! "#$%&'(#)&*+-,(. (/$ 0 1 2)34

57678:9 ';7<>= 6 ?@=BAC=@;D 6 ;7EFA 8G ;H-<>I>JK;

2 L#$ (' M%& ( &(+-, '&NMO 1'QPSR$NM (+-,(T .&N&U!V W&($ X & & 5 (&$ Y 'Z [!(Q-Q2 \']W^L-&+-,(_ _NMW ` a -$b&-!c&NM&U!0&d#-fe

(g>E 6ih ; G ;jg 6 ECk

'#E 67l e&)?@<mU 8 <onp 6qsr AC<2E Got r ; r

'&E 67l e&(;7u 6 Iv 6 uw;xI>y7A h ? q ?@< 8 6z8Fr ;2np e| q e

'#E 6xl e X AE:IK;7I G>6 -A 8 ;7E()?@~C;7E:E:;7= G A7n> fe q e

'#E 67l e;7E q AC= 6 %;7E h ;x=@J 6 ?BI>J 67r:6 "AC?A?@E;nK e q e

'#E 67l e&(= G ;E ;7I r ; 8G AE;x 6 nK e| q e

'#E 67l e#(u? r L-JK;;2np'#Joe e

0 5 (&$np 5 RQL%$

+1 777

%(Un 5 Md 'Z(" $"&) >( 6 I r E 6 =@A G A ? 8r ACu; 8 <>= r ? h ;7EF? h AC? 8 !IR

q ACE r:6z8 g 6 EHA;x=@?BuHACI r ; 6 G A2; G ; A 6G>6 A 8 =B?@~C;7I r A? 6DG A 5 ;7I2AC?@E 6 xz7 F n xzVgoe 2n q u '&'&7 X 5 n

e| q eBnZ#I7ACI>JK;xE:? ;j#= 9sr E:? q ;npxz7"A 8 A\R (I2? h ACE 8 ? G ; G A X A G AE;x= G>6 ? 6 G A

5 ;7I2AC?BE 6 nK'&'& e 6G AC= 6jG AE:A l ACECI q ? ;e&A;x=@?@ujACI r ; 6DG A 8 ; G ; e? 8Fr ACu; 8 u<>= r ? h ;7E:? h AC? 8>e 6G26HG A 8 =@?B~;7I r A e 8 ACE h ; G>6 E G AM;7= r:6 ;7I>J 6 e X AIzuHACI 6 "!$#&%&')(+*

ep''&7 X 5 F eK, r <>= 6 8:9 E:?BA-

?@?

uHACuzEF? ; G AuHAC< 8 g>;7? 8 A ! # ! #+( # A # ' # * # ' ! #! #"# ( #$

uAu%7E:?@; G26z8 ;7uH? 678 E G ;?@= h ;D 6 gZA 8 npU(A=@AIK;HL%;xE$ 6z8 ;H 6 AC=BJ 6 '*AE:A?@E;np)?B=@?@; G>678AC? 8 6 ;7E:A 8G ;-<>I2JK;2n 576 E q ?Buw;xE<>I>A 8(X AE:E:A?@E:;2nK;7E:?BIK;j'#?@I r6 AM;&S<>AC="N<>?@u;xE 7A 8

6 u;7I>y7A h ? q ?B< 8 6z8r ;'&S<>A 8 A l 6 E:;7u G <2E;7I r A 6HG A 8 AI hz6 = h ?@uHACI r:6HG A 8Fr ;j"A 8 Axe

uH?@I2JK;A 8 g 6z8 ;2np;7u; G ;A qC6 uHgK;7I>J2AC?BE; )( #*# ! ' # #+ #, #-.+ '0/ ! (12 AM; 6 I 6z8F8:63K=BJ 6 T "!, #-.+ '0/ ! (14 # !5 #"+( #$ #u G AMu;7E 6jG Axz7jA<gZA G ?"; 6 uHAC<3K=@J 6 &<2A G A 8 ACI2JK; 8:8 Af<>u; K;7=@A? ;e##I r 6 nK; 6z8r E$ 8 ;7I 6z8G A? G ; G A7n>A=@AuA G A< G A

g>E:A 8 ACI r AMA 8Fr ;H=B?@I G ; K;7=BAC?@;2e

6 =BAC? r6 E-<>uw;gZACE$7<>I r ;2k6 # #8791:;-! ( !<=/#'6 ' > / '?+!=@

?B?@?

6z8 uHAC< 8M6 EF?@ACI r ; G>6 E:A 8 A;7uH? 6z8 )?@< U 8 < A;7u 6 I e* 678Fr ;mgZAC= 6 A q A=@ACI r A*Auw; G A"A 8 A&S<>AMg2E 6 g>< 8 ACE:;7u AgZAC= 678 = 6 I 6z8 ;7I 678G A r E;K;7=@J 6jq6 IF<2I r6 e

678 &S<>AuHA(?@I q ACI r ? h ;7E;xu ;fA 8r < G ;7E#=BA r EzI>? q ;5&S<K;7I G26 A< AE; q E:?@;7IC;2kEF G ;D?@= h ; 6 gZA 8 AuHAC< 8 g2E:?@u 6z8 (I r6 I>? 6 ZAC=" 6 EFE:AC?@;?B=@A= ;Af%;7EF= 6z8 (<z< 8r6 ?B=@A= ;e

6 '#E 67l A 8F8:6 EL-AE:IK;xE G>6 A h ACE 6DG ;D?B= h ; X ?B=@J 6 &S<>AuA(?@I q ACI r ? hz6 < ;?BIzE:A 8:8 ;7E-IK; EFA;G A q6 I r E 6 =BA7e

6 ;7uH? 6DX ACEFIK;7I G26 )?@~C;7E:E;x= G A &<2A r Au 8 ? G>6 <>u 6 u qC6 I 8 AC=BJ>AC?BE 6 e 6 '&E 67l e (u? r L-JK;; g 6 E r AED?@I G ? q ; G>6 <2u ;7E r ? 6 g 6 < qC6 qC6 I>J>A q ? G>6 nuw; 8 &S<>A l 6 ?

l <>I G ;7uHACI r ;7=oI>A 8Fr ;j"A 8 A L%;x? G ; se 678 '#E 6xl A 8:8F6 E:A 8576z8F9X E;7I qC6 ; q JK; G>6G>6 u;7E:;7= nJK;x=@A 8 (= l E:A G>6G A% h ?@=@;%;7EFI>AC?BE 6 n

)<>?B~DL-? 6 I G ?A r:6 n;7EF? ; ?@; 8 L-A=@=@;7E(AI r zI>? 6 6 uHAC?BE 6 2;7g>?BACI>~C;2n 8 A q EFA r E:? ;v+@= h ?@;A7?@IK;j-A~;7E(mA G A?@E 678 A; 6z8G ACu;7? 8%q6 =BA; 8%G>6 AgK;7E r ;7uHACI r6jG A#=@A r E$zI2? q ;A"A=@A q6 Ru<>I2? q ;A 8QG ; X ; q <>= G ; G A G A&I zACI>J>;7E:?@; G ;(I>? h ACE 8 ? G ; G A G26 8Fr ; G>6G26 ? 6fG A 5 ;7I>A?@E 6&S<>AMuHA;7g 6 ?@;7E;xu IK;jEFA;x=@?@~C; 6DG A 8r ;j*A 8 Axe

678qC6 uHgK;7I>J2AC?BE 6z8 IK;f2;7=@; G A(=@<>I 678G A( 6 < r6 E:; G>6MG>6 ; 6 E:; r 7E:? 6G A 6 I r E 6 =@A)?@I5-6 ?@Ion) = h ;7E 6(6 F? !u;7? no?BIQ;xIon=BA 8:8 ;7I G E 65 ; qC6 < G '*AC? 67r6 Aj=BA 8F8 ;7I G E 6 3o; q J>?gZA= ; qC6 I h ? h CI q ? ; ;7E; G h AC=*AgZA= 6 ?@I r ACE q u >? 6G A? G>9 ? ; 8 &S<>A l 6 E;7uauD<2? r6 ?Bug 6 E r ;xI r A 8I 6jG A 8 ACI hz6 = h ?@uHACI r6jG A 8r ;H"A 8 A7e

678 uHAC< 8 '; G E:?@I2J 6z8576z8F9 6 ;7E:A 8G ;v-<2I>JK; ADU(AC=BACIK; L-;7E$ 6z8 ;v 6 AC=BJ 6 '*AE:AC?BE;2no; 6z8 6 ;7E:A 8G ;vQ<>I>JK;2n); 6z8 ?@=BAC=@;A 6 < r E 6z8 ;xu? 6z8 gZA= 6 ?@I q ACI r ? hz6 &S<>A G A=@A 8 EFA q A >?#; 6 = 6 I 6G ;Du?BI>JK; h ? G ;2e

678 uA< 8 ;7uH? 6z8 ? z<2AC=pI zAC= 6 6 =B=@?BI>A G>6 (E;7u;i 6 nS&;7= 9 E:?@;f-EF? 8Fr ?@IK;M'#?@I r6 ;;R=BJ"7A 8 AI>;;7EF? ;H=B? h A?@E:;j; q A G>6 gZAC=@;j;7uH?@~; G A7n 8 ?Bug2=@A 8 uHACI r A7e

678 uA< 8 '#E 67l A 8:8:6 E:A 8 &S<>A l 6 EFuw;7E:;7u ; K; 8 A G A 8Fr ;'#?@E uH? G A7e

? h

A 8 <>u 6vG ; "A 8 Aw;xg>E:A 8 AI r ; G ;m'&'& x X 5mq6 u 6 gK;7E r A G>6z8 EFA9&S<>? 8 ? r6z8 I>A q A 8:8 EF? 6z8gK;xE;j; 6 r ACI 6HG>6 7E;7< G AM 6 < r:6 EAuW-? I q ?@; 8 e| q e

! "#$%&'(#)&*+-,(. (/$ 0 1 2)34

57678:9 ';7<>= 6 ?@=BAC=@;D 6 ;7EFA 8G ;H-<>I>JK;

;7E$ 6 zxz7

E:?BACI r ; G26 E:A 8 kW)?@<mU 8 <(;xu 6 Im 6 u;7I>y7A h ? q ?B< 8 6z8r ;

'#E 6 zE;7u;2k &I zACI>J>;7E:?@;H#= 9sr EF? q ;

(A 8Fr ; *A 8 A 8 6 g>E 6 g 6z8Fr:6z8q6 I r E 6 = ; G>6 E:A 8 g 6 Eu 6G A= 6wG ADEFA l AE$CI q ?@; Au 6G>6 G A 8 =@? R~C;7I r A-gK;7E:; 8 ? 8Fr ACu; 8 u<>= r ? h ;7EF? h AC? 8 ?@I q ACE r:6z8 e* 8 =@A? 8"G A qC6 I r E 6 =@A 8 6 K; 8 A; G ; 8 I 6qC6 I r E 6 =@Ah A r6 EF? ;7=&<>I>? r E:? 6 n"ACucE:AC;7=B?@uHAI r ; 6G A 8 ; G ;AwIK; l 6 EFuD<2= ; 6mG>6Yq6 I r E 6 =BAw; G ;7g r ; r ? hz6g 6 Eu 6G A= 6 G AYEFA l AE$CI q ?@;2e qC6 I r E 6 =@Ag>E 6 g 6z8Fr:6 9 <>u; As r AI 8 6OG AY<2u q6 I r E 6 = ; G>6 EvA 8Fr E:< r <>E; h ;7E:? h A= &S<>A l zE:; G A 8 AI hz6 = h ? G>6 gK;7E:; 8 ? 8Fr Auw; 8 u 6 I 6ih ;7E:? h A? 8 eD 6 E G ; R 8 A 6q6 I r E 6 =BA G Aj<>u; q =@; 8:8 A G A 8 ? 8Fr Auw; 8 u<>= r ? h ;7EF? h AC? 8qC6 u^gZAE r <>EK; 6 I 6 R!=B?BI>AC;7E G AgZACIRG AI r A G>6 A 8r ; G>6 I" 6 uHA G ? G>6 e A 8r ; q ;7ujR 8 A; 8Q8 Az<2?@I r A 8Qq ;xE; qsr ACE 8r ? q ; 8 k ?) 6f8 ? 8Fr Auw; G Aq6 I r E 6 =BA 9 z= 6 K;7=BujACI r AHAsg 6 I>AI q ?@;7=BuAI r AA 8r h AC=-ACu E:AC=@; 6 ;<>ucgZA &<2ACI 6 qC6 I F<>I r:6EFA 8 ? G <K;7= ?B?); 8 ?@I q ACE r A~; 8 g 6G Au 8 AE G A 8Fq ; 8 ; G ; 8 ACu_E:A= ; 6 vACI r E; G ;mA ?B?@? (;v>I>? q ;J>?Bg r A 8 A 8:6 >EFA ;Yu; r EF?@~ G A ;xI>J 6YG A ;7= r ; l EFA9&CI q ?@; G ;Yg>=@;7I r ; 9 ;7gZACI>; 8 <2uw; qC6 I G ?R 6 U<>E ? r ~xe ^A 8 &S<>ACu; EFA 8 <>= r ;7I r A 9 =B? h E:A G26 l AIzuHACI 6 "!$#&%&')(+* n &S<>Ag 6G A 6q6 E:EFACEAu q6 I r E 6 =@; G>6 EFA 8 K; 8 A; G>678 Au 6 8 AE h ; G>6 E:A 8 e 6 3KIK;x= 8 A G A 8 ACI hz6 = h Am<2u q6 I r E 6 = ; G>6 EgK;xE; 8 ? 8r ACu; 8 u 6 I 6ih ;7EF? h AC? 8 ?BI q ACE r:6z8 K; 8 A; G26 I<>u 6 8 ACE h ; G>6 E G AH;7= r6 ;7I>J 6 =@? h EFA G>6l AIzuHACI 6 "!$#&%&')(+* e

h

8Fr E; qsr 67l J>A 8 ? 8 g>E:A 8 ACI r A Gcr6 '&'& x X 5 ; 8 ; gK;7E r ? ;x= l <>= 3K=@=BujACI r 67l r J2AEFA9&S<>?@EFACuHACI r8l 6 E r J2A G AzEFACA 6xl 6qsr6 E 6xl q ?@AI q A e| q e

('&(-R X & L- 2)$ $N`Y (O X %&**($!L% MY

57678:9 ';7<>= 6 ?@=BAC=@;D 6 ;7EFA 8G ;H-<>I>JK;

;xE q J z777

G2h ? 8F6 E 8 kV)?@<mU 8 <(;xu 6 Im 6 u;7I>y7A h ? q ?B< 8 6z8r ;

AgK;7E r uHACI r k#=@A qr EF? q ;7=)#Iz?BI>ACAE:?BI

=B? G ?BIu 6G Au 6G A=R!EFA l AE:ACI q A qC6 I r E 6 =B=@ACE 8#l 6 E<2I q AE r ;x?@Iu<>= r ? h ;7EF?@;2=BA 8 8Fr Au 8 ;7E:Ag>E 6 g 6z8 A G ?BI r J>? 8 J2A 8 ? 8 eYJ2A qC6 I r E 6 =Q=@; 8 ;7EFA K; 8 A GO6 I <>I>? rfh A qr:6 E q6 I r E 6 = n 6 < r g>< r Rl AA G >; q y;7I G u 6G A=R!EFA l AE:ACI q A; G ;7g r ? h A qC6 I r E 6 = l 6 EFuD<2= ; r ? 6 IoeJ>Ag>E 6 g 6z8 A G 8Fr E; r A7? 8;7I As r AI 8 ? 6 I 67l ; h ;7EF? ;>=BA 8Fr EF< qr <>E:A qC6 I r E 6 =B=@ACE G A h ? 8 A G l 6 E 8 ?BIz=@A R!?BI>g2< r R 8 ?BIz=AsR 6 < r g>< r > 8 8r Au 8 e q =@; 8:8D67l u<2= r ? h ;7E:?B;>=BA 8 8r ACu 8H8 < FA qrHr:6 I 6 I>=@?BI>AC;7E G ? 8Fr <2E$K;7I q A 8 J>? q J G ACgZAI G>86 I r J>A<>I>uHA; 8 <>E;>=@A 8r ; r A h ;xE:? ;>=@A 8 ? 8q6 I 8 ? G ACEFA G efJ2Aju;7?BI l A; r <>EFA 867lr J>Ag>E 6 g 6z8 A G ;7g>g>E 6 ; q J;7EFA; 8l 6 =B= 6 8 k ?) r J2A qC6 I r E 6 = 8 8Fr ACu 8 ? 8 z= 6 K;7=B=A g 6 I>ACI r ?@;7=@=8r ;>=BA ? r JDE:A 8 gZA qr&r6 ; 8 u;7=@=E:A 8 ? G <K;7= 8 A r ?B?) r J>A<2I q AE r ;x?@I r ?@A 8 q ;7I ZA<>I>u; r:q J>A G ? r JEFA 8 gZA qr%r6fr J>A?@I2g>< r ;7I G n ?B?@? &; 8 ?Bug>=BA(U(<>E ? r ~ qC6 I G ? r ? 6 I? 8 ?Bug 678 A Gw6 I r J>Ag>=@;7I r J>? 7Jl EFA9&S<>ACI q z;7?@Iuw; r E:? ]e*J2Ag>E 6 g 6z8 A GH8:q J>ACuHA? 8l E:ACA 6xlZr J2AgZA;7yS?@I g>J>AI 6 uHACI 6 Ion J2? q J? 8 < 8 <K;7=Q?@I 8:6 uHA 6 8 AE h ACE K; 8 A GOqC6 I r E 6 =@=BACE 8 e X ?@IK;x=@= zn; q6 I r E 6 =B=@AE l 6 E<2I q AE r ;x?@I >8 8r Au 8 K; 8 A G6 Im;jgZA;xyS?@I l E:ACAJ>? zJ ;x?@I 6 8 AE h AE(? 8G A h A= 6 gZA G e

h ?

"!# e FA r ? hz6 efeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e| 67r ; 8 '#E:A=@?Bu?IK;7EFA 8 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

ee 6 E:u; 8 AI 67r ;A 8 efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe ee ;7EzAu G AA 8Fr ;>?B=@? G ; G A eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe ee 6 I q AC? r6z8 8 ? qC6z88F6 >E:A 8 ? 8Fr ACu; 8 uD<2= r ? h ;7E:? h A? 8 eMeMeMefeMeMefeMeMe

e|e e 8 ACE h ;>?B=@? G ; G AAgK;7E uA r E 678G Af;7E:y 6 h eMefeMeMefeMeMe e|e e| /!I G ? q A G A 6 8 AE h ;2?@=B? G ; G A eMeMefeMeMefeMeMeMefeMeMefeMeMe $ e|e e ; r E:?B~ ')(1 ! #+4 A zE;x<mEFAC= ; r ? hz6 eMefeMeMeMefeMeMefeMeMe $

ee 6 I r E 6 =@Ag 6 Eu 6G26jG A 8 =@?B~;xI r A eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e|e >e &S<K; A 8(G ? l ACEFACI q ? ;7? 8%G A 8FqC6 I r BI<K; 8 efeMeMeMefeMeMefeMeMe e|e >e| 6 I r E 6 =@AA &S<>? h ;x=@AI r AA 8Fr ACI G ? G>6 eMeMefeMeMeMefeMeMefeMeMe

ee 6 I r E 6 =@A h A r6 EF? ;7=4<>I>? r E:? 6 efeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e 6 I r E 6 =BA G Af? 8r ACu; 8 )?@I2A;7EFA 8 g 6 EA;7=B?@uHAI r ; 6DG Af2; G ;^efeMeMefeMeMe e 6 I r E 6 =BA G Af? 8r ACu; 8 6 R$)?BI>A;xE:A 8 g 6 EA;7=B?@ujACI r ; 6DG Af2; G ; efeMeMe e 6 I r E 6 =BAL%; 8 AC; G>6 ACuW 8 ACE h ; G>6 E_efeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

e e 8 AE h ; G>6 EFA 8(G AM;x= r6 ;xI>J 6 gK;7E; 8 ? 8Fr ACu; 8 ?BI q AE r:6z8 eMefeMeMefeMeMe $ e XK6 E:u<>=@; 6G>6 '#E 6 2=@ACu;eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

e e '#= ;xI r ; efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e e U?@g r A 8 A 88F6 >E:AM;jg2= ;7I r ;^eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e e 6G A= 6jG AE:A l ACECI q ? ; eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z e e FA r ? hz6jG A qC6 I r E 6 =@A eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z

e E;7I>?B~; 6DG A 8Fr AM"As r:6 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

% & (')+*,-./ )'10 )2/ %43e 67r ; 88F6 >E:Af; r E:?@~A 8G AM"E;xI 8Fl AE$I q ?@; 8Fr EF? r ;7uHACI r AAC;7? 8 ' 6z8 ? r ? h ; 8 eMe 5$e| Auw; 8 6 >E:A 6 6 I r E 6 =BA&A r:6 E:?@;7=o(I>? r EF? 6 eMeMeMefeMeMefeMeMeMefeMeMefeMeMe e 6 I r E 6 =BA&A r:6 E:?@;7=o(I>? r EF? 6 &ACE 8 < 8 6 I r E 6 =@A H 8r E:< r <>E:;j;7EF? h AC= efeMeMe e 6 I G ? A 88:6 >E:AM;; r E:?B~ G AfNM;xI>J 6jG A= r ; X E:A9&I q ?@; eMeMefeMeMefeMeMe $

e2e 6 uHgK;7E:; 6G ; qC6 I G ? 6 U<>E ? r ~ q6 u ; q6 I G ? 6G A G A 3KI2? 6G A 8 ?BIK;7= eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $

e2e 6 I2J>A q ?@uHACI r6HG>6z88 ?@IK;x? 8G26z8 uAI 6 EFA 8 g>EF?@I q ?@gK;x? 8 eMeMefeMeMefeMeMe e2e 6 uHgK;7E:; 6G A qC6 I G ? A 8)8:6 2E:A&;u; r EF?@~ G A z;7I>J 6G A&;7= r ; l E:A & CIR

q ? ;ceMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

h ?@?

e 6 I>As 6HG>6 6 I r E 6 =@A&A r6 EF? ;7=oI>? r E:? 6Dq6 ua 8Fr ;>?B=@?B~; 6 G ;7g r ; r ? h ; e

4 '). / ) / 3 e g>E 6 ?@u; 6 g 6 E X ?@= r E 6DG Af'#EF?@uHAC?@E:;DE G ACu gK;7E;H? 8Fr Auw; 8 )?BI>A;xE:A 8 eMe e| g>E 6 ?@u; 6 g 6 E X ?@= r E 6DG Af'#EF?@uHAC?@E:;DE G ACu gK;7E;H? 8Fr Auw; 8 6 R!)?@I>AC;7E:A 8 -$ e @I r A 8 A G>6X ?B= r E 6 eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

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e e|e = q <2= 6DG>6z8qC6 A3 q ?BACI r A 8 eMeMeMefeMeMefeMeMeMefeMeMefeMeMe e e|e| ACuHg>= 6G A 8 BI r A 8 A;7g>=@? q ;7I G>6 R 8 A l 6 E:u; 8 &S<K; G E r ? q ; 8%G A

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e 6 uAI r E:? 6z8 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $ # - "!# & .' 4 ')

>e 6 I G ? A 8G Af%; 8 ;7uAI r:6 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 7>e| &S<K;A 8G>6 #E:E 6 eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z

3 & (') "! # $#% ') . 0 4& 0 ) 2#/ (' 3 e '&E 6 FA r6 AI =@? 8 A G>6 6 I r E 6 =BAg>;7E;H'#=@;7I r ; 8G ANE;7<mAC=@; r ? hz6 (I>? l 6 E:uHA

AfI>? r EF? 6 eMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e| Aug>= 6jG AM'&E 6 FA r6 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $

ee '#E 6 FA r:6 eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $ ee A 8 <2= r ; G>6z8G A 8 ?@u<>=@; 6 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $7

e '&E 6 FA r6 gK;7E;j'#= ;7I r ; 8 I>; XK6 E:u;DAz<2= ;7EefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $ e e 6 E G ;7ACu g 6 E q6 I r E 6 =BA=@?BI>A;7EeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $ e e &S<K; A 8(G>6 ACE:E 6 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $ e e 6 E G ;7ACu gZA= 6jqC6 I r E 6 =BA h A r:6 E:?@;7=o<>I2? r E:? 6 eMefeMeMeMefeMeMefeMeMe $

e g>=B? q ; 6 ; 6 6 I r E 6 =@A 6 =BACE:;7I r AM; X ;7=@JK; 8 eMeMeMefeMeMefeMeMeMefeMeMefeMeMe $ e2e FA r ? hz6jG A qC6 I r E 6 =@A eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z e2e ?BI uH? q ; G ; q ; G A? ; G A q ;xE:E 6z8 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z

e >e|e A 8:q E:? 6HG ; 8l ;7=BJK; 8 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e2e 8Fr E; r:9 z?@; G A qC6 I r E 6 =@A r:6 =@AE;7I r A; l ;7=@J>; 8 eMeMefeMeMeMefeMeMefeMeMe

e Aug>= 6jG Ag>=B? q ; 6 ; 6 6 I r E 6 =@A 6 =BACE;xI r A; X ;7=@JK; 8 eMeMeMefeMeMefeMeMe 7 e e '#E 6 FA r:6 eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e e A 8 <2= r ; G>6z8G A 8 ?@u<>=@; 6 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

e 6 uAI r E:? 6z8 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

& (') #)!1 # # ') . 0 4* / 2#/ )+ e I =B? 8 AM?Bug2=@? 3 q ; G ; efeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e| &S<K;A 8G>6z8 #E:E 6z8 <?@=B? ;xE:A 8 eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z e ; 6 E;7I r A 8G>6z8 #E:E 678 <?@=B? ;xE:A 8 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e I =B? 8 A G AM 8Fr ;>?@=B? G ; G AweMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e 8 gZA qr:6z8(G A!uHg>=@AuAI r ; 6 eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 5$

e e X <>I A 8G AMu 6G <>=@; 6 eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

h ?@?B?

e e &AE 8 6 ! eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e e A G < 6HG>6 + #? ! ')(+* eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e Aug>= 6jG Ag>=B? q ; 6 eMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

e e A 8 <2= r ; G>6z8G A 8 ?@u<>=@; 6 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e 6 >EFAf;D< 8 CI q ? ; G>6X AIzuHACI 6!6 !$#&%&')(+* I 6 SR$ AI 6 R

' & (') !# !1 / # ') .0 4* 0 2#/ ) 4 3e ';7E;7uHA r E:?B~; 6DG>6 6 I r E 6 =BA eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e| &S<K;A 8G>6 #E:E 6 eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e '&E 6 FA r6 AI =@? 8 A-gK;7E;'#= ;xI r ; 8 6 R$)?@I2A;7EFA 8 G ANE;x<jA= ; r ? hz6 (I>? l 6 E:uHA

AfI>? r EF? 6 eMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe $e '&E 6q A G ?@uHACI r6jG AM'#E 6 FA r6 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 7e Aug>= 6jG Ag>=B? q ; 6 ; 6 6 I r E 6 =@A 6 =BACE;xI r A; X ;7=@JK; 8 eMeMeMefeMeMefeMeMe

e e '#E 6 FA r:6 eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe e e A 8 <2= r ; G>6z8G A 8 ?@u<>=@; 6 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

& (') # !# !1 # # ') . 0 .* 2#/ % $2e 8r ?Buw; 6DG>6 ; 6 E:;7I r A G ;j'*ACE r <>E$>; 6 Aua'&=@;7I r ; 8 6 R$)?BI>A;7EFA 8 eMeMe - $2e| ; 6 E;7I r A G ;'*ACE r <>EK; 6 Au '#= ;7I r ; 8 6 R$)?@I2A;7EFA 8G AwNE;7< AC=@; r ? hz6

(I>? l 6 E:uHA 6 ? 8 efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - $2e X <>IA 8G Af 6G <>=@; 6 eMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - $2e 6 >EFAf; 8 '#E 6 g>EF?@A G ; G A 8G Af 8Fr ;>?B=@? G ; G A efeMeMeMefeMeMefeMeMeMefeMeMefeMeMe -

& (') 4 ' - 4 '2e XK6 E:u<>=@; 6G>6 '#E 6 2=@ACu;eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - $

2e e '#= ;xI r ; efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - $2e e U?@g r A 8 A 88F6 >E:AM;jg2= ;7I r ;^eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - $2e e 6G A= 6jG AE:A l ACECI q ? ; eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - 2e e FA r ? hz6jG A qC6 I r E 6 =@A eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe z

2e| ';7E;7uHA r E:?B~; 6DG>6 6 I r E 6 =BA eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 2e &S<K;A 8G>6 #E:E 6 eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 2e 6 I r E 6 =BA G Af? 8r ACu; 8G AfNE;7<mAC=@; r ? hz6 (I2? r E:? 6 eMeMefeMeMeMefeMeMefeMeMe 2e 6 I r E 6 =BA G Af? 8r ACu; 8G AfNE;7<mAC=@; r ? hz6 <>gZAE:? 6 E;Hu eMeMefeMeMefeMeMe 72e 8 ACE h ; G>6 E G A= r6 NM;7I2J 6 efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

2e e ; 6 E:;7I r A G>6 ACE:E 6HG AA 8r ?Buw; 6 efeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 2e e X ACI zuHACI 6 "!$#&%&')(+* eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

2e e|e "ACuHg 6HG AA r ?@I 6HG>6 g>? q6 eMefeMeMefeMeMeMefeMeMefeMeMe 2e 6 I r E 6 =@; G>6 EgK;7E;? 8r ACu; 8"G ANE;7<A= ; r ? hz6 <>gZAE:? 6 E;(u 8 ACu 6 !$#&%&')(+*

I 6z8 ?@I>;7? 8%G ;j'&=@;7I r ; efeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 2e e (I =@? 8 A G AA 8Fr ;>?B=@? G ; G A eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

2e $ 6 I r E 6 =@; G>6 EgK;7E; ? 8r ACu; 8G ANE;7< A= ; r ? hz6 <>gZAE:? 6 Ej; u )? h E:A G A6 !$#&%&')(+* I 6z8 ?@I>;7? 8%G ;H'#=@;7I r ;DA G>6 6 I r E 6 =@; G>6 EYefeMeMefeMeMeMefeMeMefeMeMe - 2e $e (I =@? 8 A G AA 8Fr ;>?B=@? G ; G A eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe -

2e Aug>= 6jG Ag>=B? q ; 6 ;D<>uWAE hz6 uHA q ;7I>? 8 u 6 eMefeMeMefeMeMeMefeMeMefeMeMe -2e e A 8:q EF? 6G26H8 AE hz6 uHA q ;7I>? 8 u 6 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe -2e e 6G A= 6jG ; G ?@I u? q ; G>6H8 ACE hz6 uA q ;7I2? 8 u 6 eMeMefeMeMeMefeMeMefeMeMe - $

?

