese esponse The paamees,, 3, 4 ae ajse n he followng scee me genealzaon of a PID conolle sce: q q q q We se fo he IFT algohm: - a cos fncon N J - a Gass-Newon appoxmaon of he Hessan fo he max R - a sep sze γ.5. The nal paamees ae:. 4 3 an he coesponng esponse can be seen n he followng fge.

49 Fg. 3.3 lose loop esponse wh he nal PID paamees fll an ese esponse oe As can be seen he ssem exhbs some qe nonlnea behavos. Applng he IFT algohm, we oban he followng esponse afe 5 eaons: Fg. 3.4 Responsefll afe 5 eaons, ese esponse oe an esponse wh he nal PID paamees ashe 47

50 The ceon eceases monooncall ng he eaons as shown below n he plo an n he able he sbscp enoes afe how man eaon he vales sem fom. Fg. 3.5 os J as fncon of he nmbe of IFT eaons eon vales J nal 56.3 J J 6.3 J J 4.7 J 5fnal.496 A compason wh he nal esponse shows ha he IFT has manage o mpove he pefomance conseabl. I shol also be noe ha he smple lnea conolle makes a spsngl goo job on hs nonlnea ssem. The coesponng fnal conolle s gven b: q 8.999q -.447q 5. q 48

51 4. Mofcaons an Impovemens o IFT 4. Mofe ceon One of he feqen paccal se of conolle esgn s o ne a conolle of fxe sce fo example a PID conolle n sch a wa ha he sep esponse of he close-loop ssem has a mnmal selng me wh a small oveshoo. The objecve n sch applcaons s o move he op of he close-loop ssem qckl fom one efeence vale o anohe; howeve, he pacla shape of he ansen esponse fom he nal efeence vale o he fnal vale s of no mpoance, pove ha oes no have lage oveshoo. In aon, who knowlege of he acal ssem whch s a majo eason fo sng IFT s no known n avance how fas a selng me can be acheve fo hs pacla ssem wh hs pacla conolle sce. B mposng he ene esponse of he close-loop ssem hogh a specfc choce of a ese esponse, ahe han js he enpon of hs ansen esponse, he classcal IFT ceon leas o conolle paamees ha ealze a compomse beween fng he ansen esponse an fng he new efeence vale, even hogh he se oes no cae abo he exac shape of he ansen esponse. Insea, b mposng a mask on he ansen esponse, he opmzaon wll ne he conolle paamees n sch a wa as o acheve he new ese efeence vale who focsng on a pacla pe-mpose ansen esponse ha s pehaps no naall acheve b he close-loop ssem. We can noce a vaan of he conol pefomance ceon 4 n whch he sgnals ~ an ae me weghe b weghngs w an w, especl. Ths he ceon N N J E L ~ λ L N s eplace b N N J E w L ~ λ w L N whee w an w ae an nonnegave nmbes. The flexbl offee b he me weghngs w an w s ha he allow one o p ffeen weghngs on ffeen pas of he me esponses. A paclal neesng applcaon s when zeo weghngs ae p on he ansen esponse of he op esponse o a sep change n he efeence sgnal. 49

52 5 In hs case he ceon becomes: N N m L L E N J ~ λ an we sa ha a mask of lengh s p on he ansen esponse of he ackng eo. B mposng a mask on he ansen esponse one oes no wase he avalable egees of feeom n he conolle paamees on he machng of a specfc an enel aba ansen esponse. Insea one can focs hese paamees enel on achevng a fas selng me. The cos acheve afe he maske neval s alwas smalle han when no mask s se. 4.. Smlaons sng weghe IFT algohm Impovng he selng me onse he plan. s s s P One wsh o ne a PID conolle n oe o acheve a bee selng me fo he close loop ssem. onse he sana fom of he PID conolle: s T T s k s G p PID ha fo he phscal ealzaon we change n: T s s s T k s G p τ wh τ an he followng nal PID paamees:.5 p T T k so ha he nal conolle s: s s s s s G 3 τ

