REPORT No. 1/9_10_01 ROBUST ADAPTIVE CONTROL OF LINEARIZABLE NONLINEAR SINGLE INPUT SYSTEMS *
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1 REPORT No. 9_0_0 ROBUT ADAPTIVE CONTROL OF LINEARIZABLE NONLINEAR INGLE INPUT YTEM * Hojn X n Peros A. Ionno Deprmen o Eecrc Enneern - ysems Unersy o ohern Corn Los Anees CA Asrc: The esn o szn conroers or nonner pns wh nnown nonneres s chenn proem. The ny o eny ecy he nonneres on-ne or o-ne moes he esn o szn conroers se on ppromons or on pprome esmes o he pn nonneres. The prce p n sch cse co e c o heorec rnees or o sy nor zero rcn or reon error sey se. In hs pper nonner ros pe conro orhm s esne n nyze or css o sne-np nonner sysems wh nnown nonneres. The conroer rnees cose oop semo sy n conerence o he rcn error o sm res se. The reon o rcon or semo sy epens on he nmer o noes n wehs n he sne yer ner newor se o esme he nnown pn nonneres. The sze o he res se epens on esn prmeers n cn e cce pror. Two empes re se o emonsre he perormnce n properes o he propose scheme. Key wors: Ape conro; nerze sysems; nonner conro; rosness; swchn ncons. * Ths wor s sppore y NF ner rn EC-98779
2 . INTRODUCTION The ron wy o esnn eec conro sysem s se on he se o Lner Tme Inrn LTI moes or he pn. O-ne reqency omn echnqes co e se o sch n LTI moe o epermen n eny s prmeers. In he cse where he prmeers o he LTI moe chne wh me n schen on-ne prmeer encon pe conro ros conro echnqes ec. re eeope oer he yers o ress sch sons. The rence on LTI moes or conro esn prposes oen ps mons on he perormnce mproemen h co e chee or he pn ner conseron. For empe he pn consss o sron nonneres s ppromon y n LTI moe my consery rece he reon o rcon n he presence o srnces n oher moen ncernes. Drn he recen yers consere reserch eors he een me o e wh he esn o szn conroers or csses o nonner pns. These eors re escre n e n recen srey pper Koooc n Arc 00 where ery een n norme hsorc perspece o he eoon o nonner conro esn s presene n scsse. Mos o he recen eors sreye y Koooc n Arc 00 on nonner conro esn ssme h he pn nonneres re nown. The cse where he pn nonneres re procs o nnown consn prmeers wh nown nonneres e rse o nmer o pe conro echnqes Kosmopoos n Ionno 999; Kh 996; Krsc e. 995; Chen n L 994; L n Chen 99; Chen n Kh 99; Kneopoos e. 99; sry n Isor 989; Tyor e In hs pper we conser css o sne np eec nerze nonner pns wh nnown nonneres. We ssme h he pn nonneres re smooh ncons n he nonner ncon mpyn he np sses scen conon h rnees h he pn s conroe. The pn nonner ncons re esme on-ne sn sne yer ner newor. A nonner pe conro w s esne se on hese esmes o ssy cern sy conons ere rom seece Lypno-e ncon. The conro w conns nmer o ros mocons h rnee sn oneness een n he cse where he esme pn oses conroy cern pons n me. The propose conro scheme rnees h or n conons rom reon o rcon whose sze epens mny on he nmer o noes n wehs o he ner newor sns re one n he rcn error coneres o res se whose sze epens on cern esn prmeers. The sze o he res cn e chosen pror y seecn hese prmeers pproprey. eer empes o nonner pns re se o emonsre he ress. Ths pper s ornze s oows: In secon he proem semen n premnres re presene. A ener meho or ppromon o nonner ncons s scsse n secon. In secon 4 new ros pe conro scheme or