2e e '#E 6 FA r:6 eMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - $2e e A 8 <2= r ; G>6z8G A 8 ?@u<>=@; 6 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe - 2e e A 8 <2= r ; G>6z8 A gZACE:?BuAI r ;7? 8 eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

2e 6 uAI r E:? 6z8 eMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

& ' !# '' 2e 6 I r ?@IS<K;A 8 (; r <2E;7? 8 gK;7E:;jA 8Fr AM*E:;K;7=BJ 6 eMeMefeMeMefeMeMeMefeMeMefeMeMe

) ' )42 ' #'. ' ( ' ! ' # ## . ! 4 & /.')+*, ./ )'10 )2 )

Le Auw;D(<?@=B? ;7E-g>;7E;j;H'#E 6ih ; G ;H'#E 6 g 6z8 ? 6 e efeMeMefeMeMeMefeMeMefeMeMe $Le| '&E 6 h ; 8G>678 Auw; 8G ;HA 6 e eMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe 7

Lee '#E 6ih ; G>6 )Auw;He| efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe Lee '#E 6ih ; G>6 6 E 6 = E:? 6 e eMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe zLee '#E 6ih ; G>6 )Auw;He efeMeMeMefeMeMefeMeMeMefeMeMefeMeMeMefeMeMefeMeMe

& ! # . & .' // * #/ )2-. ' #/ 2 . '

'44/ . ' "!# - e A 8:6 =B< 6 <>u 9 E:? q ; G AM &S<K; A 8 (? l ACEFACI q ? ;7? 8 efeMeMefeMeMeMefeMeMefeMeMe e|c= 6 E:? r u 6z8 gK;7E; 6 '#E 6 FA r6HG A 6 I r E 6 =@; G>6 EFA 8 eMefeMeMefeMeMeMefeMeMefeMeMe 77

qC6 I h ACE 8:6 E(;7I>;7= 7? qC6 g>;7E; G ? z? r ;7= A 8Fr h AC=I 6j8 ACI r ? G>6 AI r E:; G ;R 8 ; G ; - ( +! ')( -+( +! = = qC6 I h ACE 8:6 E G ? z? r ;7=]gK;7E:;H;xIK;7= z? qC6 & q6 E:EFACI r A q6 I r @IS<K; ' !+ + ! (1 G A r A q 6 A? 8:6 =@;7uHAI r6G A l ;7=@JK; # +! !+ ' &( #( ' . # ' &( 3>= r E 6jG Af;7g2E 6 2?Buw; 6G Ag>E:?BuA?@E:; 6 E G Au +! # '0/# ' &( ! & q6 I r E 6 =BA r6 =@ACE:;7I r A; l ;7=@JK; 8 # 2 ! #(1 +&(1 # ;7I2J 6HG AM;7= r ; l E:A &CI q ? ; +' *- !79"! ( + * #')( # 6 8 AE h ; G>6 E G AM;7= r:6 ;7I>J 6 +' *- * #')( -! ( ! ! = 6q ;x=@uHAI r A?BI r AzE h A= 8 Az<2I G>6 )AZA 8 z<>A

! = ; 6jG26 u 6G26jG A 8 =@?B~;xI r A? G A;7=

! G A 8 ? z<K;7= G ; G A 8 =@?BI>AC;7E:A 8 uw; r E:? q ?@;x? 8 ')( !$# /# ' ')( !79# '0 ' ! ! =B?@I>AC;7E%A?BI h ;xE:? ;xI r AI 6Dr ACuHg 6 ')( !$# '0/ ! ')( ( # ' #(1 u>= r ?Bg>=B; 8 ACI r E; G ; 8 A u>= r ?Bg>=@; 8v8 ; G ; 8 nu<>= r ? h ;7E:? h AC= / ' ')( 4/ '

-= = & qC6 I r E 6 =@A; G ;7g r ; r ? h76 g 6 Eu 6G A= 6 G AE:A l ACECI q ? ; / 9+! ! ! ! ( + ! # # ' ( !

+&(1 r A 6 E:? ; G ;jEFA;7=B?@uHAI r ; 6 &S<K;7I r ? r ; r ? h ; 79 #(1 '0 # ' ( ! ! ! #+ % "!

2?

<>u;DACI r E:; G ;jA<>u; 8 ; G ;2n2u 6 I 6 h ;7E:? h AC= ')(+* ! ')( = ')(+* !- = & q6 I r E 6 =BAg 6 Eu 6G>6DG A 8 =@?@~C;7I r A '?')(+*%/ 9+! +&(1 A 8Fr E:? r ;7uHACI r AE:AC;7=)g 678 ? r ? hz6 ' + . '0 ' ( ! !$# 0* & q6 I r E 6 =BA h A r:6 E:?@;7=o<>I2? r E:? 6 +("'0 ( !+2 +&(1 0* & qC6 I r E 6 =@AH; G ;xg r ; r ? hz6 g 6 EMu 6G A= 6mG AE:A l ACE$I q ?@;A h A r:6 Ef<>I>? r EF? 6 ("'0

( !+2 /!9+! ! ! ! ( + ! # # ' ( ! +&(1 * & q6 I r E 6 =BA jA 8Fr EF< r <>E:; h ;7E:? h A= ( # ' # ! 1+? !'+&(1 * & qC6 I r E 6 =@A; G ;xg r ; r ? hz6 g 6 E&u 6G AC= 6G AE:A l AE$I q ?@;MAA 8Fr EF< r <>E:; h ;7EF? h AC= ( #

' # ! +? !/ 9+! ! ! ! ( + ! # # ' ( ! +&(1

?@?

qC6 I r E 6 =@A;x< r6 u r ? q6r ACu G A 8 ACuHgZACI>J>; G>6 gK;7g 9 ? 8%h ? r ;x? 8 I 6 ; h ;7I 6jG ;jAIzAI>JK;7EF? ;DA G ;q ? I q ?@;2e(= 9 u G A 8 <K;As r EFACu;?@uHg 6 E r I q ? ;(Au`; h ? A 8 n h A q <>= 6z8 A 8 g>; q ?@;7? 8 n 8 ? 8Fr ACu; 8 E 6 xRr ? qC678 nAC=BA r E 6G>6 u 9C8r ? q6z8 A r ;7I r6z8#6 < r E 6z8&8 ? 8Fr ACu; 8 A=@A r E 6 uA q I2? qC678 n 6Mq6 I r E 6 =BA;7< r:6 u r ? qC6r:6 E:I 6 <R 8 A <>uw;YgK;7E r A ?@uHg 6 E r ;7I r A G A 8 ? 8r ACu; 8 ?BI G < 8Fr EF? ;7? 8 A G A uw;xI< l ; r <>E;u 6G AE:I 678 e' 6 EAsACuHg>= 6 n 6YqC6 I r E 6 =@Aw;x< r6 u r ? qC69 A 8:8 AI q ?@;7=-Au u &S<>?BIK; 8fG A q6 uw;xI G>6 I<2u 9 E:? q6 nIK; 8 ?@I G 8Fr EF? ; 8)G AQ;7=B?@uHAI r:6z8 n &S<Bu? q ; 8 n l ;xE:u; q < r ? q ; 8 n 8 ? G AE:>Ez? q ; 8 A r r ?@= z; r ; se

qC6 I r E 6 =@Ag 6G A 8 AE;xg>=@? q ; G>6 ; !4. +-/ ! =.&q6 u 6 ng 6 E-AsACuHg>= 6 n 8 ? 8Fr ACu; 8 >? 6 R= z? qC678 nSA qC6 I zuH? qC678 n 8F6q ?@;7? 8 n;xI>?@u;7? 8 A 8 AE:A 8 JS<>uw;xI 6z8 e (u !4'+-/ ! =%9(q ;7E:; qr ACEF?R~C; G>6 gZAC= 6fG A 8FqC6 I2J>A q ?BuAI r:6DG Au 6G AC= 6z8 uw; r ACu r ? qC6z8#l 6 E:u;7? 8 gK;7E; 68 AC< qC6 uHg 6 E r ;7uHACIRr:6 n g 6 E;7g>EFA 8 AI r ;7E qC6 uHg 6 E r ;7uHACI r:6 A 8Fr6q 8Fr ? q6 n#?@I r6 =BACE I q ?@;; 6qC6 I r E 6 =@Axn uD< G ;7IC; G Aq6 uHg 6 E r ;xuAI r:6 ; 6 = 6 I 6G>6Mr ACuHg 6 A?@EFE:ACg2E 6G < r ? >?B=B? G ; G AAsgZACEF?@uHACI r ;7= ; 8r E:? z?BI 5$ nA 6 e seNAE;7=BuAI r A7n !4.!+-/ ! =. ?BI r ACE;zACu q6 u 6 < r E 6z88 ? 8r ACu; 8 ; r E; hz9C8G AG ? h ACE 8 ; 8 ACI r E; G ; 8 A 8 ; G ; 8 n 6 &S<>A 6z8q ;7E; qr AE:?B~; qC6 u 6! ' !/#-</ ' ( # ' ( ! ' e

(A 8Fr ;*A 8 AM; 8:8 <2uA R 8 A &S<>A 6D6 FA r6 ; 8 ACE qC6 I r E 6 = ; G26 n G>6 E; h ;7I r A G ACI 6 u?BIK; G>6 #(1 # ng 6G A 8 AE*u 6G AC=@; G>6 u; r Auw; r ? q ;7uHACI r A#g 6 E"A &S<K; A 8G ? l ACE:AI q ? ;7? 86 E G ?@I E:?@; 8 Axn7I2A 8Fr A 8 ACI r ? RG>6 nI" 6f9 <>u !2+-/ ! = e 6 AI r ;7I r:6 n 678&h ;7= 6 E:A 8G>678 gK;7E uA r E 678&G A 8r A(u 6G AC= 68 6G A 8:q6 I>J>A q ? G>678 n"ACu 6 E:;v; 8:8 <>u;R 8 A 6vqC6 I2J>A q ?@uHACI r6vG Aj=B?@uH? r A 8 gK;7E:; 8 A< 8h ;7= 6 E:A 8 e(= 9 uG ? 8Fr:6 n 6MqC6 uHg 6 E r ;7uHACI r6G ;g>=@;7I r ;g 6G A 8 ACE h ;7EF? ;7I r AI 6r Aug 6 nSI" 6 R!=B?BI>A;xE AA 8Fr 8 < FAC? r6; gZACE r <>E$K;A 8 &S<>AI" 6 g 6G ACu 8 AEuHA G ? G ; 8j6 < A 8Fr ?Bu; G ; 8 g>EFA q ? 8 ;xuAI r Axe 8:8 <>uA R 8 A&S<>A 6z8Q8 ?@IK;x? 8&G A 8 ; G ; G ;Mg>= ;xI r ;< r ?@=B?B~; G26z8 IK;ME:AC;7=B?@uHAI r ; 68 6 uHA G ? G>6z8 g>E:A q ? 8 ;7uHACI r A7n? 8Fr6 9 n 8 Au_ACE:E 6z8 I2ACu ; r E; 8:6z8 n 6 &S<>Ajg 6G AjEFACg>EFA 8 ACI r ;7EM<2uw; 8 ?Bug2=@? 3 q ; 6 qC6 I 8 ? G AE h AC=

Au g>E 6 2=@Auw; 8HG A qC6 I r E 6 =@AEFA;7? 8G A h ? G26 ; 6 G A 8 AugZAI>J 6 =@?BuH? r ; G26 gZAC= 6z8Hr E;xI 8:G < r6 EFA 8 ng>E 6q A 8:8 ;xuAI r:6G A 8 ?BIK;7? 8 n r E;7I 8 u? 8:8 6jG A G ; G>6z8 A 6 < r E 6z88 < 8 ? 8r ACu; 8G AuA G ? 6 e

U G ? h AE 8 ; 8-r9Cq I>? q ; 8%G A qC6 I r E 6 =@A&<2Ag 6G Au 8 AE; G A9&S<K; G ; 8 ;g>E 6 >=BACu; 8QG A qC6 I r E 6 =@AAu &<2AM;Dg>=@;7I r ; r ACu A 8:8 ; 8q ;xE; qsr AE$ 8r ? q ; 8 )A h ?BI>A sk q6 I r E 6 =BA; G ;xg r ; r ? hz6 n qC6 I r E 6 =@A

H∞

n q6 I r E 6 =BA A 8Fr EF< r <2E; h ;7EF? h AC= n r A 6 E:? ; G ;EFA;7=B?@uHAI r ; 6 &S<K;7I r ? r ; r ? h ; 79 #(1 '0 # ' ( ! ! ! #+ %0"! [ X sn qC6 I r E 6 =@AI>AC<>E:;7= n qC6 I r E 6 =@AI>A ><>= 6z8:6 n qC6 I r E 6 =@Ag 6 E;7g2E:ACI G ?@~C; G>6 !$# ("')(+*8+&(1 n>A r:q ee

qC6 I r E 6 =@A MA 8Fr E:< r <>E; h ;xE:? h AC= ( # ' # ! 1+ !5+&(1 > 9 ZACu ; G A &S<K; G>6; 8 ? 8r ACu; 8 uD<2= r ? h ;7E:? h A? 8 ?BI q AE r678 e(E:? 7?@IK;x=BujACI r Af; 8 A 8Fr E; r:9 z?@; 8 > 8 6 >; 8 AC; G ; 8 IK;uHA G ? 6wG>6 A 8r ; G26 G>6w8 ? 8Fr Auw; A7e >en r y?BI 5$ sn 6 &S<>AI>Au 8 Aug2E:A 9fh ? h AC= G A h ? G26;D; 8 gZA qr:6z8r9q I>? qC6z86 <vA q6 I7u? qC6z8%q6 u 6 n2g 6 E%A 2Aug2= 6 n>;?@uHg 6z8:8 ? >?B=? G ; G A G A?BI 8Fr ;7= ; 66 < G A q6 I 8Fr E:< 6DG A;7= z<>I 8Q8 AI 8:6 E:A 8 n 6q < 8r6DG A 8 AI 8:6 EFA 8 A G A 8 ? 8r Auw; 8QG A q6 u<>I>? q ; 6AI r EFA 6z8(8 ACI 8F6 E:A 8 A 6wq6 I r E 6 =@; G>6 ECe- 8Fr:6 F< 8r ? 3 q ;;;7g>=B? q ; 6jG AfA 8Fr E:; r9 z?@; 8 K; 8 A; G ; 8 IK;EFA;7=B?@uHAI r ; 6G A 8 ; G ;n7ACu &S<>A 8 6 uHA G ? G ; 8 ;xgZACIK; 8 ; 8*h ;7E:? h A? 8 &S<>A G A h ACu 8 A7<>?@E 8 ?@IK;x? 8G AEFA l ACECI q ? ;e

g>E 6 >=@Auw; G AA 8Fr ;>?@=B?@~C; 6 E 6 >< 8Fr ;jg 6 EEFA;7=B?@uHAI r ; 6DG A 8 ; G ;DA> r ACu 8 ? G>6j6r ACu; G A G ? h ACE 8F6z8(r E;K;7=@J 6z8 IS<>uVgK; 8:8 ; G>6 EFA q ACI r A7noAxe >eBn ;7= qC67rFr ;xy "$"$n]#uHAC=;RI 6ih n (6 E 6 h ?BIon(AE 8 ? 8 ?@;7I (? 8 ACI>~ 6 I z # nS#uA=B;xI 6 h n (6 E 6 h ?BIon(AE 8 ? 8 ?@;7I (? 8 ACI>~ 6 I CnZ 8Fl ;7I G ? ;xE:? JK;7=B?@= n]J JK;7=B?@= nZ G ;7E G>8 g><>EzA 6 I "$ e(#ugK;xE r ? q <>=@;7En 8 A l 6 E G A 8 A$:; G>6 A 8 gZA q ? 3 q ;xE 6G A 8 ACuHgZACI>J 6(G>68 ? 8Fr ACu;%ACu uw;x=@JK; l A q JK; G ;2n g 6G AsR8 A%Aug2E:Az;7E 6 / 9 ! ;+! ! ! : ( + ' # n-&<2A 9 K; 8Fr ;xI r A-< 8 <K;7=I>; EFA; G A qC6 I r E 6 =@AQ; G ;7g r ; r ? hz6 2; 8Fr EF L 6G28:6 I "$ 2n) 6 ;xI>I 6 < <>I se qC6 I r E 6 =@Ajg 6 EMu 6G AC= 6mG AHE:A l AE$I q ?@;A u 6G>6z8G A 8 =B?@~C;7I r A 8 l 6 ?%;xg>=@? q ; G26 ; 8 ? 8r ACu; 8 =@?BI>AC;7E:A 8G A <>u;YACI r E:; G ; A <>u; 8 ; G ; ')(+* ! ')( = ')(+* ! = = > U 8 < 678Fr ; "$2n]U 8 <on4(E;7 6 6z8r ; 7 sn; 8 ? 8r ACu; 8 =B?@I>AC;7E:A 8G ADu>= r ?Bg>=B; 8 ACI r E; G ; 8 Au>= r ?Bg>=@; 88 ; G ; 8 /' ' ')( =4/ '- mF ; 6 6 ;7I2I 6 < "$ 2n2-J>?@AIonK<>I4n < X < sn; 8 ? 8Fr ACu; 8 I" 6 R!=@?BI2A;7EFA 82 J JK;x=@?@= n?@I U 8 <v7z72n?BI xz A; 8 ? 8Fr ACu; 8 I" 6 R=@?BI>AC;7E:A 8 G ;7E G28 g><>EzA 6 I se

g>E:?BI q ?BgK;7= 6 FA r ? hz6vG A 8Fr ;v"A 8 A 9jG A 8 AI hz6 = h ACEf;7= 6 E:? r u 678G A qC6 I r E 6 =@Ag 6 Eu 6G>6 G A 8 =@? R~C;7I r A,K; 8 AC; G>6z8 IK;E:A;x=@?BuHACI r ; 6G A 8 ; G ;gK;7E:;;x= z<>u; 8*q = ; 8:8 A 8#G A 8 ? 8r Auw; 8 I" 6 R!=@?BI2A;7EFA 8u<>= r ? h ;xE:? h A? 8)qC6 u g>;7E uHA r E 6z8 A 8 ?BIK;7? 8 ?@I q ACE r6z8 n7Au gK;7E r ? q <>= ;7EogK;7E; 8 ? 8Fr Auw; 8 =@?BI>A;xE:A 8 eG A 8 AugZAI>J 6DG>68 ? 8Fr Auw;MACu u;7=BJK; l A q JK; G ; 9 A 8 g]A q ? 3 q ; G>6 g 6 EQ<>u u 6G AC= 6MG AE:A l ACECI q ? ;2e 8 ;7= 6 EF? r u 6z8%G A 8 AI hz6 = h ? G>6z8G A h Auag 6z8:8 <>?@Eg2E 6 g>EF?@A G ; G A 8 z= 6 K;7? 8%G AA 8r ;>?B=@? G ; G Axe

6Mq6 I 8 ? G AE; G ; 8G <K; 8 ; 6 E G ;zAI 8 k ;g>E:?BuA?@E:; 9 K; 8 A; G ;MIK;M;7E &<2? r A r <>E:; G A qC6 I r E 6 =@A; G ;7g r ; r ? hz6 g 6 E u 6G AC= 6 G A E:A l ACECI q ? ;OA A 8Fr EF< r <>E:; h ;7E:? h A= ( # ' # ! 1+ ! / 9+! ! ! ! ( + ! '?')(+*;/ +! +&(1 ! SR$, U 8 <A r ;x= e xA-; 8 Az<2I G ; 9 >; 8 AC; G ;Au 6 8 AE h ; G>6 E:A 8G A ;x= r:6 ;xI>J 6q6 IF<; G>6z8 ; 6q6 I r E 6 =BAHg 6 EfA 8r EF< r <>E:; h ;7EF? h AC= J JK;x=@?@= e

"!# $ &%'( )

A h ? 8 ;R 8 AI>A 8r ; 8 A 6 ;7= z<>I 8-qC6 I q A? r678%l <>I G ;7uHACI r ;7? 8 g>;7E; 6DG A 8 AI hz6 = h ?@uHACI r6DG>6z8 ;7= 6 REF? r u 6z8-G A q6 I r E 6 =BAI>A 8r ;D*A 8 Axe% =BAC? r6 E-g 6G ACE g>; 8:8 ;xE G ?@E:A r ;7uHACI r A jA 6 e n q ; 8:6 A 8Fr A!:; l ;xu?B=B?B;7EF?B~C; G>6Dq6 u A 8Fr A 8(q6 I q AC? r6z8 A G A3KI>? A 8 e

*,+.-"+/* 021,35476,8:9;<1>=?6,@BA"9C8 6 < 8 ; G>6z86z8q6 I q A? r:6z8 AI 67r ;A 8G A 8:q E:? r6z8 ; 8 Az<2?@ECe

• u ?@u 6 A 6 u BI>?Bu 6h ;7= 6 E:A 88 ?@I z<>=@;7E:A 8G ;Mu; r EF?@~A8 6G AI 67r ; G>6z8 g 6 E

σmax(A)A

σmin(A)n>E:A 8 gZA qr ? h ;7uHACI r Axe

• u ?@u 6 A 6 u@I2?@u 6 ;7< r:6 h ;7= 6 EFA 8G A<>u;fu; r EF?@~ 8 ?Bu 9sr EF? q ;P8 6G AI 67r ; G>6z8 g 6 E

λmax(P )A

λmin(P )n>E:A 8 gZA qsr ? h ;7uHACI r A7e

• ‖x‖ G AI 67r ;;HI 6 E:u;j&< q =@? G ?@;7IK; G AM<>u h A r6 ExA ‖A‖ G ACI 67r ;;jI 6 EFuw; G ;ju; r EF?@~

A?@I G <2~C? G ; gZAC= ; I 6 EFuw; &< q =@? G ? ;xIK;2n &S<>A 9YG ; G ; gZA= 6 8 AC< u;7? 6 E h ;x= 6 E 8 ?@I z<>= ;xE

σmax(A)

se

• (u; u; r EF?@~P ∈ Rm×m

9 1. '0 ' ( # +! ("'? # n 6 < G A l 6 EFuw;A9&S<>? h ;7=@AI r A P > 0n 8 A

; l 6 E:u;&S<K; G E r ? q ; ; 8F8:6q ? ; G ; Yu; r EF?@~Pl 6 Eg 6z8 ? r ? h ; G A3KI>? G ;2n&? e AxeBn

xT Px > 0n

∀x ∈ Rmnx 6= 0

e A h A R 8 AI 6xr ;7E&S<>A; u; r EF?@~PI" 6 g>EFA q ? 8 ; 8 ACE 8 ?@u 9r E:? q ;2e A

P = P TnACI r 6

P > 08 AvA 8:6 uAI r A 8 A

λmin(P ) > 0n qC6 u 6 8 Amg 6G A qC6 I q =@<>?BE G>6

*A 6 E:Au; e G A 6 >=@A ;7I>?BAC= "$ npgoe z se( 6 I q A? r678 ;7I = 6 6z88 6h =@? G26z8gK;7E;Du; r EF?@~A 8 I>A ; r ? h ; 8G A3KI>? G ; 8 e

• I 6 EFuw; L∞e

G26H8 ?BIK;7=x(t) ∈ Rn

9G A 3KI>? G ; q6 u 6 6 ;xI>I 6 < <>I n2goe 7

‖xt‖∞ := sup0≤τ≤t

‖x(τ)‖ . e

[<>;7I G>6H6Dr Aug 6 ?BI>? q ? ;x= 9t0≤ t

n>g 6G AsR 8 A< r ?B=@?B~C;7E;DI 6 EFuw;

‖xt,t0‖∞ := supt0≤τ≤t

‖x(τ)‖ . e|

• (u; l <>I 6 qC6 I r BI<K;α[0, a) → [0, +∞)

gZAE r AI q A q =@; 8:8 A K 8 A l 6 E(A 8Fr E:? r ;7uHACI r Aq E:A 8:q ACI r AfA

α(0)=0 JK;x=@?B= n> A 3>I>? 6 e n>goe se

• qC6 IF<2I r6YG A u; r EF?@~A 8fq6 up=B?@I>JK; 8 A

mqC6 =B<>IK; 8fq < 678 A=@ACuHAI r6788 6l <>IA 8

E; q ? 6 I>;7? 8%G AsqC6 u qC6 A3 q ?@ACI r A 8 E:AC;7? 89 E:Ag>E:A 8 ACI r ; G26 g 6 E

