UV Automatique - ASI ( 1 TD9 : représentation d`état des systèmes
Transcription
UV Automatique - ASI ( 1 TD9 : représentation d`état des systèmes
"!$#&%('*)%(+-,.'-/10320.4657/98;:<+*0.2=0>8&'-,>,@?,.0.A-BC'-,EDF57/1034G/H&, IKJ-L.MON@PQN@LR>SML.T=MU3VWL.X*YOZ(YOP<[FX\] U^Y_Z(Y`\] a=XbVc-VWYd.e>Lfe>U3N@ZgXhPQi*a=L j [kP<Yml<L VcVnYOd@e>LoepU3N@ZkX=PQi*a=Lq\LqlrZtshukahML>kv kr M b F x 0 wyx{z;|&`}VON_~=U@eRZ>\] a=X=LqeRZkVOVLL.XYOMOZkXhVlQZgYPQ[kX kvyIKXbaYOP<lQPrVZkXFYmlQLoY~hU@[kMOd@e>Lf{[kX7\=Zge>L@X*Y_Zgl\LqlrZR\c-XhZge>Pri*a=Lkh\=[kX=X=L.Ml6] U.i*ahZgYPQ[kX\P<U.ML.XFYOP<L.l<lQLqMOU@ukPrVOVZkXFYylrZR\c-XhZ( e>PQi*a=L \=LoN@L VWcVWYd.epLFv= bL^YYMOL N^L@YWYLqU3i*ahZ(YOP<[FXV[kahVylrZt{[kMOepLRS ZFL.N u(t) = F (t) ẍ(t) + 2ξωn ẋ(t) + ωn2 x(t) = Ku(t) v[kX=X=L.MmlrZ>ML.T=MU3VWL.X*YOZ(YOP<[FX\] U^Y_Z(Y`\aVcVnYOd@e>Lkv1XN_~h[kPrVWPQMOZpN^[Fe>epLqU@YOZgYOVqS x1 = T7[*VWP<YPQ[kXL@Y x2 = -PYOL.VOVWLFvhZ V[kMYPQLo\=aVcVnYOd@e>LoL.VWYlQZ>T[FVPYOP<[FXv IKJ-L.MON@PQN@L L N^[FepT[kMYL.e>L@X*Ym\] ahXVcVnYOd@e>LoL.VWY`\U.N@MP<YThZgMlQL.VyU3i*ahZ(YOP<[FXhV\P<U.ML.XFYOP<L.l<lQL.VVa=PQ(ZgX*YL3VoS z¨1 + 3z˙2 + 4z˙1 + z1 = u1 (t) − 2u2 (t) z¨2 + 2z¨1 + z˙2 + z2 = u1 (t) + u2 (t) kv`X"T7[*VWL N@[ke>e>Lf(ZkMPrZg=lQL.Vy\] U^Y_Z(Y S x1 = z1 x2 = z2 x3 = z˙1 x4 = z˙2 v [kX=X=L.Ml6] U.i*ahZgYPQ[kX\] U^YOZgY`\L N^LqVcVnYOd@e>L v`a=L.l<lQLL3VnYl] U3i*ahZ(YOP<[FX\LqV[kMYPQLfVP z1 L@Y z2 V[kX*YlQL.VV[kMYPQL.V\aVWcVWYd.epL&qU@T[kX7\MLotlQZpe>@e>Lqi*a=L.VWYPQ[kX"VWPlQL.V V[kMYPQL.VyVW[FX*Y z1 L@Y z˙2 v IKJ-L.MON@PQN@L `XT7L@YP<Y`e>[gYL.a=Mm>N^[Fa=MOZkX*YyN@[kX*YPQX-a{[FXhN YOP<[FX=X=LqL@XMOU@uFP<e>Lfl<PQX=U.ZkP<MOLkv=LqVN_~=U.eRZp\=L