Tracking Control of Complementarity Lagrangian Systems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Tracking Control of Complementarity Lagrangian Systems Jean-Matthieu Bourgeot — Bernard Brogliato. N° 5384 Novembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5384--FR+ENG. THÈME 4. apport de recherche.
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T w a £ G T + T ³ e ( h dpZjefa&k _pe l dpS e ;4p]RscTgV4¤£dfeRT S _<m O9_<pT !ORZjo&O³2Z\df^j_vaiTGk I acORTo dfe9kwacs&_vZjefa l ]RscZ\eRaiORTgacsc_pe9kiAZ\TGefaUh(TGBsiZjd l ¤£_pe l o+dpe@pTGsifTGkaidby_vs l kU_IXac_peR(·)pTGefaiZ`_v^ _vhRhRscdf_focO acsc_buwTGo aidfsim _væaiT+s!U_ ;9eRZ\aiT¦dfsyZje ;9eRZ\aiT¦e2]RS(T+s!dvais&_ve9kiZ\aiZjdpehRO4_pkiTGk !¿dps!o m[o ^jTGk Ω ∪ I ∪ Ω # xyT aw£TGT+e _ve l aiO9T hRO9_fknT d[oGo ]Rs&k l ]RscZ\eR!ORZ`ocO df2uwT:oacZ\pT __'a Z`kJ]R!^ ;9^\^jT l JNPORT l _fknO9T l dfsiRZ\a AABB dfeC¨Zj9@V"siTGhRsiT:knΩTGefa&k X (·) l ]RsiZjeR_$acsc_pe9kiZacZ\dfehRO9_pkiT!Z\aiO Z\SUh9_foa&k+JNPORTkim[kwacT+S knac_pRZ\^j6Z 5+TGk"dfe (T+a¢yT+T+e A _ve l B !ORT+e acORT o dfefaiscdp^j^jT+s"ZjkUkn!Z\aco&ORT l aid_ dps&o Tgo dpe@aiscdp^Pknd acO9_ba X (·) ∂Φ _pe l X (·) SU_<m uw]RSUhaid B eaiO9T o dfefacsidf^Jkio&ORT+SUT l T:kio scZj(T l ^`_bacT+s:¤DacORT h4dfZjefa ! j Z j ^ ^ + o p d @ e p G T i s fT !Zje)_ ;9eRZ\aiTdps Z\e ;4eRZacTe@]RS(TGs dp%o+m2o+^\T:k #£aidby_ps l k D mT ;9e9_p^\^jm l 1T ;4eRT¦acORT B l o+^\d@knT M^jd2dph l T:knZjscT l ais&_<uwT:oacdpscmdv%aiO9To+dpSUhR^jT+SUT+e@ac_vscZ\a¢mgkim[kwacT+S _fk XA (·) h§e¨Zj94V9¤ X (·) Z`k§acORT o ]RscpT (CAA C) _pe l X (·) ∈ ∂Φ dpe (A C) a$Zjk$_veZjSUh9_po ai^jTGkck§ais&_buwTGoacdpscmpT a$]9k _fkiki]RSUTaiO4_ba"_gh(TGsiZjd l Z`oSUdvacZ\dfeZ`k l T:knZjscT l NPO9T+e dfe ¨Zj9¥Vdpe9^\m acORTdpscRZa&k¦dp X (·) !ZM Tf _ve l X (·) !ZM Tp AA CA) _vsc)T ;Rq[T l gNPORTdvacORT+s$aw£ddfsi9Za&k$SU_<mIv_vscm !Z\aiO acORT AA BCA) ∗ d. 2 i,nc. 1 ∗ d. 00. 00. ∗ d. ∗ d. 2k+1. k. 2k+1. ∗ d. d. i,nc. ∗ d. i,nc. ∗ d. ∗ d. d. +. d. 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(55) <. B- 9 -$. X i,nc (t) = Xd∗ (t) = Xd (t) = X i,c (t). C i,c X (t) Xd (t) . Φ. A 00 B0 B A0 Xd (t) . A. X i,nc (t) = Xd∗ (t). ¨Z\f]RscTV\JNPORTo ^jdfkiT l ^\d[dph l T:knZjscT l acs&_<uwT:oacdpscZ\T:kP_ve l o+dpefacsidf^.kiZ\fe9_v^`kG o+m2o+^\T Zje l T+q xy][adpe³_ knZjeRf^\T hRO9_fknT acOR:T ;Rq[T l h(dpZjefadv§acORT o ^jdfkiT l ^jd2dph TGsiscdps knm[knaiT+S S _<mZ\e l TGT l k(T _ kiZjpe9_p^ X ∈ ∂Φ (A AI ) !ORZ`o&O l Z\¡(TGsckscdpS X 6∈ Φ (A B ) INPO9TdpscRZ\ack _ve l aiORT$h(dfZ\e@a A fT+eRTGsc_p^\^jm b_psim sidfS dfeRT$o m[o ^jT Ω ∪ I ∪ Ω aidUaiORT$e9T q2ao m[o ^jT (AA B ) jh§eRT¦oG_ve _v^`kndZjefacT+schRsiT+aPaiORZ`k _pk l T1;9eRZjeR _ l T:knZjscT l acsc_buwTGo aidfsim dfe Ω T:_po&O o m[∪o ^jIT Ω ∪∪ΩI ∪ Ω ¤f!ORZ`o&O ZjktZ\aiTGsc_vaiT l sidfS o+m[ocpT k aido+m2o+^\T k + 1 