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5VµOQ|BIRStzVµV=[¢OviLBVUh=RSLiTBNGOtNaRSL []v [QYLOQVTBVU[¢O^Y_BNGWaNPOtI¤zYNa[§LBVU V=]zV@OBY[ TBVSvBNGTBV=W ´viOtNaWGNP[QY`OtV=vBTMYLB[yWaV¹h¾|iRSN TBVWaY ¬?RSLBh@OQNGRSLºTBV0jikMlm nopq V@OµLiV V=]zV@O¤MY[TBV h=RLBh=WGvBtV []NuRSL LBV²OQ]RSvi­SV MY[UviLBV OtV=WGWaV ¬?RSLBh@OQNaRLFLBV 9?2'F1i352'@ jikMlm nopq1d&(')(34 &(M§V=[¢OXvBLBV§¬?RSLBh@OQNaRLTBI@KMLBNaV\ RS[]NGOQNP­SV§TBRSLOuRSLOtV=[¢OQV,WZYTBI=htRNa[t[tYLBhVY`viOQRSvi%TBv RSNGLOT2I(viNaWGNa_B]VF.¦´IOtvBTBV5TiV=[zIOt|BR(TBV=[vBN( V=]zV@OtOtV=LOTiV5hRSLB[¢OQtvBNGtV vBLBV%9?2'F1i352' jikMlm nopq 1d&(')(34 &(MÁ RSvBvBLX[]([]Ot=zV TiRSLBLBIY9zROtNG­SIvBLBV WaNGO]OQIQY`OtvBtVOtt[Y_ RLBTMYLOQVh=V[UTBVtLBNG=]V=[µTBIh=V=LiLBNaV[UTBRSLOµWaY*tV­(vBV²TBI=BY[t[]V WGVh=YTBtVTBV²TBR(hvBzV=LO=F9¦V[¤¬?RStzV=[vMYTBtY`OQNGvBV=[\[tRSLO,WaV[\BWGvB[\viOtNaWGNG[tIV=[§LBROQYzzVLO§WaV[§¬?RLBhOtNaRSLB[§TBI@KMLBNGV=[\R[tNGOtNG­SV[\vBN[tRLO§TBV=[§NGLOtI=StYWaV[cB]V=zNatV[¤ªh`V[]Oa=\2 TiNatVTBRLO§WaYTBItNG­SIVOtV=z RStVWaWaV,V=[]O§L(vBWGWaV®uTBv[]([¢OQ=zVyNaTiIYWGNa[]I¹ªMY`bV VzBWaVWILBV=]SNGV OQROQYWGVT2vBL*[]([]Ot=zVyzIhY`LBNavBVNGTBIY`WaVzVLOh=RLB[tV]­`YOQNG¬M>i­SRSNaV V=zBWGVX[tvBNP­`YLO¾®@F - & '*)*"('a 21 >BA $ C & $ A>M$?) jN$Q &( %$QdO8 3*,34pBG

xe"! (*V(p"&(*V' &2!+27U)M2;

434O8P'I!5& p*V !:3I3* aYE3V!=N8'FI9?2'F1i352' 1"2'43'8' B:3C2& p*VU(x)

M**VhO8P?/="6

U(0) = 0=

Q U(x) > 0 ∀x 6= 0

=Q

U(x) < 0 ∀x 6= 0=

^U →∞

&2!‖x‖ → ∞

:76 )('43$?)*"('G 2'6!34 BG2'6!_*0c!+ !=q "1B3fU6&(B_8 'FXdO86&(4352' (3 +eBG'i35**V 2Bd(3;')&(3BGN'F2';=*,3'Fd&(3B"?/

x(t)− εx2(t)x+ x(t) = 0 28 ! 9?2B:h%$Q &(;=&(C("1N*0"!H < '34352'6!

x1 = x, x2 = x= 'F28 ! (pM'F2'6!?/

x1 = x2

x2 = −x1 + εx21x2

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿Â¾Ã@Ä

E¶ ° ± o7 p7 *jik lm nopq.n °l.l C°Z

*I"! 9 & 13*V HC(B:3 < BHO8P51"`-!-13** &(=8B &(C("1h8 'F 9?2'F1435A'X $ &(2B:43V!"!-' 'F2';*,3'Fd&(3BG & 8'5&(`%$QdO8 3*,34pB"<,c*d$Q2B:3.(3'F

(x1, x2) = (0, 0):f28 B *d$ &(')&(*,+-!- H* & !5&(;

p3*,3=L'F28 !1"23V!3BG2'6! * &`9?2'F1i352'TjikMlm nopq@1d&(')(34 &(=e!8 3C2&('=?/

U(x1, x2) =x2

1 + x22

2 1"23 YR"!f9?2') !8B 8 'F 1"2'6!34 Bd&(4352' U +-!34O8?/ 1$Q"!S8 'F 3'M-(Ba&(*VU BG35qBGa'FB(35e"1d&('34O8Pe=2 &(*V S(8 !+ !=q 34 d&(*V'1"2'6!-B:C2&(43 9L(pM'8 U28 B

ε = 0:j%

1d&(*V18 *k U

2''F?/U(x1,x2) = x1x1 + x2x2

= x1x2 + x2(−x1 + εx21x2)

= εx21x

22

2'F1U

"! 8 'F 9?2'F14352'X < '35#U-!343C(HO83"! !iB:351='# "1B"23V!"!&('=N*VL*V2' =28="! *V"! 4Bd&?"1=23BG"!SE8 !+-!MqH!3

ε < 0: $ &U)BGq"!*V"!:2BGq"!bU)B""1"d '!@= *d$Q5&(

%$QdO8 3*,34pB"(0, 0)

"! (*V(p"&(*V'b!5& p*VHU28 Bε = 0

=H"! (*V(p"&(*V'N&2!+27U)M2434O8P;'e!5& p*0 !3

ε < 0=(*V(p"&(*V' 3'6!5& p*V!3'F2' : &('6!1"aY7U)*V@= *d$ &(')&(*,+-!-"!

1"27U)*Vq= 1d&(Bc* *V & UB3V! 1G&(Ba& 1=B3V!-B * & !5& p3*,3M<(*V(p"&(*V(8 !+ !=q: 'F$Q"!U6&2! M28?28 B"!N*V 1d&2! #1"* & =U') N* &`9?2'F14352'T jikMlm nopq 1d&(')(34 &(Mh1"2 *V2'4B"N*d$QaY7U)*V7!8 3C2&(' :76 )'73$?)*"('G 2'6!34 BG2'6! S&(3'=')&(' *V !+-!Mqh < '3 U6&(B/

x1 = 2x1(x

22 − 1)

x2 = −x2(x21 + 1)

O8 3bU)B""!-'= 8 ' U23'%$QdO8 3*,34pBG, *d$Q2B3.(3' 1"2'6!34 B"2'6! * & A 9?2'F14352' jikMlm nopq@1d&(')(34 &(=hO8 U)BG"1"d '&/

U1(x1, x2) =x2

1 + x22

2 ' & &(*V2BG!?/

U1(x1,x2) = x1x1 + x2x2

= x21x

22 − 2x2

1 − x22 .

U1 < 0!3

x21x

22 − 2x2

1 − x22 < 0

: &('6!h*V`U)* &('(x2

1, x22)=`2' U8 #4Ba& 1:Bh*V 2S&E3'

! & p3*,3=N!8 BGfU6&(B1"4M 9?2'F14352'X yjikMlm nopq 1d&(')(34 &(=: ' (p435' &(*V2BG!L* &< (8 BG98&:;8&:

3 *d$Q2' 1"A'!34 qB" S&(3'=')&(' * &b9?2'F14352' jik lm nopq@1d&(')(34 &(M7!8 3C2&('M?/

U2(x1, x2) =x2

1 + 2x22

2,

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ tÄ=¾Ã@Ä

(l C°Z =kI ± °CoCMl ESE

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x12

x 22

x22=2x

12/(x

12−1)x

22=2x

12/(x

12−1)

STABLE

STABLE ou INSTABLE ?

37ESFPE 2S&(3'Fb! & p3*,3= !8 BG U6&(BU1(x1, x2)

:

'F28 !(p='F2'6!?/U2(x1,x2) = x1x1 + 2x2x2

= −2x21 − 2x2

2

N'F28 ! U28 C(2'6! 1"2'F1*,8BG_O8 *V!+ !=q"!(*V(p"&(*V'N&2!+A7U)=2i34O8'7!5& p*0 &('6!eM28 *d$Q"!=U6& 1"N%$Q5&(

(x1, x2): 'T(3 O8 *VU BG35qBG 9?2'F1i352' jikMlm nopq

U1"! 1"2'6!-B:C2&(i3C(h1d&(BH**Vh 2''F 8' BG"!8*,5&(kU"!"!33V!=7!8 BN*Vh 2S&(3'FL e!5& p3*,3ML(8!+-!Mq:

; #" "" " G@ " F " NWGVu[]([]Ot=zVcV=[]OWaNGLBIY`NGtVÁx

x(t) = Ax(t) x ∈ Rn ª¢ESFseS®YWGRSt[ÁWGV9[¢([]OQzV;V=[¢O SWaR_MYWGV=zVLO Y[¢ziOtROQNG(viV=zV=LO []OQY_BWaVXªWaV RSNGLO T´I=vBNGWaNG_BtVI@O^YLO\ WRtNGSNPLBV® []NOQRSviOtV=[;WaV[9­YWaVvBt[;B]RSB]V=[TiV

A[tRSLO[]OttNah@OQVzVLO; RS[tNPOQNP­SV(>B[]RSNGOx

Re(λi(A)) < 0 i = 1, · · · ,n

- & '*)*"('a 21 >BA $ C & )0/1347698;:7< ) ++657+ '*) + $ &> ? ) + jk!+-!=q*,3'Fd&(3BG 8&: 0T"!`&2!+A7U)=2i34O8' ! & p*V a28T*V"! C2&(*V8 BG!bU)B""U)BG"!

A!-2'G,U6&(B:435

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ QÂ@þÄ

E.e ° ± o7 p7 *jik lm nopq.n °l.l C°Z

BG"**V"!#'F- &(i3C("! =UP28 B#M28 MfS&(4B:351"!+2iB:34O8` < '35 U-!3i3C(Q=N3*baYE3V!M 8 'FcS&(iB:351"

P < '35HU-!343C( i!+A4B:34O8P !&(i3V!n9 &(3V!&(' *d$QdO86&(i352'

jikMlm nopqATP + PA+Q = 0

ª ESF®HJIzRLB[]OtQY`OtNaRSLµTBVcWaY,hRSLBTBNPOQNGRSLµ[tv [tYLOtV x h=RLB[tNGTBI=]RSLB[WZY 9?A' 1i352' jikMlm nopq 1d&(';(34 &(=

U(x) = xTPxYWGRSt[x

U(x) = xTPx+ xTPx

= xTPAx+ xTATPx

= xT (PA+ ATP )x .