3 css o nonner pns s presene n nyze. Two empes re presene n secon 5 o emonsre he properes o he propose pe conro scheme.. PROBLEM TATEMENT AND PRELIMINARIE Conser he sne-np sne op sysem whose eqons o moon cn e epresse n he cnonc orm s: n y where [ L n ] T R n s he scr conro np y s he scr sysem op re compeey nnown smooh ncons n n e. n n. The proem s o esn conro w sch h he op y rcs en esre rjecory y nown smooh ncon o me. Assmpon : s one rom eow y consn.e. n R n he sn o s nown or n R. We ene he scr ncon s he merc or escrn he rcn error ynmcs: n λ e e y y where λ s pose consn enn he nwh o he error ynmcs. The sn srce 0 represens ner eren eqon whose soon mpes h e coneres o zero wh me consn 99. Derenn wh respec o me we on: n λ one n L e n α e n n L α e n n y α n e L αe where α n L α represen he coecens n he Hrwz nom epnson o. Le n n y α n e L e α 4 Then cn e wren n he compc orm: 5
4 I n were compeey nown ncons hen he conro w [ ] 6 co e se o mee he conro ojece proe o corse h he conroy conon 0 or s sse rnee y Assmpon. Usn 6 we on whch mpes h n hereore e 7 0 n- conere o zero eponeny s. In he cse where re nnown 6 cn no oner e se. As n he ner cse we cn se he Cerny Eqence CE prncpe Ionno n n 996 o come p wh n n ess o conro w whch we cn hen moy o mee he sy n conro ojece. Le s hereore sr wh he CE conro w [ ] 8 where he nnown ncons re repce y her esmes o e enere on-ne. In he oown secons we show how o enere n moy he CE conro w n orer o rnee sy n ssy he conro ojece.. APPROXIMATION AND ON-LINE ETIMATION OF THE UNKNOWN NONLINEAR FUNCTION nce n re ssme o e smooh ncons hey cn e pprome sn or empe sne yer ner newor s: 9 0 where n re he ppromon ncons or n respecey re s n ppromon n re chosen ss ncons re he nmer o he noes n re he op wehs or n respecey. The respece ppromon errors re enoe y: 4
5 5 Here s ssme h here es se o op wehs n nmer o noes sch h he smooh ncons n cn e pprome wh ny esre ccrcy 0 n 0 oer compc se n R Ω so h: m m m m Ω 4 As shown n Lppmnn 987; Pr n ner 99; Kosmopoos 995; nner n one n he reerences heren we css o ss ncons n ner newors es o ssy he oe ners ppromon conons 4. In 9 n 0 we ssme h he esner es he nmer o noes. The weh prmeers re o e esme on ne. Le e he esmes o respecey me. Then he esmes o he ppromon ncons me re orme s 5 6 The erence eween he esme n c prmeer es ress n he esmon errors 7 8 where 9 re he prmeer errors. The esmor n prmeer errors re no e or mesremen hereore eqons 7-8 re se ony or nyss. In he oown secon we presen he pe ws h enere he prmeer esmes oeher wh he conro w.