53 Ths els he ve slggsh esponse shown n he fge below. Fg. 4. lose loop sepample esponse wh nal PID paamees We se n he IFT algohm a Gass-Newon appoxmaon of he Hessan fo he max R, a sep sze of. an he followng ese esponse : T s.5.5 Fg. 4. Dese esponse o a sep of ample The applcaon of he classcal IFT ceon sng he cos fncon J N N 5

54 els, afe 5 eaons, he esponse: Fg. 4.3 lose loop sep esponsefll obane wh he classcal IFT ceon an sng he ese esponseoe Dashe cve: esponse wh he nal PID paamees. Ths esponse s ve nsasfaco; hs s n lage pa e o an nfonae choce of nal paamees. Wh he se of a fxe mask of lengh secons, he mnmzaon of he mofe IFT ceon wh he same nal paamees leas o he followng close-loop esponses, obane afe 5 an 3 eaons especvel. Fg. 4.4 Sep esponsefll afe 5 eaonslef an afe 3 eaonsgh sng a mask of lengh, he ese esponseoe an he sep esponse wh he nal PID paameesashe Ths esponse s bee han ha obane wh a efeence ajeco. 5

55 Fnall a mask of eceasng lengh s se, wh an nal lengh of secons, an wh he same nal paamees agan. A eve eaon of he IFT scheme, he lengh of he mask s ecease b secons. Fg. 4.5 Sep esponsefll afe 5 eaonslef an afe 3 eaonsgh sng a mask of eceasng lengh, he ese esponseoe an he sep esponse wh he nal PID paameesashe Obseve he amac mpovemen of he esponse e o he se of a mask of eceasng lengh, leang o a seqence of cos cea ahe han a one-sho ceon, an o a ffeen seqence of paamee vecos han esle wh he ec se of a mask of lengh secons. Resmng, we can se he ceon J N N o compae he esls. The nal cos s J.94 an wh he mplemenaon of he classcal IFT ceon we ge J 5.. Usng IFT wh fxe mask Usng IFT wh mask of eceasng lengh J 5.69 J 5.34 J 3.7 J Fg. 4.6 The sbscp enoes afe how man eaons he vales sem fom. The nal paamees ae:

56 Afe 5 eaon of he classcal IFT ceon: Usng he mofe ceons: IFT wh fxe mask IFT wh mask of eceasng lengh.538 ; ; ; ; ; ; Fg

57 4.. Smlaons sng weghe IFT algohm Impovng he oveshoo In hs secon we wll pesen wh an example how he mofe ceon sng me weghngs can mpove he esponse pefomance n ems of maxmm oveshoo. onse he plan scee me:.493z.95 P z 3 z.743z an he scee conolle: G.6z.5z.3 z z.68z.39 Ths els a sep esponse wh a conseable oveshoo shown below: Fg. 4.8 Response o a sep of ample a wh he nal conolle paamees Usng n he classcal IFT algohm: - a paal paamezaon of he conolle nmeao - a cos fncon J N - a Gass-Newon appoxmaon of he Hessan fo he max R - a sep sze γ. 55

58 we ge, especvel afe 4 an 9 eaons, he followng sep esponses. Fg. 4.9 Sep esponsefll afe 4 eaons of he classcal IFT algohm an sep esponse wh he nal paameesoe Fg. 4. Sep esponsefll afe 9 eaons of he classcal IFT algohm an sep esponse wh he nal paameesoe 56

59 We see ha he classcal IFT ceon els a smalle maxmm oveshoo an selng me. Usng a me weghng n he cos fncon, we wll now show how we can ge bee esls. onseng he cos fncon an applng a me weghng J N [ w ] w, w 5, [,.5] [.5,].5,.5 as shown n he followng fge Fg. 4. we ge he followng sep esponse: Fg. 4. ha els a mch bee behavo wh espec o he classcal IFT ceon an was sng onl 4 eaons. 57