Rp×m(s)e- q6 IF<2I r6HG A

uw; r E:?B~CA 8 g 6 =@?BI 6 uj?@;x? 8%G A G ?BuHACI 8 6p × m

9 E:Ag>E:A 8 ACI r ; G26 g 6 ERp×m[s]

e

• G267r ;R 8 A;E:Ag>E:A 8 AI r ; 6 uH? 8r ; G>6fG26 u@I>? 6G26Mr ACuHg 6MqC6 u 6MG26 u BI>? 6G ; r E;7I 8l 6 ERuw; G ; G A";xg>= ; q A q = q <>= 66 gZACE:; q ? 6 IK;7= se 6 ACI r ;7I r:6 n2gK;xE; 8 A G A3KI>?@E-g>EFA q ? 8 ;xuAI r AA 8Fr ; 8 E:Ag>E:A 8 AI r ;A 8 n 8 AE 6 ; G>67r ; G>6z8 6z8#8 A7<>?@I r A 8&qC6 I q AC? r6z8 AI 67r ;A 8 e& 8 ?@I>;7= G A8 ; G ;

yG AM<>u 8 ? 8Fr ACu;H=B?@I2A;7EAM?@I h ;7EF? ;7I r AI 6Hr AuHg 6HqC6 uauw; r E:?B~ G A r E;xI 8Fl AE$CI q ?@;

G(s)AfACI r E; G ;

u9fG ; G>6 g 6 E

G(s)ue qC6 I hz6 =B< 6 g><>E:;

g(t) ∗ u(t)n 8 AI G>6

g(t);

E:A 8 g 6z8Fr ;M?Bug2<>= 8 ? h ; G AG(s)

n 8 ACE A 8:q E:? r ;A h ACI r <K;x=@uHACI r A qC6 u 6G(s)∗u

ng 6 E 8 ?@uHg>=B? Rq ? G ; G Axe% 6 I 8 ? G ACE:Af;jE:AC;7=@?B~; 6DG A

G(s)G ; G ;jg 6 E

x=Apx + Bpuny=Cpx + Dpu

e&I r 6 n

y(t) = G(s)u(t) = g(t) ∗ u(t) + Cp exp(Apt)x(0) , e

6 I G A 6jr ACEFu 6 A g 6 I>ACI q ?@;7= 9 ;DE:A 8 g 6z8Fr ;HJ 6 u 6 CI>AC; G26H8 ? 8Fr Auw; u(t) ≡ 0

se

• 8 Bu 6 = 6sG AI 67r ; r ;7I r:6 ; h ;xE:? h AC= qC6 ug2=@As2; G ; r E;xI 8Fl 6 EFuw; G ; G A%";xg>= ; q A &S<K;7I r:6

6H6 gZAE; G26 E G ? l ACEFACI q ? ;7= ddt

I 6HG>6 u BI>? 6G>6jr Aug 6 e

• I 6 EFuw; G>6H6 gZACE; G>6 EG(s)

9G A3KI>? G ; q6 u 6 6 ;7I>I 6 < <>I n>g4e

‖G(s)‖ := ‖g(t)‖1 =

∫ +∞

0

‖g(τ)‖dτ . e

8 A7<>?@I r A G A 8 ? 7<K;7= G ; G A 9Hh ;7=@? G ; gK;7E:;y(t) = g(t) ∗ u(t)

6 ;7I2I 6 < <>I ng>goe4$z $

‖yt‖∞ ≤ ‖g(t)‖1 ‖ut‖∞ . e

• 8 ;R 8 Af;j;>E:A h ? ; r <>E;$gK;xE; G ACI 6xr ;7E .+$# / ! (1 ! ')(1 ! * ( ! -! *+( ! !$* ! e

*,+.-"+.- 6 3,94 9 9C8 =?6 6<9 # *+!/ ! ! # ' '? #+!#9 <>u qC6 I q AC? r6fl <>I G ;7uAI r ;7=ZgK;xE;; 8# '0/#-! .;+! '0/ ! ' # +!/ ;xg>E:A 8 ACI r ; G ; 8 I 6 %;7g r <>= 6 -<>I>J>;2n 6z8r ; U 8 < 77 eY r ACEFu 6/# *+!/ +!,! # ' '? #+!pl 6 ?z< r ?@=B?B~; G26 g 6 E 6 ;7I>I 6 < 8 ;7y;x=@? 8 "$ 4g>;7E; 6z8 g z= 6z8)G A&<>u;l <>I 6mG A r E;xI 8Fl AE$CI q ? ;e 8 A z<>?BE G A3KI>A R 8 Aw;vuw;xE$zAu G AHA 8Fr ;>?@=B? G ; G A G AHu; r E:?@~A 8G Ar E;7I 8l ACECI q ? ;2n G A&u; r EF?@~CA 8 E:AC;7? 8 A G A&g 6 =@?BIzuj? 6z8 U 8 <onz-<>I>J>;2nz 6z8r ; )?B~;7EFE;7= G A&7zz # nA 6 e e

"!# u;7E$zAu G AA 8Fr ;>?@=B? G ; G A +!G(s)∈Rp×m(s) # # 1

γ0 := minj

−<(pj) , e

&( +! pj=!. .+! G(s)

$

X ; r6 k 6w8 ? 8r Auw; qC6 uVuw; r E:?B~ G A r E;7I 8l ACECI q ? ;G(s)

9 L-$L.A 8Fr h A= - ( +! ')( -+( +!-= = # ! nQ? eAxeBnQA 8Fr h AC=I 6 8 AI r ? G>6 ACI r E; G ;R 8 ; G ; 8 AmA 8:6 uHAI r A 8 A γ0 > 0 ;x?@=@; r J 5$z2nKA 6 e e e

"!# % u;7E$zAu G AA 8Fr ;>?@=B? G ; G A +!A∈Rn×n # # 1

λ0 := minj

−<(λj) , e

&( +! λj . #2 ( # !5+! A

$ !λ0 >0 ! (1 A +! ( -/ ')( # # uw; r E:?B~U(<>E ? r ~ $

"!# u;7E$zAu G AA 8Fr ;>?@=B? G ; G A +!<=/ ')(-/ ' p(s)∈R1×1[s] # # 1

γ0 := minj

−<(zj) , e $

&( +! zj #- #&! +! p(zj) = 0

$ !γ0 > 0 ! (1 p(s) +! ( -/ ')( # g 6 =B?@Izuj? 6

U<>E ? r ~ $

*,+.-"+ 1,;B9 =?1,8 ,8 1,8 8 1 3?98 8 = 9476,8 4 = 6 3 ,9 /8

';xE; ;xg>E:A 8 AI r ;xE ;7= z<>I 8q6 I q AC? r6z8 8 ? qC6z88:6 >EFA 8 ? 8r Auw; 8 u<>= r ? h ;xE:? h AC? 8 n q6 I 8 ? G ACE:;R 8 AI>A 8Fr ;wA 6 &S<>A Ap, Bp, Cp

9 <2uw;DEFA;7=B?@~C; 6DG A<>uw;Du; r E:?B~ G A r E;xI 8Fl AE$I q ?@;m × m

E:; q ? 6 IK;7= nA 8Fr E:? r ;7uHACI r Ag>E zg>E:?@;DAI 6 R 8 ?@I z<>= ;xEG(s)=Cp(sI − Ap)

−1Bp

e

% # ' # -. #4

uw;YE:AC;7=B?@~; 6 Ap, Bp, Cp9 -! ( ( ! 8 A 9 g 6z8F8 h AC= G A r AE:uH?@IK;7E G A l 6 EFuw;>I>? q ; 6

A 8Fr ; G>6x(t)∈Rn

G>6H8 ? 8Fr ACu;

x = Apx + Bpu ,

y = Cpx , e

; 8F8 <>uH?@I G>6 R 8 A 6 qC6 I>J>A q ?@uHACI r6vG ; 8 u; r E:?B~CA 8Ap

nBp

ACp

A G>6z88 ?BIK;7? 8y(t)

Au(t)

nt≥ 0

;x?@=@; r J 5$z2nKA 6 ee e# g>EF?@I q @g>? 6 n? 8r6 g 6G AE:?@; 8 ACE 6 r ? G>6 ;jg>;7E r ?@E G ;jA &<>; 6

Y(t) = Ox(t) + TU(t) , e

6 I G A

O =

Cp

CpApeee

CpAn−1p

e

9 ; /# '+! -! ( # ' '? #+! n T 9 ; /# '+!< #(/ ' 6 ACg>=B? r ~xn

T =

g0 0 · · · 0

g1 g0

e e e eeeeee e e e e e e

0

gn−1 · · · g1 g0

, e

q6 u 6z8 # > / ! .5+! # % (

g0 = 0 , gi = CpAi−1p Bp ,

e -

A 6z8%h A r6 E:A 8

Y(t) :=

y(t)

y(t)eeedn−1y(t)

dtn−1

, U(t) :=

u(t)

u(t)eeedn−1u(t)

dtn−1

. e

8 # > / ! . +! # % ( 8 6fG A3KI>? G26z8QG A l 6 EFuw;>I>? q ;g]A= ;u; r EF?@~ G A r E;xI 8Fl AE$CI q ?@;G>6j8 ? 8Fr Auw;j; r E; hz9C8G ;jA 2g>;7I 8 6

G(s) =+∞∑

i=1

gis−i .

e

A;uw; r E:?B~ G A 6 8 AE h ;>?@=B? G ; G A O g 678:8 <2?@Eg 6z8Fr:6fqC6 uHg>=BA r6 nSAI r 6 ;Mu; r EF?@~ OTO 8 AE I" 6 R 8 ?@I7<>= ;xECn 6 &S<>A 8 ? 7I>? 3 q ;&S<>A

x(t)g 6G A 8 AE 6 r ? G26 ;DgK;7E r ?@E G A e %; r E; hz98G A

x(t) =(OTO

)−1 OT [Y(t) − TU(t)] . e -

ACI G26 ; 8F8 ?@umnig 6G A R 8 A#u 6z8r E:;7E &S<>A;%E:AC;7=@?B~; 6 Ap, Bp, Cp9#6 8 AE h h AC= 8 A&A 8F6 uAI r A

8 AM;Du; r EF?@~ G A 6 8 ACE h ;>?B=@? G ; G A O g 6z8:8 <>?@E%g 678Fr6jq6 uHg>=@A r6 ;7?@=@; r J "$72n2g>g4e se

% % 4 # '

)( ' + ! +! -! ( # ' '? #!"G>6 gK;7E Cp, ApAp ∈ Rn×n, Cp ∈ Rm×n

96 uHACI 6 E ?@I r A?@E 6

νn(1 ≤ ν ≤ n)

n r ;7= &S<>A

Oν =

Cp

CpApeee

CpAν−1p

e

r ACI>JK;g 6z8Fr:6qC6 uHg>=BA r6 e @I G ? q A G A 6 8 AE h ;2?@=@? G ; G A-g 6G A 8 ACE ?@I r ACEFg>E:A r ; G26G A%<>u; l 6 E:u;K; 8Fr ;xI r Aw?BI r ACE:A 8:8 ;7I r A7k

(ν − 1)9w6 u;7? 6 EMIS>uHACE 6G A G AE:? h ; G ; 8fG>6Y8 ?BIK;7= G A 8 ; G ;

y(t)I>A q A 8:8 EF? ; 8 gK;7E; 8 A G A r ACE:uH?@I>;7E 6 A 8Fr ; G>6 ?@I>? q ?@;7=

x(0) ;x?@=@; r J 5$z2nZg>goe se(#u

6 < r E; 8 gK;7= ; h E; 8 n 6 BI G ? q A G A 6 8 ACE h ;>?B=@? G ; G A G ?BI l 6 EFuw; 68:6 >EFA; 6 E G ACu G>6z8 3>= r E 6z8&G AA 8Fr ; G>6 < r ?B=@?B~C; G>6z8 IK;fA 8Fr E:< r <>E; G>6qC6 I r E 6 =@AuD<2= r ? h ;7E:? h A=pg 6 EQu 6G AC= 6fG AE:A l ACECI q ? ; h ? G A%;xg r <>= 6 e

% / . ') .

< r E 6wq6 I q AC? r6 ?@uHg 6 E r ;7I r AgK;7E:; 6wq6 I r E 6 =BAMuD<2= r ? h ;7E:? h A=g 6 Eu 6G A= 6HG AfE:A l AE$I q ?@; 9f6* # ! # ' ( #G>6(8 ? 8Fr Au;2n &S<>AQEFA9&S<>ACE 6(qC6 I q AC? r6(G Auw; r E:?B~ ')(1 ! #+2 gK;7E:;(; 8 <K; G A 3>I>? 6 eau; r EF?@~ ')(1 ! #+4 ; 8F8:6q ? ; G ; ; G(s)

9 <>u; u; r E:?@~ r E:? ;xIz<>=@;7E?BI l ACEF? 6 E q6 u ; l 6 E:u; 6 = 6ih ? q J 1X ;7=

ξ(s) = H(s)G ? ;

sn∗

1 , . . . , sn∗

m

, e $

H(s) =

1 0 . . . 0

h21(s) 1 . . . 0eee eee eee

hm1(s) hm2(s) . . . 1

, e

6 I G A 6z8 g 6 =B?@Izuj? 678hij(s)

8 6 I<>= 6z86 < G ? h ? 8 h AC? 8 g 6 Esn r ;7= &S<>A

Kp = lims→∞

ξ(s)G(s) ex

$

8 A$:;%3KI>? r:6 AI 6 8 ?@Iz<2= ;7Ee A h AsR 8 ADI 67r ;7E &S<>AD;uw; r E:?B~ξ(s)

G A 3>I>? G ;wAu e $ 9 >I>? q ; A=@=BACE N 66G ?@I x*Au 6z8Fr E;R 8 AAu 6 = 6 h ? q J X ;7= &<2A

ξ(s)9 ?@I h ;7EF? ;7I r A

qC6 uHgZACI 8 ; 6jG ?@I u? q ;2e u; r EF?@~Kp∈Rm×m

Au e|7 9G ACI 6 u?BIK; G ; * #(1+! # # !71: ( + ' # +' *- !79"! ( + * #')( U X N )A%; &S<>?

(n∗1, . . . , n

∗m)9Q6 * # ! # ' ( ( !4 ' # G A

G(s)e-A

n∗i = n∗

n ∀i∈1, . . . ,m n G ?@~ R 8 A<&S<>A G(s)r ACu * # ! # ' ( ( !2 ' # ("' / !

n∗e# 6 E:; h ;7I r A7n * # ! # ' ( ( !2 ' # ("' / !#8 ACE EFA l ACEF? G>6 ;>E:A h ? ; G ;xuAI r A qC6 u 6 * #

! # ' ( +("' / ! e';7E; 8 ? 8Fr Auw; 8q6 u zE:;7< E:AC=@; r ? hz6 <>I>? l 6 E:uHA r AuDR 8 A

ξ(s) = sn∗ nAI r 6m6 =B?@uH? r A e|7 g 6G A 8 ACEE:ACA 8:q E:? r:6HqC6 u 6

Kp = lims→∞

sn∗

G(s) . e

1gK;7E r ?BE G ;wAsgK;7I 8 6 G ;uw; r E:?B~ G A r E:;7I 8l ACE$I q ?@;G(s)

< r ?B=@?B~C;7I G>6 R 8 A 678 gK;7E uA r E 6z8G Af;xE:y 6 h e 6 r9 ujR 8 A

Kp = lims→∞

sn∗

+∞∑

i=1

gis−i = lim

s→∞

n∗

∑

i=1

gisn∗−i ,

e|z

G A 6 I G A 8 A q6 I q =@<>? &S<>A;ju; r EF?@~U X N G A 8 ? 8r Auw; 8q6 u 7E;7<mE:A= ; r ? hz6 <>I>? l 6 E:uHAn∗≥1

9G ; G ;jgZA= 6 gK;7E uHA r E 6jG Af;xE:y 6 h

Kp = gn∗ = CpAn∗−1p Bp .

e|

6 I q =@<>? R 8 A r ;7u 9 u &<2A7ngK;7E; 8 ? 8Fr ACu; 8MqC6 u zE:;7< E:A= ; r ? hz6 <>I2? l 6 EFuAn∗ ≥ 2

n 6z8 gK; RE uA r E 6z8G AH;7EFy 6 h

gi = CpAi−1p Bp

r u &<2A 8 ACEIS<>= 6z8 ∀i∈1, . . . , n∗ − 1 gK;7E; &<2A 6=B?@uH? r A e|z 8 A$:;3KI>? r6 e

*,+.-"+ 1,; = 351 91,3 471 <1 <9C8 6,; = 9

8 A 8Fr E:; r9 z? ; 8MG A q6 I r E 6 =@Ajg>E 6 g 6z8Fr ; 8 I2A 8Fr ;*A 8 A 8 6 K; 8 A; G ; 8 I 6 / 9+! ' # ( !9$ / 98+! '-#(1 ! g 6G A 8 ACEzAE; G26 gZA= 6 +&(1 ! ! ? # ( # ' ( ! AgZA= 6 +&(1 ! ( !4 ' # ("'0 ' &S<>A 8 6Or ACu; 8 K; 8Fr ;7I r A As r AI 8:678 e. 8:8 <>uHAsR 8 A &S<>A 6 =@A? r:6 E A 8Fr A$:;l ;7uH?B=B?B;7EF?B~; G>6YqC6 u 6 qC6 I r E 6 =@A ! ? # ( # ' ( ! &S<>A 9 ZACu ;7g>E:A 8 AI r ; G>6 I 6z8 =@? h E 6z8

r yS?BI 5$2n- r y?BI e0(= 9 u G ? 8F8:6 n(;&S<>? 8 A ;7g>=B? q ; 6 +&(1 ! ( !4 ' # ("'0 ' L%;7? G ; n> G ;7E G>8 g><>EzA 6 I 5$ sn&S<>A 9 ;7g>EFA 8 ACI r ; G>6 I 6 %;7g r <>= 6 e

% ) <2E;7I r Af; 6q6 E:ECI q ? ; G>6 u 6G>6jG A 8 =B?@~C;7I r A7n 6j8 ?@I>;7= G A q6 I r E 6 =BA 9G A 8:qC6 I r BI< 6 I 6jr Aug 6 euw; h A~8&<2A I>A 8r ; "A 8 A 8 6Yr E:; r ; G26z88 ? 8r ACu; 8qC6 u 8 ?@IK;x? 8G A 8:q6 I r @IS< 6z8 I 6Yr ACuHg 6 n 9I>A q A 8:8 EF? 6G A3KI>?@EMg>EFA q ? 8 ;7uHAI r A 68 ? zI>? 3 q ; G>6mG ; 8F6 =@< 6G AA &<>;A 8G ? l ACE:AI q ? ;7? 8qC6 u8 ?@IK;7? 8vG A 8:q6 I r @IS< 6z8 e &S<>? 8 A ; G>67r ; ; G A 3>I>? 6 G A X ?@=B?g>g 6 h e A h A R 8 A I 67r ;xE&S<>A 6 8 ?@I>;7= G A q6 I r E 6 =BA

uI" 6 9 I2A q A 8:8 ;xE:? ;xuAI r A <2uw; l <>I 6 G>6Or Aug 6 I 6 8 AI r ? G>6

< 8 <K;7= &<>;7I G>66 u 6G>6mG A G A 8 =@?B~;7uHACI r6m9 ; r ?BIz? G>6 e';7E;mA h ? r ;7E 6 A q A 8F86YG AHI 67r ; 6 nu(t)

G ACI 67r ; l <2IA 8 = 6q ;7=BuHAI r A?@I r A zE h AC? 8 &S<>A 8 6 A9&S<>? h ;7=@AI r A 8 ;unI 6O8 AI r ? G>6OG A

+&(1 ! !79 ' ( # ! (1 ! r yS?@I 5$ nxgK;xE; &<>;7= &S<>AE 8F6 =@< 6(G A X ?@=B?Bg2g 6 h x(t)G>68 ? 8Fr ACu;(Au

u;7=BJK; l A q JK; G ;2e A h A 8 AEAI l ; r ?B~; G26 &S<>A; 8:6 =B< 6x(t)

9 n 1 +! ("' n; 8:6 =@< r ;7uAI r Aq6 I r @IS<K;2e* 8Fr ; G A 3>I>? 69 u 67r ? h ; G ;%gZAC= ;E:Ag>E:A 8 AI r ; 6 ; G A9&S<K; G ; G>6qC6 uHg 6 E r ;7uHACI r6G A8 ? 8Fr ACu; 8l 8 ? q6z8 &<>;7I G>66 uA q ;7I2? 8 u 6fG A q JK; h A;7uHAI r6 EFA;7= r AI G A;<2u G ? 8 g 6z8 ? r ? hz6 ? G A;7=q6 u l E:A9&I q ? ; G A q JK; h A;7uHACI r6 ?BI3KI>? r ; &<2Avg 6G A E:A;x=@?@~C;7E 6*/ 9 +! '-#(1 !)'?+!$# n6 &S<>A 6 E:? 7?@I>; ; A &<>; 6 G ? l ACE:AI q ? ;7= G A 8FqC6 I r BI<K; X ?@=B?Bg2g 6 h "$"$ng2goe 7 + n r yS?@I n4 r yS?@I A r ;7= e seM#I r 6 nogK;7E:;!&<>;7= &S<>AE 8:6 =@< 6 n

ug 6G AE 8 ACE 8 < 8r ? r < G26 g 6 E

u(t)I 6 =@; G>6jG ?BE:A? r6DG ;jA &<>; 6wG ? l ACEFACI q ? ;x= &<2AfEFAzAM; G ?@I uH? q ; G>6H8 ? 8Fr Auw;e&u 6 E:; 6

q6 I r E 6 =BAA9&S<>? h ;7=BACI r Au(t) = ueq(t)

I 6 8 A$:; G ?@EFA r ;7uAI r A G ? 8 g 6 I h AC= n]gK;7E:; 8 ? 8Fr ACu; 8 ; 3KI 8;53K= r E:;zACu G A

ug 6 EQ<>u;fu; r E:?B~ G A r E:;7I 8Fl AE$I q ?@;

G(s)A 8Fr E:? r ;7uHACI r A(g2Ezg>EF? ;MEFA 8 <2= r ;fAu

G(s)u = G(s)u(t) = G(s)ueq(t)e (= 9 u G ? 8F8:6 n 8 A

G(s)l 6 ED<2u 3K= r E 6 gK; 8F8 ;R K;7? 2; 8jq6 u

;xI>J 6 <2I>? r E:? 6 G(0)=I

&A K;7I G ;gK; 8F8 ;7I r A 8 <3 q ?@AI r ACuHACI r A=@;7E$;nSACI r 66q6 I r E 6 =BAA &<2? h ;x=@ACI r Ag 6G A 8 ACE;7g>E 6 ?@u; G>6 gZA= 6"+&(1 !</ ' ueq(t)≈uav(t)=G(s)u

e

% % & .'+ ##' +&(1 ! !79+' ( # ! (1 ! ACu 8 ? 8r ACu; 8QG ?@I u? q6z8qC6 u u 6G>678-G A 8 =B?@~C;7I r A 8-9 < 8 <>;7=@uHACI r A G AsR3KI2? G>6G <>E;7I r A; l ; 8 A G A G A 8 =@?@~C;7uHACI r6 r yS?@I 5$n2%;7g r <>= 6 : e> +&(1 ! !79+' ( # ! (1 !! ! ( '?l 6 ? G A3KI>? G26 g 6 E U 8 < 6z8r ; nZ (A 3KI>? 6 e| qC6 u 6 <>uw; zACI2ACE;x=@?@~C; 6&S<>A 8 Aw;7g>=B? q ;v; 6 u 6 h ?@uHACI r6vqC6 uHg>=BA r6mG268 ? 8Fr ACu;2n*? eAxeBn G ACI r E 6 A l 6 E; G ; 8 <>gZACE l q ?BA G A

"!$#&%('*)+#,.-0/% 12'43&)516#8769;:=<>.@?BADC E FHGJI>KLNMMMPO5?Q!$#&%('*)+#,.-0RS-'4#&)516T&,U769V:+<>. :BEXWYLNMMMPOZ[RS-'4#&)\/%(1]'43&)516#[7_^`abF$c;GdI>eC F+fghLiMMgjgO5W

G A 8 =B?@~C;7uAI r:6σ(x(t), t)=0

e 6 I 8 ? G ACEFA%; q = ; 8F8 A G A 8 ? 8r Auw; 8 I 6 R!=B?BI>AC;7E:A 8 ;-3KI 8 I 6qC6 I r E 6 =@AG A 8:q E:? r ;jg 6 E r yS?@I 5$n>A9&pe e|

x = f(x, t) + Γ(x, t)u , ex

6 I G Af : Rn × R+ → Rn

AΓ : Rn × R+ → Rn×m

8 6Yl <2IA 8D8 <K; h A 8 e A$:;x(t)

<>u;8F6 =@< 6G A 8r A 8 ? 8r ACu;gK;7E;

t ∈ [0, T )e #I r 6 n 6 +&(1 ! ! 79+' ( # ! (1 ! ! ! ( '?j9 <2uw;

l <>I 6 = 6q ;7=BuHAI r A?BI r A zE h AC= G A3KI>? G ;jACu &S<K; 8 A r6G>6j6 ?BI r ACE h ;7= 6[0, T )

n 6 r ? G ;j;DgK;7E r ?@EG ; G ACEF? h ; G ; r:67r ;x= d

dtσ(x(t), t)

A G A ex sn>A 9G ; G>6 g 6 E

ueq = [JΓ(x(t), t)]−1

[d

dtσ(x(t), t) − Jf(x(t), t) − ∂

∂tσ(x(t), t)

]

, e|

6 I G AJ := ∂

∂xσ(x(t), t)

A 8 AH; 8:8 <2uA'&S<>Adet[JΓ(x(t), t)] 6= 0

n ∀x∈Rnn ∀t∈ [0, T )

eH 8:8 ;A 2g2E:A 8F8 6 A 8r ZAu G A3KI>? G ;O<>u; h AC~ &<2A ; 8F6 =@< 6

x(t)9 ; 8:6 =@< r ;7uHACI r A qC6 I r BI<K;

g 6 E G A3KI>? 6 A7n g 6 E r ;7I r6 n r Au G AE:? h ; G ; 8 ACu &S<K; 8 A r6G266r ACuHg 6 e 68 ACI r ? G>6 < 8 <K;7= n6 +&(1 ! !79 ' ( # ! (1 !H9YG A3KI>? G>6 I>; 8 <2g]AE l q ?BA G A G A 8 =B?@~C;7uHACI r:6 σ(x(t), t) = 0

nACI r 6ddt

σ(x(t), t)=0ACu e se

*,+.-"+ 1,; = 351 9 9= 1 3 6 ; = 3 1

uw;; 6 E G ;zACu 8 ?Bu?B=@;xE ; 6 > 96Mq6 I r E 6 =BAI" 6 R!=@?BI2A;7EE 6 >< 8Fr6 g2E 6 FA r ; G>6 gZA= 6 u 9r6G26G Afo;7g><>I 6ih N< r u;7I )AC? r u;7I>I nZN< r u;7I K; 8 A; G26 I 6 +&(1 ! ( !2 ' # +("'0 ' ('0 ( !+4 +( (,sn%ACu 8 < 8Fr ? r <>? 6 +( ')( # sgn(·) %&S<>A< 8 <K;7=BuAI r AEFA;7=B?@~C; 6q J>; h A;xuAI r:6jG26Dq6 I r E 6 =BAI 6 >e2 q ;7E; qr ACE$ 8Fr ? q ; qC6 u<>u G A 8F8 ; 8; 6 E G ;7ACI 89 &S<>A A 8Fr 6Yl <>I G ;7uAI r ; G ; 8G ?BE:A r ; 6 < ?@I G ?@EFA r ;xuAI r AI 6 u 6G>6G A 8 =B?@~;xI r Axe 8:8 ?BunI 6 &S<>A 8 A7<>A7n 8 A(< r ?@=B?B~;;&S<>? 6r AE:u 68+&(1 ! 1 / 9 +! '-#(1 ! '?')(+*/ +!+&(1 >Y,%gK;7E:;jEFA l ACEF?@E r ;7I r:6j6 2 &S<K;7I r:6H6 e