RNGOQvBLBVXzY`OQ]Nah=V%TiIKMLBNGVX RS[tNPOQNG­V(>B[]N

PV=[¢Ow[tRWaviOtNaRSLµ R[tNGOtNG­SVuTiVEFs YWaRt[

U(x) > 0 ∀x 6= 0VO

U(x) = −xTQx ⇒ U(x) < 0 ∀x 6= 0 .HJRLBh(>MTYBt[WaV%Ot|BI=Rt=zVXI=LBIQYWi>BWaVu[¢[¢OQzVcV=[¢OJY`[](zOQROtNavBV=zV=LO;[]O^Y`_BWaV`FHJIzRLB[]OtQY`OtNaRSLµTBVcWaY,hRSLBTBNPOQNGRSLµLBI=hV=[][QYNa]VÁx RSvBvBL\h=RSviBWaV

(A,Q)vBVWahRSLBvBVTBRSLBLiI(>

WI(vBY`OQNGRSL ESFsT´NaLih=RSLiL(vBVP

V=viO§LBVBY[§YTBzVO]OQ]VvBLBV[tRSWGviOQNGRSL viLBNavBV`F YNG[c[tNAV[]Ow[¢O^Y_iWaVXYWaRt[W ´I=vMY`OtNaRSLµTBV\jik lm nopq YTizV@O;vBLBVX[tRSWGviOQNGRSLµvBLBNG(viV x

P =∫ ∞

0eA

T tQeAtdt .ª¢EF~®

LV /VOx

ATP + PA =∫ ∞

0AT eA

T tQeAtdt+∫ ∞

0eA

T tQeAtAdt

=∫ ∞

0

d

dt

(eA

T tQeAt)dt

= eAT tQeAt

∣∣∣∞

0

= −Qª¢EFf¥S®

h=YeAt → 0

vMY`LBTt→∞

[tNAV[]O[]O^Y`_BWaV`F

H%Y`LB[;WaY§h^|MYBNPOQtVJ[tvBNP­`YLO?>i RSvBTBIzRLOQ]V=WGV=[tI[tvBWPO^Y`Ot[TBVuWaY§h=RSzzYLBTBV%¦¨ >iRSLviOtNaWGNG[tV;vBLBV­`Y`tNZY`LOQVTiVJWaYXh=RSLBTiNGOQNGRSLLBIh=V=[][QYNGtVJTBVw[]OQY_BNGWaNPOQIÁxS[tRSNPO

AvBLBVwzyY`OttNGh=V;[¢O^Y_BWGV

YWGRSt[P[tRSWGviOQNGRSLUTBVESF V=[¢OJTBI@KMLBNGVc RS[tNPOQNP­SVX[tN

QV=[¢OJTiIKMLBNGVc RS[]NGOQNP­SVªI@­(NaTBVLOwTYB]=[

ESF´~®¾F

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ ¾Ã@Ä

E

F 5RSLB[]NaTBItRSLi[vBL[]([]Ot=zVcWaNGLBIYNG]VÁx

x(t) = Ax(t) +Bu(t) x ∈ Rn; u ∈ Rm

z(t) = Nx(t) z ∈ Rq .

ª eFPE.®5 3 M.- (+ ) WZYMY`NatV

(A,B)V=[]O[]OQY_BNGWaNG[QY`_BWGV(>h´V=[¢OaM\ TBNGtVµv´NaWL2´Y²MY[,TBVUzRTiV

NaLi[]O^Y`_BWaV%VONGLBSRSvi­SVtLMY`_BWaV%TMYLB[;WaVu[¢[¢OQzV`F

& + $ > .+ ¦Yh=RzzYLBTiVcMYtV@OQRSviwT´IOQY`OJvBN[]OQY_BNGWaNG[tVXWGVc[]([]Ot=zV§VOJzNaLiNaz NG[]VXWaVh=]NGOt=tVX¦¨x

J =∫ ∞

0(zT (t)Qz(t) + uT (t)Ru(t))dt

ª eFseS®

=∫ ∞

0(xT (t)Qxx(t) + uT (t)Ru(t))dt

Y­V=hR > 0

>Q ≥ 0

ªQx = NTQN

®5[=´I=htNPO7xu(t) = −Kcx(t)

Y­SVhKc = R−1BTPc

ª eF®VOPc

[]RSWavOQNaRLµ RS[tNPOQNG­SV ª +65 '*& ) ® TiVXW ´I=vMY`OtNaRSLµTBV,uCli7xPcA+ ATPc − PcBR

−1BTPc +Qx = 0.ª eF´~®

L¹YY`WaRS][xJ zNGL = xT0 Pcx0

ªx0

xMI@O^Y`O;NaLBNPOQNaY`W^\t = 0

®@F

$ &(' + > . ¦7Y¤TLBYzNGvBV\TBv²[]([]Ot=zV,V=L _ RvBh=WGV§¬?V=]zIV\[tviuWaYyWaRNTBV\hRSz ¿¿ ¼ !"$# ¼%%&s¿'&)(s¿* +,$5¿'.¿-/.10)235476/8*:9;.¿<$= ¼>"%(!&?½$ ¼(¿

&< !(>>¿(¿'%>@>%¿<&A¼ B<!C&EDGF!H IKJ:%¼ L ¼%%/)MN@ ¿O@¢¼< ¿P%(>&i½ ¼ (QJ(s¿@]¼)RS&1&?7 ¼ !?T@¿$U@¢¿(>&7¿V(f¼= ¢½&(s¼ >WM=+ >XY(>!Z¿[.¿$\]<^ ,¿$O(>>]¼! ¿$¼:F@¿:@N&@ !, ¿VRS&¼< ¼TRS&.¿:_;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ ]À=Â@þÄ

E~ jl p±u±%lo *j

zYLBTBVu(t) = −K x(t)

RS_ I=NPO`\ WIvMY`OQNGRSLux

x(t) = (A−BK)x(t) .ªeFs¥S®

¦7Y²]I= RSLi[tVYviOtRSLBRSzV¤TBVx(t)

[Ih=]NGOYWGRSt[xx(t) = eAf tx0

Y­SV=hAf = A − BK

V@Ox0 = x(t = 0)

F¦VXhtNGOt=]V

JTBV@­(NaVLOx

J =∫ ∞

0(xT (t)Qxx(t) + uT (t)Ru(t))dt

ª eF®

=∫ ∞

0xT (t)(Qx +KTRK)x(t)dt

ª eFsdS®

= xT0

(∫ ∞

0eA

Tft(Qx +KTRK)eAf tdt

)x0

ª eFr®

= xT0 Px0

ª eF³®Y­SV=hÁx

P =∫ ∞

0eA

Tft(Qx +KTRK)eAf tdt.

¦7YhRSLOtQYNGLOtVAf

[]OQY_BWGVcV=LOQtY ZLiVcvBVP­SItNGKBVbWI(vBY`OQNGRSLTBV,jikMlm nopq ªC­SRSNa9Ot|BI=R2

]=zV ESF®xATf P + PAf +Qx +KTRK = 0

ªeFPE¶® Y;YNaWGWGVvBt[

P ≥ 0hY

J = xT0 Px0

VOJ ≥ 0 ∀ x0

ª?h=tNPOQtVu(vBYTBQYOQNavBV®@F RNGO

KcWaY­`YWaVvBRSiOtNazYWPV TiV

KvBNizNaLBNGz NG[]V

JV@OPc

WZYw[tRWaviOtNaRSLbh=RS]tV=[] RSLBTMY`LOQVTBVXWI(vBY`OQNGRSLUTBV\jikMlm nopqeFPE¶6>B[]RSNGOx

(A−BKc)TPc + Pc(A−BKc) +Qx +KT

c RKc = 0ªeFPESE.®

5RSLi[tNaTiI=tRLB[;vBLBVu­`Y]NZY`OtNaRL∆K

YviOtRSvB9TBVKc

[]RSNGOK = Kc +∆K

> Y`WaRS][9NGW2V=LtI[tvBWPOQVvBLiVX­`Y`tNZYOQNaRL

∆PY`viOQRSvi;TBV

Pc [tRNGO

P = Pc +∆PvBN­SI=tNPKMVÁx

(A−B(Kc +∆K))T (Pc +∆P ) + (Pc +∆P ) (A−B(Kc +∆K)) + · · ·

ª eFPE.e®· · ·Qx + (Kc +∆K)

TR(Kc +∆K) = 0

KcV[]O§WZYµ­YWaVvBcRSiOtNazYWGV Yv0[tVLB[§TBv*htNPOQ=]V

J>7[t[tN WGVh=]NGOt=tVY`vBSzV=LOQV RSvBXOQRSvOQV

­`Y`tNZYOQNGRSL∆K

YviOtRSvB9TBVKc

[]RSNGOx

∆P > 0 ∀ ∆K / A−B(Kc +∆K)[]OQY_BWGV

ª?[tN∆K

V[]O;OtV=WGWaV%vBVA−B(Kc +∆K)

V[]ONaLi[]O^Y`_BWaVA>BYWaRt[WGVuhtNGOt=tVuTiV­(NaVLONGLiKMLBNZ®@F NRSLUtVOtQYLih¾|BVXW ´I=vMY`OtNaRSLeFPESEXTBVXW ´I=vMY`OtNaRSLeFPE.e>BLBRvB[;RS_iOQVLBRSLB[x

(A−B(Kc +∆K))T ∆P −∆T

KBTPc +∆P (A−B(Kc +∆K))− PcB∆K + · · ·

· · ·∆TKR∆K +∆T

KRKc +KTc R∆K = 0

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ ¾Ã@Ä

7CoCm E.¥

(A−BK)T ∆P +∆P (A−BK) + · · ·

· · ·∆TK(RKc −BTPc) + (RKc −BTPc)

T∆K +∆TKR∆K = 0

J´V=[]OvBLBV9I=vMYOQNaRL,TBVJjikMlm nopq FA−BK

IOQYLO []OQY_BWGV(>∆P

V[]O RS[tNPOQNP¬ [t[]N7ª?OQ|iI=RS]=zVTBVc¦YBvBLiR.­i®x

∆TK(RKc −BTPc) + (RKc −BTPc)

T∆K +∆TKR∆K > 0 ∀ ∆K

RS∆TKR∆K > 0 ∀ ∆K

h=YR > 0

>MNaW¬YviO;TBRLBhX(viV x

RKc −BTPc = 0 ,[tRNGOx

Kc = R−1BTPc .ªeFGE®

NRSLº]V= RS¢OQV¹h=V@OtOtV I=vMY`OtNaRSL TMYLB[W ´I=vMY`OtNaRSLºeFGEEE>LBRSvB[RS_iOtV=LBRLB[ $ + & > .? ) 3 9 ) S '*'*> ) x

PcA+ ATPc − PcBR−1BTPc +Qx = 0.

ªeFPE~®;V=zYtvBVÁx.RSLb V=vOtVOttRSv­SV=TBNGtV=h@OQVzVLOWI(vBY`OQNGRSLcTBV;jikMlm nopqUVL§_ RSvBhWaV¬?V=]zIVeFPESEe\§MY¢OQNG TBVJW ´I=vMYOQNaRLyTBVbuCli7V=L¤_RvBh=WGVJRSvi­SV]OtVXeFGE=~\V@OTBVXe(FGEF W[]v O9TBVQY . RSvOQV=9V@OTBVXtVOtQYLBh^|BV

PcBR−1BTPc

TMYLB[weFPE~BF

1 `- &('*> ) 3 3 )W !"

#$&%

' (*)+ ,+

+

1

+

-

Ge(FGE 1"S& 'FBa&(*k N* &_1"2S&(') e*,3'Fd&(3B"LO8 & (Ba&(434O8P: QYLi[]¬?V=¢OwTBv[¢[¢OQzVÁx

Gzu(s) =ZU(s) = N(sI − A)−1B

F

QYLi[]¬?V=¢OwTBVX_ RSvih=WaVÁxLc(s) = Kc(sI − A)−1B

F7ªRLh=RvB VcV=L ="6 ®

;RS_BvB[¢OQV=[][tVbTBVXWaY,hRSzzyYLiTBVX¦¨\F LTiI=zRSLOttVÁx

(I + LTc (−s))R(I + Lc(s)) = R +GTzu(−s)QGzu(s)

ªeFPE.¥S®H%YLi[WaVuh=Y[;zRLBR.­YtNaY_BWGV

(TBNGz

u = 1)>BRSLRS_OQNaVLOx

|1 + Lc (jω)| ≥ 1RSv

|Su (jω)| ≤ 1 ,∀ω .

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ tÃÂ@þÄ

E jl p±u±%lo *j

¦7YXKBSvBtVJeFseXNaWGWGvB[¢OQ]V;h=V@OtOtVwBtRSitNaI@OQI;TMYLB[ WaV;BWaYLTiV vBNG[]O=F W V=LtI=[]vBWGOtVwWGV=[zyY]SV=[TBVX[¢O^Y_BNGWaNPOtI x

Y`tSVXTiVcSYNaL[0.5; +∞],

Y`tSVXTiVcB|BY[tV[−60; +60]

ªeFPE®5V@OtOQV%BtRBtNGIOQIJL2V[]O;iWavB[ ­SI=¢KMI=Vu RSvB5WaV[[¢LOQ|B[tV=[¦¨ TBRSLO9WGVJh=]NGOQtVwOQNGV=LOh=RziOtVTBVX RSLiTBI=tY`OQNGRSLB[9h=tRNa[tIV=[;IO^YO © hRSzzyYLiTBVÁx

J =∫ ∞

0(xTQxx+ 2xTSu+ uTRu)dt .