60 58 4. The BFGS meho fo he seach econ The max R n he paamee pae law eemnes he pae econ an s heefoe ccal fo he pefomance of he algohm. We have seen ha a goo choce s o le R be an appoxmaon of he Hessan. Especall f ~ s small, he Gass-Newon econ T T N N R λ s a esable chose an he naal appoxmaon s: T T N es es es es N R λ Anohe goo chose s he Boen-Fleche-Golfab-ShannoBFGS meho, one of he qas-newon mehos. One of he mes of he qas-newon mehos s ha he goo esmaon of Hessan max s gven fom he gaens of he cos fncon J an he esgn paamees. BFGS meho s well known as a goo opmzaon meho [Hamamoo e al, 3]. The enewal law o esmae he Hessan base on BFGS meho s gven as follows. k k T k k T k k k k T k T k k k k s B s B s s B s z z z B B whee ' ' k k k k k k k k k k J J J J z s B B The spescp enoes he k-h eaon. The nal max vale of B s an aba posve efne max. Usall B s chosen o be an en max. The followng facs ae well known abo he BFGS meho: a f B k s smmec hen B k s smmec; b f B k s posve efne an z k T s k >, hen B k s posve efne. When z k T s k > s no sasfe, le B k B k >. Howeve sch a case selom occs. Noce ha he BFGS meho ses he same aa as he Gass-Newon meho. Ths he IFT paamee pang law becomes: J B γ.

61 4.3 The Hamamoo-Fka-Sge IFT Appoach We conse he wo.o.f. conol ssem epce n he followng fge. Fg. 4.3 lose loop ssem wh wo-egee-of-feeom conolle Snce an wo.o.f. conol ssem can be ansfome n o hs confgaon [Sge & Yoshkawa, 986], hee s no loss of geneal hee. In fges, P s he SISO plan, an K an F enoe he conolles o be esgne. The scala, an ae he plan np, s op an he efeence, especvel. The sgnal enoes an exenal es sgnal whch s se fo he esmaon of he pefomance of he close loop ssem an we assme ha we can choose abal. Noe ha all sbssems ae known excep fo he plan P, an all he sgnals ae obsevable. As we have seen befoe, he ssem T s he ese close loop moel whch s gven n avance, an T s he ese ajeco whch shol ack. One sngshng feae of he conol sce shown n he fge s as follows. Le T enoe he close loop ansfe fncon fom o, hen s known ha he so-calle cononal feeback pope hols, ha s, T T K s sasfe wheneve F T P hols. In ohe wos, K oes no pla an ole fo ackng pope n he nomnal case, an he man ole of K s o sppess he effec of sbances an plan nceanes [Sge & Yoshkawa, 986]. Whle appaenl, he ole of F s o specf he ackng pope. The conolles K an F ae sppose o be nqel eemne b he esgn paamees a an b especvel. We se he smbols K a an F b n oe o ncae he epenence on he paamees, explcl. Fo I/O sgnals, we se he smla smbols sch as a an b. Fom he above obsevaon, wol be naal o ne K a an F b o as o acheve low sensv an ese ackng pope, especvel. In he followng we assme ha one sablzng conolle pa K a, F b s gven n avance. An each expemen s pefome n he fne me neval [, f ]. 59

62 The saeg popose b Hamamoo, Fka an Sge HFS-meho s he sepaee nng of he feeback an feefowa conolles. In pacla, as fo he feeback conolle, he concenae on nng o acheve low sensv nsea of ackng pope, whle he feefowa conolle s ne fom he vewpon f comman ackng pope. - Feeback conolle nng In hs secon we concenae on he nng of he feeback conolle K a. In oe o acheve low sensv, hs new meho o mnmze he weghe sensv fncon WsSs n a cean case, whee Ws s he gven weghng ansfe fncon an s he sensv fncon. S s P s K s To achve hs goal he HFS meho pefoms he followng wo expemens A,B. Expemen A Se f an n fge 4.3. Then he conolle ssem s shown as he fge below. Fg. 4.4 Injec he es sgnal whch s calclae fom s W s n s whee n s a val whe sgnal whch has zeo mean an an appopae covaance. Then, le a an a be he coesponng I/O sgnals of he plan. Fo hese I/O sgnals, we solve he followng poblem. 6