V>Y 9 g>E 6 FA r ; G26 gK;7E;&S<>A 6 u 6G>6G A 8 =@?@~C;7I r A 6qC6 EFE;Au ;x= z<>u; h ;xE:?@A G ; G A G>6A 8 gK; 6G AQA 8r ; G>6 Asg>EFA 8:8 ;g 6 E

σ(x)=0n 6 I G A

σ : Rn → Rm9 ;x= z<2uw; l <>I 68 <K; h A7e#'*;7E;

8 A;7g>EFA 8 ACI r ;7E 678#qC6 I q AC? r6z8 g2E:?@I q ?@g>;7? 8*G A<2uw; l 6 E:u; 8 ?@uHg>=BA 8 n qC6 I 8 ? G ACE; R 8 A;g>= ;7I r ;=@?BI>A;7E e sn 6 I G A

x∈Rnnu, y∈Rm

n2A 6 J2?@gZACEFg>= ;xI 6G A 3KI2? G>6 g 6 Eσ(x)=Sx=0

e&(u; q6 I G ? 6I>A q A 8:8 EF? ;gK;xE;&S<>A 6 u 6G>6HG A 8 =@?B~;xI r A 6q6 E:E:;HACu

Sx=09 g 678Fr6

(SBp)=mnp? e A7enpg 6z8Fr:6

q6 uHg>=@A r6 ev#u 6 < r E:; 8 gK;7=@; h E; 8 n G ?@~ R 8 A&<2A 6 * # ! # ' ( jG ;u; r EF?@~ G A r E;xI 8Fl AE$CI q ?@;

G Aug>;7E;

σ8 A$:;m<>I>? l 6 EFuADAH<>I>? r EF? 6 eHd l q ?@= 8 ; r ? 8Fl ;x~CACEA 8r ; q6 I G ? 6m8 A 6 A 8r ; G26G ;

g>=@;7I r ; l 6 E G ? 8 g 6 I h AC= e 6 AI r ;7I r:6 n &S<K;xI G>6 8 ADuHA G AD;7gZAIK; 86 8 ?@IK;7= G A 8 ; G ; G ; g2= ;7I r ;y n]A 8r ; qC6 I G ? 6w8 AD;7g2=@? q ; r ;7u 9 u wu; r E:?@~

Kp =CpBp

n &S<>A G A h A r ACE r ;7u 9 uag 6z8Fr6q6 uHg>=@A r6 n? eAxeBn(; u; r EF?@~ G A r E:;7I 8Fl AE$I q ?@; G ; g>= ;xI r ; G A h A r AE zE:;7<E:A= ; r ? hz6 <2I>? l 6 E:uHAAv<>I2? r E:? 6 n 6 &<2A 9 A 8Fr < G ; G26 uw;7? 8G A r ;7=@J>; G ;7uHACI r A I 6 %;xg r <>= 6 eO!I l AC=B?@~uAI r AxnAuG ? h ACE 8 ; 8 ;7g>=B? q ;A 8 g>E r ? q ; 8 nQA 8Fr ; qC6 I G ? 6 I" 6 98 ; r ? 8Fl A? r ; A 6 g>E 6 FA r:6 G>6 >Y g 6 EEFA;7=B?@uHAI r ; 6vG A 8 ; G ; 8 A r:6 E:IK;v<>u g>E 6 >=@Auw; G ? l q ?B= ngK;7E r ? q <>=@;7E:ujACI r A 8 Aw;mg>= ;7I r ; l 6 E?BI q AE r ; U 8 <ono-<>I2JK;2no 678Fr ; ?B~;xE:E;x= G AM7z # n]A 6 euw; l 6 E:u; G A 8 A;xg>=@? q ;7E 6> g 6 EEFAC=@?BuHACI r ; 6jG A 8 ; G ;HAuWg>=@;7I r ; 8 ACu &S<>Afg 6z8r6

(CpBp)<m9 ; r E:; hz9C8G>6 < 8:6

G A 6 8 ACE h ; G26 E:A 8 n q6 I l 6 E:uHA 9 ?@I r E 6G <2~C? G26 I>;A 6 e e

% C !# &C $ &% ( !# $ %

8 ;xE r ? 6z8 G ;7E G>8 g><>EzA 6 I n2g2<>E$zA 6 Ion> G ;7E G>8 XK6z8Fr AE n G ;7E G>8 g2<>E$zA 6 I -;7g>=B? q ;xu <>u 6 8 ACE h ; G>6 Eg 6 Eu 6G>6jG A 8 =B?@~C;7I r AgK;7E:;jA 8Fr ?Buw;xE 6 A 8Fr ; G>6HG Ag>=@;7I r ; 8 u<>= r ? h ;7EF? h AC? 8&G A zE;x<wE:A= ; r ? hz6 <>I2? l 6 EFuAA(<>I>? r EF? 6 ;<3Ku G A(E:AC;7=@?B~;7E 6qC6 I r E 6 =@Ah A r6 EF? ;7=2<>I2? r E:? 6 gK;7E:; 8 A z<>?BE#<>u; r E; FA r zE:?@; G AEFA l ACECI q ? ;e##u q6 I r E; 8Fr Axn 6z8 %;7g r <>= 6z8 A G A 8r ;m"A 8 A7n G A l 6 EFuw; 8 ?Bu?B=@;xE; *; 6 6 ;7I>I 6 < "$ 2n)-J>?@AI A r ;7= e sn 8 A z<>ACu6 G A 8 AI hz6 = h ?@uHACI r6 G>6 qC6 I r E 6 =BA ; G ;7g r ; r ? hz6 g 6 Eu 6G A= 6G A EFA l ACECI q ? ; / 9+! ! ! ! ( + !# # ' ( !%+&(1 , &S<>AjI 6 < 8 ; 6 8 AE h ; G>6 E G AjA 8Fr ; G>6 n &S<>A 9D8 < 8Fr ? r < G>6 g 6 E3K= r E 6z8#G AACI r E:; G ;A G A 8 ; G ; ; 6 6 ;7I>I 6 < "$5$ sn 6 ;7I2I 6 < <2I nA 6 2e e se 8:8 ?Bun2A 8Fr ;j; 6 E G ;zAu I" 6D8 A K; 8 AC?@;DI<2u u 6G A= 6G AA 8Fr ; G>6 A 2; r6 A7n>g 6 E r ;7I r:6 n2gK;7E:A q A8 ACEuw;7? 8 IK; r <>E;7=ogK;xE; 8 ? 8r Auw; 8 ?BI q AE r678 e

W; 6 E G ;7ACu ;&S<>?%; G>67r ; G ;YgZAE:uH? r A ; G Au 6 I 8Fr E:; 6 G A g>E 6 g2E:?@A G ; G A 8DG A A 8Fr ;>?@=B?RG ; G A z= 6 K;7=(A 2g 6 I>ACI q ? ;x= n 6 &S<>A I" 6O9Yl A? r6 ACu ; 6 6 ;7I2I 6 < 5$ 2n%-J>?@AI A r ;7= e n" G ;7E G>8 g><>E$7A 6 I n*g><>EzA 6 IOA r ;7= e n" G ;7E G28 g><2E$zA 6 I se(= 9 u G ? 8:8:6 n-; 6 E G ;R 8 A r ;7I r6 6 q6 I r E 6 =BAvg 6 Eju 6G26 G A 8 =B?@~C;7I r A G A 8 ? 8r ACu; 8 =B?@I>AC;7E:A 8DG AzE:;7< EFAC=@; r ? hz6 <2I>? l 6 EFuA A <>I>? r EF? 6 U 8 <onf-<2I>JK; 6z8Fr ;7z n-<>I2JK;2nMU 8 <onM 6z8Fr ; )?@~C;7E:E:;7= G A xz 5&S<K;xI r6 G A 7E;7<OEFAC=@; r ? hz6 <>I>? l 6 E:uHA ;xE$>? r E EF? 6 U 8 <4n--<2I>JK;2nQ 6z8Fr ;

B] [)+T&/)+- ,.T'*12R&# /% 1 #) -; [I>! LW#"W $W $W

?B~;7EFE;7= G Aj7z # noU 8 <4n -<>I>JK;n 6z8Fr ; )?@~C;7E:E:;7= G AH7zz sn &S<>Aw;xI r AE:? 6 E:uHACI r A l zE:;; 6 E G ; G26 ;xgZACIK; 8 g 6 E -J>?BACI A r ;7= e ew r E;K;7=BJ 6 g>E:A=@?Bu?IK;7E -J>?@AI X < q6 I 8 ? G AE 6 < ;7gZAIK; 86vqC6 I r E 6 =@A G A 8 ? 8r Auw; 8 =B?@I>AC;7E:A 8q6 u 7E;7< E:A= ; r ? hz6 <>I>? l 6 E:uHA? 7<K;7= ;<>u 6 < G>6 ? 8 e (A h A R 8 AmEFA 8:8 ;7= r ;7E&<2A 6 qC6 I r E 6 =@A G A 8 ? 8Fr ACu; 8DG A zE:;7< EFAC= ; r ? hz6 <2I>? l 6 E:uHA;7E>? r E EF? 6 9 K; 8Fr ;xI r Amuw;x? 8Dq6 uHg>=@A 6 G>6 &<2A 6 qC6 I r E 6 =BA G A 8 ? 8Fr ACu; 8jG A 7E;7< EFAC= ; r ? h76<>I2? l 6 EFuAwAv<>I>? r E:? 6 e 6 AI r ;xI r6 n 6 A 8r < G>6 G>6 q6 I r E 6 =BA G A 8 ? 8Fr ACu; 8DG A zE;x< EFAC= ; r ? h76<>I2? l 6 EFuAjAH<>I>? r E:? 6v9 ?@uHg 6 E r ;7I r AjgK;xE; 8 AE h ?@E G Aw?BI r E 6G < 6 ; 6q ; 8:6 u;7? 8 zACE:;7=&Ag 6 Er ACEj;7g2=@? q ;A 8 g>E r ? q ; 8q6 u 6 6 qC6 I r E 6 =@A G Av<2u J>AC=B? q zg r AE 6 g><>EzA 6 I A r ;7= e sn 6q6 I r E 6 =BA G Aw<>u l 6 E:I 6 G ;7E G>8 g><>E$7A 6 I A 6qC6 I r E 6 =@A r:6 =@AE;7I r A; l ;7=BJK; 8MG A<>u; q ; G A? ; G A q ;7EFE 6z8 g>E 6 g 6z8r6 IK;HA 6 e G A 8Fr ;j"A 8 A -<>I2JK;2nZ 6z8Fr ; U 8 <7z # e

(u; G ; 8 g2E 6 g 6z8r ; 8G A 8Fr ; "A 8 A 9 6YqC6 I r E 6 =@A h A r:6 E:?@;7=Q<>I2? r E:? 6 g 6 Eu 6G AC= 6G A E:A l AsRECI q ? ;2n"u 6G>6mG A 8 =@?@~C;7I r AHAHE:A;x=@?BuHACI r ; 6vG A 8 ; G ; ("'0 ( !+2 / 9+! ! ! ! ( + ! '?')(+*/ 9+!<+&(1 0 ! (R$,g>;7E; 8 ? 8Fr ACu; 8 =B?@I>AC;7E:A 8 u<>= r ? h ;xE:? h A? 8 ?@I q ACE r:6z8 e >I>? q ;?BI l 6 E:u; 6G A8&S<>A 8 A G A h A G ? 8 g 6 E # ' '&8F6 >EFAv;Yuw; r E:?B~wU X N G ; g2= ;7I r ;

Kp

9 6q6 I>J>A q ?BuAI r6YG Aw<>uw;uw; r E:?B~

Sp

r ;7= &S<>A −KpSp

8 A$:; U(<2E ? r ~7eY 8Fr:6 E:A= ;2; 6 E:A &S<>? 8 ?Rr:6mG A G A3KI>? 6mG A 8 ?@IK;x=

KpSp > 0< 8 <K;7=BuAI r AH; 8:8 <>u? G>6 ACu 6 < r E 6z8 u 9sr6G>678 *; 6

6 ;xI>I 6 < 5$"$2n&*; 6 6 ;7I2I 6 < "$2nQ-J>?BACI\A r ;7= e sn 6 ;7I>I 6 < <>I nAsR 6 2e e se

&u A 8 &<2ACu; 8G A qC6 I r E 6 =BA G A 8 ? 8Fr ACu; 8 u<>= r ? h ;7E:? h AC? 8 g 6 Efu 6G26G A 8 =B?@~C;7I r AwAE:A;x=@? RuHACI r ; 6DG A 8 ; G ; G A 8 AI hz6 = h ? G>6z8 ;7I r ACEF? 6 E:uHACI r A G ;xE G>8 g><>EzA 6 I n>g2<>E$zA 6 IA r ;7= e n4 G ;7E G>8 g><>E$7A 6 I n ;7I 7z n)-J 6 ?&7z(I" 6 J E:A 8r E:? A 8 A SRg>= q ? r ; 8%8:6 2E:Af;juw; r E:?B~U X N G ;jg>=@;7I r ;DAf;juw; r E:?B~ G A G ? 8r E:? ><>? 6DG>6HqC6 I r E 6 =BAg 6G A 8 ACE?BI q AE r ;

Bp = Bp + ∆Bp

n 6 I G AB

p

9 ; u; r EF?@~I 6 uH?@IK;x= G A G ? 8r E:? ><>? 6 G>6vqC6 I r E 6 =@AA

∆Bp

9 ; ?@I q ACE r AC~C;2e 6 ACI r ;7I r6 n; 8:8 <>uHAsR 8 A&S<>A; uw; r E:?B~ G A G ? 8r E:? ><>? 6 G A 8 ; G ;Cp

8 A$:; q6 I>J>A q ? G ;2e(&u g2<>E$zA 6 IYA r ;7= e nZ G ;7E G>8 g><>EzA 6 I n]-J 6 ? 77 ;?@I q ACE r AC~C; 9wq ; 8 ; G ;2n ? eAxeBn

∆Bp = Bp F (t)

6 I G A ;I 6 EFuw; G ;u; r E:?B~F (t)

9 =@?BuH? r ; G ;g 6 E ‖F (t)‖ ≤ kf < 1

eA 8r E:? A 88 ?Bu?B=@;xE:A 8M8:6 >E:A∆Bp

8 6 ; 8:8 <>uH? G ; 8 ACu G ;xE G>8 g2<>E$zA 6 I n ;xI\7z se (A l 6 E:u; q6 I r E; 8r ;7I r A7n 6 R I" 6 E:A &S<>ACEH; =@?RuH? r ; 6mG A ‖∆Bp‖

n &S<>Ag 6G A 8 AE<2uw;v?@I q ACE r AC~C; G A 8Fq ; 8 ; G ;2em A l ; r:6 n"gK;7E:;m<>uw; G ; G ;u; r EF?@~

Kp

I 6 uH?@I>;7= K

p

%AM;7= 7<>u;u; r EF?@~Sp

r ;x= &S<>A −Kp Sp

8 A!:;U<>E ? r ~xnKAs? 8r A

B] [ [T&'X)51 [I LgW "W $ W $ W F= c+c :+i[cU fgF=CF+iE:+ C _gF+ C :F=[:=4c

K "!p = C #!p B #!p

7P]%$be'&.:+CE)(gO5W

-

<>u; h ?B~C?@I2JK;7IC; ;ZACE r ; Au r6 E:I 6 G AK

p

r ;7= &S<>A −KpSp

8 A!:; U<>E ? r ~xe 8F8 ; h ?B~C?RI>J>;7IC;I" 6 g>E:A q ? 8 ; 8 AE%=@?BuH? r ; G ;MI2ACu g>E:A q ? 8 ; 8 ; r ? 8Fl ;x~CAE%;7= 7<>u; q6 I G ? 6G A q ; 8 ;7uAI r:6 n; 8F8 ?Bu 8 ACI G>6 ng 6G AsR 8 A r E;K;7=@J>;7E q6 u ?BI q AE r A~; 8 u;7? 8 zACE:;7? 8 e A l ; r:6 n#g>E:A 8 <>uHAsR 8 A 8 ACEG ? l q ?B= 6 r ACEEFA 8 <>= r ; G>6 uHAC=BJ 6 EIK; ;xg>=@? q ; 6G>6 q6 I r E 6 =BA h A r6 E:? ;x="<>I>? r EF? 6 npg 6 ? 8 ; qC6 I G ?R 6 U<>E ? r ~ 9 I2A q A 8:8 E:? ;A 8 <=3 q ?BACI r AgK;7E:;D;; r E:; r ? h ? G ; G A G ; 8 <>gZACE l q ?@A G A G A 8 =@?B~;7uHACI r6 h ? G A L%;7? G ; n>U 8 <on]-<2I>JK;2nZ 6z8r ; ?B~;7EFE;7= G Axz # QAfA 6 e G A 8Fr ;H"A 8 A-se

% C !# $ &% ) !# $ %

'

A q ACI r ACuHACI r Axn G ;7E G>8 g><>EzA 6 I &g>E 6 g2< 8 AE;7u <2u q6 I r E 6 =@; G>6 EQg 6 E-EFA;7=B?@uHAIRr ; 6 G A 8 ; G ; gK;xE; 6 E:; 8Fr E:A;xuAI r:6 G A<2u u 6G A= 6 G AEFA l AE$CI q ? ; gK;7E:; 8 ? 8Fr ACu; 8 RY I" 6 R=@?BI>AC;7E:A 8 ;7g2=@? q ;7I G>6 q6 I r E 6 =BA h A r6 E:? ;x=<>I>? r E:? 6 AD<>u 6 8 ACE h ; G>6 Eg 6 Eu 6G>6 G A 8 R=B?@~C;7I r AgK;7E; A 8Fr ?Buw;xE 6 A 8Fr ; G>6 G ;Og>=@;7I r ;2e0#u q6 I r E; 8r A7n; 8 ; 6 E G ;zACI 8 G A ?BI U 8 <w7zznS?BI7z nzU 8 <on 6z8Fr ; -<>I>J>;7z 8 A7<>ACu 6MG A 8 AI hz6 = h ?@uHACI r:6MG26MqC6 I r E 6 =@A; G ;7g r ; r ? hz6 g 6 E(u 6G AC= 6HG AEFA l AE$CI q ?@; 8 Au 6 < 8:6wG A 6 8 ACE h ; G>6 E:A 8G AA 8r ; G26 n4u; 8 3K= r E 6z8G AAI r E:; G ;jA G A 8 ; G ;&<2AMI" 6 E:A qC6 I 8r E 6 Au 6 A 8Fr ; G>6HG ;jg>=@;7I r ;DA g>=@? q ? r ;7uHAI r Axe

8 ;7E r ? 678 U 8 <on 6z8Fr ; -<>I>J>;7zz Cn U 8 <on 6z8Fr ; -<>I>JK;7z # n U 8 <A r ;7= e77 q6 u >?@I>;7u ; 8 ? G>9 ?@; 8 ?BI r E 6G <>~? G ; 8 Au ?@I U 8 <Y7z7 -gK;7E:; r E:; r ;xEgZAE r <>E$K;A 8 I" 6 R=B?@I>AC;7E:A 8 Au 8 ? 8Fr ACu; 8 > qC6 u 6 (R gK;7E; 8 ? 8r ACu; 8 u<>= r ? h ;xE:? h AC? 8 =@?@I2A;7EFA 8(G AzE:;7< EFAC= ; r ? h76 <>I2? l 6 EFuAA <>I2? r E:? 6 G A Q<>I>JK;2nU 8 <on 6z8r ; ?B~;xE:E;x= G A 7z se. 8 R8 A r E;K;x=@J 6 n;&S<>? G A 8:q E:? r6 I 6 %;7g r <>= 6 ng 6z8F8 ? >?@=B? r ; ; ;7g>=B? q ; 6 G>6 q6 I r E 6 =BA; <>u;q = ; 8F8 A G A 8 ? 8Fr Auw; 8 q6 u gZAE r <>E$K;A 8 I" 6 R!=B?BI>A;7EFA 8 &S<>AI" 6m8 6 I>A q A 8:8 ;7E:?@;7uAIRr A <>I>? l 6 E:uHACuHAI r AY=B?@uH? r ; G ; 8 e. =@?BuH? r ; 6 <>I>? l 6 E:uHA G ;OgZACE r <>E$>; 6 9 ; 8:8 <2u? G ;OAu U 8 < A r ;7= e7z nU 8 <on-<>I>J>;2n 6z8r ; )?@~C;7EFE;7= G A 7z # nU 8 <on-<>I2JK;2n 6z8r ; )?B~;7EFE;7= G A 7z Cn -<>I>J>;2nU 8 <4n- 6z8r ; )?B~;7EFE;7= G A 7z MA I 6z8 %;xg r <>= 678 A G A 8 Rr ;j"A 8 A7e

(A 8Fr ;w"A 8 Axn);uw; r E:?B~ G A G ? 8Fr E:? 2<>? 6HG>6 q6 I r E 6 =BA Bp

sn4;u; r EF?@~ G A G ? 8r E:? ><>? 6HG ;8 ; G ;

Cp

A7n qC6 I 8 A9&>ACI r ACuHACI r A7n);uw; r E:?B~ G A ;7I>J 6wG A;x= r ; l E:A9&I q ?@; G ;wg2= ;7I r ; Kp

g 6G Au 8 AE*?@I q ACE r ; 8 e* g2E:?@I q ?@g>;7=?BI l 6 E:u; 6 &S<>A 8 A G A h A qC6 I>J>A q ACE # ' '78:6 >E:A-;u; r EF?@~ F= c+ fiF=EY c+ c+F$F :+F= A [I>! LW#"W $W $W

Kp

9 <>u; u; r E:?B~Sp

r ;x= &S<>A −KpSp

8 A$:; U(<>E ? r ~xe 8Fr6 EFA 8 <2= r ;YACu qC6 I G ? A 8 uAI 6z8EFA 8Fr E:? r ? h ; 8G>6 &S<>A 6 < r E; 8 < 8 <K;7=BuAI r AjAI qC6 I r E:; G ; 8 IK; =@? r ACE:; r <>E:;2n qC6 u 6 g 6 EA 2Aug2= 6 n;J>?Bg r A 8 A G A5&S<>A;j?BI q AE r A~;HI>;Hu; r EF?@~ G A G ? 8Fr EF? ><2? 6DG>6q6 I r E 6 =BA 9 E:A 8r E:? r ;Dg 6 E;7= 7<>uu; 6 E;7I r A Axe 2eBn G ;7E G>8 g><>EzA 6 I "$2nK%;7g r <>= 6 $se

&u G ;xE G>8 g><>E$7A 6 I "$2nZ%;xg r <>= 6 qC6 I 8 ? G ACE; R 8 A 6 E:; 8Fr EFA;xuAI r6HG A r E; FAsRr zE:?@; 8MG AwEFA l ACECI q ? ; K; 8 AC; G>6 I 6qC6 I r E 6 =BA h A r:6 E:?@;7=&<>I>? r E:? 6 AwAu 6 8 ACE h ; G>6 Eg 6 Eu 6G>6G A 8 =B?@~C;7I r A7e- 8F8 <>uHAsR 8 A 6qC6 I2J>A q ?@uHACI r6G ;HgK;7E r AM=B?@I>AC;7EI 6 uH?@IK;x= G ;Hg>=@;7I r ;'&<2A G A h A r ACEzE:;7< E:A= ; r ? hz6 <2I>? l 6 EFuAjAw<>I>? r EF? 6 e 6G ; 8 ; 8 ?@I q AE r A~; 88 6 ?@I 8 ACEF? G ; 8 I<>u;gZACE r <>EK;R 6 I" 6 R=@?BI2A;7E &S<>A G A h A 8 ACE q ; 8 ; G ; qC6 u ; ACI r E; G ; G ; g>= ;7I r ;YA =@?BuH? r ; G ;g 6 E l <2IA 8q6 I>J>A q ? G ; 8 &S<>A G ACgZAI G ACuW;7gZACI>; 8G ; 8 ; G ;DA G ;jAI r E; G ; G ;jg>=@;7I r ;2e

A 6 e G A G ;7E G>8 g2<>E$zA 6 I "$ q6 I 8 ? G AE; 6 q6 I r E 6 =BA h A r6 E:? ;x=*<>I>? r E:? 6G Ag>= ;xI r ; 8 Au &S<>A;MgZAE r <>EK; 6 I" 6 R!=B?BI>AC;7E#g 6G A(A 8Fr ;xE G A 8:q ; 8 ; G ; G ;MACI r E; G ; G ;fg2= ;7I r ;A 9G AgZACI G ACI r A G26 A 8r ; G>6 e 6 ACI r ;7I r6 n2I>A 8:8 ; 8 A 6j8 A; 8:8 <>uHA;&S<>A 6 A 8r ; G26jG ;g2= ;7I r ; 9uHA G ? G>6 AI" 68 A q6 I 8 ? G ACE:; 6 E; 8r E:AC;7uHACI r6G A r E:;FA r zEF? ; 8 e ;7E r ? 6G A U(<>? ;7y q6 I 8 ? G AE;<>u g>E 6 >=@Au; 8 ?@uH?@=B;7EnSAs q A r6 &S<>A 6qC6 I r E 6 =BA 9 >; 8 AC; G>6 ;xgZACIK; 8 IK;E:AC;7=@?BuHAI r ; R 6G A 8 ; G ;2n7A 8:q6 =@J2ACI G>6 R 8 A; 8 <>gZACE l q ?BA G A q JK; h A;xuAI r6G A l 6 EFuw;&S<>A 6 zE:;7<E:AC=@; r ? hz6G>68 ? 8Fr ACu; 8 A!:;D<>I>? l 6 E:uHAA<>I>? r E:? 6 e 6 ACI r ;7I r6 n>;DEFAz? 6G AA 8r ;>?B=@? G ; G A 9 =@?BuH? r ; G ;2n<>u;h A~ &S<>A 8 A(; G>67r ;<>uw;;7uHg>=@? r < G A q6 I 8r ;7I r Ag>;7E;; l <>I 6MG Au 6G <>=@; 6G26MqC6 I r E 6 =BAg 6 Eu 6G>6jG A 8 =B?@~C;7I r A7e

; 8 ; 6 E G ;zACI 8 g2E 6 g 6z8r ; 8 I 6z8 %;7g r <>= 6z8 A $ G A 8Fr ;("A 8 A-gK;xE; 6q6 I r E 6 =BA G A 8 ? 8Fr ACu; 8I" 6 R!=B?BI>AC;7E:A 8fG A zE:;7< E:A= ; r ? hz6 <2I>? l 6 EFuAH;7E$2? r E E:? 6 n r ;7I r66z8fr ACEFu 6z8 =B?@I>AC;7E:A 8 &S<K;7I r6Y6z8I" 6 R!=B?BI>AC;7E:A 8 g 6G ACu 8 ACE?BI q AE r:6z8 AD; G u? r A R 8 A5&<2ADA 8:8 A 8r ACEFu 6z8(G ACgZAI G ;7u G ; 8(h ;7EF? h AC? 8G AA 8r ; G>6 I" 6 uHA G ? G ; 8 e (= 9 u G ? 8Fr6 n>; 8 gZACE r <>E$>;A 8 g 6G Au 8 AE G A 8Fq ; 8 ; G ; 8 Au EFAC= ; 6DACI r E; G ; G ;jg2= ;7I r ;2e