G eFfe j 358 %kn R N* &`9?A' 1i352' N4Bd&('6!n9?B: hp 281*Vej :

" ! " ! #"$" !&%' <?

¦ ´I=vMY`OtNaRSL¹eFGE=~ [=Ih=]NGOwY`vB[t[tN)(7x

[Pc − In]

[A −BR−1BT

−Qx −AT

] [InPc

]= 0 .

H =

[A −BR−1BT

−Qx −AT

] V=[]OXYB V=WGI=V\WZYzyYOQtNGh=Vb|MYzNaWPOtRSLiNGV=LBLiV\Y[t[]R(h=NGI=V \yW ´I=vMY2OtNaRSLµTBV\XCli7eFPE~BF

) . 3 & & )H

WaV=[2n

­`YWGV=vB][;BtRSitV=[;TBVH

[tRSLO7x WaV=[

n­`YWGV=vit[;B]RSBtV[[¢O^Y_BWGV=[;TiVcWaY,_ RSvih=WaV%¬?V=]zIVX[t Vh

(A−BKc)>

<In

<:s½.¿V(f¼=9¼ T@ ¿VT¿ !".¿[<>¿!n

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ =¾Ã@Ä

.pÁ°n77CpÁo ° Án7li7CpÁo uCli7 E.d

WGV=[n­YWaVvBt[ B]RSB]V=[5RB RS[tIV=[5MY tYB RS¢O#\cW Y VwNGzyYNGLMY`NG]VbªTBRLBhwNaLB[¢O^Y_BWGV=[Q®@F

$ &(' + > . /V=h@OQvBRLB[vBLBVXOtQYLi[]¬?RS]zyY`OtNaRLµ]I=SvBWGNatV%[tvBHx

H = M−1HMY­SVh

M =

[In 0

Pc In

] VOM−1 =

[In 0

−Pc In

],

YWGRSt[x

H =

[In 0

−Pc In

] [A−BR−1BTPc −BR

−1BT

−Qx − ATPc −AT

]

=

[A−BR−1BTPc −BR−1BT

−PcA− ATPc −Qx − PcBR−1BTPc −A

T + PcBR−1BT

]

=

[A−BKc −BR−1BT

0 −(A−BKc)T

].

HJRSLihÁx

det(sI2n − H) = det(sI2n −H) = det(sIn − (A−BKc)) det(sIn + (A−BKc)T )

& += $ 7.? 3> $) + += + ) + 3> ) + > > .+ ¦7YTBI=zRSLB[¢OQQYOQNaRLuB]I=hI=TBVLOQV LBRvB[zRSLOttV vBV [

InPc

] V=[¢O§WaYUzY`OttNahV,TBV=[X­SV=h@OQV=vit[µª?TBVWaRSLiSvBV=vi2n®%vBNV=LiSV=LBTitV=LO§WaV

[tRvB[wV=[tBYh=V§NGL­`Y]NZY`LOY[t[]R(h=NaIbYv n­YWaVvBt[BtRBtV[J[¢O^Y_iWaV=[TBV

Hª?[t Vh

(A− BKc)®¾F

RSNPOΛWZYzY`OQtNGh=V

n × nªTBNaYSRSLBYWGV¤RSv LBRL ®\TBV[

n­`YWGV=vB][ BtRSitV=[ []O^Y`_BWaV[ VO[tRNGO

XWaYzY`OttNahVcTBV=[J­SVhOtV=vB][uBtRSitV=[JTBV

HY`[t[tR(hNaI[ \

Λh=YWahvBWaI[JBYJV V=zBWaV§BYJviLBV

TBIh=RSz RS[tNPOQNGRSLµ[t VhOQtYWaVXRvvBLBVXTBI=hRSz RS[tNPOQNaRLµTBV n ²TBVHx

HX = XΛ,

[tNWRLUMY¢OQNGOtNaRLBLBVX

V=L¹TiV=v [tRSvi[r zY`OttNahV=[n× n X1

VOX2

x

X =

[X1

X2

],

YWGRSt[RSL VviOwIh=]NatVÁx[

A −BR−1BT

−Qx −AT

] [In

X2X−11

]=

[In

X2X−11

]X1ΛX

−11

VOTiRSLBhPc = X2X

−11 .

PcV[]OviLBNavBVuhY9NGW2L´V Na[¢OQVXv´vBL¹[]V=vBWVLB[tVz,_iWaVuTBV

n­`Y`WaV=vit[9BtRSitV=[;[¢O^Y_iWaV[;TBV

HF

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ C@Â@þÄ

Er jl p±u±%lo *j

:9 F F $ " ">@ ! "¦Vwh=]NGOQtVJ¦¨ V=vO OttVw(vBYWaNPO^Y`OtNP­SV=zVLO V= v¤h=RSzzVvBLyh=RSzBtRzNG[VLOQ]VJ V=M

¬?RS]zyY`LBh=V§ªTBIh=tNPOQVwMY WZYc RSLBTBIQYOQNaRLy[tvBz® V@Oh=RLB[tRSzzY`OQNGRSL ªTBIh=]NGOQVwMY WZYc RSLiTBI

tY`OQNGRSLµ[tvBu®¾F ¨%vBVXhVcMY`[t[tVXO=NGW RSvi;WaV[;]=SWaYSV[;V OQtzV[TBVXh=VXh=RSzB]RSzNa[SF

- & '0)²'"K - & '*) & & >M$= 2'6!34 B"2'6!e8'!+-!=qL*,3'Fd&(3BG?/x(t) = Ax(t) +Bu(t) x ∈ Rn; u ∈ Rm

z(t) = Nx(t) z ∈ Rq .

ªeFGEdS®

j^&_1"2S&(') U &(B BGM28 BH%$Q5&(O83I3'33V!- *V 1B:3MqBG j /J =

∫ ∞

0(zT (t)z(t) + ρuT (t)u(t))dt

ªeFGEr®

=∫ ∞

0(xTNTN(t)x(t) + ρuT (t)u(t))dt

&(C("1ρ!-1d&(* &(3BG#U-!343 9 = !$Q"1B:3 /

u(t) = −Kcx(t)&(C("1

Kc =1

ρBTPc et Pc = Ric

[A − 1

ρBBT

−NTN −AT

].

ªeFseE.®

28 !H&(C(2'6!L&E*0ABG!?/8limρ→0(Pc) = P0

aYE3V!= limρ→0(J) = xT (0)P0x(0)8 ' < ' = !3I*V"!e +"U2nq"!-"!e!8 3C2&('="! !-2' C(B:3 < ""!?/

∗ dim(z) ≤ dim(u) (q ≤ m),

∗*VL4Bd&('6!9?B

Z/U(s) = N(sIn − A)−1B'F$ &LU6&2!h 2B"-!

&('6!N*VH 3;VU)* &('T(BG23,

ªeFfeeS®

&(*V2BG!P0 = 0.

¦7YwTBV=]LBNatV5| R`OQ|B[tV9NGWaWGvB[¢OQ]V WGV ¬YNPOvBV(>`TMYLB[WaV hY[TBV5[]([]Ot=zV \JTiI=B|BY[QYVLBRSL\zN LBNGzyY`Wª?h´V=[¢OaM\ TBNGtVwY­SV=hwTBV[]=ItR[ NaLB[¢O^Y_BWGV=[ ¡¢®>SRSLLBVw VviO5MY[ RSvB[t[]V= ¡wNGLBTBIKBLBNazV=LOWGV=[ V=¢¬?RStzYLih=V=[ z =zVU[]N;WRL [sY`viOQRS]Na[]VTiV=[h=RzzYLBTBV[ [tYLB[YvBhvBLBVWaNGzN O^YOQNGRL

Àt»L& ¼ %%¿$( JX = >@

[H]<$½.¿W(s¼ (>&<> ]1 TRS&.¿ <¿ %/!>F@¿W.¿W(Q# :R&`¼> ¿

0 XA + ATX −XRxX + Qx = 0

D _¼ @>¢¿ M=(f¼=9¼ T@ ¿V+¼=>(!¿$.¿

H =

[A −Rx

−Qx −AT

].

D < ¾Ä_

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ H=¾Ã@Ä

2pm 7 M l=kÁ±umM7p7 Án E³

h=Vh=NV[]OwOQRSviO7\ ¬YNGOWGRSSNGvBVcBviNa[tvBVbWaV[b]=ItR[,ªTBv [¢[¢OQzV§h=RzzVcTivhRSt]V=h@OQV=vi^®wY`OrOQNGtVLOXWGV=[u WaV[XVL0_ RvBh=WGV,¬?VtzI=V vMY`LBT0WGV,YNGL TiV_ RvBh=WGVYviSzVLOtV¹ª?RED WGRS][tvBV WZY RSLBTiI=QYOQNaRLU[tvB

uTBNGzNGLvBV%TMYLi[WGVuh=tNPOQtVc¦¨X®¾F

RSvBYLMY`WG([tVvMYWaNPO^Y`OtNP­SVzVLO WaV=[ BtRSitNaI@OQI[yY`[](zOQROtNavBV=[TBVWZY hRSzzYLiTBV¦¨ >MRL V=viOviOtNaWaNP[tV=vBLRSvOQNaW2StYB|BNGvBVÁxMWGV ¤M+ > )²$ + F

¤M+ > )²$ + JV[]OJWGVXWaNGV=vUTBV=[;­`YWGV=vB][BtRBtV[wTBVcWaY zY`OQ]NahVu|MYzNaWPOtRSLBNPV=LBLiVH

WaRt[tvBVρ­`Y`tNGV%TBVX¶ \\W ´NaLiKBLBN F¦7Y\MY]OtNaVJ[]O^Y`_BWaV%TBVuhVXWGNaV=v¤zRSLOttV%WI@­SRSWGviOQNGRSL¤[]V=WGRSL

ρTBVXWZY\Ti(LMYzNavBVXVL¹_ RvBh=WGVu¬?V=]zIVF

H =

[A − 1

ρBBT

−NTN −AT

] ª e(Ffe`®

=

[A 0

−NTN −AT

]−

[B

0

]1

ρ

[0 BT

] ª e(Ffe~®

= A− BkCª eFseS¥S®

¦V BG 2I!O86&(BGL*V-18 !hRSt]V=[t RLBT¹TBRSLBhcYvUWaNaVvµTBV=[tYh=NGLBV=[9TBv¹[¢([]OQzVXKBhOQNP¬x

S =

[A B

C 0

] ª e(Ffe`®

RSvB9vBLYNaLµTBVX_ RSvBhWaVk = 1/ρ

­`Y`tNZY`LO9TBVcW∞

\0F

R`OQRSLB[xGzu(s) =

Z

U(s) = N(sI − A)−1B (= Ge

∏nz0 (s− zi)∏n0 (s− pi)

VL¹zRSLBR­`Y]NZY_iWGV).