63 FB onolle Desgn Poblem Fn he paamee a * whch mnmzes wh J x fb a a λa a x, x f x T τ x τ τ hogh eaon of expemens, whee λ a s a posve consan weghng scala. Noe ha he ansfe fncon fom o s eqal o he sensv fncon S. Theefoe, f f an λ a, he above cos s eqal o he sqae of he H nom of WS becase of he wheness of he val sgnal n. Now we appl he HFS meho, ha eves fom he sana IFT algohm base on Hjalmasson an Bkelan 998 an Hamamoo an Sge 999. The paamee a s pae b γ B ' a a a fb a ' whee enoes he vale of a a he -h enewal, an J s he gaen a of J a wh espec o a. As fo he sana IFT meho, he max B a gves he pae econ an can be fo example a Gass-Newon appoxmaon of he Hessan of J o a max fon wh he BFGS meho. In o scsson an followng example he scala γ s consee as a fxe sep-sze. ' Fs, we calclae J fom he I/O aa of expemens who an moel of P. The j-h en of J fb a ' J fb s gven b ', λ, ' ' fb, j a j a a a j a a whee ' a enoes he evave wh espec o a an he sbscp j means he j-h elemen of he veco. ' In oe o calclae J, we pefom he followng expemen. fb a J fb a Expemen B Se an n fg. 4.3 an njec he sgnal f a whch s obane b Expemen A, an le f a an f a be he coesponng I/O sgnals of he plan. 6

64 6 Snce PK a a a a a K hol fom Expemen A, he evave wh espec o a gves s ' ' ' ' ' a a j a a a j a a a j a a j a j f K PK P K PK PK P K PK PK ' ' ' ' ' a a j a a a j a j a a a j a j f K PK P K K K whee we se he elaons a a a PK P f a a a PK f fom Expemen B. Theefoe we can calclae ' a fb J fom he aa,,, ' ' a a a a hogh Expemens A an B. The pocee of feeback conolle nng s soppe when, fo a gven scala > ε n avance, ε < k a fb k a fb J J an we ega k a as he sb-opmal paamees * a.

65 - Feefowa conolle nng k Now we fx he feeback conolle as K K, an ne he feefowa conolle F b. The objecve hee s o make ack moe accael fo he gven efeence. Fo hs ppose we pefom he followng expemen. a Expemen Injec o he wo.o.f. conol ssem n fg. 4.3 wh F b an K, an le b an b be he coesponng I/O sgnals of he plan. The ssem s shown n he fge below. Fg. 4.5 Fo hs ssem we solve he followng poblem. FF onolle Desgn Poblem Assme he conolles K an F whch sablze he ssem shown n fg. b * 4.3 ae gven. Fn he paamee b whch mnmzes he cos fncon J ff b a λb b hogh eaon of expemens, whee λ b enoes he posve consan weghng scala. The IFT pocee s almos he same as n he case of FB conolle nng b eplacng J fb J ff, K F an a b. The j-h en of he gaen of he cos fncon s gven b: J ', λ, ' ' ff, j b j b b b j b b To calclae we nee he followng expemen. 63

66 Expemen D Se an n fg. 4.3 an njec he sgnal f, an le b an b be he coesponng I/O sgnals of he plan. Fom Expemens an D, he followng elaons hol: PK P PK K b F b T PK PK P b b Whle, fom he evave wh espec o b, we have: ' ' ' j b F j F j b PK ' ' P ' j b F j Fj b PK ' ' Theefoe, he aa,,, Expemens an D, an we can calclae b ae obane hogh b b b ' J ff fom hese aa. A SIMULATION EXAMPLE Le s noce a smple moel of a obo jon expesse b he Laplace fncon: G whee J s he jon momen of nea. s J s The example s aken fo [Scalamogna, ]. In ha wok, he ppose was o mpove he pefomance of he conolle ssem b ang o he ssem a sable exenal conol sgnal sng Ieave Leanng onol IL. We wll examne he same example n ems of IFT. 64