% C !# C $ 'C (

A h ? G>6 ; G ? 3 q <>= G ; G A 8 g>E r ? q ; 8 ACI hz6 = h ? G ; 8 IK;uHA G ? 6HG A r:6G ; 8 ; 8h ;7EF? h AC? 8(G AfA 8r ; G>6 nl 6 E:;7u G A 8 ACI hz6 = h ? G ; 8G ? h ACE 8 ; 8 A 8r E:; r9 z? ; 8 K; 8 AC; G ; 8 Au EFA;7=B?@uHAI r ; 6G A 8 ; G ;(g>;7E;2Y h ? G A J JK;x=@?@= n"J<4n 8 ;7? n < -JK;xI 7zz $sew A 8F8 AwE:A 8 gZAC? r6 n 6 ;7E r ? 6g>? 6 I>A?@E 6G A L 6 I G ;xE:A h n>L 6 I G ;xE:A h n (6z8Fr =@A h ; r yS?@I 5$ &?@I r E 6G <>~C?B< 6 < 8F6DG A 6 8 ACERh ; G>6 EFA 8 ; 8:8 ?@I r r ? qC6z8 ;7?B=@; r J "$z I 6 >0gK;7E:;mE:A 8F6 = h ACEf<>u g>E 6 >=@Au; l <>I G ;xuAI r ;x=

G>6 >Yn 6 < 8 A$:;2nZ;HEFA;7=B?@~C; 6jG>6 / 9 +! '-#(1 ! )'?+!$# IK;jg>E r ? q ;2eQ ? G>9 ?@; q JK; h A G A L 6 I G ;xE:A h A r ;7= e "$ h ? G A r ;7u 9 u r yS?@I "$ n] r yS?BI A r ;7= e $ 96 g>E 6 FA r6 G>6q6 I r E 6 =BAgK;7E:; 8 A 6 r AE 6 u 6G>6wG A 8 =@?B~;7I r AI 6 A 8Fr ; G>6 6 8 ACE h ; G>6

x U 8 < n)A 6 n

U 8 <4n]-<2I>JK;2nZ 6z8Fr ; )?B~;7EFE;7= G A77 # nKA 6 e se

';7E;?@=@< 8Fr E:;7E&;M;7g>=@? q ; 6G A 6 8 ACE h ; G26 E:A 8 ACu >n qC6 I 8 ? G ACE:A R 8 A 6f6 8 ACE h ; G>6 E-; 8F8 ?@IRr r ? qC6 gK;7E;j;Dg>= ;7I r ; e G ; G>6 g 6 E

˙x = Apx + Bpu − Ly , e|

y = Cpx − y ,

I 6 &S<K;7= 8 A; 8:8 <>uHA 6qC6 I2J>A q ?@uHACI r6G>6z8h ;7= 6 EFA 8G ; 8 uw; r E:?B~CA 8Ap

nBp

ACp

G ;Hg>= ;7I r ;2e-u; r EF?@~ G A EFA;x=@?@ujACI r ; 6G26 AE:E 6 G A 8 ; G ; G26 6 8 AE h ; G26 E

L G A h A 8 ACEA 8FqC6 =BJ>? G ;gK;xE;

&S<>Aj; u; r EF?@~(Ap − LCp)

8 A!:;vU(<2E ? r ~7n 6 &S<>A 98 Aug>EFADg 6z8F8 h AC= 8 A 6 gK;7E Cp, Apl 6 E

6 8 ACE h h AC= ;7?@=@; r J "$z2npA A 8 >e A e eQ 8F8 ?@umn";7E:;7I r AsR 8 A5&S<>Af; G ?BI uH? q ; G>6 AE:E 6G AA 8Fr ?Buw; 6DG>6 A 8Fr ; G>6

x(t) := x(t) − x(t)n G ; G ;Dg 6 E

˙x = (Ap − LCp)x , e|

y = Cpx ,

8 A$:;; 8:8 ?BI r:67r ? q ;7uHACI r AA 8Fr h AC= e# h ;7EF?@A G ; G A(ACu &S<>A 6qC6 EFE:A 6 u 6G>6fG A 8 =@?B~;7I r A 9(G ; G ;fAur ACE:u 6z8G A

σ(x) = Sx , e $

q <:; G AE:? h ; G ;DACu E:A= ; 6 ; 6jr ACuHg 6H6 r ? G ;j;DgK;7E r ?@E G A e 9

σ = SApx + SBpu − SLCpx , e|

6 I G A 6 r AE:u 6SLCpx

G A q ;x?MA 2g 6 I>ACI q ? ;x=@uHACI r A ; ~CAE 6 e 8F8 ?Bun; qC6 I G ? 6 G A 1.2;7g2E:A 8 ACI r ; G ;wIK;wA 6 e|e gK;7E:;;HAs? 8r I q ?@; G>6 u 6G>6HG A 8 =@?B~;xI r A 8 A;7g>=B? q ;H; 6 g>E 6G < r6

SBp

nz? eAxeBnz;u; r E:?@~Sr ACu &S<>A 8 AE A 8FqC6 =BJ>? G ;g>;7E;&<2A

SBp

r ACI2JK; 1.4 +-/ !4 e* A h A R 8 AI 6xr ;7E &S<>A

σ9 <>u ACEFE 6 ;7<?B=@?@;7E &S<>Ag 6G A 8 ACE q ;7= q <>=@; G>6 gK;7E; &<>;7= &S<>AE

SA 8:qC6 =@J>? G>6 n<>u;

h A~5&S<>Ax9r:67r ;x=@uHACI r A qC6 I>J>A q ? G26 e

-

&I r 6 n 6jG A 8 =@?B~;7uHACI r6 $? G A;7= (g 6G A 8 ACE 6 r ? G>6 < r ?B=@?B~C;7I G>6 R 8 A 6 8 AE h ; G>6 E:A 8(G AMA 8Fr ; G>6; 8F8 ?BI r r ? q6z8 n?BI G Ag]AI G AI r ACuHACI r A G>6 zE:;7<jEFAC=@; r ? hz66 E:? 7?@I>;7= G ;uw; r E:?B~ G A r E;7I 8Fl ACECI q ? ; G ;g>=@;7I r ;

Gyu(s)e

A 6 g>E 6 >=BACu;m; 8 ACEfE:A 8F6 = h ? G>6ml 6 Ef;mA 8Fr ;2?@=B?@~; 6vG ;g2= ;7I r ; 8 ACu 8 AE:A &<2ACE:AE 6q ;R8 ;7uAI r:6vqC6 u_<2u_u 6G A= 6 G AjEFA l ACECI q ? ;nAI r 6

Gyu(s)I" 6 g>E:A q ? 8 ; 8 ACE G A l ; 8 ADu@I>?BuH;2e

8Fr A E:A &S<>? 8 ? r6 8 A ;xg>=@? q ; ;xgZACIK; 8 u; r E:?B~ G A r E:;7I 8l ACE$I q ?@;Gσu(s)

h ? G A X ? 7<>E; e n<>u; h AC~ &S<>A 6fG A 8 =@?B~;7uHACI r6f9 6 h AE:IK; G26 gZA= 6z8 ~CACE 6z8-G A r E:;7I 8 u? 8:8 6fG A

Gσu(s) 6 <>I>n

(6 y 67r6ih ? r y?BI e

S

8 ACE h ; G>6 E

'#=@;7I r ;-

-

?

?

?

6

y=Cpx

−

+

y

y=Cpx

xu

σ

$%

>Y

6 I r E 6 =BA g 6 E u 6G>6OG A 8 =@?B~;7I r A < r ?B=@?B~C;7I G>6 R 8 A <2u 6 8 ACE h ; G>6 E G A A 8r ; G>6 e -?BI G ? q ; 6 # % 9 ! '-#(1 ! +! # e

g 6 I r6 g>EF?@I q ?@gK;7= 9 &S<>A 6f8 ? 8Fr Au; K; 8 A; G26 I 6f6 8 AE h ; G>6 EQg 6z8F8 ? 2?@=@? r ;;MA 2? 8Fr CI q ? ; G A<>u;fuw; r E:?B~ G A r E:;7I 8l ACE$I q ? ;=@?@I2A;7E qC6 u 7E;7< EFAC=@; r ? hz6 <>I2? l 6 EFuA

n∗=1A G A l ; 8 Au@I2?@uH;

l A q J>;7I G>6 <>u = ; 6 n G AI 6 uH?@IK; G26 # 9 ! '-#(1 ! +! # sn ACu r6 EFI 6G>6 >= 6qC6 >Y qC6 I r E 6 =@A G A 8FqC6 I r BI< 6 n]?@I G AgZACI G ACI r ACuHACI r A G>6 zE;7<EFAC=@; r ? hz6G ;g>=@;7I r ;2nq6 I l 6 EFuA 9 G A 8Fr ; q ; G>6 IK; X ? 7<>E; e eY A ? 8Fr CI q ? ; G A <2u 9 I>A q A 8F8 EF? ;YgK;xE;Y;EFA;7=B?@~C; 6G>6 / 9%+! '-#(1 !8)'?+!$# r yS?@IvA r ;7= e npA 6 $2e se

*,+"+/* 8 9 3 6 1 359C8 9 6 = 1 ,6,; 1 6 356 8 8 =?9476,8 ;B9 3 = 1,8

#u L 6 I G ;xE:A h A r ;7= e "$ HA IK; 8 A 6 ;xI r AE:? 6 E 8 A ; 8:8 <2uA 6OqC6 I>J2A q ?@uHACI r:6 q6 ug2=@A Rr:6 G ; g>= ;7I r ;e ';7E; 8 A z;7E;xI r ?BE E 6 >< 8Fr A~ ; ?BI q AE r A~; 8 I>; g>=@;7I r ;2n l 6 E;xu G A 8 AI hz6 = h ?RG>6786 8 AE h ; G>6 E:A 8 g 6 Eu 6G26z8G A 8 =@?@~C;7I r A 8 = 6xr ?@I2A7n4U(A G E:? q y ? 8 ; ; "$ n ;7= q67r:r ;7y "$"$ 6 < G A%;7= r6 ;xI>J 6 #uA=B;xI 6 h A r ;7= e # n #uA=B;7I 6 h A r ;7= e Cn 8Fl ;7I G ? ;7EF? JK;7=B?@= nKJ J>;7=@?B= n>)< g><>EzA 6 I "$2nK)< g2<>E$zA 6 I se

6z8r E;K;7=BJ 6z8 ;7I r ACEF? 6 EFA 88F6 >EFA 6HqC6 I r E 6 =@A K; 8 A; G>6 ACu 6 8 AE h ; G>6 E:A 8 E 6 2< 8Fr678 n 6H6 RFA r ? h76 u;7? 8 < 8 <K;7= 9 ;A 8Fr ;2?@=B?@~; 6 ; r E; hz9C8%G ;E:A;x=@?BuHACI r ; 6G ; 8 ; G ; Axe 2eBn ;7= q67r:r ;7y 5$"$2nZ#uHAC=;7I 6ih A r ;7= e z # nZ#uHAC=;7I 6ih A r ;7= e CnZ 8Fl ;xI G ? ;xE:? J>;7=@?B= n ;7y U(<>? n>J JK;7=B?@= n2)< g><>E$7A 6 I "$2n2)< g2<>E$zA 6 I $ nuw; 86 ;7E r ? 6 J JK;7=B?@= %; 6 E G ; 6 E:; 8Fr E:AC;7uAI r:6HG A r E; FA r zE:?@; 8 e

(u_A 2Aug2= 6m8 ?@uHg>=@A 8 Aj?@=B< 8Fr E; r ? hz6 G AjA 8Fr ;>?B=@?@~C; 6 K; 8 A; G26 ACu >Y < r ?@=@?~;7I G26 R 8 A<>u 6 8 AE h ; G>6 E G A;x= r6 ;xI>J 6 +' *- * #')( -! ( ! UNM 9 ;7g>E:A 8 AI r ; G>6 ACu J JK;x=@?@= e#A 8F8 ;D; 6 E G ;zAun 6 ')( ( ! .G>6 ;7I2J 6G266 8 ACE h ; G>6 E 9 EFACg>EFA 8 ACI r ; G>6 gZAC= 6gK;xE uHA r E 6 gZA9&S<>AI 6

εAM<>uaE:A 8 <>= r ; G>6Hr @g>? qC6j9 ;HAs? 8Fr I q ?@; G Af<>u; qC6 I 8Fr ;7I r A

ε∗ >0r ;7=

&S<>A 8 A0<ε<ε∗

nAI r 68 A(g 6G A ;7E;xI r ?BE 6Ml <>I q ? 6 I>;7uHACI r:6MG>6fq6 I r E 6 =@; G>6 Ee ' 6 EA ACuHg>= 6 nq6 I 8 ? G AE:AsR 8 Af;Dg>= ;xI r ;

x1 = x2 ,

x2 = u + d(x1) , e

y = x1 .

&I r 6 n qC6 I 8 ? G AE;7I G>6 R 8 A<&S<>Ad(x1)

9 <>u;DgZAE r <2E$K; 6 n><>uaUNf g 6G AE:?@; 8 AE J JK;x=@?@=

˙x1 = x2 +1

ε(y − x1) ,

˙x2 = u +1

ε2(y − x1) .

e

A l AC? r6G ;gZAE r <>E$K; 68:6 >EFA 6 A 8r ; G26 A 8r ?@u; G>6 x1, x2

9 E:A G <>~C? G>6 I>;uHA G ? G ;&S<>Aε → +0

e 8r AfUNM0;7g>E:A 8 AI r ;Hg>E 6 >=BACu; 8 ?@I>AE:AI r A 8qC6 u 8 ACI 8 ? >?B=@? G ; G A;jE:< G>6 nK;7= 9 u G Ag 6G AEA ? >?BE 6 ?BI G A 8 A$ h AC= l AIzuHACI 6"! #&%&')(+* h ? G A(A 6 2e e I 6z88 ?@IK;x? 8#G26M6 8 AE h ; G>6 E

$

A G>6 qC6 I r E 6 = ; G>6 E 8l ;7I G ?@;7E:? JK;7=B?@= n4J JK;7=B?@= e 6 %;7g r <>= 6 G A 8r ;"A 8 A 9 G A 8 ACI hz6 = h ? G>6 <2u A 8 &<2ACu; > K; 8 A; G>6 IS<>u UNMc=B? h E:A G A "!$#&%&')(+* gK;7E:; 6q6 I r E 6 =BAg 6 Eu 6G A= 6DG AEFA l ACECI q ? ; G A 8 ? 8Fr ACu; 8 >0=B?@I>AC;7EFA 8 A?BI q AE r678 e

>$ !& '!# $

*,+)+/* 6,; =?6A 8Fr ;H"A 8 A q6 I 8 ? G AE;R 8 Ag>= ;xI r ; 8 u<>= r ? h ;7EF? h AC? 8 n 6 8 ACE h h AC? 8 A q6 I r E 6 = h A? 8G ; l 6 EFuw;

xp = Apxp + φ(xp, t) + Bpu ,

y = Cpxp , e

6 I G Axp ∈ Rn

9H6 A 8r ; G>6G ;mg2= ;7I r ;2nu ∈ Rm

9H68 ?@IK;7= G AAI r E; G ;ny ∈ Rm

9H68 ?@IK;x= G A8 ; G ;A

n≥me1gZAE r <2E$K; 6

φ : Rn × R+ → Rn9 ?@I q ACE r ;wAg 6G A 8 AE G AgZACI G AI r A G>6

A 8Fr ; G>6 eQ A 8r ; G26xp

I" 6H9 uHA G ? G>6 e 8 u; r EF?@~CA 8Ap

nBp

ACp

8 6 ?BI q AE r ; 8 e u; r EF?@~ G A r E;7I 8l ACECI q ? ; G>6j8 < 8 ? 8r ACu;j=B?@I>AC;7E 9

G(s) := Cp(sI − Ap)−1Bp .

e

*,+)+.- >= 9C8 9C8 8 1 359 6 6,; =?6 8:8 <2uA R 8 A5&S<>AM;jg>=@;7I r ; 8 ; r ? 8Fl ;7~; 88 Az<2?@I r A 8qC6 I G ? A 8 k

8 ~ACE 678G A r E:;7I 8 uH? 8:8 6DG AG(s)

r Cu g>;7E r AE:AC;7=)A 8Fr EF? r ;7uAI r AI>Az; r ? h ;2e

%G(s)

r ACu g 6z8r6Hq6 ug2=@A r:6 A 9 A 8Fr E:? r ;7uHACI r Ag>E7g>E:?@;2e

BI G ? q A G A 6 8 ACE h ;>?B=@? G ; G AνG A

G(s)9q6 I>J2A q ? G>6 e

uw; r E:?B~ ')(1 ! #+2 ξ(s) 9G ? ; 6 IK;7=>A G(s)g 6z8:8 <>? zE:;7<HE:A= ; r ? hz6h A r6 E:? ;x=K<>I2? l 6 EFuA

n∗e

3 ';xE;v; u; r E:?@~ G A ;xI>J 6vG AH;7= r ; l E:A9&I q ? ;Kp

; 8:8 <>uHAsR 8 A'&S<>A 9DqC6 I>J2A q ? G ; <>u;uw; r E:?B~

Sp ∈ Rm×mG A l 6 EFuw;&S<>A −KpSp

8 A!:;U<>E ? r ~xe

$ OgZACE r <>E$>; 6φ(xp, t)

9&q6 I r @IS<K;g 6 Eg>;7E r A 8 ACu E:A= ; 6 ;tA&= 6q ;7=@ujACI r A#?Bg 8:q J>? r ~

ACu E:A= ; 6 ;xp

e

' r ACEFu 6 I" 6 R=@?BI2A;7Eφ8 ; r ? 8l ;7~ ‖φ(xp, t)‖ ≤ kx‖xp‖ + ϕ(y, t) , ∀xp, t

n 6 I G Akx ≥ 0

9 <>uWA 8:q ;7= ;xE q6 I>J>A q ? G>6 Aϕ : Rm × R+ → R+ 9 <>u; l <>I 6qC6 I>J2A q ? G ;2n q6 I r @IS<K;

g 6 Eg>;7E r A 8 ACu E:A= ; 6 ;tA qC6 I r BI<K;DACu E:AC=@; 6 ;

ye

\J>?@g r A 8 A G A l ; 8 A-u@I>?BuH; A%;J>?Bg r A 8 A 8 6 gK; G E 6 I 6q6 I r E 6 =BA-; G ;7g r ; r ? hz6g 6 E"u 6G A= 6G A-EFA l AE$CI q ?@; 6 ;7I>I 6 < <>I n ; 6 6 ;xI>I 6 < "$2nx-J>?BACIA r ;7= e se

.J>?Bg r A 8 A g 6G A 8 AE; l E 6 <2; G ; G A l 6 E:u;w; 8 AEFA9&S<>ACEFACE;7gZAIK; 8 <>uVuw; 6 E:;7I r AgK;xE;

νn q6 u 6m9Hl AC? r6 ACu 6 ;7I>I 6 < <2I e 6 AI r ;7I r:6 n*? 8Fr:6 ; q ;7E:EFA r ;xE:?@;vI 6 ;7<R

uHACI r6vG ; 6 E G ACu G>6z8 3K= r E 6z8G AjA 8r ; G26 < r ?B=@?B~C; G>6z8 IK;vA 8Fr E:< r <>E; G>6vqC6 I r E 6 =@Ag 6 Eu 6G AC= 6G AEFA l ACECI q ? ;jAI 6 ;7<>uHACI r6jG>6 IS>uHACE 6jG AgK;7E uHA r E 6z8G>6jqC6 I r E 6 = ; G26 ECe

J>?Bg r A 8 A 8F6 >EFA;Mu; r EF?@~ ')(1 ! #+4 9(G ? 8:q < r ? G ; G A r ;x=@JK; G ;7uAI r AI 6 &S<>A 8 Az<2A U 8 <4nQ<>I>JK;2n) 6z8r ; ?B~;xE:E;x= G Axz # n4A 6 e| se 6 q ; 8:6 u 6 I 6 h ;7E:? h A= nZ;wJ2?@g r AsR8 A (A9&S<>? h ;7=BA; 6 q6 I>J>A q ?BuAI r:6 G>6 7E;7<E:A= ; r ? hz6G A

G(s)n 6 &<2A 9 < 8 <>;7=*I 6 q6 I r E 6 =BA

6 ;7I>I 6 < <>I nZA 6 e e se-0=B? r ACE; r <>E; 8:6 >EFA qC6 I r E 6 =@AM; G ;7g r ; r ? hz6H6 <> u<>= r ? h ;7EF? h AC=>< 8 <>;7=@uHACI r A(; 8:8 <2uA(;7= z<>u qC6 I>J2A q ?@uHACI r:6 g>E 9h ? 6f8F6 >EFA;Mu; r E:?B~ ')(1 ! #+4 ; 6 6 ;7I>I 6 < "$"$nx*; 6 6 ;7I>I 6 < "$2nz-J>?BACI X < znz-J2?@ACIDA r ;x= e seuw; h A~&S<>AA 8r 8 ACI G26q6 I 8 ? G AE; G>6M6Mq6 I r E 6 =BA G A 8 ? 8Fr ACu; 8 ?BI q AE r678 n;7= z<>u; 8 J>?@g r A 8 A 8g 6G Au 8 AEM?@IK; q AC? r h AC? 8 IK;mg2E r ? q ;eA 8F8 A 8 AI r ? G>6 ;mJ2?@g r A 8 A (sn r ;7u 9 u ; 8F8 <>uH? G ;g 6 E 6z8r ;2n U 8 <on!uw;x? (6 y 67r:6 h ? w77 sn"g 6G AgK;7E:A q AEu<>? r:6 E:A 8Fr EF? r ? h ;2e 6 AI r ;7IRr:6 ng 6G AsR 8 A ;xE$z<>uHACI r ;7E!&S<>A 8 ACEF? ;Og 6z8:8 h A=M;x= q ;xI;xE <>u;Ou; r EF?@~ ')(1 ! #+2 G ? ; 6 IK;7=q6 u 6 < 8:6 G Av<>u g2E 9 R qC6 ugZAI 8 ; G26 E; G A9&S<K; G>6 e\ A l ; r6 nIK; uw;x? 6 EF?@; G>6z8Hq ; 8:678 I<>u8 ACI r ? G>6 zAI 9 EF? qC6 sn 6 )Auw;e G A *; 6 6 ;7I>I 6 < "$5$ ;7E;xI r A;A 2? 8Fr CI q ? ; G A<2u.g>E 9 Rq6 uHgZACI 8 ; G>6 E

Wp(s)G A l 6 E:u; &S<>A

G(s)Wp(s)r ACI2JK;Yu; r EF?@~ ')(1 ! #+4 G ? ; 6 IK;7= h ? G A

r ;7u 9 u N? ZACI 8 n* q J ;xE r ~ X < n q J ;7E r ~ Q;7I se= 9 u G ? 8:8F6 nWp(s)

I" 6fG Ag]AI G A G>678 gK;xE uHA r E 6z8&G AG(s)

e#(u; h AC~&S<>A;Mu; r EF?@~ ')(1 ! #+4 8 A!:; qC6 I>J>A q ? G ;2nG ?@; 6 IK;7="A 8 A 6 zE;x< EFAC=@; r ? hz6wG A q ; G ;wAC=BACuHACI r:6 G A

G(s) 6 < G A

G(s)Wp(s) l 6 E q6 I>J2AsR

q ? G>6 nAI r 6ξ(s)

g 6G AE 8 ACE G A r AE:uH?@IK; G ; 8 Au &<>;7= &S<>AE qC6 I>J>A q ?@uHACI r6 g>E 9h ? 68F6 >E:A 6z8gK;xE uHA r E 6z8 G A

G(s) 6 = 6 h ? q J X ;7= se';7E:; 8 A 6 r ACE 6 7E;7<jEFAC=@; r ? hz6h A r6 E:? ;x=><2I>?R

l 6 EFuAxng 6G A R 8 A 8 Az<2?@E 6 u 9sr6G>6G A -J>?BACI A r ;x= e &S<>A;xg>=@? q ;f<2u g>E 9 R qC6 uHgZACI 8 ; G>6 EEFA 8 <2= r ;xI G>6 IS<>u zE:;7<vE:AC=@; r ? hz6h A r:6 E:?@;7=o<>I2? l 6 EFuAA? z<K;x=);

n∗=maxi n∗i e

7

(A 8Fr ;"A 8 AACuHg>E:A ;R 8 A 6qC6 I r E 6 =@A h A r6 E:?@;7=Z<>I>? r EF? 6 n 6 &S<>AE:A &S<>ACE 6q <>uHg>E:?BuAI r:6G ;J>?Bg r A 8 A AjA 8Fr ACI G A 8 ? zI>? 3 q ; r ? h ;7uHACI r Aj; q =@; 8:8 A G AHg>= ;xI r ; 8 &S<>A 8 Ag 6G A qC6 I r E 6 = ;7Ee"E;K;7=@J 6z8 ;7I r AE:? 6 E:A 8 E:A9&S<>AE:Au &<2A ;Ou; r EF?@~

KpSp

8 A$:; g 678 ? r ? h ; G A 3KI2? G ;OA ;x= z<>u; 8; 6 E G ;7ACI 8r ;7u 9 uTE:A &<2ACE:Au &S<>A

KpSp

8 A!:; 8 ?@u 9sr E:? q ; 6 ;7I2I 6 < <>I n4*; 6 6 ;xI>I 6 < "$"$2n; 6 6 ;7I>I 6 < "$ 2n7-J>?@AI X < nz-J>?BACIDA r ;7= e e* 8 ?BuA r EF? ; G A

KpSp

9 <>u;Mg>E 6 g>EF?@A G ; G AI" 6 zAI 9 EF? q ;5&S<>A(g 6G A 8 AE l ; q ?B=@ujACI r A G A 8Fr E:< G ;fg 6 E?@I q ACE r A~; 8;7E>? r E;7EF?@;7ujACI r A(gZA &S<>ACI>; 8 Au

Kp

e = 9 u G ? 8F8:6 n 8 AKpSp

l 6 EQg 6z8 ? r ? h ; G A 3KI>? G ;2nACI r 6 ? 8Fr6?Bug2=@? q ; &S<>A −KpSp

9 U(<2E ? r ~7e 6 ACI r ;7I r:6 n;YE:A q @g2E 6q ;YI" 6 9wh ACE G ; G AC?BE;2n q6 I l 6 E:uHA; '#E 6 g 6z8 ? 6 ee A h A R 8 A E:A 8F8 ;7= r ;7E%&S<>AY; J>?Bg r A 8 A jg 6G AI" 6 8 AE r E:? h ?@;7=BujACI r A8 ; r ? 8Fl AC? r ;2n g 6 ? 8 ; 8:8 <2uA R 8 A &<2A-;uw; r E:?B~

Sp

9qC6 I 8Fr ;7I r A7n7u; 8 ;uw; r E:?B~Kp

g 6G A 8 AE*?BI q AE r ;n<>u; h A~&S<>A 6M8 ? 8Fr ACu; 9 ?@I q AE r6 e (; 8 A A 8 e Ae 8 6fG ? 8:q < r ? G ; 8 ; 8&qC6 I G ? A 88:6 2E:A;mu; r EF?@~ G A z;7I>J 6G A;7= r ; l E:A & CI q ? ;gK;xE; 8 ? 8Fr ACu; 8 ;mA 8Fr E:< r <>E; h ;7E:? h A= K; 8 AC; G>678 IK; +( ')( # AYg>;7E; 8 ? 8Fr Au; 8wqC6 u ; G ;7g r ; 6 gK;7E;xu 9sr E:? q ; K; 8 AC; G>6z8 IK; l ; r6 E:; 6 G ;u; r EF?@~ G A ;7I>J 6HG AM;7= r ; l E:A &CI q ? ;2e