N^Loe>[gYL.a=ML.VWYmMOL@T=MOU.VL@X*YUoVa=MlQZ s7uka=MOL v i(t) R L Cr(t) u(t) e(t) J f ω Cm(t) wyx{z| }VN_~hU@eRZt\=a"e>[kYL@ahMmpN@[ka=M_ZgX*YmN^[FX*YPQX*a Z>VW[FMWYOP<L L3VnYmlrZpT7[*VWP<YPQ[kXZgXhuka=lrZgPQMLo\Lol] ZkM=MOLo\aep[kYL.a=M.v=Lqe>[gYL.a=ML@X*YOMOZ(rX=Lfa=X=L N_~7ZgMOukLfe>ZgYU.MPrZglQPQVU@LfThZkMlQL N@[ka=Thl<LqMOU.Vθ(t) PQVWYOZkX*Y cr (t) v7L.VZkaYMOL.VyXh[gYOZgYPQ[kXhVV[kX*YT=MOU.N@PQVU@L3VN@P\L3VV[kahV.v S=MOU.VPrVnY_ZgXhN@LL@YmP<Xh\=ahN Y_ZgXhN@L \abN^PQMON@a=PYm\] PQXh\a=P<Y\a"e>[gYL.a=M3 e(t) SGN^L.ePQXh\a=P<YLF kb S=N@[kXhVWYOZkXFYOLq\LoGN^L.e"v R, ShLPQX=L.MWYOP<LoYO[gYOZkl<L6ZgMO=MOLte>[gYOL@a=M SN^[-L^ N^PQL@X*Y`\L {M[kYWYL.e>L@X*Ym-PQVOi*a=L@aJ ki S7N^[kX7VnY_ZgX*YL \LtN^[kahT=l<LF ω Sh-PYOL.VOVWL J f ZkX=ukahlQZkP<MOLkv X¡Va=T=T[FVL@M_Z¢i*a=L"Y[FaYL"lQZ£T=ahPQVOVZkXhN^[email protected]^YMOPQi*a=L\a$ep[kYL.a=MRL@X$N^[kX-FL@MYPQLL@X$T=a=PrVOVZkXhN^L"e>U.N@ZkX=PriFahL"Nk] L3VnY¤g¤\PQMOL v=¥¦ZgMN@[kXhVU.i*a=L.XFY ki = kb = k v Pm (t) = cm (t)ω(t) = e(t)i(t) kv§¨ThZkMWYOP<M\=LblQZ©lQ[kP`\]ª~=e«ZgT=Thl<Pri*a=U@L©l] PQXh\ahPYL@Y\LblrZ©l<[FP{[kX7\=Zge>L@X*Y_ZglQL\=LlQZ¬\=c*X7Zge>PQi*a=LFK\U.\=a=P<MOLlQL.V U.i*ahZgYPQ[kXhV\=c*X7Zge>PQi*a=L3Vyuk[Fa=kL.MXhZkX*Y;lQL N^[ke>T[kMYL.epL.X*Y\=ae>[gYOL@a=M3v vyI¦JT=MOP<e>L.MflrZ"T7[*VWP<YPQ[kX£ZgX=uFa=lrZgPQML Θ(s) V[ka7VlrZ{[kMOepL Θ(s) = H(s)U (s) + Hcr (s)Cr (s) v&ahL@lL.VWYql6] [kM_\MOLt\a Vc-VWYd.e>Lkv =v`XFL@aYU3N^MOP<MOL`[email protected]_Z(YOP<[FX\] U^YOZgYy\=ae>[gYOL@a=M3vh[email protected][FX*YlQL.VL@X*YOMU.L.V7f¥M[FT7[*VWL.M\L3VZkMPrZghl<L3V;\] U^Y_Z(Y T[FVOVWPQ=lQL.V.v=®kahVWYP<shL.MmN^L.VN_~h[kP<J1v ¯ v[kX=X=L.MlQZpMOL@T=MOU.VL@X*Y_Z(YPQ[kX"\] U^YOZgY`\abVWcVWYd.epLFv