67 65 Fom G c s we oban Gz ha s he scee-me veson of G c s nclng he Zeo Oe Hol ZOH: z z G z z J T z G gan s whee T s s he samplng me. The jon s conolle b a PD feeback conolle Kz an b a fee-fowa conolle F f z. z k T k k z T k T k z z T k k z G s p s p s p s p hence z F z F z K zeo gan an z z F z z T J z F gan f s f wh J.94 Ns k p.7 k.4 T s. s The followng fge shows he scheme we ae conseng noce he sbance ae. Fg. 4.6 We choose he followng paamezaons of he wo conolles: z z z K z z z z F f

68 wh he nal paamees: The efeence ajeco chosen n he expemen s sn π. The nal behavo s shown n he pces below. Fg. 4.7 Fg. 4.8 We wsh o mpove he pefomance of he conolle ssem b sng he HFS- IFT appoach. 66

69 We pefom 3 eaon of he IFT algohm o ne he feeback conolle K wh a Gass-Newon appoxmaon of he Hessan fo he max B, a sep sze of. an λ a λ b. The fnal paamees ae: Fg. 4.9 an he coesponng ackng eo: B 3 eaon we pass fom an nal cos -5 J fb J fb.439 o a fnal cos The followng fge shows he anson of he cos J fb b eaons. Fg. 4. Fom hs fge we can see ha he cos J fb eceases b he meho popose b Hamamoo, Fka an Sge. 67

70 The followng fge shows he op esponse coesponng o he es sgnal. Fg. 4. Fll: esponse afe feeback nng Doe: esponse wh he nal conolle Ths fge shows ha lowe sensv s acheve va IFT. The gan an phase plos of sensv ae shown n he fge below. The obane conolle acheves a sasfaco pope n he low-feqenc oman. These fges show ha he popose IFT meho woks well n oe o acheve low sensv. Fg. 4. Sensv fncon Fll: fnal conolle afe feeback nng Doe: nal conolle 68

71 We now fx he feeback conolle as fon afe he pevos 3 eaons an we pefom eaons wh he ppose o ne he feefowa conolle F f. B en eaons, we pass fom an nal cos -3 J ff J ff.4 o a fnal cos The followng fge shows he anson of he cos J ff as a fncon of eaon nmbe. Fg. 4.3 The followng fge shows he ackng eo afe he nng of he feefowa conolle F f. Fg. 4.4 Tackng eo -Fll: afe feefowa nng - Doe: afe feeback nng - Dashe: nal 69

72 The fnal paamees ae: Smmazng, he HFS-meho se sepaae nng of he feeback an feefowa conolles. As fo he feeback conolle, he pon s focse on nng n oe o acheve low sensv nsea of ackng pope, whle he feefowa conolle have been ne fom he vewpon of enhancemen of ackng pefomance. The expemenal esls show he effecveness of he popose IFT meho. 7

73 5. Applcaons on an Insal Robo 5. The ABB nsal obo Ib- bef escpon The obo se n he expemens s an ABB Ib- nsal obo. The obo has seven lnks whch ae connece b sx jons, as shown n he followng fge. Fg. 5. Fg. 5. I s bl p b wo bg ams an a ws. Jon axs B n he fge s se o move he lowe am back an foh, wheeas jon 3 A moves he ppe am p an own. Jon 4 D s se o n he ws n an jon 5 E bens he ws n aon s cene. The sxh jon F s se o n he obo en effeco, whch s mone on he p of he ws he en effeco s no shown n he fge. Fnall, jon ns he ene obo aon s base. The obo ssem has ffeen bl-n conolles, one fo he conol of each jon angle. These conolles ae cascae PID conolles. Fg