J>?@g r A 8 A 8:6 >E:A;fI" 6 R=@?BI>AC;7E:? G ; G A ;xE;7I r A;As? 8r I q ?@;= 6q ;x=ZA;<>I>? q ? G ; G A G ;8F6 =@< 6G A e gK;7E:;

u ≡ 0e gZACE r <>E$>; 6 I" 6 R!=B?BI>A;xE

φ9fG A 8FqC6 I2J>A q ? G ;2n]ACI r EFA r ;xI r6 n

;J2?@g r A 8 A &A 8Fr ;ZAC=BA q Au; 6 E;xI r A 8 gK;7E:;; 8 <K;I 6 E:u;2e &S<>?]I" 68 A?@uHg A(I2ACI>JS<>u;EFA 8Fr E:? 6l 6 E r A 8F6 >E:A

ϕe#I r 6 ng 6G ACE:?@; 8 A r ACE

ϕ(y) = ‖y‖2n g 6 EA ACuHg>= 6 e 8:8 ?Bumn 8 A

; G u? r A r E:; r ;7E 8 ? 8Fr ACu; 8 &S<>A-;7g2E:A 8 ACI r ;7u 6l AIzuHACI 6G AQA 8:q ;xgZA-ACu r Aug 6 3KI>? r:6 ACu uw;x=@JK;;ZAE r ;2e& 678 %;7g r <>= 6z8 A n>; 8 J>?@g r A 8 A 8 QA 8 6 EFA l 6 EFu<>= ; G ; 8 g>;7E; 6Dq ; 8:6DG A8 ? 8Fr ACu; 8 =@?BI>A;7EFA 8 n>I 6z8 &S<K;7? 8

φ9l <>I 6 ;7gZAIK; 8G>6jr Aug 6 A 9 <>I2? l 6 EFuAuAI r A=B?@uj? r ; G ;e

d?@uHg 6 E r ;7I r AMEFA 8F8 ;7= r ;7E &<2AD; 8 ?@I q AE r A~; 8 AuAp

nBp

nCp

Aφg 6G Au 8 AE G A 8Fq ; 8 ; G ; 8

G ; ACI r E; G ;2n qC6 u 6 Au *; 6 6 ;7I2I 6 < 5$ 2n-U(<>? ;7y ng><>E$7A 6 I ; h ?BA 8 n"-J>?@AI A r ;7= e se 8r ; 9 <>uw; h ;7I r ;zAu ACu E:A= ; 6 ;; 6 E G ;zACI 8 Au &<2A 8 A; 8F8 <>uHA &S<>AY; 8 ?BI q AE r AC~C; 8 A 8Fr 6 q ; 8 ; G ; 8qC6 u ; ACI r E:; G ;2n%? 8Fr6 9 n%A 8Fr 6 Asg>=B? q ? r ;xuHACI r Aq6 I q ACI r E; G ; 8 I<>u;HgZACE r <>EK; 6 G AAI r E; G ; ;7= q67r:r ;7y 5$"$2n ;7I 7z n ;xy U<>? nK G ;7E G>8 g2<>E$zA 6 I nK G ;xE G>8 g><>EzA 6 I e

*,+)+ 1 9 1 92359 9 3C; 6

u 6G A= 6 G ADE:A l ACE$I q ?@; 9DG A 3>I>? G>6 qC6 u 6 U 8 <on"-<>I>JK;n* 6z8r ; ?B~;7EFE;7= G AD77 # n)AsR 6 e

yM = WM(s) r , e

6 I G Ar, yM ∈ Rm

A7n 8 AuagZAE G ; G A zAI>ACE:;7=@? G ; G Axn

WM(s) =G ?@;

1

s + γ1

, . . . ,1

s + γm

L−1(s) , γj > 0 , (j = 1, . . . ,m) , e

L(s) = L1(s)L2(s) · · ·LN(s) , N := n∗ − 1 , e

Li(s) = (s + αi)I , αi > 0 , (i = 1, . . . , N) . e

8:8 <>uHAsR 8 A&S<>A 6 8 ?@I>;7= G AmE:A l AE$I q ?@;r(t)

9mqC6 I r BI< 6 g 6 EHgK;7E r A 8 Am<>I>? l 6 E:uHACuHACI r A=B?@uH? r ; G>6 e 8r Au 6G A= 6G AE:A l ACE$I q ?@; 9 ; G A9&S<K; G>6 ;g>=@;7I r ; 8%q6 u uw; r E:?B~ G A r E;xI 8Fl AE$CI q ?@;

G(s)G A zE;x< EFAC=@; r ? hz6 <>I2? l 6 EFuA

n∗n 6 &S<>AA &<2? h ;x=@A uw; r E:?B~ ')(1 ! #+2 8 AE G ; l 6 E:u;

ξ(s)=sn∗

Ie';7E;&S<>A%; G A 3KI2? 6G A

L(s)8 A!:; q6 ug2=@A r ;2n 8 A

n∗=1n7AI r 6

L(s)=Ie* 8:8 ?Bu

8 ACI G>6 nWM(s)

r ACu 6 uHA 8 u 6 zE:;7<mE:A= ; r ? hz6n∗ G A G(s)

A 8 AC< ;7I2J 6HG AM;7= r ; l EFA9&I q ?@; 9;Duw; r E:?B~? G ACI r ? G ; G A ? e A7en

lims→∞ ξ(s)WM(s)=Ise

';7E; 8 A< r ?@=B?B~;xEu 6G AC= 6z8G AE:A l ACECI q ? ;uw;x? 8 7ACE;x? 8 g 6G AsR 8 Ag>E 9 R!g>E 6q A 8:8 ;xE 6w8 ?BIK;7= G AEFA l ACECI q ? ;

r(t); r E; h79C8%G A<>u 3K= r E 6 A 8Fr h AC=o;7g>E 6 g>E:?@; G>6 n q6 I l 6 EFuA 98 < zACEF? G>6 Au 2; 8Fr E

L 6G28:6 I "$2npA 6 e e se

*,+)+ 9= 1 9 1,; = 351 9

6 FA r ? hz6j96 r AE; q6 I h AE$ I q ?@;; 8:8 ?BI r r ? q ; G>6 ACEFE 6HG A 8 ; G ;

e(t) := y(t) − yM(t) e $

gK;xE;D~CAE 6H6 <mgK;xE;D<>uw;DgZA &S<>ACIK; h ?@~C?BI>JK;xI; G A~CAE 6 I 6 A 8 gK; 6G>6 AE:E 6 ez

% "

8Fr A r A r69(6 E;7I>?B~; G>6Mq6 u 6f8 A z<>AxeQ 8 %;7g r <>= 6z8 f;M;xg>E:A 8 ACI r ;xu qC6 I q A? r:6z8 A)ACu; 8 8 ? q6z8 gK;7E:; 6jG A 8 ACI hz6 = h ?@uHAI r6jG ; 8 A 8r E; r9 7? ; 8G A qC6 I r E 6 =@Ag>E 6 g 6z8Fr ; 8 I 678q ;7g r <>= 678%8 A Rz<2?@I r A 8 e- q6 I r E 6 =BA h A r6 E:? ;x=o<>I>? r E:? 6D9 EFA h ? 8 ; G>6 I 6 %;7g r <>= 6 n 6 I G A 9Mq6 uHgK;7E; G>6jqC6 u6q6 I r E 6 =@A;A 8r EF< r <>E:; h ;7EF? h AC=>; 8 AC; G>6 I>; ( ')( # e %;7g r <>= 6 ;xg>E:A 8 ACI r ;M;;7g2E 6 R?@u; 6 A; 8 BI r A 8 A G A 3K= r E 6z8QG Ag>EF?@uHAC?BE; 6 E G ACu < r ?B=@?B~C; G>6z8 I>; l <>I 6DG Au 6G <>=@; 6G>6q6 I r E 6 =BA h A r6 E:? ;x=p<2I>? r E:? 6 e* gK;xE;7uHA r E:?@~C; 6MG26q6 I r E 6 =BAg 6 Eu 6G AC= 6G A(EFA l AE$I q ?@;fgK;xE;8 ? 8Fr ACu; 8 =B?@I>AC;7E:A 8 u<>= r ? h ;7E:? h A? 8-qC6 u gZAE r <2E$K; 6HG AACI r E; G ; 9 E:A h ? 8 ; G ;jI 6 %;7g r <2= 6 >nI 6 &S<K;7= G A 8r ; q ;7uDR 8 Av; 8Dq6 I G ? A 8DG A q ; 8 ;7uHACI r6 G ; g2= ;7I r ;ACu uw;x=@JK; l A q JK; G ; qC6 u 6u 6G A= 6DG AE:A l ACE$I q ? ;jA 6HG A 8 AI hz6 = h ?@uHACI r6HG ; 8 A9&S<K; A 8(G ; G ?@I u? q ; G>6 ACE:E 6 e

%;7g r <2= 6 ;xg>E:A 8 ACI r ; 6 G A 8 ACI hz6 = h ?@uHACI r6 G>6 qC6 I r E 6 =@A h A r:6 E:?@;7=<>I>? r EF? 6 gK;xE; 6q ; 8:6 u;7? 88 ?BuHg>=@A 8 n&? 8Fr:6 9 n 8 ? 8Fr ACu; 8 =B?@I>AC;7E:A 8DG A zE;x< E:A= ; r ? hz6 <2I>? l 6 EFuA Av<>I2? r E:? 6 e zAI>AE;7=B?@~; 6DG>6Hq6 I r E 6 =BAgK;7E; 8 ? 8r Auw; 8 =@?BI>A;xE:A 8G A zE:;7<mEFAC=@; r ? hz6 <>I>? l 6 E:uHA;xE$>? r E E:? 69l A? r ;fI 6 %;xg r <>= 6 8 A z<>?BI G>6 R 8 A 6G A 8 AI hz6 = h ?@uHACI r6G A U 8 < A r ;x= e 7>nU 8 <on>)?@~;xE:E;x= G A (E;7 6 QgK;7E; 6Hq ; 8F6 2De

q6 I r E 6 =BA G A<2uw; q =@; 8:8 A G A 8 ? 8Fr ACu; 8 I" 6 R!=B?BI>A;xE:A 8G A zE;7<DEFAC=@; r ? hz6 <>I2? l 6 EFuAQA<>I2? r REF? 69G A 8 ACI hz6 = h ? G>6 I 6 %;7g r <2= 6 A 9 A 8r ACI G ? G>6 gK;7E;<>uw; q = ; 8F8 A G A 8 ? 8Fr Au; 8 I" 6 R!=@?BI2A;7EFA 8G A zE:;7<vE:AC=@; r ? hz6 <>I>? l 6 E:uHA;7E>? r E EF? 6 I 6 %;7g r <>= 6 $2e

^%;xg r <>= 6 EFA h ? 8 ; 678D6 8 AE h ; G>6 E:A 8HG Av;7= r:6 z;7I>J 6 gK;7E;nAu 8 A7<>? G ;nQ;7g2E:A 8 ACI r ;7E6 G A 8 ACI hz6 = h ?@uHACI r6 G A <>u A 8 &S<>ACu; 2 K; 8 AC; G>6 I<>u UNfW=B? h E:A G A !$#&%&')(+* gK;7E:; 6q6 I r E 6 =BAg 6 Eu 6G AC= 6MG A(E:A l ACECI q ? ; G A 8 ? 8r Auw; 8 >nS=@?BI>A;xE:A 8 A?@I q AE r6z8 e# 6 3KI>;7= G>6fq ;Rg r <>= 68 6 ;7g2E:A 8 ACI r ; G>678 EFA 8 <>= r ; G>6z8 AsgZACEF?@uHACI r ;7? 8 gK;7E; q6 ug2E 6 h ;7E ;A 3 q q ?@; G>6 A 8 &S<>Auw;g>E 6 g 6z8r6 Au qC6 I G ? A 8 K; 8Fr ;7I r AE:A;x=@? 8r ; 8 e

0%;xg r <>= 6 ;7g>EFA 8 ACI r ; qC6 I q =@< 8 A 88F6 >E:A 6r E;K;x=@J 6 E:AC;7=@?B~; G26 I2A 8Fr ;D"A 8 AAg>E 6 g A;7= z<>u; 8%G ?@E:A A 8 Au &S<>AMA 8Fr ;j=@?BI>JK; G AMgZA 8 &S<>? 8 ;Hg 6G ACE 8 A z<>?@Ee

8 g><2=@? q ; A 8 E:A;x=@?@~C; G ; 8(G <>E;xI r A 6 G A 8 ACI hz6 = h ?BuAI r:6 G A 8Fr A r E;>;7=@J 6G A G>6 < r:6 E; G>68 6 E:A= ; q ? 6 IK; G ; 8 I 6 g CI G ? q AMfe

8 g>E 6ih ; 8HG A;7= z<>I 8 )Auw; 8 ;7g2E:A 8 ACI r ; G>678 ; 6 = 6 I 6 G>6 r A r:6 g>E:?BI q ?BgK;7= ZACu q6 u 6;7= z<>I 8 Auw; 8 l 6 E:;7u ?@I q =@< G>6z8 I 678 g CI G ? q A 8 LaA e1 8r6Ol 6 ? l A? r6 gK;7E:; l ; q ?B=@? r ;7Ew;6 E;7I2?@~; 6 A ; qC6 ug>EFACAI 8 6OG ; 8 g>E:?BI q ?BgK;7? 8qC6 I r EF? ><2? A 8 G A 8Fr ;O"A 8 Axn &S<>A A 8Fr 6 I 6r As r6 g>EF?@I q ?@gK;7= e

ag CI G ? q Av 8 <>uw;xE:?@~C;Y;7= z<>I 8 ; 8 gZA qsr6z8DG>6z8 u 9r:6G>6z8 IS<>u 9 EF? qC678 < r ?@=B?B~; G26z8 I>A 8r ;"A 8 AfgK;xE; 6 g>E 6 FA r6jG A qC6 I r E 6 = ; G>6 E:A 8 A 8 ?@u<2= ; 6 e

x

+

α Ω !"#

$%&' )(*+,-#./%&10324156787$9: '0;2=<2

A 8Fr ; "A 8 A; G>67r ; R 8 A ;Y=@A? G A qC6 I r E 6 =@A h A r6 EF? ;x=-<>I>? r EF? 6 +("'0 ( !+4 +&(1 (,6 EF? 7?@I>;7=BujACI r Ag>E 6 g 6z8r ;Mg 6 E N< r u;7I )AC? r uw;xI>I h ? G A r ;7u 9 u N< r u;7I 2n 6 <>I>nK r y?BI ?> ~ >I>ACE $

U = −ρ(x, t)v(x)

‖v(x)‖ , e

6 I G Ax9%6h A r6 E G AA 8Fr ; G>6 n

v(x)9 <2uw; l <>I 6h A r6 E:? ;x= G>6 A 8Fr ; G>66 < G A%gK;7E r A G A 8Fr A A7e >en

8 ; G ; Aρ(x, t)≥0

n ∀xn ∀t≥0

e l <>I 6ρ(·) n G ACI 6 uH?BIK; G ; +( !+!/ 9 # n G A h A

8 ACE 8 ?BI r A r ?B~; G ; G Awu 6G>6 ;?BI G <2~C?@E 6G A 8 =B?@~C;7uAI r:6 IK; h ;7E:?BA G ; G Av(x) = 0

e 6 E; h ;7I r A7n; 8F8 <>uHAsR 8 A<&<2A

U =08 A

v(x)=0nK;3Ku G A 8 A G A 3>I>?@E qC6 uHg>=BA r ;xuAI r A;D=@A? G A qC6 I r E 6 =@Axe

'#E 6 g 6z8 ? 6 e n*;7g>E:A 8 AI r ; G ;; 8 A z<>?BECn"A 8r ;ZA=@A q Aw; 8 g>E 6 g>E:?BA G ; G A 8 8 ? q ; 8MG AHA 8 Rr ;>?@=B? G ; G AA G A q6 I h AE$ I q ?@; G>6 q6 I r E 6 =@A h A r6 E:? ;x=-<>I2? r E:? 6 e 8:8 ; g>E 6 g 678 ? 6 E:AC=@;2; ;J>?Bg r A 8 A G A&S<>A

ρ(x, t)I<>I q ; 8 A$:; I2A; r ? hz6 no? e A7en 8 ADEFA9&S<>ACE;7gZACIK; 8 &S<>A ∃tp ≥ 0

r ;7=#&S<>Aρ(x, t)≥ 0

n ∀xn ∀t≥ tp

eM 8r6 9 r ?@=*I>; ;7I =B? 8 A G AD;7= 6 E:? r u 678 K; 8 AC; G>678 I 6 ( gK;7E; 8 A?BI q =B<>?@E 6 A l AC? r6 G26 r E:;7I 8 ? r 7E:? 6wG>6z8 3>= r E 6z8 ;xg>=@? q ; G26z8 IK; zACE:; 6 G A

ρnoEFACg>EFA 8 AI r ; G>6 gZAC= 6

8 ?@IK;7=π2(t) ∈ R

?BI r E 6G <>~? G>6 IK; G A 8 ? z<K;7= G ; G A e se 6 &S<>A 8 A z<>A7n 6z8r ACEFu 678πi(t)

8 68 ?@IK;7? 8 A g 6 I>ACI q ?@;7=@uHACI r A G A q EFA 8Fq ACI r A 8 &S<>A 8 ; r ? 8Fl ;7~ACu ; G A 8 ? 7<K;7= G ; G A

‖πi(t)‖ ≤ R exp(−λt) , ∀t ≥ 0 , e|

gK;xE;j;7= 7<>uw; 8%qC6 I 8Fr ;7I r A 8 E:A;x? 8R≥0

Aλ>0

e

. "!# % &( '?+! ! ' !/#

e(t) = AMe(t) + K [U + dU(t) + π1(t)] , e

U = −ρ(e, t)e

‖e‖ , e

&( +!AM , K∈Rm×m dU(t)

!ρ $ / ! -!79 !. ')( #'

π1(t)!

π2(t) !

! 1&( ! ( + ' # / ! (1 !5+!+ !.+ ! (1 !9$ ! −K '0 !

ρ(e, t) ≥ δ + ce‖e(t)‖ + (1 + cd)‖dU(t)‖ + π2(t) , ∀t ≥ 0 , e

&( +!ce, cd ≥ 0

*+&( #(1 ! #+!79 # #- !δ ≥ 0 =/# +&( #(1 ! # '0 ' # ! (1

∃k1, k2, λ1 >0 #' 79"! #8+! ' * # #+!

‖e(t)‖ ≤ (k1‖e(0)‖ + k2R) exp(−λ1t) e

# ' ! '0 # ∀t ≥ 0$ '0/ 8 ' !/# * # / ! (1 ! ! 1&( ! ( + ' # / ! ( ! ! ( ! 79 #(

π1(t)≡ 0!

π2(t)≡ 0$ / ' . -! δ > 0 ! (1 / 9 +! '-#(1 ! ( 1&(14 e = 0

')("' + ' # !/ # *=/ !/ 1 ("'02 ts≥0$

6 ( # uw; h AC~&S<>A −K9 U<>E ? r ~7n4As? 8r Au^u; r EF?@~A 8

P = P T > 0A

Q = QT > 0&S<>A 8 ; r ? 8Fl ;7~ACu ;jA &<>; 6G AMo;7g><>I 6 h

KT P + PK = Q . e

&I r 6 n q6 I 8 ? G ACEFAsR 8 Af; l 6 E:u;&S<K; G E r ? q ;

V (e) = eT Pe , e $

&S<>AMg 678:8 <2? G ACEF? h ; G ; r ACuHg 6 E;x= G ; G ;jg 6 E

V = −ρeT Qe

‖e‖ +eT(AT

MP +PAM

)e+2eTPK(dU +π1)

e

≤ −ρλmin(Q)‖e‖+λmax

(AT

MP +PAM

)‖e‖2+2σmax(PK)‖e‖ (‖dU‖+‖π1‖) .

e

8FqC6 =BJ>AI G>6 R 8 AM<>u; l <>I 6HG Au 6G <>=@; 6ρ&S<>A 8 ; r ? 8Fl ;C;j; G A 8 ? z<K;7= G ; G A e qC6 u

ce ≥ max

λmax(A

TMP +PAM)

λmin(Q)+ δ , 0

, cd ≥ 2σmax(PK)

λmin(Q)− 1 ,

e

6 I G AM; q6 I 8Fr ;7I r Aδ >0

A 8r ;ZA=@A q Af;ju;7EzACu G AA 8Fr ;>?B=@? G ; G A G A 8 A!:; G ;2n 6 r:9 uDR 8 A

V ≤ −λmin(Q)[δ+δ‖e‖−(2+cd)R exp(−λt)

]‖e‖ .

e

6 E;2n r:6 u;7I G>6 R 8 A; G A 8 ? 7<K;7= G ; G A G AM;i=@A? 7JR$? r ~

λmin(P ) ‖e‖2 ≤ V (e) ≤ λmax(P ) ‖e‖2 , e -

A G A 3KI>?BI G>6 R 8 A; 8(qC6 I 8Fr ;xI r A 8cQ1 =λmin(Q)/

√

λmax(P ) (>0)ncQ2 =λmin(Q)/λmax(P )

(>0)A

cD =(2+cd)λmin(Q)/√

λmin(P ) (>0)nS; G A 8 ? 7<K;7= G ; G A e *g 6G A 8 ACE&E:AA 8Fq E:? r ;

q6 u 6

V ≤ −δcQ1

√V − δ cQ2V + cDR exp(−λt)

√V .

e

A 3KI2?@I G>6r :=

√Vn 6 r:9 uDR 8 A

2r ≤ −δcQ1 − δ cQ2 r + cDR exp(−λt) . e -

6 I 8 A &>ACI r ACuAI r A7np;7g>=B? q ;7I G>6f6 )Auw;Le h ? G AA 6 Le I 6 (g CI G ? q AL,g 6G AsR 8 Aq6 I q =@<>?BE &S<>A

r(t) ≤ [r(0) + cR] exp(−λ1t) , e -

6 I G Ac > 0

9 <>u; qC6 I 8Fr ;7I r A ;xg>E 6 g2E:? ; G ;mAλ1 < min(λ, δ cQ2/2)

e(g>=@? q ;xI G>6 R 8 A; G A 8 ? Rz<>;7= G ; G A G A;i=BAC? 7JR$? r ~ e - ACu e - n 6 r9 uDR 8 A 3KI>;7=@uHACI r A; G A 8 ? 7<K;7= G ; G A e seA

δ>0ACu e n G>6 )ACu;jLe g 6G AsR 8 A q6 I q =@<>?BECn r ;7u 9 un&S<>A ∃t1≥0

r ;7= &S<>Ar(t)=0

n

∀t≥ t1n; 8:8 ?@umn 6 u 6G26G A 8 =B?@~C;7I r A(ACu

e=0qC6 uA ;fACu ;x= z<2u r Aug 6 3>I>? r:6

tsn0≤ ts≤ t1

e

& 4(')2#/ ) % . "!# % !AM = −γMI γM > 0 ! (1*. ! #. #

6 1. ' $ ( '?. +-/ ce =0( # ( =%+!</ 9 # $$

6 ( # 6 u AM =−γMInγM >0

n r ACuDR 8 A<&S<>Aλmax(A

TMP +PAM)=−2γMλmin(P )

A ; G A 8 ? 7<K;7= G ; G A e 9w8 ; r ? 8l AC? r ; q6 uce = 0

Aδ = 2γMλmin(P )/λmin(Q)

e g2E 6 h ;g 6G A 8 ACE qC6 uHg>=@A r ; G ; 8 Az<2?@I G>6 R 8 A 678 gK; 8:8F6z8G ;Dg>E 6ih ; G ;H'#E 6 g 6z8 ? q ; 6 e e

6 E 6 = E:? 6 e ;7g>E:A 8 AI r ;<>u; qC6 I G ? 6 Au &S<>A 8 Ag 6G AA 8FqC6 =@J>AEce =0

I 6 =@?BuH? r ;xI r A?BI l ACEF? 6 E G ; l <>I 6G A&u 6G <>=@; 6 e sn 6 &S<>A&g 6z8:8 ? >?B=@? r ; 8 ?@uHg>=@? 3 q ;xE); 8 <K;?@uHg>=BAuHACI r ; 6 e

d ?@uHg 6 E r ;7I r AmE:A 8F8 ;7= r ;7E%&S<>A ; J>?@g r A 8 A G A"&S<>A 6 ;7I2J 6 G AY;7= r ; l EFA9&CI q ?@;K

G>68 ? 8Fr ACu; 8 A!:; 1 '0 ' ( #/ ! (1 ! ! ( ! −K

8 A$:; U<>E ? r ~-; 8F8 <>uH? G ;YIK; '#E 6 g 6z8 ? 6 e 9<>u; q6 I G ? 6 I>A q A 8F8 EF? ; A 8 <3 q ?@ACI r AYgK;xE; ; ;7g2=@? q ; 6 G>6 q6 I r E 6 =BA h A r:6 E:?@;7=<>I2? r E:? 6 nq6 I l 6 EFuA 8 A q6 I q =B<>? l E:AI r AI>; A 6 e e0E:A=@A h I q ? ; G A 8Fr ; qC6 I G ? 6w9 ; h ;7=B? ; G ;jIK; 8A A 8 e Afe >e

(C >% % C

$ % & C # #(

_'#E 6 g 6z8 ? 6 e q6 I 8 ? G AE; 8 ? 8r ACu; 8 Y qC6 u uw; r E:?B~ G A r E;xI 8Fl AE$CI q ? ;WM(s)K

n6 I G A

WM(s)= (sI − AM)−1A −K ∈Rm×m

9 U<>E ? r ~xe(';7E:; 8 ? 8Fr Auw; 8 >n 8 AAM

l 6 EU<>E ? r ~xn A 8Fr AwE:A 8 <>= r ; G>6 g 6G A 8 ACEA 8Fr AI G ? G>6 ; r E:; hz98G>6 )Auw; G A # /#( #&%- ( ' + !! gK;7E;'&S<K;7= &<2ACE WM(s)

A 8r EF? r ;xuAI r AEFA;7=)g 6z8 ? r ? hz6 ' + 1. '0 ' ( ! !$# 2'& n<>u; h AC~&S<>A

WM(s)kk ∈R

8 ACE 2'&0gK;7E:;!&S<K;7= &<2ACEk > 0

U 8 < A r ;7= e e 8r6 9;7g2=@? q ; G>6 I 6 )ACu;e G A 8Fr ?BIK; G26 ; 8 ? 8Fr ACu; 8 > (g CI G ? q A , ev 6q ; 8:6 Yn 8 A

WM(s)∈Rm×m(s)l 6 E-<>u;u; r EF?@~ G A r E;xI 8Fl AE$CI q ? ;j2'& A −K

l 6 E%<2uw;uw; r E:?B~U(<>E ? r ~zAI 9 E:? q ;2n ACI r 6

WM(s)Kg 6G A#I" 68 AE*2'&n qC6 u 68 A#g 6G A q6 I q =@<>?BE);%gK;7E r ?@E G>6 )ACu; 2e

G A J>;7=@?B= n>g>g4eK7 z sn qC6 I l 6 E:uHA 9 ;7g>EFA 8 ACI r ; G>6 ; 8 A z<>?@Ee

! % ! ' ' ! #Z(s) ∈ Rm×m(s)

$ / ! -! 79"!det[Z(s) + ZT (−s)]

( -! # '?+! (1 ' +$#/ ! (1 ! (1 $ (1 Z(s) A 8Fr E:? r ;7uHACI r A E:AC;7=g 6z8 ? r ? h ; -! !;.-/ ! (1 ! -!

Z(s) '0 - -! # . .+!29.'. ! !/ ! (12. +! Z(s) ' ( ! !/ # !

!$# ! '0 #/ ! (1 ! ( ! * # ' ( #

Z(jω) + ZT (−jω) > 0 , ∀ω∈ R ,!