ª e(FfedS®;=SWGV=[9TBVXh=RSLi[]OQ]vBhOtNaRSLTBvWGNaVvXx TMYLi[WaV h=Y[zRSLBR­`Y]NZY_BWPV(>=hV [tRLOWaV[7tSWaV[7h=WaY[t[]NavBV[YBiWaNavBIV=[Y`v[]([]Ot=zV

ShRSziOQV%OQVL(v¹TiV=[]V=zYtvBV[[]vBNG­`Y`LOQV[

T2sYit=[;WZY\BtRSitNaI@OQIuTBVHx

TBVO(λI −H) =

TBVO(λI − (A−BKc))

TBV@O (λI + AT −KT

c BT) ª e(Ffe`r®

hVWGNaVv TBV[ QY`h=NaLiV=[V[]O [¢(zI@OQtNGvBV;MYtYB RS¢O \XW Y V9NazYSNPLMYNPtVÁx⇒

WaV9[]NaSLiV;TBvSYNaLµT ­YLB[V=[]Oh=VWavBN RSvB9WaV=vBVWW Y VXNGzyY`SNGLBYNG]V%LBVu¬YNPO9MY[;MY`]OQNGVuTBvWaNGV=v^>

WGV=[2n

WaV[JVL²_ RSvih=WaVbRSvi­SV]OtV\TBVS[tRSLO%WaV[

n WaV[JTBV

Gzu(s)hRSziWaI@OQI=[wMY[

WGV=vB][;RSi RS[tIV=[> WGV=[ ]=ItRS[TiV

S[tRLOWGV=[f]I=]RS[TBV

Gzu(s)h=RzBWGIOtI=[9MY[;WaVvBt[9RSB RS[]I=V[=F

H%YLi[cWGV,hY`[cz\vBWPOQN ­YtNaY`_BWGVA>2RL LBV VviOcBWavi[XBYtWGV=%TBv [tNGSLBV,TBv YNGL²T ­YLB[yªh`V[]OvBLBV§zY`OttNahVcTBVbYNaLi[^®9zyY`Na[WaYBtRBtNGIOQIcTBV§[](zIOttNaVbMY`wQYB R]Oe\WsY VbNGzyY`SNGLMY`NG]VV=[¢O%OQRv . RSvB][w­SI=]NGKMIV\VO%RSL VviOXY ]zVw(viV,WGVbWaNGV=v TiNatVhO,ªh=VWavBNvBNLBRvB[%NGLOQItV[t[tV®V=[¢OwhV=WGvBN(viN7LiVchRSLOQNaVLO;MY[;WsY VXNGzyY`SNGLBYNG]V%V@OWaV%WaNaVvµh=RSzBWGI=zV=LOQYNGtVJV=[¢OwhV=WGvBNvBN;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ =Â@þÄ

e¶ jl p±u±%lo *j

hRSLOtNaVLO W Y VwNGzyY`SNPLMYNG]VF LiKML^>(WZYXMY¢OQNaV;TBvyWaNGV=vvBN LBRSvB[ NaLOQItV[t[tVJ RvB5YLBYWG([]V= WaYTi(LMY`zNG(viV\VL _ RvBh=WGV§¬?VtzI=V§V=[]OuhV=WaWGV§TBv²TiV=zN BWaYL SYvBh^|BV\hY`%WZYhRSzzYLBTiV\¦¨ V[]O 7`++@ >BA $ + > ) F

5V[w]=SWGV=[9TBVX[](zzOttNGVuhRSLBTBvBNG[tVLOwYv hRSLBh=WGvB[tNGRSLB[9[tvBNP­`YLOQV=[F

+ )ρ→∞

ªhRSzzyY`LBTBVN\,I=LiV=tNaVXzNaLiNazYWPV® WaV=[J­`YWGV=vB][uBtRBtV[X[¢O^Y_BWGV=[uTiV

ALBV,[tRSLOXMY[uzR(TBNGKBI=V=[%MY`uWaY¤hRSzzYLBTiV,¦¨\F

NWaVu[¢([]OQzVcV[]O[]O^Y`_BWaVuV=L_ RSvBhWaVXRvi­SV=¢OQVbY`WaRS][Kc = 0

> WaV=[­`YWGV=vB][itRSB]V=[NGLB[]OQY_BWGV=[TBV

A[tRSLOiWZYhI=V[V=Lc_ RSvBh=WGV ¬?VtzI=V[tviWaVvBt[NazYSV[

[]O^Y`_BWaV[→ Kc 6= 0

F

`+ )ρ → 0

ªYvih=vBLBV§WaNGzNPOQY`OtNGRL¹[]vBJWZYhRSzzYLiTBV®9[tRSNPOnzf

WaVbLiRSz\_BtVbTBVh]=I=]RS[TMY`LB[WGV%OQtYLB[]¬?V]O

Gzu(s) = Z/U(s)> YWGRSt[x

q­`YWGV=vB][µB]RSBtV[UTiV²WZY*_ RSvBhWaV ¬?V=]zIV²[]RSLOµBWZY`h=I=V[µ[tvBµWGV=[ ]=ItRS[µ[]OQY_BWGV=[µTiV

Gzu(s)VO[tviWGV=[9NazYSV[9[¢O^Y_BWGV=[;TiV=[`]I=tR[NaLi[]O^Y`_BWaV[9TBV

Gzu(s)>

WaV=[n − q

­`YWGV=vit[\BtRSitV=[\tV=[¢O^YLOQV[ [tRLO,tV . V@OQIV=[S\WNGLiKMLBN TMYLi[,WaVyTBV=zN5iWZYLYvBh^|BVA>

Pc = 0

[tNGzu(s)

V=[¢O7\,TBI=B|BY[QYVczNGLBNGzY`WÁªMY[;TiVL]I=]RS[NaLi[]O^Y`_BWaV[^®>

76 )'73$?) ' 7 > >BA$)W 5RSLi[tNaTiI=tRLB[WGVu[]([]Ot=zVÁx

x =

[0 1

−1 0

]x+

[0

1

]u

z =[1 −0.1

]x

.ªeFse³®

V@OWaVuhtNGOt=]V xJ =

∫ ∞

0

[z2 (t) + ρu2 (t)

]dt

Y­SVhρ > 0.

¦7Y9[]V=[][tNaRL Y`OtWZY_u[]vBNG­YLOtV VtzVO7TiV OtQYhV=WGV BG-2!O8 &(BG *0 18 ! Y`[t[tR(hNaI \h=VitRS_BWG=zVªC­SRSNa5KMSvB]VbeFsS®@F !#"$%'&!)(*&+(,-$./&01$'&)(2(#3546-78:9<;"$=,-$=>1?$=/5@

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ @Ä=¾Ã@Ä

2pm 7 M l=kÁ±umM7p7 Án eE

−20 −15 −10 −5 0 5 10 15 20−6

−4

−2

0

2

4

6

Real Axis

Imag

Axi

s

3eF _o`-2I!O86&(BGL*V-18 ! !8 BN8' aY7U)*V 2'F2C2&(B:34& p*V:

76 )('43$?) ' $ . < > >BA$)Y 5RSLB[tNGTBItRSLB[;WaVu[¢[¢OQ=zVÁx

x =

0 1 0 0

−1 0 0 0

1 0 0 1

0 0 0 0

x+

0 0

1 0

0 0

0 1

u

z =

[1 −0.1 0 0

0 0 1 0.2

]x (t)

.ªeFs¶®

VOWGVuh=]NGOt=tVÁxJ =

∫ ∞

0

[zT z + ρuTu

]dt

Y­SV=hρ > 0.

OQNPOQtV¤T2V VthNahVRSLTBI­SVWaRSi V=QY vBLiVµ¬?RSLBh@OQNaRL 3 946-708:9 V=]zV@OtOQYLO,TBVµOQtYh=V,WaVBG-2%!O8 &(BGb*V-18 !JTBYLB[ WGVh=Y[ SILBI=tYWª?WZYu¬?RLBhOtNaRSL 3'4678:9 L2sYhh=ViOQVJvBVwTBV[5[¢([]OQzV[zRSLBR­`YtNaY_BWPV=[^®¾F2¦Y¤[](LO^Y VT2sYi V=WÁ[tV=tYUWaY¤z =zV\vBV hV=WGWaV\TBV,WaYy¬?RSLBh@OQNGRSL 43 F LRS_iOtNaVLOYWGRSt[WZY\[tV[t[tNGRSL¹[]vBNG­YLOtVXVOWZY§KMSvitVbeF´~BF >%? '%>%>! >% ?#3 946-708:9 ; ==(&)=; =5@@<

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ QÂ@þÄ

eSe jl p±u±%lo *j

−20 −15 −10 −5 0 5 10 15 20−10

−8

−6

−4

−2

0

2

4

6

8

10

Real Axis

Imag

Axi

s

3eF´~ _of 2I!O86&(BGL*V-18 ! !8 BN8' aY7U)*V 8 *,i3;=C2&(B:34& p*V:

F " 6 "!$#%&%

5RSLi[tNaTiI=tRLB[WGVu[]([]OQzV x

x (t) =

[0 1

0 0

]x (t) +

[0

1

]u (t) +

[0

1

]e (t)

z (t) =[1 −1

]x (t)

.ªeFiE.®

RSvB[;tV=h^|BVth^|BRSLB[wWGVutISvBWaY`OQVvB(e (t) = 0)

vBNzNGLBNGzNP[tV%WaVuhtNPOQ=]V

J =∫ ∞

0

[z2 (t) + ρu2 (t)

]dt

Y­SV=hρ = r2 > 0.

YWahvBWaV9WaVu]VOtRSvB;T2I@O^Y`OKc

[QY`LB[tI[tRSviTBtVcWIvMY`OQNGRSLUTiVb;NahhY`OtN5> OQQY`h=V=W ´I­SRWaviOtNaRSLyTBVuWaY\TLBYzNGvBV%V=LU_ RSvBhWaVJ¬?V=]zIVuWGRSt[]vBV

ρ­`Y]NaV%TiV

0\∞

> (viV=WaWGV­YWaVvB TBV

ρ V=tzV@O T2sY­SRSNaÁVL_ RSvBhWaV¬?V=]zIV9vBL[¢[¢OQ=zV;TBv[tVh=RLBTRStTitV

YzR]OQN)\yE SF' Y`WaVvB5TBVJWZYbBvBWG[QY`OtNaRSLyhRS]tV=[] RSLBTMY`LOQV c¦NaV=v¤TiV)(5WZY`h+* h=RS]tV[t RSLTMYLO

(viV=W7V[]OWaV%OQtYLB[¢¬?V=]OZ/E(s)

VL¹_ RSvih=WGVu¬?V=]zIV YWahvBWaV

PcWaYX[]RSWaviOtNaRLTiVwWI(vBY`OQNGRSLTBVw;Nah=h=Y`OQNBV=L¬?RLBhOtNaRSLTBV

ρVO

limρ→0(Pc)F

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ ¾Ã@Ä

C e

" % &

Lh=RLB[tNGTB=]VcWaVu[¢([]OQzVcTiv¹[]V=hRSLBT¹RStTB]VÁx

G(s) =Y

U(s) =

s+ 3

(s+ 2)(s− 1)

QYhV=;(vBYWaNPO^Y`OtNP­SV=zV=LO9WaVw tR(RO9[tvMY`tVcWGRhvB[¢¡uY[t[]R(h=NGIcYvhtNGOt=]Vc¦¨x

J =∫ ∞

0(y2 + ρu2)dt.

V=vOa RL²tVLBTBtV\WGV=[w­`YWGV=vB][%B]RSBtV[%TBv²[]([]Ot=zV,V=L²_ RSvih=WaVb¬?VtzI=V\Yt_iNGOQtYNGtVzV=LOJQY BNGTBV=[;VL . RSvBYLO;[tvi

ρF

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ ¾À=Â@þÄ

eS¥

9 F F 5RSLB[]NaTBItRSLi[WaVu[¢([]OQzVcWGNaLBI=YNGtVJTRtTB]V

n[tvBNP­`YLO7x

x = Ax+Bu+Mw

y = Cx+Du+ v

z = Nx

ªiFPE.®

REDw

VOv]V=B]I=[]V=LOtV=LO¹TBV[_itvBNPOQ[µ_BWaYLBh[>TBV zR.V=LBLBV²L(vBWGWaV(> NaLBTiI= V=LiTMYLOt[>;Y­SVh

tV[t VhOQNP­SV=zV=LOw RSvB;zY`OttNahV%TBVXh=R­`YtNaYLBhVW

V@OVF

E[w(t)w(t+ τ)T

]= Wδ(τ) E

[v(t)v(t+ τ)T

]= V δ(τ) E

[w(t)v(t+ τ)T

]= 0 .Y­SVh

W ≥ 0VO

V > 0 ªiFseS® LLBROtVcYvB[][tN

Wx = MWMT WZY\zyY`OttNGh=V%TBVXh=R­`Y]NZYLih=VXTBv_B]vBNGO9TI@O^Y`O=F MY¢OQNa9TBv¹­SVhOQVvB

yTiVbzV=[tvitV=[w_B]vBNGOtI=V=[§ª]VOQRvBTBVb[]RS]OtNaV®:>MLBRSvB[w]V=h^|BV=]h^|BRSLB[

vBLBVXWaRNTiVchRSzzyY`LBTBVuvBNzNaLBNGzNP[tV%WGVuh=tNPOQtV

J = limT→∞

E

[∫ T

0

(zTQz + uTRu

)dt

] ªiF®RED

z = NxTBI[tNaLBV¤WaV­SVhOQVvB \]I=SvBWGV=§VO

QVO

RTiV=v zyY`OttNGh=V=[bTBV¤ RSLBTBIQY`OtNaRSL

Y­V=h(>Mh=RSzzVuB]I=h=ITBVzzV=LO?>

Q = QT ≥ 0VO

R = RT > 0 .