I¦Y_Z(YOV ShN^[Fa=MOZkX*Yy\=ZkXhVyl] PQXh\a=P<Y x1 = i T7[*VWP<YPQ[kXbZkX=uka=lrZgPQMOL x2 = θ h*P<YL3VVLqZgX=uFa=lrZgPQML x3 = ω ° IKX*[email protected]<[FX"\] ZglQPQepL.X*YOZ(YOP<[FX u1 = u =N^[Fa=T=lQLoMU3VWPrVnY_ZgX*Y {TL@MYa=MOhZgYPQ[kX7 x2 = Cr v j [FMWYOP<L>S=T[FVPYOP<[FX"ZkX=ukahlQZkP<MOL y = θ v IKJ-L.MON@PQN@L ¯ ± [FXhVPQ\U.M[FXhVl<LqN@P<M_N^a=P<[email protected]^YMOPQi*a=LqVa=PQZkX*Y.v L iL iC i C VC VR R wyx{z|1E} ± PQMON@a=PYmU.l<L3N YOMPri*a=Lqm ± kv[kX=X=L.M`lrZ>{[kX7N YPQ[kX\=LqYM_ZgXhVW{L@MY\aN@P<M_N^ahPYVOZkN_~7ZgX*Y`i*a=L l] L.XFYOMU.L u(t) = i(t) L^Y`lrZV[kMYPQL y(t) = VR (t) L3VnYflQZ YL.XhVP<[FX"ZkaJ[kMOX=L.V\LqlrZpMU3VWPrVWYOZgX7N^Lkvha=L.lL.VWYl6] [kM_\MOLq\abVWcVWYd@e>L vy¥KMO[kT[FVL@M\L3V(ZgMOPQZk=l<L3Vy\] U@YOZgYmT7[Fa=Ml<LqN@P<M_N^a=P<Y.v7®kahVWYP<shL.Mk[gYOMLoN_~=[FPJv =vyIKXb\U3\a=PQMLqlQLoep[\d.l<L \] U@YOZ(Y`\=a"N@P<M_N^a=P<[email protected]^YMOPQi*a=LFv IKJ-L.MON@PQN@LE² j [kP<YKl<LyhZFN;\LmVL.N^YPQ[kX A ML.T=MU3VWL.X*YUyVWa=MlrZshuka=MOL ¯ vgLyhZFNL.VWY¦Zkl<PQe>L@X*YUyThZkMa=X=LyT7[Fe>T7Lm\[FX*YlQL\U@hPY\LN@[ke>eRZgXh\L L.VWY;(ZgMOPrZg=lQLkv-LhZkNVWL*Pr\LfqYM_ZkL.MOVKa=X=L`(ZkX=X=Lo\LVL.N^YPQ[kX vZ lQ[kP1\] U.N@[ka=lQL@e>L@X*Yy\LV[kMYPQLmL.VWYyX=[kX¤lQP<X=U3ZgPQML qe (t) Gl<[FP\L ³L@MOX=[Fa=P<lQlQPrS qs (t) = α√2gh ZkL3N g l] ZkN@N@U@lQU@M_Z(YOP<[FX\LolrZ>T7L3αVZkXFYOL@a=M.v qe(t) qs(t) h(t) S wyx{z;| ¯ }VON_~=U.e>Z>\LoMOL@e>T=lQPQVOVOZgukLo\] a=XhZkN kvyIN@MPQMLl<Lfe>[\d@lQL\] U^Y_Z(YyX=[kX=lQP<XhU.ZgPQMOL`\aVcVnYOd@e>Lkv=XT=MOL@X7\MOZtN^[Fepe>LZkMPrZghl<L`\] U@YOZ(YylrZ ~hZkaYL.a=M h vK] L@X*YMOU@L L.VWYmlQLo\=U@=P<Y qe (t) L^YmlrZpV[kMYPQLoL.VWYml<Lq\U.