74 5. The expemenal plafom The expemenal plafom consss of: - econfge Ib- obo ssem obo an conol cabne; - VME base boa compe ssem age ssem; - Hos compe ssem conssng of Sn woksaons hos ssem; - Ehene connecon beween hos an age. The Ib- s conolle fom VME-base embee compes. Sn woksaons ae se fo sofwae evelopmen an conol engneeng, as well as fo obo opeao neacon. PU-boa Powe P Fg. 5.4 The above fge shows he Ib- pa of he laboao. Sgnals fom nenal sensos of he obo o he VME ssem go va he senso neface o he DSP boa connece o he VME bs. The mase compe n he VME compe s base on Powe P pocesso. Spevson an safe fncons ae mplemene on a M683 boa, well sepaae fom he es of he ssem o peven amage of he obo. Dgal Sgnal PocessosDSP ae se fo low-level conol an fleng of sensos sgnals. Sensos eqng ve hgh aa banwhs ae connece ecl o he DSP boas. An aonal DSP boa belongs o he foce-oqe senso. 7

75 5.3 The Malab-obo connecon B he Sn woksaons on hos-level s possble o efne an we he pogams o conol he obo o sen efeences o be acke. Tha can be one nse he Malab envonmen. A Malab pogam calle Exc_hanle excaon hanle s avalable fo smple excaon expemens on he obo. Ths pogam can be se o efne veloc an poson efeences o he obo sevos. The nps can be seps, amps, snsos, nose an ohe aba sgnals fom he Malab wokspace. Fg. 5.5 A lo of sgnals can be ecoe ng he excaon. These ncle np oqes, poson measemens, ffeenae poson veloc, an foce an oqe measemens fom he foce senso. The ecoe sgnals can be expoe o he Malab wokspace fo plong an aa pocessng. Fg

76 5.4 The Smlnk conol neface A Smlnk conol neface s avalable o hanle he connecon beween he obo an he Malab envonmen. The fge below epesens one obo jon. Fg. 5.7 Wh hs neface, s possble o connec an log sgnals wh he Excaon hanle Exc_hanle an also o bl o own conolle ha we wll connec o he oqe_ef oqe efeence sgnal. Afe we have bl o conolle, he Smlnk scheme can be aomacall anslae no coe an ownloae ecl no he obo conol n. To be able o se o own conolle fo jon k we nee o acvae he new conolles fo he obo. Ths opeaon can be one sng he malab pogam acvae_smjk. To eacvae he ognal conolles fo he jon we se acvae_smj. Wh he malab pogams RegOff an RegOn we can emove an ensall, especvel, he ognal conolle fo he specfe -h jon. 74

77 5.5 Applcaon of IFT o a obo jon moel Befoe we come o he eal applcaon we wll s some esls of he IFT algohm fo nng a conolle fo a moel of he base jon jon fo he obo. The jon moel sce we conse s he followng: G s s a s s s ωξ s ω ω ξ s ω The vales se n he smlaons ae: a 7 ω 5 ω 7 ξ.4 ξ.45 Noce he pesence of an negao n he pocess moel. We conol he ssem wh a sana PID conol sce: - PID Robo jon Fg. 5.8 G s k p T s T s Fo he ealzaon we choose: wh τ.. G s k p T s τ s T s hoce of he nal PID paamees - AMIGO nng les We choose o sa wh nal paamees Kp, T an T fon sng he AMIGO consevave nng les [Åsöm & Häggln, 3]. 75

78 The AMIGO les, n he case when an negao s pesen n he pocess, can be esme n he followng wa. We make an open loop sep esponse expemen n whch we mease he qanes K v an L of he fge, ha ae he slope an he nesecon wh he me axs of he sagh lne, especvel. K V L Fg. 5.9 Open loop sep esponse of he ssem Gs Then he AMIGO les gve he followng qanes of he PID paamees: k T T p.45 K L 8L.5L In o specfc case, we mease he qanes: V K V L. So ha we ge he nal PID paamees fo he obo jon conolle: k T T p