¦7Y§[]RSWavOQNaRLTBVJh=VJBtRS_iWa=zVw[sY`BBvBNGVJ[tvB5WaV 3 3 ) ) + & 3> > .? vBN2I@O^Y_iWaNPO vBVWZY\h=RzzYLBTiVuRSiOtNazYWGVJV=[]ORS_iOtV=LvBV

YS®V=L tVh¾|iV=th^|MY`LO W ´V=[¢OQNGzI5RiOQNGzyY`WxªYv,[tVLB[TBV9WaYJ­YtNaYLBhVTVttVvB zNGLBNGzY`WGV®

TiVXW ´IOQY`OxMY`9WZY\zI@OQ|BR(TBVuTBvgNGWGOttVJTBV§¯§l°(±%loI>ih´V=[¢OaM\ TBNGtV%RSLUV=[¢OQNazV%W ´IOQY`O

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ @ÃÂ@þÄ

e ÁjlºpÁ±u±%lo j

+-

x

u

x

^

x y

Retour d’état

Estimateur

Système

+ -+

+

+

++

e

yε

3 FGE j^&!iB:81i8 BGS(8 1"2B:BG"1M8 BNj * & iBa&('6!3V!"!352'X(3B""1=D

'F$Q"!%U6&2!HBG;U)BG"!-'="#U28Bh "!NBa&(3V!-2'6!h 1* &(B:M 0:

xMYUW ´I=vMY`OtNaRSL h=WZY`[t[tNGvBV TBv KBWGOQ]V TBV¯§l°±%lom\ hRSLBTBNPOQNGRSL (viV0WaV²OQ]NaBWGVO

(A,MW 1/2,C

) [tRNGO;TBI@OQVhO^Y`_BWaVXVO;[]O^Y`_BNaWGNa[tY_BWGV`F

˙x = Ax+Bu+Kf (y − Cx−Du)

Y­SVhKf = PfC

TV −1 R(D Pf RS_ INGO`\,W ´I=vMY`OtNaRSLµTBV\XCli7[]vBNG­`Y`LOQVÁx

PfAT + APf − PfC

TV −1CPf +MWMT = 0ªF~®

Y­SVhPf = P T

f > 0. _ ®V=L\V=zBWGR.YLOhVO V[]OtNazI h=RSzzV [=´NaW(I@O^YNPOWZYzV=[tvitV5V YhOtVTBv§­SV=hOtV=vBÁT´IO^YO?> RSvButI[tRSviTBtV WaV\B]RS_BWG=zV\TBV,h=RSzzyY`LBTBV\RSOQNazYWPV§WaNGLBIYNG]V§TBIOtV=]zNGLBNa[¢OQVUª?zI:OQ|BR(TBVc¦¨X®

[tRSNPObª[]N

(A,B,Q1/2N)V[]OwTiIOQVhOQY_BWGVcV@O[]OQY_BNaWGNG[QY_BWGV®x

u = −Kcx

Y­SVhKc = R−1BTPcPcA+ ATPc − PcBR

−1BTPc +NTQN = 0.

ªiFs¥S®¦7YwKMSvB]V;FGEtVBtI[tV=LOQV;WaYJ[¢OQtvBh@OQvB]V9TBv,h=RttVhOQVvB¦¨ TMY`LB[ WGV5_ RSvBhWaVTiV9]I=SviWZY`OtNaRSL2F

¦7Y ]V=B]I=[]V=LO^Y`OtNaRSLT´IO^YOTBvh=RS]tVhOQVvBw¦¨ [=´I=htNPOx[

˙x

u

]=

[A−BKc −KfC +KfDKc Kf

−Kc 0

] [x

y

].

ªiF®

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ =¾Ã@Ä

2pm 7 M eSd

9 F F $ " L VviOyzRSLOQ]V= vBV¹WZY ]V=B]I=[]V=LOQY`OQNGRSLºT2I@O^Y`OTBv OQQY`LB[]¬?V]OV=L _ RSvBh=WGVU¬?V=]zIV

V=LOQ]VeVOεy

ª?­SRSNGKBSvBtVciFPE.®5[Ih=tNPOx

εxεy

=

A−BKc BKc B

0 A−KfC 0

0 C 0

x

εxe

.

ªiFsdS®

RSvεx = x− x

TiI=[tNGSLBVXWVttVvB9TV[]OtNazY`OQNGRSLµTBVXWI@O^Y`O=F MY`]OQNG;TBV§hVO]OQV§]V=B]I=[tVLOQY`OQNGRSL^> WGV 3 3) ) + & 3> > Y`BMYtY aOhWZYNGtV

zV=LOV@O V=viO OttVu¬?RStz\vBWGI%TBVXWaY\¬Y RSL[]vBNG­`Y`LOtV x WGV=[­`YWGV=vit[ÁB]RSB]V=[ÁTBVWZYw_ RvBh=WGV5¬?VtzI=V5 V=vi­SVLO OttV[tIMY]I=V[[tVWaRSL\WaV[

n­`Y`WaV=vit[

itRSB]V=[ Tiv tV@OQRSvi,TI@O^YO ª?[t V=h(A − BKc)

®§V@OWGV=[n­YWaVvBt[,B]RSBtV[ TBv KMWGOttV

T2V[]OQNGzyYOQNaRL ª[] V=h(A−KfC)

®> WGV=[

n­`YWGV=vB][;BtRBtV[TBvUKMWPOQtVuT2V[]OQNGzyYOQNaRLµ[tRLONGLBSRSvi­SVtLMY`_BWaV[;MY`

e>

WGV=[n­`YWGV=vB][;BtRBtV[TBvtV@OQRSvi;T´IO^YO[tRSLONaLBR_B[tV]­`Y`_BWaV[;MY

εyF

¦VuOQtYLB[]¬?V]OTiVe­SVt[

εyV[]OTBRSLBhuOQRv . RSvB][;L(viW F

9 :9 ; CB "? F !" F G 9 F ¦ R_ . VhOQNP¬BTBV WZY;B]R(h=ITBvBtV TBV ]I=SWaYSV ¦ V=[]OTiV tV[]OQYvBtVWaV[zyY]SV=[7TBV [¢O^Y_BNGWaNPOtI

NGTBIY`WaV¾¡TBv¤tVOtRSvB5T´IO^YO9TBVJWZYbh=RzzYLBTiV%¦¨ªRSv¤TBvMY`WaV=zVLO5TBv¤KMWPOQtVwTBVb¯§l°±%lo®¾F5 3 M.- (+ ) +b,+ > 3 3$ > .? WGV[¢([]OQzVw[]OQ]Nah@OQV=zV=LO B]RSBtVTBI@KMLBNGO BY WZYX]V=B]I=[tVLO^YOQNaRL0ª

A>B>CV@OD® ­SI=tNPKMVÁx

D = 0

ª[¢OQ]NahOtV=zV=LOitRSB]V®:>C(sI − A)−1B

\ zNaLiNaz§vBz TBVXB|MY`[tVyª?MY[;TBVN]=ItRS[;NaLB[¢O^Y_BWGV=[Q®> LiRSz\_BtVuI=SYWT2VLOQ]I=V[wVOTiVc[]RS]OtNaV=[F

) S )('*) ) ) &) ¦7Y itR(h=ITBvB]VcTBVc[¢(LOQ|i=[tV§¦¨ b© ¦ h=RSLB[]Na[¢OQVL\9x> (LOQ|iIOQNG[tV> TMYLB[viLBV¹BtVzNG=tVUIOQY V(> WGVhRSt]V=hOtV=vBy¦¨ MY`viLºh¾|iRSN Y

B]RSBtNGI¤TBV[ RSLBTiI=QYOQNaRLB[Qx

V@OR

RS_ I=NG[t[tYLO Yv V NaSVLBh=V[TivhY|BNGV=\TBV[ h^|MY]SV=[F¦V=[Y[]VhOt[TBVJh=V@OtOQVJBtVzNG=]Vw[](LOQ|B[tVuh=RSLih=V=]LBV=LO9WaVwhRSz RS]OtV=zV=LO _MY[][tVJ¬?]I=vBV=Lih=VTBV[ ­`Y`WaVvBt[ []NaLBvBWaNG=]V=[\TBvOQtYLB[¢¬?V=]O

−Kc (sI − A)−1 B>ÁWaV=[\¬?tIvBV=LBhV=[TBVUhRSvBBvitV=[

h=RttV[t RSLiTMYLOtV=[ > W YA/2YNG_BWGNP[t[tVzV=LO9|MYvOQVu¬?tI(viV=LBhV(> V@OQh FGFPFA H%YLB[uviLBV,[tVh=RLBTBV I@O^Y VA>\¤BY]OtNawTviL tISWZYV§LBRSzNaLBYW

Wx0

V@OV0

TBvKMWGOttVTBV\¯§l°(±%lo >BRSLYvBSzV=LOQVQY,WaVuMY`QYzOttV

qTBvLBRSv­SVYv¹]I=SWaYSVÁx

Wx = Wx0 + qBBT , V = V0

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ 6C@Â@þÄ

er ÁjlºpÁ±u±%lo j

. vB[]vW\ch=VvBVwWGV9OQtYLB[¢¬?V=]O TBV_ RSvBhWaVK(s)G(s)

TBvhRSt]V=hOtV=vB ¦¨ tVh=Rvi­(tV(>([]vB vBLBV_MY`LBTBVXTBVu¬?tI=vBVLBh=Vb[]v [QYzzV=LO;WZY]SV(>iWGV%OQQY`LB[]¬?V]OTBVX_ RSvBhWaVuTBVXtVOtRSvB9TI@O^Y`Ow¦¨ x

limq→∞

K(s)G(s) = −Kc(sI − A)−1B .

) S )('0) ) += . )W> (LOt|BIOtNa[tV5WaVKBWGOQ]VwTiVc¯§l°(±%loV=L¤zYLiNaBvBWaYLO WaV[ h=R­`Y]NZYLBhV=[

WxVOVY`KML

vBVcWGV%OQtYLB[]¬?V]O−C (sI − A)−1 Kf

[tRSNPO;[tY`OQNG[]¬YNG[QYLO`\,WZY\[tR]OQNGVuTBv[]([]Ot=zVFAI H%Y`LB[;vBLiVX[]V=hRSLBTBVXIO^Y` V(>\\MY]OtNa5T´vBLUtI=WZYSVJLBRzNGLMYW

Qx0

VOR0

TBvU]VOtRSvBT´IOQY`Ow¦¨ >MRL¹YvBzVLOQVQY,WaVuBYQYz@OQtV

qTBvLBRvi­SVYv¹]I=WZYSVÁx

Qx = Qx0 + qCTC, R = R0

. vB[]vW\ch=VvBVwWGV9OQtYLB[¢¬?V=]O TBV_ RSvBhWaVG(s)K(s)

TBvhRSt]V=hOtV=vB ¦¨ tVh=Rvi­(tV(>([]vB vBLBV_MY`LBTBVwTBV¬?tI=vBVLBh=VJ[tv [QY`zzV=LO5WZY]SVA>WGV;OQQY`LB[]¬?V]O5TBVw_ RSvih=WGVwTBvKMWPOQtVTiVc¯§l°±%lox

limq→∞

G(s)K(s) = −C(sI − A)−1Kf .

¦7YJKMSvB]V;iFsew­NG[tvMY`WaNa[]V5W ´V /VO TBv MY`QYzOttVqª?LBROtI

α[tvi WZYwKMSvB]V®[]vB WaV5WaNaVv\TBV9_BWaYh+*

TBvY`NaLµTBVX_ RSvBh=WGVK(s)G(s)

ª]V=hRSvi­(tVzVLOwVL¹VLOQ]I=V®@F

9 ; CB "? " F $ Lh=RLB[tNGTB=]VcWGVu[]([]Ot=zVcWaNaLiIYNG]V%TBNG[th=]VOx

xn+1 = Axn +Bun +Mwn

yn = Cxn +Dun + vnzn = Nxn

R WaV=[c_BtviNGOQ[wn

V@Ovn

[tRSLO§TBV=[§B]R(h=V[t[tvi[,[¢O^Y`OtNaRSLiLMYNa]V=[XTBVzRSV=LiLBVyLvBWaWGV(>B[]V=vBTBR2_BWaYLBh[;VOTiVczY`OttNGh=V%TBVXh=R­`YtNaYLBhV

W ≥ 0VOV > 0

x

E[wk vTl ] = Wδk−l, E[vk v

Tl ] = V δk−l, E[wk v

Tl ] = 0 .