kid¦acO9_baZ\a£o+Xdpe@p(.)TGsifTGk acd<y_ps l k X (·) tNPO9T¦SUZ\q@ac]RscT¦(T aw£TGT+egaiORT i Xg_pe l o+dpefacZ\e2]Rdp]9k l m2e9_vSUZ`o+k!o+^\T:_vsc^\m_vhRh(T:_vs&k+ eki]RSUS _vscm aiO9To dfefacsidf^tknais&_bacT+fmI_pe l knac_pRZ\^jZ\a¢m_pe9_v^jm[kiZjk$_psiT ^jT l !Z\aiO df]Rs l Z¡DT+scT+efaais&_b uwTGo aidfsiZjTG+k \ X (·) ZjeacORT"o dfefacsidf^¥ZjeRhR]RaG¤ X (·) Z\eacORT"¥mf_phR]ReRdb]Re4oaiZjdpe¤ X (·) _ve l X (·) X@acZ\^j^scT TGsiscZjeR"acd¨Zj99#V \t!ORTGegacORTkim2knaiTGSÃZ`k Z\eRZ\aiZ`_v^jZ`knT l dpe X (·) (T+a¢£TGT+e C _pe l A !ZM Tp£dfe #¤9aiORTGe X (t) = X (t) dfe (CA ) _ve l X (t) ∈ ∂Φ dpe (A C) JZjeRZacZ`_v^j^\m X(0) 6= X (0) Ωl l Z\¡(TGsck§_ve l Z`kknT+a acVd 5+T+scd Z\eaiORT"¥mf_phR]ReRdb]9e9oaiZjdfe _va§_ _pe Y<dfs X(0) ¤9acORT+e acZ\SUT"o+dpscsi˙T:knh(dp6=e l XZj˙eRUai(0)d acORXT ;9s&knXaZjS(·)h9_foa: NPORZ`kZjk acORTS _<uwdfs l Z`kio+siT+h4_ve9o+m o+dpSUh9_vscT l acd]9e9o dfe[ knais&_vZjeRT l SUdpaiZjdpeo+dpe@aiscdp^9Zje!ORZ`ocO _v^j^9dp]Rsais&_buwTGo aidpscZjTGk_vscT!aiORTki_pSTf¤f]9ki]9_p^\^jm l T+eRdpaiT l _pk X (·) !>knTGT¦siTGS _vsc.} # 0. 2k+2. 0. 0. k+1. 0. d. 0. ∗ d. 2k. k. 2k+1. 2k+3. 2k i,c. k. i,c. d. i,c. d. i,c. 0. ∗ d. 2k+1. ∗ d. 0. k 00. 00. i,nc. i,c. d. 00. i,c. d. d. , g+ &(' %
(56) $# NPORTkwa&_vRZj^jZawmo scZacT+scZ\dfe]9knT l ZjeacORZ`k$h9_vh(TGs¦Z`k$_peT q2aiTGe9kiZ\dfedp£aiORTmf_vhR]Re9d<IknT:o dfe l SUT+aiORd l _ l _vh[acT l acd$o ^jdfkiT l ^jd2dphST:ocO4_veRZ`o+_p^Rknm[knaiTGS !Z\aiOU]Re9Z\^`_bacT+s&_v^Ro dfe9kwacsc_pZ\e@ackJ_ve l O9_pk%(TGT+eUhRsidfh(dfkiT l Zje:A xysidfp^jZj_vaid "L6E\¤(+ O,OpoPF9_ve l A xysidfp^jZj_vaid "L!Ej¤[vpfF #¥T+a x(·) l TGeRdvacTPacORTkwa&_bacT!dv(aiO9To ^jdfkiT l ^jd2dph knm[knaiTGS Zje !n #y!ZacOkndfSTTGT l 9_foc o dfefaiscdp^j^jT+s u(X, X,˙ t) ª"!$# _ _ ®% # Ω¬i"ª - « +:[ - ª « G ª a U"!0/" $ !+9 /%&0/%b" 6/ Ω D P>&)/%b=16g 71 9,
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(58) t > 0
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(74) t ∈ Ω
(75) R < +∞ E ¥T a£]9k l T1;9e9T!acORT§o ^jdfkiT l ^\d[dphZjSh4_poa£rdfZ\e4o+_vs !S _vhUacO9_bayo dfsiscTGkih(dpe l k%aid$acORT§knT:oacZ\dfe Σ = ¤[!ORZ`o&OgZjk!_O@m2h(TGscki]Rsi_fo Tdv%o d l ZjSUT+e9kiZjdpe o+_ps l {x|F (X) = 0, X˙ ∇F (X) < 0, i ∈ I} NPORT h9siT ZjSh4_poapT+^jd[o Z\aiZjTGk_vscTo&ORdfkiT+e aid l 1T ;9e9T P dpsU_ scTG_fkndfepZjpT+e _bæacT+sUo α^`_v=ZjSÀ}R](I) mT l "T ;9eR,T \ t∈Ω. . − I. T. i. i. ΣI. PΣ I :. !bo #. − Σ− I → ΣI. xΣI (k) 7→ xΣI (k + 1).. ! ORTGsiT Z`kacORT¦knac_bacT§dp JT aP]9kyZ\e@aiscd l ]9o+TaiORT¦h(dfkiZacZ\pT]Re4oaiZjdpe acO9_baP!Zj^\^.kiT+scpTZje acORTkn]R4xknTf]9T+efa_ve9_p^\m[kiZjkG%PT a V l TGeRdvacTacORT$siT:kwacsiZ`oacZ\dfe dv V acd Σ V (·) ª"!$# _ _ ®% # .G ® # - « GG[ - ª « G ª a K/9&0/%$1 ?/k/" /% ,>&]/%b=1!K ' Q
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(87) % '&(
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