79 These paamees el he followng nal sep esponse: Fg. 5. Inal sep esponse of he obo jon moel Now we appl he sana IFT algohm wh he am of mpovng he obo jon esponse. We cone he followng ese op esponse o he efeence sgnal chosen as a sep:. T. s 3s Fg. 5. Jon - Dese op 77

80 The applcaon of he classcal IFT ceon sng he cos fncon an conseng: J N N - a Gass-Newon appoxmaon of he Hessan fo he max R - a sep sze γ. els, afe especvel,, 3 an 4 eaons, he esponses: Fg. 5. Fll lne: fnal Jon sep esponse afe eaons Dashe: nal esponse - Doe: ese op Fg. 5.3 Fll lne: fnal Jon sep esponse afe eaons Dashe: nal esponse - Doe: ese op 78

81 Fg. 5.4 Fll lne: fnal Jon sep esponse afe 3 eaons Dashe: nal esponse - Doe: ese op Fg. 5.5 Fll lne: fnal Jon sep esponse afe 4 eaons Dashe: nal esponse - Doe: ese op 79

82 We can plo he gaph of he cos fncon J espec o he IFT eaons an smmaze he vales n he followng able. Fg. 5.6 eon as fncon of he nmbe of IFT eaons eon vales J nal.8744 J.56 J.98 J 3.6 J 4fnal.958 Fg. 5.7 The fnal vales of he PID paamees ae: k T T p

83 We can follow he paamee pae ng he IFT algohm lookng a he followng gaphs: Fg. 5.8 Fg. 5.9 Fg. 5. 8

84 5.6 APPLIATION OF IFT ON THE REAL ROBOT PROESS 5.6. THE JOINT EXPERIMENT We mplemen a PID conolle n he Smlnk obo neface as he followng he sana conolle s eacvae an he conol sgnal s ecl on he oqe. Fg. 5. The ppose s o mpove he esponse of he obo jon. The efeence ses s ef 4 a s a, wh a.5, n sep. 4 We sa wh he nal paamees: K T T p.4. We se he ceon N J an N - ese op as he efeence sgnal; - a Gass-Newon appoxmaon of he Hessan fo he max R ; - a sep sze γ.. We pefome hee ffeen IFT schemes: sana no weghs, mofe wh oble weghng an mofe wh vaable mask. The esls ae shown n he followng secons. 8

85 Sana IFT no weghs Fg. 5. Fll: nal esponse - Doe: ese op Fg. 5.3 Fll: fnal esponse afe eaons Dashe: nal Doe: ese We can noce how he selng me s no ve sasfaco. 83

86 In he followng pces he paamee paes ae shown. Fg. 5.4 Fg. 5.5 Fg

87 Mofe IFT fxe weghng We se now a mofe ceon sng me weghng as shown n he fs pce. wegh wegh 5.5 s wegh 3.75 s wegh 5 Fg. 5.7 Fll: nal esponse - Doe: ese op Fg. 5.8 Fll: fnal esponse afe 8 eaons Dashe: nal Doe: ese We can noce how hs echnqe els o a bee esponse n ems of selng me. Howeve we noce oscllaons ng he oveshoo an he esl fom eaon nmbe 9 became ccal sabl poblem. 85

88 The paamee paes ae shown n he pces below. Fg. 5.9 Fg. 5.3 Fg

89 Mofe IFT Weghng wh a vaable mask A mask wh nal leng of.5 secons s se as shown n he fge below an a eve eaon s ecease b.5 secons. wegh wegh wegh wegh Fg. 5.3 Fll: nal esponse - Doe: ese op Fg Fll: fnal esponse afe eaons Dashe: nal Doe: ese 87