ªFsrS® RSvB[;h¾|iV=th^|BRLB[wvBLBVXWaRNTiVchRSzzyYLiTBVuMY`;tV@OQRvB;TBVX[tR]OQNGVuvBN7zNaLiNGzNG[]V%WaVuhtNGOt=tVÁx

J = limN→∞

E

[N∑

n=1

zTnQzn +N−1∑

n=0

uTnRun

]Q ≥ 0

V@OR > 0,

ªiFs³S®

TBRLBhUviL B]RS_BWazV¤TBVU]I=SviWZY`OtNaRSLK\|BRStN0]=RLWGNa_B]V ª?h=RSzzV¤TMYLi[ WaV¤h=Y[ hRSLOtNaLv ®@FÁ¦7Y[]RSWaviOtNaRSL,[=YBBviNaV;I=YWGV=zV=LO[tvB WGV 3 3 ) ) + & 3> > >h`V[]OrM\2 TBNa]V;vBLKBWGOQ]V;TiV¯§l°±%lo0VO;vBL¹]VOtRSvB;T2I@O^Y`OV[]OQNGzIuRSvB]I=TBNPOF

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ /H=¾Ã@Ä

k o7 0j . M e³

%!$# $

L¹TBNG[thtV@O?>MRSLTiNa[]OtNaLBvBVÁx WI@O^Y`O;V=[]OtNazI

xn/n\WNGLB[]OQYLO

nhRSzOQVXOtV=Lv TBVcWZY zV[tvBtV

ynVOwWZY\­`YtNaYLBhVcTBV

WVt]V=vB9T´V=[]OtNazY`OQNGRSLUY`[t[tR(hNaIVÁx

Pe = E[(xn/n − xn)(xn/n − xn)T ]

zyY`OttNGh=V%[](zIOttNavBVn× n,

WI@O^Y`O,itI=TiNGOxn/n−1

\²WNGLB[]OQYLOn. vB[]OtVY­YLO WaYzV[tvB]V

ynV@OWaY ­`Y]NZYLih=VµTBV

WVt]V=vB9TBVXBtITBNah@OQNGRSLY[][tR(h=NGI=VÁx

Pf = E[(xn/n−1 − xn)(xn/n−1 − xn)T ]

zY`OttNahV%[](zIOQ]NavBVn× n,

¦V=[;I=vMY`OtNaRSLi[;TBvUKMWGOttVuTBV\¯§l°(±%lo0V=L & '*)3) '*> ) [Vh=tNP­SV=LOJYWaRt[x

xn+1/n = Axn/n−1 +Bun +Kf (yn − Cxn/n−1 −Dun)ªiFGE¶®

xn/n = xn/n−1 +Ke(yn − Cxn/n−1 −Dun)[tRSNPO;VLBh=RtVÁx

xn/n = (I −KeC)xn/n−1 −KeDun +KeynªFGEE.®

Y­V=hÁxKf = AKe

YNGLUTBVXB]I=TBNGhOtNaRSLF> ªFGEeS®Ke = PfC

T (CPfCT + V )−1 YNGLUT´V=[¢OQNGzyY`OtNaRL^> ªFGE®

Y­V=hPf

[]RSWavOQNaRLµTBVcWVvMY`OtNaRSLµTBV\XCli7TBNG[th=]OtV x

Pf = APfAT − APfC

T (CPfCT + V )−1CPfA

T +MWMT ªFGE=~® L¹YY`WaRS][x

Pe = (I −KeC)PfªFGE¥S®

)('*> ) VL¹B]I=[]V=LBhV§TiVc_itvBNPOQ[9h=RS]tI=WGI=[;VLOttVXW ´I=vMY`OtNaRSLµTI@O^Y`O;VOTBVXzV=[tvitVÁx

E[wkvTl ] = Tδk−l

WaV[yI(vBY`OQNGRSLB[¤iFPE¶6> FGEEE> iFPE0VOµiFPE.¥ [tRLO¤NaLih¾|MY`LBSI=[yV@O¤WaV=[I=vMY`OtNaRSL iFPE.e VOµiFPE~TBV@­NGV=LBLiV=LOJ]V=[]VhOtNG­SVzVLOx

Kf = AKe +KpcY­SVh

Kpc = T (CPfCT + V )−1 SYNaLµTBVJ B]I=TBN hRSt]V=hOtNaRSL(¡">

Pf = FPfFT−FPfC

T (CPfCT+V )−1CPfF

T+MWMT−TV −1T T Y­SV=hF = A−TV −1C .

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ =Â@þÄ

S¶ ÁjlºpÁ±u±%lo j

% ! %& " $

¦VXYNGLUTiVc]VOtRSvB;T´IOQY`O[Ih=tNPOxKc = (R +BTPcB)−1BTPcAY­SVh

Pc = ATPcA− ATPcB(R +BTPcB)−1BTPcA+NTQN.

ªiFPE®

L]VOttRSvi­SVXvBLBVXTivMYWaNPOQIJMY]¬Y`NGOQV%VLOttVXh=V[I=vMYOQNaRLB[;V@OWaV[9I=vMY`OtNaRSLB[9TBVXB]I=TBNGhOQNGRSLB[iFPE.e\¤iFPE~BF YJQYi RS]OuYv²h=Y[%h=RSLOQNGL(v^>RSL V=viOuTBNG[]OQNGLBSvBVJTBVv ¬?RtzV=[JT´NazBWZY`LOQY`OQNGRSLµTBvh=RS]tV=h@OQVvBw¦¨ TBNG[th=]VOxMWaY\¬?RS]zV%BtITBNah@OQVvB;VOWaY\¬?RS]zV%V=[¢OQNazY`OtV=vBF

!$ "$

LtV@OQRSvitLBVXWaV[;I@O^Y`Ot[;BtITBNGOt[;TMYLB[;WaVuSYNaLKc

x

un = −Kcxn/n−1,

h`V[]Oa=\2 TiNatVuvBVX RSvB;h=YWahvBWaV9V@OwYBBWGNavBV=WZY\h=RSzzYLBTBVN\,W ´NaLi[]O^Y`LOnRSLLiVchRSLBLMYaO

MY`[ynFM¦Y,tVBtI[tV=LO^YOQNaRL¹T´IOQY`OTBvhRSt]V=hOtV=vB[Ih=]NGOwYWGRS][x[xn+1/n

un

]=

[A−BKc −KfC +KfDKc Kf

−Kc 0

] [xn/n−1

yn

].

ªiFPE.dS®

5V@OtOQVXtV=itI=[]V=LO^Y`OtNaRSLV=[¢OJY`LMYWaRSvBV \ hV=WaWGVuRS_iOtV=LvBVcTMY`LB[;WaVuhY`[hRSLOtNaLvF

$ " "$

5V@OtOQVu¬?RStzVuV[]OBWavB[9vB[tviV=WaWGVF LtV@OQRSvitLBVXWaV[;I@O^Y`OQ[;V[]OtNazI=[x

un = −Kcxn/n,

h`V[]Oa=\2 TiNatV;vBVwW RSLhRSLBLMY GOyn

YvzRzVLO TiVwhY`Wah=viWaV=V@O5T2sYiBWaNG(viV= WaYXh=RzzYLBTBV`F LUWsY_i[tV=Lih=VuTBV%OtQYLi[tzNa[t[]NaRSLyTBNGtV=h@O

DªvBNYWaRvBtTBN h=RSLi[tNaTiI=QY`_BWaVzVLOWaV[LBRO^YOQNaRLB[^®:>

WaY,]V=B]I=[tVLOQY`OQNGRSLTI@O^Y`OTiv¹hRSt]V=h@OQV=vi[=´I=htNGOwY`WaRS][x[xn+1/n

un

]=

[(A−BKc)(I −KeC) (A−BKc)Ke

−Kc(I −KeC) −KcKe

] [xn/n−1

yn

].

ªiFPEr®

Y QY`B RS]O \uWaYJ¬?RStzV9itI=hI=TiV=LOtV(>(RLhRSLB[]OQY`OQV;vBV;h=V;h=RS]tVhOQVvB itI=[]V=LOQVvBLBV9OQtYLB[MzNa[][tNaRLµTBNGtVhOtVXVLOQ]V

ynVOunF YNG[9TMYLi[;WaV[9TBV=v h=Y[>BWaV%BtNGLBh=NG VuTBVu[tI=BYQYOQNaRLUV=[¢O

­SItNGKBIÁxMWaV=[2n

­`YWaVvBt[9BtRSitV=[;TBVXWZY\_ RSvBhWaVu¬?VtzI=VX[]RSLO7x WaV=[

n­`Y`WaV=vit[BtRSitV=[9TBvµtV@OQRSvBT´IOQY`Ocª[t Vh

(A−BKc)® vBN[tRLO9NaLBRS_i[tV=¢­`Y_iWaV=[

MY9W ´NaLBLBR­`Y`OtNGRSLεy = yn − (Cxn/n−1 +Dun)

F WaV=[

n­`YWGV=vB][ÁB]RSB]V=[ÁTBv\KBWGOQ]V5T´V=[¢OQNGzyY`OtNaRSL¤ª?[t V=h

(A−KfC) =[] V=h

A(I−KeC)®

(viN7[]RSLONGLBSRSv­SV=tLBY_BWaV[;BYen

ªWGVu[tNaLMYWTiVchRSLB[]NaSLBVu[tvBun®>

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ ÀÄ=¾Ã@Ä

C iE

¦V=[TBVv hRS]tV=h@OQVvBt[;iFPE.dbV@O;iFPEr6>(_BNaVLµv´NGWa[5[]RSNaVLOTBN /2ItVLOQ[ >iTBRLBLBV=LO9WaV=[2n

z =zV­`Y`WaV=vit[XB]RSBtV[cV=L _ RSvBhWaV\¬?V=tzI=V`F 5Vh=NV=[¢ObNGWaWGvi[]OQ]I,BYXWV VthNahViFs¥FiF LiKML^>TMYLB[WaV[;TBVv ¬?RtzV=[ >BW ´IO^YO;TBvh=RttVhOQVvBhRSt]V=[t RLBT\ W ´I@O^Y`O9BtITBNGO;VOLiRSLV=[¢OQNazIF

)('*> ) L§tISNazV VtzYLBVLO>.RSLb V=viOTBRSLBh NazBWaYLOQVWaVh=RttVhOtV=vB¦¨ TBNa[]h=]VOMY9[QY tV=itI=[]V=LO^Y`OtNaRSLT´IO^YOF H%YLi[WaVuhY[;LBRSL[]OQY`OQNGRSLBLMYNGtV%Rv¹VL¹]I= RSLB[]VcOtQYLi[tNGOtRSNa]V\ WaY,hRSLBTiNGOQNGRSLµNaLBNPOQNaY`WGVÁx

x0/0V@O

Pe0,ªiFGE³®

WaV[I=vMYOQNaRLB[T2NGzBWaYLO^YOQNPRSLcTBV WZYh=RzzYLBTiV []RSLOh=VWaWGV=[Tiv§KMWPOQtV TBV¯§l°±%lo¤TBNa[]h=]VOLBRSL[¢O^Y`OtNaRSLiLMYNGtVÁx

xn+1/n = Axn/n +Bun ]I=TBNGhOQNGRSL

xn+1/n+1 = xn+1/n +Ken+1(yn+1 − Cxn+1/n) Correction

un+1 = −Kcn+1xn+1/n+1

ª?RSv−Kcn+1xn+1/n) Commande .

ªFfe`¶®

Y­V=hwWaV[ I=vMYOQNaRLB[ TiVwB]RSMYSY`OQNGRSL TBV=[ h=R­`Y]NZYLih=V[ VO TBvYNGLvBN []RSLO NaLiTBI= VLBTMYLOQV=[TBV

ynV@O V=v­SV=LOwTBRSLih @OQtVXNGLOQIS]I=V=[;|BRt[;WaNGSLBVÁx

Ken = Pf nCT (CPf nC

T + V )−1

Pf n+1= APenA

T +MWMT

Pen = (I −KenC)Pf n .