6 6 4. ROBUT ADAPTIVE CONTROL LAW The CE conro w 8 cnno e se o sze he cose oop sysem or nmer o resons. Frs cnno e enere on-ne recy ony cn e se n he conro w. econ he esmes my er consery rom he c ones en o he wron conro con ny. Thr here s no rnee h w no ssme es cose o zero. In sch cse he esme pn s cose o ose conroy en o posse re es or. In orer o e cre o hese proems he CE conro w 8 s moe o: [ ] s s 0 where 0 s esn consn re sm esn consns n < s The prmeers n respecey re pe s oows: sn where [ ] s s 4 s 5 0 re he pe ns chosen y he esner 0 s sm esn prmeer sn s he sn ncon sn 0 n sn- oherwse n s connos swchn ncon en y: < < 0 6
7 where 0 s esn prmeer se o o sconny n s shown rphcy n Fre. A connos swchn ncon shown n Fre nse o sconnos one s se n orer o rnee he esence n nqeness o he soon o he cose-oop sysem Poycrpo n Ionno 99. F.. Connos wchn Fncon By esn he conro w n 0 w neer ecome snr snce 0. Thereore he propose conroer oercomes he cy enconere n mpemenn some pe conro ws where he ene moe ecomes nconroe some pons n me. I s so neresn o noe h 0 wh he sme spee s 0. Ths when he esme o pproches zero he conro np remns one n so reces o zero. In oher wors n sch cse s poness o conro wh ppers o he conroer s nconroe pn. The conro w 0- s esne sn sy n Lypno ype rmens n s properes re escre y he oown heorem. Theorem: Conser he sysem he conro w 0 n he pe ws. I ssmpon hos n sses he conon hen en sm pose nmers n here es pose consns * * * * < < < sch h or * * * * * * m 0 0 Ω n 0 Ω where Ω R n Ω Ω R sns n he cose-oop sysem re one n he rcn error n s eres re one rom oe y n e λ 0 L n. Proo: Le s conser he oown Lypno-e ncon: V 7 7
8 8 The me ere V s hen en y V 8 where 0 or n or. In ew o he pe ws n 0 V or. Thereore he remnn o hs proo es srcy wh he cse o. Frs we nyze he rs erm n V n 8. Le s rewre he conro w n 0 s 9 where s en y: s s 0 In ew o eqon 5 n or cn e wren s: By ssn he conro np 9 n 0 no we on: ] [ ] [ s s ] [ ] [ ] [ Usn he enes 4 ecomes: s s 5 The s erm n 5 represens he eec o he esn consn n he conro w n he ppromon error.
9 9 Dene: 6 6 6c Then s shown n he Appen he soe e o he s erm n cn e epresse s: 7 nce we he 8 n 7 cn e rewren s: 9 Then n ew o 5 n sn 5 n s he rs erm n V s epresse s: 40 Usn 9 n 40 n he men whe nocn 4 we on: 4 In ew o he secon erm n V cn e epresse s: 4 Fny sn he s erm n V cn so e epne s:
10 0 sn 4 Here we he se he eny sn. nce n 0 ony mpes h hen or 0 he sn o s wys he oppose sn o 0. Comnn 4 4 n 4 V sses: V 44 By choosn sch h he oown conons re sse < c 45 m m 45e we on 0 V or 46 0 V or 46 The ress 46- re proe -4 ho. nce -4 ho on compc se.e. Ω ses nee o remn n hs compc se or 0 n orer or he ress o e. Conser he se 0 V V M 47
11 where V 0 V 0 n V 0 s chosen s he res consn or whch M Ω Ω where Ω Ω. Then or 0 Ω n 0 0 Ω oows rom h V s one rom oe y V 0 or 0 whch mpes h Ω Ω 0. Ths mpes h n re one or 0. nce V s one rom eow n s non-ncresn wh me hs m.e. m V V. Usn 46 n he c h 0 or we he m 0 τ τ 0 V 0 V < whch mpes h L. From L oows h sns re one whch mpes h L. From L n L we he 0 s Ionno n n 996. Ths mpes h coneres o he reon whch n rn mpes h he rcn error coneres o sm res se whose sze s chrcerze y he sze o he esn prmeer. We cn so essh h he rcn error n s