5$

=/#! #-5 :+&( ' -! #-! *+' # ' ! '0 #

Z(∞) + ZT (∞)>0 ;

%Z(∞) + ZT (∞)=0

!limω→∞ ω2[Z(jω) + ZT (−jω)]>0 ;

Z(∞)+ZT (∞)≥0

')(+* # /#-( (1 # !,! ' !/ +&( #(1 ! 1. '0 ' ( #-σ0

!ω0

#' 79 !ω2σmin[Z(jω) + ZT (−jω)]≥σ0 , ∀|ω|≥ω0 .

IK;x=@?@~C;7I G>6 R 8 Am; 8qC6 I G ? A 8G>6 )ACu; e n q6 I q =@<>? R 8 A &S<>A 8 AZ(s)

l 6 Ew2'&MnAI r 6< r ?@=B?B~;7I G>6 R 8 Af;wu; r EF?@~ G A ;7I2J 6

K = kIn qC6 u

k∈RA

k > 0n]E:A 8 <>= r ;7E &S<>AD;wu; r EF?@~ G A

r E;7I 8l ACECI q ? ; Z(s)K

r ;7u 9 u 8 ACE 2'&e' 6 E 6 < r E 6 = ; G>6 n*gK;xE;u 6z8Fr E;7E5&S<>A −K

U<>E ? r ~g 6G ACE E:A 8 <>= r ;7EMACu Z(s)K

I" 6 R2'&n qC6 I 8 ? G ACE:;R 8 Ag 6 EA 2Aug2= 6 ;ju; r E:?@~ G A r E;xI 8Fl AE$CI q ?@;

Z(s) =

1s+1

0

0 1s+2

, e

&S<>A 9 2'& qC6 u 6D8 Ag 6G A qC6 I q =B<>?@E h ACE:? 3 q ;xI G>6 R 8 A<&S<>AZ(s)

8 ; r ? 8Fl ;7~; 8qC6 I G ? A 8 ?) n ?B?)A ?@?@? e G>6 )ACu;He e# 8F8 <>uH?@I G26 R 8 A

K =

1 −10

0 1

, e $

8 ACI G>6 −KU<>E ? r ~xnKACI r 6

Z(s)K =

1s+1

−10s+1

0 1s+2

. e

A 8Fr AMAsACuHg>= 6jr ACuDR 8 A

Z(jω)K + KT ZT (−jω) =

2ω2+1

−10jω+1

−10−jω+1

4ω2+4

, e|7

&S<>A 9 <>u;uw; r E:?B~ G A 8 ?@I>;7=Q?@I G A 3>I>? G>6 gK;7E;ω = 0

n 6 &S<>A qC6 I r E:;7E:?@;; q6 I G ? 6 ?@?) G26)Auw; e n q6 I 8 A &>ACI r ACuAI r A

Z(s)KDI" 6 9 2'&MeQ 6 I q =@<>? R 8 A; gK;7E r ?@E G A 8r A qC6 I r E:;R

A 2AuHg>= 6 &S<>A #-/ ')( -!79"!#/# ';+! #( ! : ( + ' # Z(s)-! # 6 #+&( ' +!

79"! −K-! # '0 ( + ' ! (1 ! # #!79"! Z(s)K

-! # 6;$

< r E; 6 8 ACE h ; 6 ?@uHg 6 E r ;7I r A h Au G26l ; r6G A &<2A; 8 A9&S<K;A 8 e "A e "g 6G ACEF? ;7u8 ACEE:ACA 8:q E:? r ; 8qC6 u 6

U = −cee(t) − ρd(t)e

‖e‖ , e|

ρd(t) ≥ δ + (1 + cd)‖dU(t)‖ , e|z

6 I G A 8 A(AI l ; r ?@~C;<&<2A;M?BI q =B< 8 6G>6Mr ACEFu 6(ce‖e(t)‖)

IK; l <>I 6fG Au 6G <2= ; 6 e A9&S<>? Rh ;7=@A ?@I q =@< 8 6G26r ACEFu 6G AE:AC;7=@?BuHAI r ; 6 g>E 6 g 6 E q ? 6 IK;7=

(cee(t))ACu e| e r ACE:u 6

(ce‖e‖)ACu e g 6G A 8 ACEQ; G ? q ? 6 IK; G26 ; 6r AE:u 6

(AMe)ACu e n 6 &S<>AE:A 8 <>= r ;I>;Mu; r E:?@~

G A r E:;7I 8Fl AE$I q ?@; G ; G ;Hg 6 E(sI − AM + ceK)−1K

n &S<>A 9 A 8Fr h A= q6 uaE:AC;7=@?BuHACI r ; 6 g 6 E;xI>J 6 A 8Fq ;7=@;7E

U = −ke , ∀k ∈ R+ ,G A 8:G A&S<>A −K

8 A!:; U(<>E ? r ~mAce

8 A!:; 8 <3 q ?@AI r AsRuHACI r A zE;xI G A7e-d\?@uHg 6 E r ;7I r A 6 8 AE h ;xE&S<>Af; 8l <>I A 8G A r E;xI 8Fl AE$I q ?@;w2'&np< r ?B=@?B~C; G ; 8IK; 8 ; 6 E G ;zAI 8 > n r ;xu 9 u 8 6 A 8r h AC? 8-qC6 u E:AC;7=B?@uHAI r ; 6 g 6 E;7I>J 6 A 8:q ;R=@;7ECeQ 6 I q =@<2?R 8 AAI r 6 &S<>AM<>uw;Dg>E 6 g>E:?BA G ; G AACu q6 u<>u IK; 8 ; 6 E G ;zAI 8 >0Ag 6 E%u 6G>6z8-G A 8 =@?B~;xI r A 8%9 ; ! # ' '? #+! # # # !$# '0/ ! (1 # 1 * #(1 !.+$# # e

$ % C !& ! % #( &

_A 6 2e G A U 8 <on(-<2I>JK;2n 6z8Fr ; )?B~;7EFE;7= G Axz # D;7g>E:A 8 AI r ; G ? h ACE 8F6z8 )ACu; 8&S<>A 8 ACE h Au q6 u 6wl ACE:E:;7uHACI r ; 8 g>;7E; 6 G A 8 ACI hz6 = h ?BuAI r:6 G ; 8 =@A? 8 gK;xE; 6 qC6 I r E 6 =@A h A r6 EF?R;7=&<>I>? r E:? 6mG A 8 ? 8Fr ACu; 8 u<>= r ? h ;7E:? h AC? 8qC6 u zE:;7< E:A= ; r ? hz6vh A r6 E:? ;x=<2I>? l 6 EFuAH;7E$2? r E E:? 6 e 8:8 A 8 )ACu; 8 nAI<>I q ? ; G>6z8 ; 8 A z<>?BECn 8 6 7ACI>AE;7=B?@~;A 8 gK;xE; 6 q ; 8F6 u<>= r ? h ;xE:? h A= G>678)Auw; 8fqC6 EFE:A 8 g 6 I G AI r A 8 g>;7E; 8 ? 8r Auw; 8 >W;xg>E:A 8 ACI r ; G26z8 Au U 8 <OA r ;7= e h ? G A(g CI G ? q Af,se

! % % &( '?+! ! ' !/#

ε(t) = M(s) [U + dU(t) + π1(t)] , e

&( +!M(s)∈Rm×m(s) +! #--!/ )("'0/# !/ * # ! # ' ( ( !4 ' # +("' / ! ! ("'0 '

!K # #/# '+! * #(1+! # # !7 : ( + ' #$ / ! -!<79"! −K '0 ')( #

dU(t) !. ')( #' π1(t)!

π2(t) ! ! 1&( ! ( + ' # / ! (1 !<+!+ !.+ ! (1 !9$ !

U = −ρ(ε, t)ε

‖ε‖ , e|x

ρ(ε, t) ≥ δ + cε‖ε‖ + (1 + cd) ‖dU(t)‖ + cfεf + π2(t) , ∀t ≥ 0 , e|

&( +!ρ ')( # εf *+! #"!

εf = −γ0εf + ‖ε‖ , e

δ≥0 =/# +&( #(1 ! # '0 ' # ! cf γ0 cε cd

%+&( #(1 ! ( ( ! * # ' ( #-# ' # #- ! (1 ∃ c1, c2, λ1 >0

#' 79"! #!+! ' *# #+!

‖ε(t)‖ ! ‖xea(t)‖ ≤ [c1‖xea(0)‖ + c2R] exp(−λ1t) e|

# ' ! '0 # ∀t ≥ 0 &( +! xea := [xTe , εf ]

T ( !4 ! ! # +-/ !4 ' !/#!/ /# # !+ # # -! ( xe

! # +!%79 # 79"! ' !$# '-# ! # ' ' ( ! !%+! !+ ( ! +! $ . ' ( ! / ! (1 ! ( 4/ )("'0/# $ / ' . -! δ > 0 ! ( ε(t)

-! 2 ( #'?+! (1 ' +$#/ ! (1 ! (1 # . # */ !/ 1 ("'04 ts≥0

$6 ( # ? G A 8 A 6 Le|e I 6 g CI G ? q AfLe(u; h ACE 8 6 uw;x? 8 A 8 gZA q ? ;7=B?@~; G ; G26 )ACu;He 96H8 A7<>?@I r Af 6 E 6 = EF? 6 e

& 4(')2#/ ) % % ! % % -! -!579 !

M(s) = diag

1

s + γ1

, . . . ,1

s + γm

K , e|5$

( !/#$ $ (1 4 .5. ! #.< !/#$ '= ( '?.<+-/ 79 # 79 ! +( +!</ 9 # 79"!;-! # ! 79"!;# ' # # #8+! ' * # #+!

ρ(ε, t) ≥ δ + cε‖ε‖ + (1 + cd) ‖dU(t)‖ + π2(t) , ∀t ≥ 0 . e|

/ ' . -! γj =γ>0 (∀j∈1, . . . ,m) ! (1 19+! -! !.+ "! cε =0( # +( %+!

/ 9 # $ $6 ( # ? G A 8 A 6 Le|e|DI 6 g CI G ? q AfLe

8 A z<>?BI r AYAuw; 9 <>uw; h ACE 8 6 E:A h ? 8 ; G ; G>6 Auw;O Au U 8 <on-<>I>J>;2n 6z8Fr ; )?B~;7EFE;7= G A#7zz # se"';xE;g2E 6 h ;7E)A 8Fr A&Auw; l 6 ?7I2A q A 8:8 E:? 6 ; G ? q ? 6 IK;7E4;I 6ih ;E:A 8Fr E:? 6 e 8F6 >E:A%; l <>I 6G A%u 6G <>= ; 6 e\?@uHg 6 E r I q ?@; G A 8r ;E:A 8Fr EF? 6G A h ACE 8 ACE E:AC; h ;x=@?@; G ;2n<>u;h A~5&S<>AMI" 6 gK;7EFA q ?@;DI>A q A 8F8 E:?@; 6 E:? 7?@I>;7=BujACI r Axe

! % &( '?+! ! ' !/#

˙ε(t) = −αε(t) + K [U + dU(t)] , e

ε(t) = ε(t) + π1(t) + β(t) ,

%79# !/ ! # ! (1 # # #? # # # 1

ε(t) = L−1(s)K [U + dU(t)] + π1(t) + β(t) , e

&( +!L(s) = (s + α)I I # /# ' '?+! (1 '? #+! m × m ε, ε, U, dU , β, π1 ∈ Rm α > 0

−K ∈Rm×m '0 π1(t) ! 1&( ! ( + ' # / ! ( !+!+ !.+ ! (1 ! ! . ')( #' π1(t)!

β(t) # . #/ ! (1 ! +&(1)(1 . ∀t ≥ 0

$ / ! -! +&("!+ '0/ ! (12 +!/ /# #(1 !dU(t)≥‖dU(t)‖ ∀t≥0 # # # "! # dU(t) -! ( dU(t)

!dU(t)

$ !

U = −ρ(t)ε

‖ε‖ , e

ρ(t) = ρ(t) + π2(t) , e

&( +! ')( # !.+$# # π2(t) ! ! 1&( ! ( + ' # / ! (1 !<+!+ !.+ ! (1 ! ! ρ !;# ' #

ρ(t) ≥ (1 + cd)dU(t) , e

ρ(t) ≤ (1 + cd)dU(t) , e

∀t≥0 ( ! cd /#'+&( #(1 ! #+!79 # #5 # 79 ! 0<cd≤ cd <+∞ ! (1 ∃ c1, c2, c3 >0 #' 79"!. ')( #'

ε(t)!

ε(t)'/# #. 1

‖ε(t)‖ ! ‖ε(t)‖ ≤ c1‖ε(0)‖ exp(−αt) + c2R exp(−λ1t) + c3‖βt‖∞ , e

&( +!λ1 <min(α, λ)

$

6 ( # ? G A 8 A 6 Le|e I 6 g CI G ? q AfLe

% C !# ! % #( & ' % C !& 2

C "!

? h ACE 8:6z88 ? 8Fr ACu; 8(G A q6 I r E 6 =BA A 8Fr E:< r <>E; h ;xE:? h AC= > ( # ' # !; 1+ !%+&(1 r E; G ? q ? 6 IK;7? 8%8 6 K; 8 A; G>678 IK; +( ')( # x ∈ Rm

sgn(x) :=

sgn(x1)eee

sgn(xm)

, sgn(xi) :=

18 A

xi > 0,

08 A

xi = 0,

−18 A

xi < 0.

e

';7E; 8 A qC6 uHgK;7E:;7E; 8 g>E 6 g>E:?BA G ; G A 8G AMA 8r ;>?B=@? G ; G A G>6 2 q6 ua; 8 g2E 6 g>EF?@A G ; G A 8G>6n4AI<>I q ?@;R 8 A 6 8 A z<>?@I r A*A 6 E:Au; g>E 6 h ; G>6 ACu U 8 <on ; 8 ~CyS<>E:A ? q ~ L-JK;;v77z2n"A 6 E:ACu;e e

1. % (= '?+! ! ' !/#!+!.+&(1)(1

x = Ksgn(x) , K =

k11 k12

k21 k22

, x ∈ R2 . e $

(1x(t) = 0 /#. * # / ! (1 ! #- ')(14- ' +$#/ ! (1 ! ! ( ! -! !.-/ ! (1 !'-!

/#8 #- -! *+')(1 !+&( ' -! # ' ! '0 #

k12k21 =0 k11 <0!

k22 <0$

k12k21 6=0 k11

|k12|+ k22

|k21|<0 # '0/ +-/ det(K)>0

$ E:A l ACECI q ? ; X ?@=B?Bg>g 6ih "$"$2n"g>goe 2 $ ;7g>EFA 8 ACI r ; qC6 I G ? A 8 A9&S<>? h ;7=BACI r A 8 8fG>6

"A 6 E:ACu;je e&';xE; 8 ? 8Fr Auw; 8%G A 6 E G ACu ;7E$2? r E E:?@; 8 6DqC6 I2J>A q ? G ; 8 ;xgZACIK; 8q6 I G ? A 88 <3Rq ?@AI r A 8 n qC6 u 6 ; qC6 I G ? 6 ACI<2I q ?@; G ;jI 6 *A 6 E:Auw;j;K;7? 6 e

1. % % (= '?+! ! ' !/#!+!.+&(1)(1

x = Ksgn(x) , x ∈ Rm . e

! ∃P =diagp1, . . . , pm>0 # 79 ! PK <0 ! (1 x(t)=0 / #. * # / ! (1 !

#- ')(14- ' +$#/ ! (1 ! ! ( ! $

6 ( # g2E 6 h ; G ; 8 <3 q ? I q ?@; G A 8r ; q6 I G ? 6DG A 8 ACI hz6 = h ? G ;jg 6 E'*ACE 8 ? G28 yS?@? 9 ;7g>E:A R8 ACI r ; G ;jACu U 8 <mA r ;7= eK7zznK'#E 6ih ; G>6 "A 6 E:Auw;ese

d E:A=@A h ;7I r AI 67r ;xE &S<>A; 8 u; r E:?@~A 8K&S<>A 8 ; r ? 8l ;7~CAu ; q6 I G ? 6 gK;7E:;;;7g>=@? q ; 6fG>6

"A 6 E:ACu;e 8 6 U(<>E ? r ~xnKu; 8 I>Au r:6G ;Duw; r E:?B~U<>E ? r ~ 8 ; r ? 8Fl ;7~A 8r ; qC6 I G ? 6 e';7E; 6Hq ; 8:6 gK;7E r ? q <>= ;xE%ACu &S<>A 8 AM?Bug A

P =II 6 "A 6 EFACu;e|n r yS?@I n>goe

q6 I q =@<>? &<2A<>u; qC6 I G ? 6D8 <3 q ?@ACI r AgK;xE;;A 8r ;2?@=@? G ; G A G>6D8 ? 8Fr ACu; e 9 &S<>AK <0

e A h AsR 8 AfE:A 8:8 ;7= r ;7E &<2AMI" 6j8 AAs? 7A<&S<>A

K8 A!:;H<2uw;Du; r E:?@~ 8 ?Bu 9sr E:? q ;2e

' 6G AsR 8 Awu 678Fr E:;7E5&S<>A 8 ? 8Fr Auw; 8fqC6 u u; r EF?@~ G A z;7I>J 6G Aw;7= r ; l EFA9&CI q ? ;K = kij

r E:? ;xIz<>=@;7En 8 <>gZAE:? 6 E 6 <\?BI l ACEF? 6 ECn qC6 ukii < 0

n ∀i ∈ 1, . . . ,m n r ;7u 9 u 8 ; r ? 8Fl ;7~ACu ;J>?Bg r A 8 A; 8:8 <>uH? G ;I 6 "A 6 EFACu; ee.;7I =@? 8 A G A 8r A 88 ? 8Fr ACu; 8 g 6G A;7g>=B? q ;xE 6 qC6 I q AC? r6G A +&(1 ! +' ! 79+' + n &S<>A 9 u<>? r:6 < r ?@=B?B~; G26 gK;7E:; 8 ? 8Fr ACu; 8 u<2= r ? h ;7E:? h A? 8 r yS?BI 5$2nA 6 :F e se ';7E;Y;7g2E:A 8 ACI r ;7E 6YqC6 I q A? r6YG A +&(1 !+' ! 79 ' + n qC6 I 8 ? G ACE:;R 8 A 6Yq ; 8F6Au &S<>Aw;muw; r E:?B~

K9Hr EF? ;7I z<>=@;7E?BI l ACEF? 6 Eng 6 ? 86q ; 8F6G ;muw; r E:?B~ r E:?@;7Iz<2= ;xE 8 <>gZAE:? 6 E 9

;7I = 6 6 e*#I r 6 g 6G AsR 8 AA 8:q EFA h ACEQ; 8 A7<>?@I r AA9&S<K; 6MG ? l ACEFACI q ? ;7=gK;7E;;g>EF?@uHAC?BE; h ;7EF? h AC=G AA 8Fr ; G>6 k

x1 = k11sgn(x1) , t ≥ 0 , ez

&S<>A 9 ?@I G ACgZACI G ACI r A G ; 8G ACu;7? 8(h ;7E:? h AC? 8G ADA 8Fr ; G>6 eM 8:6 =@< 6 G ; A9&S<K; 6vG ? l ACE:AI q ?@;7=6 ZA G A q A |x1(t)| = |x1(0)| + k11t

n#ACI &S<K;xI r6x1(t) 6= 0

e 6 u 6 8 A ; 8:8 <2u?B<*&S<>Ak11 < 0

nq6 I q =@<>? R 8 A &S<>A

x1(t)qC6 I h ACE$7A Au r ACuHg 6 3KI>? r6

t1 = |x1(0)|/|k11| ≥ 0g>;7E; ~ACE 6 e

A &<>; 6 gK;7E:;H; G ACEF? h ; G ; G ; 8 A z<>I G ; h ;7E:? h A= G AMA 8Fr ; G>6H9G ; G ;Dg 6 E

x2 = k21sgn(x1) + k22sgn(x2) , t ≥ 0 . e

6 u 6D6Dq6 I r E 6 =BAA9&S<>? h ;7=@AI r A q6 E:EFA 8 g 6 I G AI r AM; 68 ?BIK;7=sgn(x1)

6 ZA G A q A(sgn(x1))eq =

0n ∀t≥ t1

n>g 6G AsR 8 AE:ACA 8:q E:A h ACE(A 8:8 ;HA &S<K; 6 IK; l 6 E:u;A9&S<>? h ;7=@AI r A

x2 = k22sgn(x2) , t ≥ t1 , e S

q6 I q =@<>?BI G>6 R 8 A&S<>Ax2(t)

qC6 I h AE$zAAu ;7= z<>u r Aug 6 3>I>? r:6 gK;xE;f~CAE 6 n G A l 6 E:u;f;xI = 6 ; h ;7E:? h A=

x1(t)e# 8F8 Ag>E 6q A G ?BuAI r:6 g 6G A 8 ACE;7g>=B? q ; G>6 E:A q <>E 8 ? h ;7uHACI r A 8&G ACu;7? 8h ;xE:? h AC? 8

G AvA 8r ; G26 n-gZAE:uH? r ?BI G26 qC6 I q =@<>?BE&S<>A ∃c1, c2 > 0r ;7? 8 &S<>A; I 6 E:u; G>6 h A r6 E G AmA 8Fr ; G>6

q6 uHg>=@A r6 8 ; r ? 8Fl ;x~ ‖x(t)‖ ≤ −c1t + c2‖x(0)‖ ; r9 &<2Ax = 0

8 A!:;\; r ?@I z? G26 Au ;x= z<>ur ACuHg 6 3KI>? r6 e 8:8 ?@u qC6 I q =B<>?R 8 A &S<>AY; 6 E:? 7ACu 9 <>u g 6 I r6 G AA &S<>?@= >EF? 6 z= 6 K;7=BujACI r A; 8F8 ?BI r6xr ? q ;7uHACI r AA 8r h A= e

!I l AC=B?@~uAI r Axn*I" 68 A q6 I>J>A q A I>AI>J<2uw; qC6 I G ? 6 I>A q A 8:8 EF? ;A 8 <3 q ?@AI r AwgK;7E:;;A 8 Rr ;>?@=B? G ; G A G26 8 ? 8Fr ACu; e q6 u G ?@uHACI 8 6 8 <>gZACEF? 6 Ej; G>6 ? 8 eO A l 6 EFuw; q6 I r E; 8r ;7I r Axng>E 6 hz6 <R 8 AACu L-;7? G ; 6 8 A z<>?BI r Am"A 6 E:ACu; gK;7E:; 8 ? 8Fr ACu; 8Hq6 u qC6 I r E 6 =@A h A r6 EF? ;7=<>I2? r E:? 6G A G ?BuAI 8 6 ;7E$2? r E E:?@;e

1. % (= '?+! ! ' !/#

x = K(x)x

‖x‖ , e

&( +!x ∈ Rm K : Rm → Rm×m

!det(K(x)) 6= 0, ∀x

$ / ')( -! 79"!K(x)

!

#-;+! ' ( # #- '0/ '0 # #- #! ' *+!/ ! # +! ! #5 ' !/# $ ! ( ! #- ')(14- ' +$#/ ! (1 ! ! ( ! ')( ( ! -! ! .-/ ! (1 ! -! ' !/#

z = K(z)z , z ∈ Rm , e z

! ( ! #- ')(14- ' +$#/ ! (1 ! ! ( ! ')( ( ! $

';7E r ? q <>= ;7EFuHACI r Axn 8 AK(x)

l 6 EH<>u; u; r EF?@~ q6 I 8r ;7I r A7n q6 I q =@<>? R 8 A &S<>A; 6 EF? zAu G268 ? 8Fr ACu;M e 8 ACE z= 6 K;7=BuHAI r A; 8:8 ?@I r67r ? q ;xuAI r AA 8r h A= 8 A(A 8:6 uAI r A 8 A(;u; r EF?@~G A ;7I2J 6G A;7= r ; l E:A &CI q ? ;

Kl 6 EQU(<2E ? r ~7e#d l q ?B=pACI qC6 I r E;xE-u; r EF?@~A 8

K&<2AEFA 8 <>= r ;7u

IK;YA 8Fr ;>?B=@? G ; G A ; 8F8 ?BI r r ? q ; G Am;7u >; 8 ; 8 =@A? 8G A qC6 I r E 6 =@A7n#g 6 EDA ACuHg>= 6 kK = −I

e 6AI r ;xI r:6 nJ u; r E:?@~A 8 &S<>AE:A 8 <>= r ;7u IK;A 8r ;>?B=@? G ; G A G>68 ? 8Fr ACu;Au0uw;7=BJK; l A q JK; G ;;7gZAIK; 8gK;xE;D<>u; G ; 8 =@A? 8%G A qC6 I r E 6 =@Axn>g 6 EA ACuHg>= 6 ;Duw; r E:?B~

K1 =

−1, 5 −2

1 1

, e

(a) (b)

x x

xx

0 0

2

1 1

2

'#= ;7I 678G A l ; 8 Axk ; qC6 I r E 6 =@A h A r6 EF? ;7=*<>I2? r E:? 6 A qC6 I r E 6 =@AgZA= ; l <2I 68 ?BIK;7= e uw; r E:?B~ G A ;7I>J 6G Aw;7= r ; l E:A9&I q ?@; 9

K = −IgK;7E;m;7u K; 8 ; 8

=@A? 8%G A q6 I r E 6 =@A7e

EFA 8 <2= r ;DIK;DA 8Fr ;>?B=@? G ; G A;7gZAIK; 8G>6 AxnKACu q6 I r E;7g>;7E r ? G ;2n>;Duw; r E:?B~

K2 =

−0, 5 1

−5 1

, e

EFA 8 <2= r ;IK;A 8Fr ;>?B=@? G ; G A;7gZAIK; 8G>6 2e ';7E; 8 A qC6 uHgK;xE;7E 6YG A 8 AugZAI>J 6 G ; 8 =BAC? 8fG Aq6 I r E 6 =BA7n; 8&X ? 7<>E; 8 e|Ae ;7g>E:A 8 AI r ;7u.E:A 8 <>= r ; G>678#G A 8 ?@u<>= ;A 8 qC6 u.;=@A? G A qC6 I r E 6 =@A e (A qC6 u_; =@A? G A qC6 I r E 6 =@A> e $ gK;7E;

K = K1

ADgK;xE;K = K2

eA 8F8 ; 88 ?@u<>=@;A 8 n; q6 I G ? 6 ?BI>? q ?@;7= 9

x(0) = [1, 2]Te 67r ;R 8 AwI>; 8X ? z<>E; 8 ee|;A e e &S<>A

;7u >; 8 ; 8 =BAC? 8*G A q6 I r E 6 =BAQE:A 8 <>= r ;7u IK; qC6 I h ACE CI q ? ; G>6 A 8Fr ; G>6xg>;7E;; 6 E:? zACu ACu r ACuHg 6

3KI2? r6 n G A 8:G A5&S<>A 6j8 ? 8r Auw;DAu uw;x=@JK; l A q JK; G ; 8 A!:;; 8:8 ?BI r:67r ? q ;7uHACI r AA 8r h AC= e(u; G ? l AE:ACI ;uw;7E q ;7I r AMACI r E:AfA 8:8 ; 8 =@A? 89 &S<>A 6 u 6G>6G A 8 =B?@~C;7I r AfIK;=@A?> g 6G A

6q6 E:EFACEAu &<>;7= &S<>AE 8 <>gZACE l q ?BA G A q JK; h A;7uHACI r6 ?@I G ? h ? G <K;x= xi = 0

;xI r A 8G AACI qC6 I r E:;7E;j?@I r ACE 8 A 6

x=0npAI &S<K;7I r6 &S<>AfI 6 ( 6 u 6G>6HG A 8 =B?@~C;7I r A 6q6 E:EFAf;7gZAIK; 8 I>; 6 E:? zAue