ªFfe(E.®

9 F " %

Lh=RLB[tNGTB=]VcWaVu[¢([]OQzVcTiv¹B]V=zNaV9RS]TBtVÁx

Y

U(s) = G(s) =

s

1− s

ªFfeeS®

TiRSLBLBVvBLBV;]V=B]I=[tVLOQY`OQNGRSLT´IOQY`OwªA>B>C>D®TBv []([]Ot=zVVOhY`WahvBWaV WGV9]VOtRSvB

T2I@O^Y`OKc

[tRWaviOtNaRSLµTBvB]RS_BWG=zVX¦¨ \ ILBV=]SNaVuzNGLBNGzY`WGVA> h=YWahvBWaVXWGV YNGL TBV gXYWGzyY`L

KfVL0¬?RSLihOQNGRSL TiV

ρxWZY¤­`Y`tNZY`LBh=V TBv0_BtvBNPOcT´IOQY`O

ªE[w(t)w(t)T ] = ρδ(t)

®WGV­`Y]NZYLih=V Tivc_B]vBNGOTBV zV[tvB]VI@O^YLO7vBLiNGO^Y`NatV9ª

E[v(t)v(t)T ] =

1®@F

h=YWahvBWaVJWZY¬?RSLihOQNGRSLTBV\OtQYLB[¢¬?V=¢OXTiv h=RS]tVhOQVvBuV=L²¬?RSLihOQNGRSLTBVρF¨%vBV=WGWaV§V[]OXWZY

BY]OQNGh=viWZY]NPOQIwTBVuhVuh=RS]tV=h@OQVvB\¨%viVuTBV­(NGV=LOWaYb¬?RSLBh@OQNaRLµTBVJOQQYLi[]¬?V=¢O;TivUh=RS]tV=h:OtV=vB;WGRSt[](viV

ρOtV=LBTi[;­SV=][wW´NaLiKMLiN F LLBROtV=tY

KLTRhVchRSt]V=h@OQV=vi=F

OtQYhV=9WaV[9WaNGV=v TBVbokÁn ²TBVKc(sI −A)−1B

VOKLTRG(s)

F 5RLBh=WGvB[tNGRSLB[9[tviWGV=[9BtRSitNaI@OQI[;TBVXtVh=RSvi­(]V=zV=LO SF

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ À QÂ@þÄ

e ÁjlºpÁ±u±%lo j

%

L ]V=B]V=LBT NGh=NJW ´V V=]h=NGh=V [tviUWZYh=RSzzYLBTBV²¦¨ ª?­SRNa¤[tVhOQNGRSL eF´~BFPE.®yvBV W ´RSLhRSziWa@OQV MYWaYwBtNG[tV5V=L,hRSziOQV5TiVW ´I=vMYOQNaRL§TBVzV=[]vBtVV@O TiV=[Á_BtvBNPOQ[[tvBÁWGV5[¢([]OQzV

[]RSNGOxx(t) =

[0 1

0 0

]x (t) +

[0

1

]u (t) +

[0

1

]w (t)

ªiFse®

y(t) =[1 0

]x (t) + v (t)

ªFfe~®

z(t) =[1 −1

]x (t)

ªiFseS¥S®RED

w (t)V@Ov (t)

TBI[tNGSLBV=LOTBV=v _B]vBNGOt[9_BWZYLih=[;NGLBTBI V=LBTMY`LOQ[ > TBVXzR.SVLBLBVXLvBWaWGVuV@OTBV­`Y`tNZY`LBh=V

E[wwT

]= α2, E

[vvT

]= 1 .

[](LOt|BIOtNa[tVWaVutISvBWaY`OQVvB9zNaLBNGz NG[QY`LO9WGVuh=]NGOQtVÁx

J = limT→∞

E

[∫ T

0

[z2 (t) + r2u2 (t) dt

]] ªiFfe`®

¬YNa]VuWGVu[th^|BI=zY,¬?RSLBh@OQNaRLBLBV=WTiVcWaY,_ RSvih=WaV%¬?V=]zIVª[¢([]OQzV ]I=SviWZY`OtV=vBQ®> TBI=zRSLOQtV;WaY,B]RSBtNGIOtIc¦ 0x

limα→∞

K(s)G(s) = −Kc(sI − A)−1B.ªiFseSdS®

[tRSvB[¹Y`OQWaY_^>­SItNPKMV=5hVOtOtV%B]RSBtNGIOtIJ[tviWGVwWaNaVvyTBV(5WaYh * ª?TMYLB[5WaVwh=Y[r = 1/2

®@F

Lh=RLB[tNGTB=]VcWGVuzR(Ti=WaV%TBNG[th=]VO[]vBNG­`Y`LO§ª?NaW2[=YSNPO;T´vBLNaLOtI=StY`OQVvB^®xxn+1 = xn + un + wn

yn = xn + vn

REDwn

V@Ovn

[tRSLObTiV=[c_B]vBNGOt[XT´IOQY`OcV@ObTiVzV=[]vBtV LBRS]zyYWGNG[]I=[ª?h=VLOQ]I=[>7NGLBTBI= VLBTMYLOQ[V@O,TBV­`Y]NZYLih=VyvBLBNPO^YNGtV®@F L []RSvB|MY`NGOQVhYWGh=vBWGV=bWaY¹h=RSzzYLBTBVy¦¨ vBN5zNaLBNGz NG[tVWaVhtNGOt=]V x

J = limN→∞

E

[N∑

n=1

x2n +

N−1∑

n=0

u2n

]

YWGh=vBWGV=ÁWaV[ ¬?RS]zV[ itI=TiNahOtV=vB ¡9V@OÁ V[]OtNazY`OQVvB¢¡5h=RttV[t RSLiTMYLOtV=[ \XhV;BtRS_iWa=zVF QY2hV=;VOhRSzzV=LOQV9WaV[;TBNaYSQY`zzV=[TBVwp ¹TBV[TBVv hRSt]V=h@OQV=vit[=F

;I=SviWZY`OtNaRLU¦¨ © ¦¨ »M¼¾½@¿ À ¾Ã@Ä

C S

−360 −315 −270 −225 −180 −135 −90 −45 0−40

−30

−20

−10

0

10

20

30

40

PHASE

dB

0.2338

0.7974

5.484

37.62

0.06014

0.1334

0.3379

1.117

3.736

7.301

0.1699

0.4081

1.168

4.747

22.79

113.1

α=1 α=1000

LQ

3 iFse _j 358AYc * & 1B"* &(43 9G!h&(8(Y_aYB"1351""!L "!e!-"1i352'6! 0 : :;8c :&: 0 :

;I=SvBWaY`OQNGRSLµ¦¨ © ¦¨ »M¼¾½¿ À@À=Â@þÄ

¥

"!$#&%(')#&*,+.-0/1%/2!$354687:9;7=<>7@?BADC><FEHG>7@CFEHIKJLNM,OQPSRUTWVXGZYN<([\7]G^C><FY h C>YN<ZM_6a`cb h IKJNGZEdGZCFMfe:MDgh9N[HIKEHCFM<i[dM

9N<ZEdJ h Ed9MjLNM2GZA9;7=<F7@C>EHIKJkAJNIKJ h A29;7=lKMnmKo)pq[dMrC><>7@JNGZstM&<ZCSMJuC><FM2[d7 h I=JNGFEHlKJNMeMDCv[xwyEHJNJNIz@7@CFEdI=J

εyA&C>7=JWCSJ|YN[x@I=JQ9M&Y~Cv<FMI=Y h [HM< h M)C><F7=JNGZstM&<ZCSGFYN<SJwEd?9IK<^C>MjWYNM&[N9;7@<>7=?&CF<FM

Q(s)GZC>7=N[HM

L~EHCj9;7@<>7=?&CF<FMLNMOPSRaT|VUvGFM[HIKJ[\7;lKYN<ZM~HG>7=JNG)WYNM h M[d70JNM9M<^C>YN<FM h MC><F7=JNG^stM<ZC M&CZC>MiC>M h JNEdWYNM9M&<F?M&CQLNM9N7=<>7@?BADC><FEHGFM&< 7zKM h C><ZIKEHGk9;7=<>7@?BDC><FM&Gp

Kc

Kf

MDCQ(s)

[wMJ~GFM?~[dMLNM&G h IK<Z<FM h C>M&YN<FGGZC>7=NEH[dEHG>7=JuCn[HMGZ|GZCF?M=

F

F +-

+

¡

¢ £¤v¥

¦

§v¨ª©^«

¬

­®S¯

°&±

e=0

x

u

x

^

x y

Retour d’état

Estimateur

paramètre Q

Système

+ -+

+

+

++

²j³´nµ¶N·¹¸º¼»c½~»=¾F»=¿ÀDÁ¾ÄÃdÅ»=ÁxÃÆqÇ,È|É0OPSRaTWVÅDÊh¾cË·»ÅDÁx¾ÄÊ;ÌDÁxÊ|¾>Écº2ÍÎ1Ï

Ð1AlKYN[d7@C>EHIKJ6a`5Ñ@6U`cb Ò;ÓÄÔ&ÕvÖ&×&ØD×ÄÙ

µS²SP SRa³|V$T

354&'4 +"! * 3!$3$# %&¼!U+"! IKJ~GFEdL~A<FI=JNGj?B7=EdJuC>M&J;7=JWC YNJBGZ|G^C>?MLwIK<FLN<ZM

nWYNMJNIKYNGrGFY~9N9IKGFM&<FIKJ~G)G>7=JNGr9M<ZCFM

LNMclKA&JNA<F7=[HEHC>A¹GZCF<FE h C>M&?BM&JWC19N<ZIK9N<FMSpx = Ax+Bu,

y = Cx,

N·

MDCYNJ h IK<Z<FM h C>M&YN<LwIK<FL~<FMnpxK = AKxK +BKy

u = CKxK +DKy.

N mK

')(+*-,/.1024365jC><ZIKY~zKM&<[HMG 9;7=<>7@?BDC><ZMGKc

Kf

MDCQ GZC>7@C>EH|Y~M h 7=<[HMGZ|G^C>?MMDC[dM

h IK<F<ZM h CFMYN<IKJWC[dM?87?MI=<FLN<FMCFM[HGWYNM[HM h I=<F<FM h CFMYN<56a`cb A&WYNEHz@7@[dM&JWCcL~A&;J~Ev9;7=<¹[dM&GA&|YN7@C>EHIKJNGp

˙x = Ax+Bu+Kf (y − Cx)

u = −Kcx+Q(y − Cx)

N GZIKEHCnAWYNEHzq7=[dM&JWC7=Y h I=<F<FM h CFMYN<~ mLNY9IKEdJuC1LNMz|YNM5L~Y h IK?9IK<ZCFM?MJWCnMJWCF<FA&MÑ@GZIK<ZCFEdM@9 *-.;:=<+>1*-?@5¼M[H[dM h IKJNGZEdGZCFMe h 7=[ h YN[HM<[d7?B7@C><ZE h McLNM09;7=GZG>7=lKM

TMJuC><FM0[dMGLNMYhg <FM9~<FAGZMJBA

C>7@C>EHIKJNGn~ mM&C~ pxK = T x .