eres re one rom oe y n e λ 0 L n one n L 99; one n Coesee 986. Remr : I oows rom 45 h he e zone wh n s n mporn esn prmeer o he propose ros pe conroer. nce he rcn error coneres o he res se λ n s 0 he ccrcy o rcn epens on he wh o he e zone. Howeer rom 45 we he.e. epens on he es o n. To on sm rcn errors we nee re n sm n. Dene 4 he prmeer cn e rewren s 4 4. nce 45 reqres < whch n rn reqres < < 4. Ths reqremen cn e sse. A sm e o cn e one y eepn 4 sm. Frhermore snce 4 he es o n 4 epen on he ros. Howeer sm es o reqre rer nmer o noes whch mpes hher orer ner newor. A sm e my mpy rer conro np when s sm. Anoher wy o rece he rcn error s o ncrese he n. Howeer he se o re n s nesre snce w reqre
12 re conro np. Ths here s reo eween he sze o he rcn error he conro n he ppromon error n ros conro eor h he cor cn ow.. The choces o he n he e zone wh epen on he mmm Remr : As we emonsre n secon 5 sn wo empes he esn prmeers cn e chosen o mee he erc neqes en y he heorem. Remr : In he conro w 0 he erms s n s cn e ewe o he roe. Frs hey cnce he ncern prs n e o mocon n he conro w n ppromon errors. econ hey c s spec -mocons Ionno n n 996 n he pe conroer preenn he ncerny erms n rom ecomn none. 5. IMULATION In hs secon we emonsre he properes o he propose pe conro w sn wo empes. Empe : conser he oown secon orer nonner sysem sn4π sn π 4 sn 0.0sn00 π π 48 y where he nonner ncons n prmeers re nnown. The op y n re ssme o e e or mesremen. The op y s reqre o rc esre rjecory ene y sn π. The mne o he y rcn error e y y sey se s reqre o e ess hn A one hen yer r Gssn newor wh T ss ncon ep[ π ξ ξ ] s se o pprome n whch n hs cse re he e nme smooh ncons on compc se Ω Ω Ω where Ω Ω 55. The men ξ s he cener o he r Gssn represenn he smpn r n s he rnce represenn mesre o he wh o he r Gssn. By choosn smpn r wh mesh sze 0.5 n rnce 4π sm norm ppromon on o cn e chee nner n one 99. Frhermore
13 0.99. Noe h 0.0sn00 s consere s srnce erm n s no esme. The es o re chosen o e respecey. Usn 6-c The consns n 0. 0 re chosen sch h conons 45c-e re sse. By seecn λ n he reqremen o rcn error ess hn 0.05 s chee. Gen conon 45 he n s chosen o ssy. 05. Thereore wh e o he rcn error remns one rom oe y Fres n show he smon ress or he rcn error n connos swchn ncon. F. Trcn error rn he rs 0 secons The she nes nce he reqre error on F.. The connos swchn ncon n he pe w rn he rs 4 secons
14 Empe : Conser he oown nonner sysem ep.5cos 49 y where nonner erms n prmeers re nnown. The sysem s ny We e o ree he op y cose o zero ess hn sy. In hs empe one hen yer r Gssn ner newor s se o pprome he nnown nonneres on compc se Ω Ω Ω where Ω 88 Ω 88. The ppromon errors re one y o-ne smons n rnn. In hs empe. The es o re chosen o e respecey. Usn 6-c we on The consns n 00 re chosen sch h conons 45-e re sse. By seecn λ he mne o he reon error s rnee o e one rom oe y A n es or 0 0 re en zero. Fre 4 n 5 show he smon ress o he reon error n connos swchn ncon respecey. F. 4. The reon o op rn he rs secon Fre 5 emonsres h swchn sops er n n ernn se n coneres o consn wh no rher swchn. 4