90 The paamee paes ae shown n he pce below. Fg Fg Fg

91 5.6. THE FLEXIBLE BEAM EXPERIMENT The ppose s o esgn a conolle fo he beam eflecon an o mpove he behavo of he ssem sng he IFT algohm. We se p he obo wh a flexble beam mone on he jon 6 of he obo see fge below. Fg The flexble beam We have se a foce/oqe senso JR3 fo measng he beam eflecon oqe appoxmael popoonal o he eflecon. The beam was also eqppe wh a san gage whch also col be se fo esmang he beam eflecon b n hese expemens we onl se oqe measemens. The ssems s songl elae wh he wo-mass-pocess seen n he pevos chape, whee we col conse he fs mass as he obo mass an he secon mass as he beam mass. The pocess col be compae wh he pce below, whee m >>m. Fg The wo-mass-pocess 89

92 Ienfcaon Expemen The fs sep s o enf a SIMO moel Sngle Inp Mlple Op fom he poson efeencesgnal pos ef o he obo poson an beam efleconsgnal m whch s o be se fo conol of he beam eflecon. In hese expemens we have he sana poson conolle acvae. Sep esponse expemen Fg Fg. 5.4 We see ha he poson conol of he obo jon s sasfaco b hee ae lage pool ampe oscllaons fo he flexble beam ~6 Hz. 9

93 Feqenc esponse expemen To esmae a goo moel fo he conol we se a Pseo Ranom Bna Seqence PRBS as excaon sgnal. Fg. 5.4 Response of he jon poson wh a PRBS as efeence Fg. 5.4 Response of he flexble beam evaon sgnal m wh a PRBS as efeence 9

94 Usng a sb-space esmaon meho N4SID fom Ssem Ienfcaon Toolbox Malab, we ge goo esls fo a sae space moel of oe 6: loss fncon.394, FPE.445. Fg Feqenc esponse of he 6 oe moel fon wh he N4SID meho pos_ef o m We o ece o oe 4 o cape essenal namcs b, wh he ognal aa seqence we have a poblem of machng he coec fs esonanc feqenc an we have a lowe peak fo he gan. We ge he followng vales: loss fncon.93784, FPE.44. Fg Feqenc esponse of he 4 oe moel fll lne fon wh he N4SID meho pos ef o m an of he 6 oe moel oe 9

95 Fnall, sng a low pass fle on he sgnals befoe ong he enfcaon n oe o mach he fs esonance feqenc, we ge, mplemenng agan he N4SID meho, a sae space moel of oe 4 wh loss fncon.94e-6 an FPE.436e-6 an he followng feqenc esponse. Fg Feqenc esponse of he 4 oe moel fll lne fon wh he N4SID meho pos ef o m sng flee aa Doe: pevos 6 oe moel Dashe: pevos 4 oe moel no pe-fleng Fg Flee veson of he aa se se fo he enfcaon of he foh-oe moel wh pefleng 93

96 The fnal scee sae-space moel enfe oe 4, pefleng an chosen fo he conol mplemenaon s: x s Ax B x D wh he samplng me s.5 an he maces: A B D The poles of he enfe moel ae: p p p p j j.547 j j.736 Fg

97 ONTROL DESIGN We choose a sae feeback conol sce wh he ppose o amp he flexble beam eflecon m an sll ge a goo fas sep esponse. Pole placemen Fg Fo he close-loop ssem we choose o keep he spee an he ampng of he wo poles coesponng o he poson an move he wo pool ampe poles elae o he flexble beam keepng he same egenfeqenc b nceasng he ampng. Fom he enfe scee me moel he poles wee ansfee o he coesponng connos me poles wee we have an eas nepeaon of he ampng. z a z a s ωζs ω ps Poles ae mappe accong o z e p connos me pole, z scee me pole [Åsöm & Wenmak]. In o case we ncease he ampng of he complex poles fom appox. o.74 We make a pole placemen esgn wh he new scee me poles. p p p p j.6 j.6 j.3 j.3 Fg