=P<Y`\L VW[FMWYOP<L qs(t) v vU^YL.Me>PQX=L@MlQLqT[kPQXFY`\=L{[FXhN YOP<[FX=X=L.epL.X*Y`VnY_Z(YPQ[kXhXhZgPQMLo\ahZFN (x, u) v =vyIKXb\U3\a=PQMLqlQLoep[\d.l<L \] U@YOZ(YmlQPQX=U.ZkMPrVWUoZga=Y[kahMm\aT7[FP<X*Y (x, u) v IKJ-L.MON@PQN@L ´ `XbVWcVWYd@e>L \LolQU@-P<YOZ(YOP<[FXe>ZkukX=U@YPri*a=LoL.VWYVON_~=U.e>ZgYPrVWUqVa=MlrZEs7uka=MOL ²v µlL.VWYN^[FepT[FVUb\] a=X¡[email protected] YOM[k¤ZkP<eRZkXFYR\LbMOU.VPQVWYOZkXhN^L R L^Y\] P<X7\ahN Y_ZgXhN@L L Zkl<PQe>L@X*YUThZkMa=X=LYOL@XhVP<[FX u(t) L^Ya=X N@[ka=M_ZgX*Y vZ\L@aJPQd@e>L>ThZgMYPQLR\a¬VWcVWYd.epL>L3VnYo{[kMOe>U@L>ThZkMqa=X=L>T=PQd.N^L>e>U^Y_ZglQl<Pri*a=LRe>[k=PQlQL>Va=MqlrZki*a=L.l<lQLRV@] L^JL@M_N^L ahX=Lo{[kM_N^Lqi(t)\] Z(YYM_ZkN YOP<[FX[Fa"\=LqMU.T=a=lrVWPQ[kX F YOL@lQl<L i*a=L>S F =k i2 e2 ZpT7[*VWP<YPQ[kX e \LqlrZpT=PQd.N^Lqe>[F=P<lQLoL.VWYme>L.Va=MU.LT7ZgM[email protected]=M[kT=YPriFahLo\=LquFZgPQX"a=X=P<YOZgPQMOLkv i Electro−aimant R, L u F e Piece mobile P=mg wx{z;|² }VN_~=U.eRZpVPQX=T=lQPshU \] a=XbVcVnYOd@e>Lq\Lql<U.-PY_Z(YPQ[kX"eRZguFX=U^YOPQi*a=L kvyIN@MPQMLol] U3i*ahZ(YOP<[FXU.l<L3N YMOPriFahLo\=Lql6] [email protected]^YMO[gZgPQe>ZkX*YL@Yyl] 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X(t) = X2 (t) = e(t) i(t) X3 (t) K ] L@X*YOMU.LfL3VnYlrZpYL.XhVWPQ[kX\] ZglQP<e>L.XFY_Z(YOP<[FX u(t) L^YmlrZ>VW[FMWYOP<LoL3VnYlrZtT[FVP<YPQ[kX e(t) v =vLRT[kPQX*YEVnY_Z(YOP<[FX=XhZgPQMOL>\=a©VcVnYOd@e>LRL.VWYEN@ZkMOZFN YOU@MOPQVUtT7ZgMqa=X=LRT[FVP<YPQ[kX¬\] U.i*a=PQl<PQ=MOL \] U.i*a=PQl<PQ=MOL (X̄, ū) L@X"{[kX7N YPQ[kX\L e0 v ¯ v[kX=X=L.Ml<Lqe>[\d.l<LolQP<X=U3ZgMOPQVUq\abVWcVWYd.epLqZkaY[Fa=Mm\LqN^LqT[kPQX*YVWYOZgYPQ[kX=X7ZgPQMLFv ē = e0 vIKJ-ThMPQe>L@M lQLRT7[FP<X*Y