15 F. 5. The swchn ncon rn he rs secon 6. CONCLUION In hs pper we conser he conro proem o sne np eec nerze nonner sysem wh nnown nonneres. The nonneres re ssme o e smooh ncons n s sch cn e pprome n esme onne sn sne yer ner newor. A ros pe conroer scheme s esne h ses he esme nonner ncons n empoys nmer o ros mocons n orer o compense or ncernes n he esmon. The conro scheme rnees semo sy n conerence o he rcn error o sm res se whose sze epens on cern esn prmeers. emo sy s chrcerze y reon o rcon or sy whose sze epens on he noes o he ner newor se o pprome he nonner ncons o he pn. Or ress presen mehoooy or choosn ros esn prmeers so h he rcn error s rnee o conere n remn whn esre ons sey se. The eenson o hese ress o wer css o nonner sysem s crreny ner neson. REFERENCE Chen F.-C. n C.-C. L 994. Apey Conron Nonner Connos-Tme ysems Usn Myer Ner Newors IEEE Trnscon on Aom. Conr Chen F.-C. n H. K. Kh 99. Ape Conro o Nonner ysems Usn Ner Newors In. J. Conro Ionno P. A. n J. n 996. Ros Ape Conro Prence H Upper e Rer NJ. 5
16 Kneopoos I. P. Koooc n A.. Morse 99. ysemc Desn o Ape Conroers or Feec Lnerze ysems IEEE Trns. Aom. Conr Kh H. K Ape Op Feec Conro o Nonner ysems Represene y Inp-Op Moes IEEE Trns. Aom. Conr Koooc P. n M. Arc 00. Consrce Nonner Conro: A Hsorc Perspece Aomc Kosmopoos E. B. M. M. Poycrpo M. A. Chrsooo n P. A. Ionno 995. Hh-Orer Ner Newor rcres or Iencon o Dynmc ysems IEEE Trns. Ner Newors Kosmopoos E. B. n P. A. Ionno 999. A wchn Ape Conroer or Feec Lnerze ysems IEEE Trns. Aom. Conr Krsc M. I. Kneopoos n P. Koooc 995. Nonner n Ape Conro Desn New Yor: Wey. Lppmnn R. P An Inrocon o Compn wh Ner Nes IEEE AP Mzne 4-. L C.-C. n F.-C. Chen 99. Ape conro o nonner connos-me sysems sn ner newors ener ree eree n MIMO cses In. J. Conro Pr J. n I. W. ner 99. Appromon n R-Bss-Fncon Newors Ner Compon Poycrpo M. M. n P. A. Ionno 99. On he Esence n Unqeness o oons n Ape Conro ysems IEEE Trns. Aom. Conr nner R. n J. E. one 99. e Ape Conro n Recrse Iencon Usn R Gssn Newors In Proc. 0 h Conerence on Decson n Conro Brhon Enn 6-. nner R. n J. E. one 99. Gssn newors or Drec Ape Conro IEEE Trns. Ner Newors nner R. n J. E. one 994. Fncon Appromon Ner Newors n Ape Nonner Conro In Proc. Amercn Conr. Conerence 5-. sry.. n A. Isor 989. Ape Conro o Lnerze ysems IEEE Trns. Aom. Conr one J. E. n J. A. Coesee 986. Ape n Conroer ynhess or non-ner ysems In. J. Conro one J.E. n W. L 99. Appe nonner conro Prence H Enewoo Cs NJ. Tyor D. P. V. Koooc R. Mrno n I. Kneopoos 989. Ape Reon o Nonner ysems wh Unmoee Dynmcs IEEE Trns. Aom. Conr
17 7 APPENDIX-PROOF OF INEQUALITY 7 In hs ppen we proe neqy 7 se n he proo o heorem. Le s sr wh he eqy A. From 4 0 he ncon cn e wren s: A. Then A. nce A. cn e wren s A.4 sn no A.4 we on A.5 Usn A. oows h A.6 From A.6 he conro w cn e epresse s A.7 Usn he c we he A.8 nce A. cn so e wren s A.9
18 8 we he A.0 Thereore cn e epresse s A. n A. Usn A.8 n A. he soe e o he secon erm n A. cn e wren n he oown orm: A. The rs erm n A. cn e wren s: A.4 nce 0 ony we on: A.5 From A. n A.5 7 oows.e.
19 9 A.6 where re s ene n 6-c.
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Frs-order pecewse-lnear dynamc crcus. Fndng he soluon We wll sudy rs-order dynamc crcus composed o a nonlnear resse one-por, ermnaed eher by a lnear capacor or a lnear nducor (see Fg.. Nonlnear resse one-por