~yWC J9M&Y~C?IKJuC><FM&<nWYNM

TM&GZCGFIK[HY~C>EHIKJLNMc[wAWY;7@CFEdI=J(LNMEDc³GF=FVH8³ b)IKJ¹ÐKLr p

[−T I]

[A+BDKC BCK

BKC AK

] [I

T

]= 0

~NMK

LNI=JWCr[d7?7qC><FE h M 7=?Ed[·CFIKJ~EHMJ~JNMnMG^Cj[\7¹?7@CF<FE h M2L~|J;7=?EdWYNM1M&JBIKY h [dMnstM&<F?AMAcl

M&CZC>MA&|YN7@C>EHIKJ 9MY~C¹7=[HIK<FG"7&CF<FM0<FA&GFIK[HYNM59;7=<[\7<FM hÄ M< h MkLwyY~J GZIKYNGMGF9N7 h MkEHJWz@7=<ZE\7=JuC7@GFGFI h EdAeQYNJMJNGZM?N[HMcLNM

nz@7=[HMYN<ZG19N<FIK9~<FMGPO h wMG^CQA e+A LNEH<FMSp

¸ <FM h M&< h M<jYNJGZIKYNGRA MGZ9;7 h MEHJWzq7=<FEd7=JWC S = S ? (U)

SLNM1LNEH?BM&JNGFEHIKJnLNM[d7?B7@CF<FE h M

Acl;GZIKEHC¼p

AclU = UΛ . N

M¹GZIKYNGTA MGZ9;7 h MM&GZCn7=GZGFI h EHA¹e0YNJiM&JNGFM&?N[dMLNMnz@7=[HMYN<ZG)9N<ZIK9N<FM&GhGZ9M h

(Λ)|9;7=<F?E

[dMG2n

z@7=[dM&YN<FG19~<FIK9N<ZMG1LNMAcl

¸ 9;7=<ZCFEHCFEdI=JNJNM<$[HMGSzKM h C>M&YN<FG

UWYNE;M&JNlKMJ~LN<FM&JWC h M1GFI=YNGRA MGF9N7 h MLNMn[\7sÂ7VU&IKJGFY~EHz@7@JWC>MSp

U =

[U1

U2

], U1 ∈ R

n. N oK

¸ h 7=[ h YN[HM<n[d7QGZIK[dYhC>EdI=JcpT = U2U

−11

Ð1AlKY~[\7@CFEdI=J(6a`cÑ=6a`cb Ò;ÓÄÔDÕvÖ ØÄ×DÙ

µ ¼VH8³GSR o

6¼MG 9;7@<>7=?&CF<FMGnGFIKJuC7=[dI=<FG h 7=[ h YN[HAGn9;7=<n[dM&G1<FM&[\7@CFEdI=JNG2GFYNE·z@7=JuC>MSp¸Q = DK

¸Kc = −CKT −DKC

¸Kf = T−1BK −BDK

32 ( P:3 5 h MDCFC>M¹C>M hÄ JNEdWYNMc9M&Y~CQA 7&CF<FMlKAJ~A<>7@[dEHGFAM7=Y|g h 7=GnLNMG h IK<Z<FM h C>M&YN<FG1LwIK<ZLN<FMGZYN9A<ZEdM&YN<nIKY

?87H?M0EHJ~stA&<FEdM&YN<eB[xwyI=<FLN<ZM0L~Y ?I|LN[HM=¶68M0[dM h CFMYN<GFM0<FM9I=<ZC>M&<>77=Y h 7=9NE·C><ZMBLNM[d7Q<ZA&stA&<FMJ h Mª¼9IKYN<n9N[dYNGnLNMcLNA&C>7=Ed[HG

[\7,<FA&GFIK[HY~C>EHIKJ LNM([wAWY;7@CFEdIKJ LNM Dc³GF=FVH8³1~NM sÂ7@EHC7@9N9;7=<F7dCF<FMYNJ~M h IK?NEHJ;7@CFIKEH<FMiLNMGZIK[dYhC>EdI=JNG1GFM[HIKJ [dM h IKEªgLNM&Gnzq7=[dM&YN<FG9N<FIK9~<FMG9;7=<Z?BE¼[HMG

2nzq7=[dM&YN<FG9N<FI=9N<FM&GLNM

Acl

L;7=JNG[dM h 7=[ h YN[jLNY:GFIKYNGTA M&GF9;7 h M(EdJuz@7=<ZE\7=JuC M&CFCFM h I=?NEdJ;7qC>IKEH<ZMBLNMiGFI=[dY~CFEdIKJNG9M&Y~CQA 7&C><ZMc<FAL~YNEHCFMcGFM&[dIKJ([dM&G h I=JNGFEHLNA<F7@C>EHIKJNGnGFYNE·z@7=JuC>M&G¼p¸ h IKEdGZEd<kYNJNM(M&JNGFM&?Q~[dMLNM

nzq7=[dM&YN<FGQ9N<ZIK9N<ZMG7=YhC>I A h IKJ ZY~lKAM&G9IKY~<C><FI=Y~zKM<

YNJNMc9;7=<F7=?BADC><ZEdG>7qC>EHI=J<FA&M[H[dM=¸ [dM&G

nzq7=[dM&YN<FG9N<FIK9~<FMG h IKEdGZEdM&G 7=GFGZI h EHAMG7=Y,GFI=YNGRA MGF9N7 h MQEdJWzq7=<FEd7=JWC9IKYN<[dM

<FA&GFIK[HY~C>EHIKJL~M~NMh h IK<F<ZMGF9I=JNLNMJuCnec[\7cLhhJN7=?EdWYNMLNM h I=?B?B7=JNL~M GZ9M h(A−

BKc)Ä

¸ YNJNMczq7=[dM&YN<19~<FIK9N<ZM5EHJNlKI=Y~zKM<ZJ;7=N[HMcLN7=JNG[HMGZ|GZCF?M0LNI=EHC1LNI=J h 7DC><ZMcC>I=Y ZIKYN<ZGGFA&[dM h C>EHIKJNJNA&McL;7=JNG h M hÄ IKE·g

¸ YNJNM¹z@7@[dMY~<29N<FIK9~<FMEHJNIKNGZM<^z@7=N[HM¹L;7=JNGn[HM¹GZ|G^C>?McJNML~IKEHCF7=?B7=EHG 7DC><ZMGFA&[dM h AC>EHIKJNJNA&McL;7@JNG h M h I=E·g

¸ MJ~NJ=IKJ hÄ IKEdGZEHC h IKY~<>7=??BM&JWCU[\7L~|J;7=?EdWYNM)L wyM&GZC>EH?B7@C>EHIKJ GZ9M h(A−KfC)

9N[dY~G1<>7@9NEdLNMWYNMc[\7L~|J;7=?EdWYNMLNM h IK??7@JNLNM GF9M h

(A−BKc)Ä

%&8/2! &! *!S3&8#!8+"!6¼Mi9N<FEHJ h EH9;7=[rEdJuC>A&< 7&CLNMi[\7 GZCF<FY h C>YN<ZM(6U`cb <FAGZEdLNMiLN7=JNG[dMBsÂ7=E·Ck|Y~M[dM&Gkz@7@<FE\7@N[dM&G

LwA&C>7@CaL~Y h I=<F<FM h CFMYN<8IKJuC8YNJ5GFM&JNGa9 |GFEH|Y~MrA&z|EdL~MJuCa9NYNEHGFWYNM h M GZIKJWC8[dM&G¼MG^C>Ed?AG¼L~MG8A&C>7@C>GLNY,GZ|GZCF?M h IK??B7=JNLNA@ 7=[HIK<FGWYNMQ[HMG¹ADC7qC>G¹LwYNJ h IK<F<ZM h CFMYN<EHGFGZY,LwyY~JNMQGZ|JuC &GFM

H∞IKYLwyY~JNMµA G^hJuC &GFM0JwyI=JWC9N7=GLNM5GFEHlKJNE· h 7qC>EdI=J(EH?B?A&LNEd7@CFM=#"hE8[HMGnz@7=<ZE\7=~[dM&G1LwA&C>7@CLNY

?I|LN[HM)GFY~<v[HMWYNM[~M&GZC LNADzKM[HIK9N9A&M1[d7stIK<Z?BM)6U`cb AWYNEHzq7=[dM&JWCFMnI=JWCvL~MGvYNJ~EHC>A&GS9 |GZEdWYNMG&7=[HIK<FG1EH[¼M&GZC9IKGZGFEHN[dMcLNM0L~IKJNJNM&<YNJNM5YNJNEHCFA59 |GFEH|Y~MeCFIKYNG[dMG1lu7=EHJNG1|Y~EaLNAD;JNEHGFGFM&JWC[dMh I=<F<FM h CFMYN<&%$kwyI'& [xwyEHJWC>A&< 7DC¹LNM h MDCFCFMC>M hÄ J~EdWYNMQLNY,9IKEdJuC¹LNMkz|YNMQLNM[wEd?9~[d7=JuC7qC>E·IKJLNM&Gh I=<F<FM h CFMYN<ZG¼p

¸ Y~JNMstIKEdG[HM h I=<F<FM h CFMYN<GZ|JWC A&CFEdGZA9;7=<QYNJNMCFM h JNEH|Y~MWYNM[ h I=JNWYNM= GZIKJ 9;7=GZG>7=l=MGZIKYNG[\7stIK<Z?BMc6a`cbX9M&<F?M&CFCF<>7LNMcz|EdGZY;7=[HEdGFM&<nMJ [dEHlKJNM¹IKY IK<FG[HEdl=JNMYNJNM5M&GZC>EH?B7 ACFEdIKJLNMGjADC7@CFGrLNYG^|GZC>&?BM zKIKEd<vGFM h C>EHIKJBNWD($M? 7&?BM@u[wEdJ~E·C>EH7=[·EHG>7qC>EHI=JLNM&GjA&C>7@C>GL~Y h IK<F<ZM h C>MY~<7@Y LNA&?7=<Z<>7=l=M,LNM [wy7=GZGFM<^z|EdGFGZM?MJuC IKY [dI=<FGLNM&G h IK?B?YhC7@C>EHI=JNGL~M[dI=EdG0LNM h IK??B7=JNLNM@UGwy7=9N9NY~EdM<F7 GFYN<[dM&GA&C>7@C>G9 |GZEdWYNMGLNY:GZ|G^C>?M9N<FEHGMJ

Ð1AlKYN[d7@C>EHIKJ6a`5Ñ@6U`cb Ò;ÓÄÔ&ÕvÖ)DØD×ÄÙ

µS²SP SRa³|V$T

h IK?9~C>ML;7@JNG[d7?BI|LNA&[dEHG>7@CFEHI=JUIKY ZIKYN<FG19I=YN<1LNM&G<>7@EdGFI=JNGnLwyEHJWCFM<Z9N<FADC7@CFEdIKJ9 AGFEdWYNM0LNMGA&C>7@C>GLNY h I=<F<FM h CFMYN<& h M&CZC>MkGZCF<FY h C>YN<FMkI <FM0A&lu7=[HM?MJWCLNM&G¹7=[HCFM<FJN7@C>E·zKMGEdJWCFA<ZMGFGF7=JWCFMGGFY~<[d7sÂ7VU&IKJL wyEHJ ZM h C>M<1[HMG h IKJ~GFEdl=JNMG1L;7=JNG1[d7QIKY h [dM@

¸ [\7G^C><FY h C>Y~<FMB6a`5b sÂ7@EdGF7=JWC57@9N9;7=<F7dCF<FM[dMQ?BI|L~[dMLNY_G^|GZC>&?BM@aEH[SGFM&<>7(Alu7@[dM?MJuC9N[dYNG7=EdGZAL w 7=LN7=9~C>M&<i[HM h IK<Z<FM h C>M&YN<7=Y9IKEHJWCBLNMstI=J h CFEdIKJNJ~M?MJuCi9N7=<[d7_GFEH?B9~[dM7=L;7=9~C>7@C>EHIKJ_LNY]?BI|LN&[dMGFYN9N9I=GFA h IKJNJWY C J]9MYhCQA&lu7=[dM&?MJWC0MJWz|EHG>7=l=M<k[\7 9~<FEdGZMMJ h IK?B9hC>MLNMJNIKJBA [dEHJNA7@<FE·CFAGQLNYG^hG^C>&?BM GFE1M[d[HMGGFI=JWCB?BI|LNA&[dEHGFA&MG G>7qC>YN<F7@C>EHIKJLw 7 h C>EHIKJNJNM&YN<N9;7=<nMDghM&?B9N[HMÄ

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3 s

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Position système Vitesse système Correcteur: état 1Correcteur: état 2

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1Réponse à des CI − Correcteur ss forme LQG

Position systèmeVitesse système Position estiméeVitesse estimée

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µS²SP SRa³|V$T

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x =

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y =[1 0

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~ uJ h IK<Z<FM h C>M&YN<

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K(s) =U

Y(s) = −

20s+ 8

s2 + 7s+ 18

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P (s) =

[P11(s) P12(s)

P21(s) P22(s)

].

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Gzw(s) = Fl(P,K) = P11(s) + P12(s)K(s)(I − P22(s)K(s))−1P21(s) . Mh

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x = Ax+B2u+ wx

y = C2x+D22u+ v,

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T [C1 D12] =

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Mh MK

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M&CT = GKS

zKIKEH< h IKYN<FG1GFY~<[d7QG^hJuC &GFMH∞

G k

−

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