8r6H9M6 8 ACE h ; G>6 I 678 g>=@;7I 678G A l ; 8 AIK; X ? z<>E;je e';7E:;H;j=BAC?4> X ? >eKe e %I 67r ;R 8 A&S<>A

x1 =0A

x2 =08 6H8 <>gZACE l q ?@A 8(G A G A 8 =B?@~C;7uAI r:6 np; 8:8 ?Bu q6 u 6 ; 8 <>;?BI r ACE 8 A 6

x=0e

6 ACI r ;xI r6 nKgK;7E:;H;D=@A?)( X ? >e>e e|; 6 u 6G>6jG A 8 =B?@~;xI r A 9 E:A 8Fr E:? r:6 ; 6 g 6 I r:6x=0

e

x

x

1

2

Est

ado

t (s)

+

0 3 6 9 12 15 18 21 24 27 30

−4

−3

−2

−1

0

1

2

3

4

5

−5

;

x

x

Est

ado

1

2

t (s)

+−8

−6

−4

−2

0

2

4

6

8

10

12

0 6 9 12 15 18 21 24 27 303

?@u<>=@;A 8mqC6 u 6\qC6 I r E 6 =@Ah A r:6 E:?@;7=o<>I>? r E:? 6 e q6 u ;

K = K1

A K = K2

e 8 ? 8Fr ACu;vAu u;7=@JK; l A q J>;RG ; 9 ?@I 8r h AC= qC6 u ; u; r EF?@~K2

e

x

x

1

2

Est

ado

t (s)

+

0 1 2 3 4 5 6 7 8 9 10

−4

−3

−2

−1

0

1

2

3

4

5

−5

; E

stad

o x

x

1

2

t (s)

+

1 2 3 4 5 6 7 8 9 10−5

−4

−3

−2

−1

0

1

2

3

4

5

0

?Bu<>= ;A 8q6 u 6\qC6 I r E 6 =@AgZAC=@; l <>I 68 ?@I>;7= e $ qC6 u ;

K = K1

A K = K2

e 8 ? 8r ACu;vAu u;7=BJK; l A q JK;RG ; 9 ?@I 8Fr h AC= q6 u ; u; r EF?@~K1

e

% C (C % !#

%

-+<+/* 1 4 6 3?6 @,1 6 1,; @ ,1 <3 = 1 4 6 1,; @ ,1 <9 9 ; @ ,1 9 8 ; 6

';xE; 8 A ; h ;7=@?@;7Ew;OE:A=@A h I q ? ; G ; q6 I G ? 6 U<>E ? r ~ 8F6 >E:A ;Ou; r E:?@~ G A z;7I>J 6OG A ;7= r ;l EFA9&CI q ?@;ACu 8 ? 8Fr ACu; 8 &S<>A;7g2=@? q ;7u 6q6 I r E 6 =BA h A r:6 E:?@;7=<>I>? r EF? 6 n7;xg>E:A 8 ACI r ;R 8 A; 8 A z<>?BI r Ag>E 6 g 6z8 ? 6 &<2A%A 8r ;ZA=@A q A &S<>A; q6 I G ? 6 U<>E ? r ~xn; 8F8 <>uH? G ;(I>A 8r ;"A 8 A7n 9 uHACI 678 E:A 8Fr EF?Rr ? h ; G>6 &<2A-;As? CI q ? ; G A &S<>A-;u; r EF?@~ G A;xI>J 6G AQ;7= r ; l E:A9&I q ?@; 8 A$:;g 6z8 ? r ? h ; G A 3KI2? G ;; 8F8 <>uH? G ;DAu 6 ;7I>I 6 < <>I n2*; 6 6 ;7I>I 6 < "$"$2n2*; 6 6 ;xI>I 6 < "$2nK-J>?@AI1X < nK-J>?BACImA r ;x= e se

. "!# % % / ')( -!K∈Rm×m ( ! #+! ' #- #- #- # /#-! # #'=

!K >0 ! (1 −K '0 K ( ( !+ !# ' #/ ! (1 ! '0/ ' +$#$ ,5 ! (1 #(12 # # /# !+ .+$#( 5( ! #+! ' #$

!K =KT ! (1 −K '0 -! !;.-/ ! (1 ! -! K >0

$

6 ( # ';xE; 8 Afg>E 6 h ;xE;Hg>EF?@uHAC?BE;Dg>;7E r A G ;; 3KEFuw; 6 ?)snK< r ?@=B?B~;R 8 AM;HA9&S<K; 6wG Ao;7g2<>I 6 h

PK + KT P =Qn q6 u

P = In &S<>AE:A 8 <>= r ;wACu

Q=K + KTeu; h AC~&<2Aj;

l 6 EFuw;%&S<K; G E r ? q ; ; 8F8:6q ? ; G ; u; r EF?@~K9 g 6z8 ? r ? h ; G A 3KI2? G ;2n4ACI r 6 ; l 6 E:u;%&<>; G E r ? q ;

; 8F8:6q ? ; G ; u; r E:?@~Qr ;7u 9 u 8 ACE g 678 ? r ? h ; G A 3KI2? G ;2n qC6 I q =@<2?@I G>6 R 8 A&S<>A −K

9 U(<2E ? r ~7e 8 Az<>I G ;DgK;7E r A G A 8:8 ;j; 3KEFuw; 6 g 6G A 8 AEg>E 6 h ; G ;< r ?B=@?B~C;7I G>6 R 8 Ag 6 E-AsACuHg>= 6 ;u; r EF?@~

K3 =

1 10

0 1

, e

&S<>Awg 6z8:8 <>? 6z8fG>6 ? 8 ;7< r6ih ;7= 6 E:A 8 ? z<K;7? 8 ;+1ng 6 E r ;7I r:6 −K3

9 U(<2E ? r ~7e 6 AI r ;xI r:6 n#;l 6 EFuw;&S<K; G E r ? q ;j; 8:8:6q ? ; G ; Duw; r E:?B~

K3

x2

1 + 10x1x2 + x22

r ACu 8 ?@IK;x=o?@I G A 3KI>? G>6 e ;-3KE:u; 6 ?@? G A q6 E:EFA G26 l ; r:6 G A&<2A

K = KT > 08 AYA 8:6 uAI r A 8 A r6G26z8 678

;7< r6ih ;7= 6 E:A 8G AKl 6 EFACu A 8r E:? r ;xuAI r Ag 6z8 ? r ? hz6z8 n 6 &<2AMA9&S<>? h ;7=BAM; −K

8 ACEU(<>E ? r ~7e

$

';7E; 8 AM; h ;x=@? ;xE; 8qC6 I 8 A9&I q ? ; 8G ;H'#E 6 g 6z8 ? 6 enK;Duw; r E:?B~

K2×2 =

k11 k12

k21 k22

e -$

9fr:6 u; G ; qC6 u 6 A 2Aug2= 6 gK;7E:; q6 uHgK;7E;A 8 e0uw; r E:?B~ −K2×2

8 AE U(<>E ? r ~ 8 A; 8G <>; 8q6 I G ? A 8 ;>;7? 6Dl 6 E:ACu 8 ; r ? 8l AC? r ; 8 k

tr(K2×2) = k11 + k22 > 0

A

%det(K2×2) > 0 .

' 6 E 6 < r E 6 =@; G>6 nK2×2 >0

A &<2? h ;x=@A; 8 AME:A &S<>ACEFACE &S<>AK2×2 + KT

2×2 >0n 6 &S<>AMAs? zA 6

q <>uHg>E:?BuAI r6DG ; 8G <K; 8qC6 I G ? A 8 ;K;x? 6 k

tr(K2×2) = k11 + k22 > 0

A

%det(K2×2) > (k12 − k21)

2/4 (≥ 0) .

6 uHgK;7E;xI G>6 R 8 A; 8#qC6 I G ? A 8 ; q ?Bu;2n 6 8 ACE h ;R 8 A &S<>A U 9 ? G CI r ? q ; q6 I G ? 6 ' se 6 AI r ;7I r:6 n; qC6 I G ? 6 ' #g 6G A 8 ACE ZAu uw;x? 8 EFA 8r E:? r ? h ; G>6 &<2A; qC6 I G ? 6 U ng 6 ? 8; E:A 8r E:? 6 8:6 >E:A 6 G A r ACEFu?BIK;7I r A 9 uw;7? 8j8 A h AE; ACu ' e ';7E; u; r E:?B~CA 8j8 ?@u 9r E:? q ; 8 n6 = ; G>6 G ?BE:AC? r6 G ; G A 8 ? z<K;7= G ; G AmAu ' 9 IS<>= 6 nQAI r 6 ; 8HqC6 I G ? A 8 U DA 'Q 8 6A &<2? h ;x=@ACI r A 8 n qC6 I l 6 E:uHA 9G A 8 AA 8 gZAE;7E;jgK;7E r ?BE G ;H; 3KEFuw; 6 ?B?)IK;H'#E 6 g 6z8 ? 6 e|e

-+<+.- 1,;<9 /479C; = 1 <1 8:8 ;<6 8 <1,8 479C;<1,3?9C8 3 /; 6 8(= z<2I 8r E;K;7=@J 6z88F6 >EFA 6 qC6 I r E 6 =BAD; G ;xg r ; r ? hz6 G A 8 ? 8Fr ACu; 8 u<>= r ? h ;7E:? h AC? 8 K; 8 A; G26z8 Aul ; r:6 E;A 8 Axe >eBnK Mnp2 G ;Du; r EF?@~ G A ;7I>J 6DG A;7= r ; l EFA9&I q ?@;H; 8:8 <2uAu 6jq6 I>J>A Rq ?@uHACI r6 G>6z8 ')( #' ./ ! ( ! ')( + ' #' (G A 8:8 ;muw; r E:?B~ 6z8Fr ;vA r ;7= e)77 eD 6vq ; 8F6G ;wu; r EF?@~ G A ;xI>J 6 G AD;x= r ; l E:A9&I q ?@; G ;wg>= ;7I r ; 8 ACE

Kp = kij∈Rm×mn4? 8F8:6 8 ? zI>? 3 q ;

&S<>A G A h Au 8 ACE qC6 I2J>A q ? G>6z86z88 ?@IK;7? 8G26z8G A r ACE:uH?@I>;7I r A 8

∆1 = k11 , ∆2 =

∣∣∣∣∣∣

k11 k12

k21 k22

∣∣∣∣∣∣

, · · · ∆m =

∣∣∣∣∣∣∣∣∣

k11 · · · k1meee e e e eee

km1 · · · kmm

∣∣∣∣∣∣∣∣∣

, e

A &S<>AA 8:8 A 8 uHACI 6 E:A 8 I 6M8 6 I<>= 6z8 e& q ; 8:6 ACu &S<>A(;7= z<>u0uAI 6 E 8 A!:;MIS<>= 6 g 6G A 8 AE q6 IRr:6 E:IK; G>6 g 8 R!u<>= r ?Bg>=? q ;7I G>6 R 8 A

Kp

g 6 E<>u;uw; r E:?B~ G AgZACE:u< r ; 6jG A qC6 =@<>IK; 8 ; G A &S<K; G ;gK;xE;&S<>A r6G26z8%6z8 uAI 6 E:A 88 A$:;xu I" 6 I<>= 6z8 n 6 &<2Ag 6G A 8 AEg 6z8:8 h A=o<>u; h A~<&S<>A

Kp

9I" 6 R 8 ?@I7<>= ;xECe

qC6 I2J>A q ?@uHACI r6vG>6788 ?@IK;x? 8G>6z8 uAI 6 EFA 8G AKp

A9&S<>? h ;7=BAj; 8 A q6 I>J>A q AEM<>u; u; r EF?@~Sp

r ;7=#&<2A r6G>6z86z8 uHACI 6 E:A 8G AK = KpSp

8 A$:;7u^g 6z8 ? r ? hz6z8 ef 8:8F6 g 6G A 8 ACE 6 r ? G26 gZAC=@;u; r EF?@~

Sp = diag sgn(∆1), sgn(∆1∆2), . . . , sgn(∆1 . . . ∆m) . e

&I r 6 n qC6 I 8 ? G ACE:;7I G>6 R 8 A&<2A r6G>6z8678 uAI 6 E:A 8G AK8 A!:;7uTg 6z8 ? r ? h76z8 n4g 6G A R 8 A qC6 uDR

gK;xE;7EA 8Fr ;jJ>?Bg r A 8 A qC6 ua; q6 I G ? 6 U<>E ? r ~A q6 ua; qC6 I G ? 6jG A<&<2AK >0

e 6wq ; 8:6 ACu &S<>A

K = KTn 6 "A 6 EFACu; e G A 6 >=BA ;xI>?@A= "$ npg>goe z

A 8Fr ;ZAC=@A q A8&S<>A <>u; qC6 I G ? 6 I2A q A 8:8 E:? ;YA 8 <3 q ?@AI r A gK;xE;K > 0

9 &S<>A r:6G>6z86z88 A< 8uHACI 6 E:A 8 g>E:?BI q ?BgK;7? 88 A!:;7uag 6z8 ? r ? hz6z8 e0;-3KE:u; 6 ?@? G ;'#E 6 g 6z8 ? 6 e|jA 8Fr ;ZAC=BA q A&S<>A8 A

K = KTnACI r 6

K >08 AA 8F6 uHACI r A 8 A −K

l 6 E%U<>E ? r ~xe# 8F8 ?@u 8 AI G>6 n q6 I q =@<>? R 8 A&<2A; 8%r E$ 8q6 I G ? A 88 6 A &<2? h ;7=BACI r A 8 gK;7E:;ju; r E:?B~CA 8%8 ?@u 9sr E:? q ; 8 e

';7E; ;uw; r E:?B~K2×2

n; qC6 I G ? 6 G A g 6z8 ? r ? h ? G ; G A G>6z8 uAI 6 E:A 8 g>E:?BI q ?BgK;7? 8 g 6G A 8 ACEA 2g2=@? q ? r ; G ; q6 u 6 k

k11 > 0

A

%det(K2×2) > 0 .

6 uHgK;7E;xI G>6 R 8 AY; 8 G ? h ACE 8 ; 8 qC6 I G ? A 8 A 8r ;ZA=@A q ? G ; 8 gK;7E; ; uw; r E:?B~K2×2

nI 67r ;R 8 A&S<>A ; qC6 I G ? 6 Y 9 ? z<>;7= qC6 I G ? 6 Usn &S<>A 8 6 uHACI 678 E:A 8Fr EF? r ? h ; 8 G>6 &S<>A ;q6 I G ? 6 ' e ' 6 E 6 < r E 6 =@; G>6 n&; 8DqC6 I G ? A 8 U A ' 8 6 ? G CI r ? q ; 8 u; 8G ? l ACE:AuG A se

-+<+ 1 4 6 3?6 @,1 9 1,; @A"9C8 8 1 3?9 6 476>= 3 <9 ,6,;<1 9 6 = 63?9 C; 6

';xE; 8 A qC6 uHg>=@A r ;xE; q6 ug>;7E; 6 G ; 8 &S<K; r E 6 qC6 I G ? A 8w8:6 >EFAY; u; r E:?@~ G A ;7I2J 6 G A;7= r ; l EFA9&CI q ? ; qC6 I 8 ? G ACE:; G ; 8 I>A 8r A q ;7g r <>= 6 n; ;ZAC=@;e ;xg>E:A 8 ACI r ;;x= z<>I 8 E:A 8 <>= r ; G>6z8

6 r ? G26z8fq6 u uw; r E:?B~CA 8 ?BI q AE r ; 8 e' 6 E 8 ?Bug2=@? q ? G ; G A7n q6 I 8 ? G AE;R 8 A ;xgZACIK; 8 <2ucgK;xE uHA r E 6?BI q AE r6

α∈Rse# u; r E:?@~ qC6 I 8 ? G ACE; G ; 9G ; G ;jgZAC= 6 g>E 6G < r:6

K =KpSp

e

[<>; G E 6 q6 ug>;7E; r ? hz6G A u; r E:?B~CA 8G A z;7I>J 6 G Av;x= r ; l E:A9&I q ?@;Kp

qC6 u<2u g>;7E uHA r E 6 ?BI q AE r:6

αe d ?@I G ? q ; G ;;u; r EF?@~ G A g2E 9 R qC6 uHgZACI 8 ; 6

Sp; G>67r ; G ;DgK;7E:; q ; G ;Kp

eQ 6 uHgK;7E:;7uDR 8 A; 8l ;7?2; 8G A h ;7= 6 E:A 8G>6 gK;xE uHA r E 6αg>;7E;&S<>Af;ju; r E:?B~

K =KpSp

; r ACI G ;H; 8qC6 I G ? A 8 A 8 gZA q ? 3 q ; G ; 8 e

Kp Sp K >0 −K "!$# %'& (*) ∆1, ∆2 >0 +-, $.

/ 0

1 2α

−2 α

1 −2

2 1

α>0 α>0 α>0 α>0

/ 0

1 2α2

−2 α

I 0, 525<α<1, 490

−1<α<−0, 25

ou

α>0

α<−0, 25

ou

α>0

−1<α<−0, 25

ou

α>0

/ 0

α −2

2α2 1

I 0, 525<α<1, 490

−1<α<−0, 25

ou

α>0

α>0

−1<α<−0, 25

ou

α>0

/ 10

α −2

2α2 1

0 1

−1 0

α<−0, 333

ou

α>0, 2

α<−0, 25

ou

α>0

α<−0, 25

ou

α>0

α<−0, 25

ou

α>0

/ 10

α −1

4 −α2

I @α 0<α<1 0<α<1, 587 0<α<1, 587

/ 1 0

cos(α) sen(α)

−sen(α) cos(α)

I −π

2<α< π

2−

π

2<α< π

2−

π

2<α< π

2−

π

2<α< π

2

.324 5 +-, % 6 # % 6 # 798:;( % ) −K;(* '<=> > '? % >7 # (*@ % 9A'(*B C D 9E( % C 7F9G( %# ( #%

1; % ?1ECHB > :< # 5>' CH6 # I5>'(*BC J@KH % 8:DLNM OM

';7E; 6Mq ; 8:6 ? < r ?@=B?B~ 6 <R 8 A<2uw;u; r E:?@~Sp

; G A9&S<K; G ;gK;7E;;&S<>AKpSp =(KpSp)

Te#uw;

h A~A 8Fr ;ZA=@A q ? G ;j; 8 ?@uHA r EF? ; G ;u; r EF?@~xn2; 8 &S<K; r E 6jq6 I G ? A 8%r Cu E:A &<2? 8 ? r678 ? G I r ? qC678 n 6 <8 A$:;2n 6 gK;7E uA r E 6 ?BI q AE r6jG A h AfA 8Fr ;7EIK; l ;7? 2;

α>0e

';7E; ; 8 uw; r E:?B~CA 8Kp

q6 I 8 ? G ACE:; G ; 8 ACu ?B?) h ? n g 6G AsR 8 Avu 6z8Fr E;7E &<2A@Sp ∈ R2×2

n

det(Sp) 6= 0n r ;7= &S<>A

KpSp = (KpSp)Tn ∀α ∈ R

ev 8Fr ; 9 <>uw;vE:A 8Fr EF? 6 ?@uHg 6 E r ;7I r Ajg 6 ? 8A q =@<2? 678 ;x= 6 E:? r u 678HG A q6 I r E 6 =BA &S<>AAs? 7ACu ; 8 ?@uHA r E:?@; G ; uw; r E:?B~ G A ;7I>J 6 G A;7= r ;l EFA9&CI q ?@;HgK;xE; ;7E;xI r ?BE; 88 <K; 8 g>E 6 g2E:?@A G ; G A 8 e

';7E; 6z8q ; 8:678 ?@?)A ?@?B? noI 6xr ;R 8 A&<2A; qC6 I G ? 6K > 0

E:A 8Fr EF?@I7A 6 gK;7E uA r E 6α;

l ;7? 2; 8 =@?BuH? r ; G ; 8 A u<>? r:6 A 8r EFAC? r ; 8 e ';xE; 6Yq ; 8F6 h fI 6 A ? 8Fr Aα ∈ R

&S<>A EFA 8 <2= r A AuK >0

e&' 6 E 6 < r E 6 = ; G26 n2gK;7E:;H; 8G Auw;x? 8%qC6 I G ? A 8 A qC6 I 8 ? G ACE:;7I G>6 R 8 A 6z8q ; 8F6z8 ?@? Q; ? h n; 8l ;7? 2; 8 ; G u? 8:8 h A? 8 g>;7E;

α8 6 ?B=@?Buj? r ; G ; 8 ; 6 uAI 6z8 <>I2?@= ; r ACE:;7=BujACI r A7e&';7E; 6Hq ; 8:6 ?@?

; qC6 I G ? 6YG A uHACI 6 E:A 8 g2E:?@I q ?BgK;7? 8 g 6z8 ? r ? hz678f8 Y?@uHg A&S<>A 6 g>;7E uHA r E 6 A 8Fr A$:; l 6 E; G ;l ;7? 2; −0, 25≤α≤0

nKACI &S<K;7I r6 &S<>Af; qC6 I G ? 6 U<>E ? r ~AM; q6 I G ? 6 2 r ;7u 9 u I" 6; G u? r Aua; l ;7? 2;

α≤−1e';7E; 6q ; 8:6 ?@?B? -; qC6 I G ? 6G AMuAI 6 E:A 8 g 678 ? r ? hz6z8 ?Bug A;&<2A

6 gK;7E uA r E 6 A 8r A$:;jIK; l ;7? >;α>0

n>ACI &S<K;7I r6 &S<>Af; 8qC6 I G ? A 8 U<>E ? r ~A> r ;xu 9 u

; G u? r AuW; l ;7? 2; −1<α<−0, 25n 6 &S<>A 9 <2uw; h ;7I r ;zAu ZACuVu 6G A 8r ; 8 A q6 uHgK;7E; G ;

; 678 E:A &S<>? 8 ? r678 EFAC=@; r ? hz6z8 ; 6z8 uHACI 6 E:A 8 g>E:?BI q ?BgK;7? 8 g 6z8 ? r ? hz6z8 e

q ; 8F6 ? h qC6 I 8 ? G ACE:;j;DuHA 8 uw;u; r E:?@~(?@I q ACE r ;Kp

< 8 ; G ;DI 6Dq ; 8F6 ?@?B? nu; 8 ; G>67r ;R 8 A<>u; u; r E:?B~

Sp

&S<>AgZAE:u< r ; ; 8Hq6 =@<>I>; 8jG AKp

Am?BI h AE r A 6 8 ?@IK;x= G ; 8 A7<>I G ; qC6 =B<>IK;2e 8:8 ?Bu r:6G ; 8 ; 8(qC6 I G ? A 8 E:A 8 <>= r ;7uaIK;HAs q =B< 8 6wG A l ;7? >; 8 K; 8r ;7I r AfA 8r E:A? r ; 8(G A h ;x= 6 EFA 8G A

αe* 8 EFA 8Fr E:? A 8 ?Bug 678Fr ; 8 gZA= ; q6 I G ? 6 U<>E ? r ~xn7g]A= ; qC6 I G ? 6 > A-gZAC=@; qC6 I G ? 6

G AjuAI 6 EFA 8 g>E:?BI q ?BgK;7? 8 g 678 ? r ? hz6788 6 ? G I r ? q ; 8 AI r E:A 8 ? eH 6 AI r ;xI r6 n*; q6 I G ? 6K > 0

?Bug A=@?BuH? r A 8 gK;7E;α<>u g 6 < qC6 u;7? 8 EFA 8Fr E:? r ? hz678G26 &S<>AM; &<2AC=@A 8 ?@uHg 6z8Fr:6z8 gZA= ; 8G ACu;7? 8

q6 I G ? A 8 e

A l 6 E qC6 I 8 ? G AE; G26 &<2A −KG A h ; 8 ; r ? 8Fl ;x~CACE; 8qC6 I G ? A 8 I>A q A 8F8 E:?@; 8 A 8 <3 q ?BACI r A 8

gK;xE; ; A 8Fr ;>?@=B? G ; G A G>6 qC6 I r E 6 =@A A 8r E:< r <>E:; h ;7E:? h A= K; 8 A; G>6 IK; l <>I 6 8 ?@I>;7= h ? G A"A 6 E:ACu;me snoI 67r ;R 8 A&<2AH; 8l ;7?2; 8 ; G uH? 8F8 h AC? 8 gK;7E:;

α8 6 ? G CI r ? q ; 8 8l ;7?2; 8 gK;xE;v;

q6 I G ? 6 U(<2E ? r ~gK;xE; 678fq ; 8F6z8 ? ? h A h ?)sev 6 ACI r ;7I r6 n 6q ; 8:6 h E:A 8 <>= r ;IS<>u;l ;7? 2;Hu;7? 8 ;7uHg>= ; G A

αgK;7E:;w; q6 I G ? 6 > G>6 &S<>A; l ;x?2;HgK;7E:;w; q6 I G ? 6 U(<2E ? r ~7n

q6 ?@I q ? G ?@I G>6qC6 u ; l ;7? >; G AαACu &S<>Af;xu 6786z8 uHACI 6 EFA 88 6 g 6z8 ? r ? hz6z8 e

u; r E:?B~ G AE 6xr ; 6 I 6jq ; 8F6 h ?) 9 K; 8Fr ;7I r A?Bug 6 E r ;7I r A(ACu q6 I r E 6 =BAg 6 E 8 ACE hz6 h ? 8 6 6z8Fr ;vA r ;x= e)xz e 67r ;R 8 A&S<>A r:6G ; 8 ; 8qC6 I G ? A 88 6v8 ; r ? 8Fl AC? r ; 8 gK;7E:; <>u; E 67r ; 6G AM; r:9G>6 ? 8 &S<K; G E:;7I r A 8 e

6 I q =@<>? R 8 AI2A 8:8 ; 8Mq6 ug>;7E; A 8 &S<>A # !.+ # #+!79 # # # /# ' Sp + 1+ ' # # # ! ' #- ! ' -! - ')( + ! ! -#-9$ A As? 8Fr ?BE<>u;uw; r E:?B~ Sp

322;"&S<>A 8 ?BuA r EF?@~CAw;u; r EF?@~

(KpSp)gK;7E:;'&<>;7? 8 &<2ACE h ;7= 6 EFA 8G>6z8 gK;xE uHA r E 6z8 ?BI q AE r:6z8 nKACI r 6 ; 8 &<>; r E 6qC6 I G ?R

A 88 AE 6 A9&S<>? h ;7=@AI r A 8 eD A l ; r6 ; q6 I G ? 6vG A'&S<>AK8 A!:;vg 6z8 ? r ? h ; G A 3KI>? G ; 9 ; u;7? 8

EFA 8Fr E:? r ? h ; G A r:6G ; 8 e ' 6 E 9 umnQI 6 8 Amg 6G Am; 3>E:u;7EI 6 q ; 8:6 zAE;7= &S<>A;7= z<>u; G ; 8Dr E 8q6 I G ? A 8 U(<>E ? r ~xn-> 6 < 8 ?@I>;7? 8jG>6z8 uHACI 6 EFA 8 g>EF?@I q ?@gK;x? 8 8 A$:; ; uHAI 6z8 EFA 8Fr E:? r ? h ;2ng 6 ? 8 ; 8 E:A 8Fr E:? A 8-G ACgZACI G ACu G ; q =@; 8:8 A G ;fuw; r E:?B~?@I q ACE r ;

Kp

q6 I 8 ? G AE; G ;fA G ;u; r EF?@~Sp

A 8:qC6 =@J>? G ;2e

S

% % C !# ! % ( & $ 2

&!& " ( (

&S<>? 8 A G ? 8Fq < r A ; qC6 I>As 6 AI r E:A 6 aA ; ! # ' ' # # # # ' ( #YG A LQEFI>A 8 ?B=@=BAu 8 "$72np!= q J>u;7I>I AI 8 z qC6 I l 6 E:uHA l 6 ? l AC? r6 Au U 8 <ono-<>I2JK;2no 6z8r ; )?B~;7EFE;7= G A7zz # nKA 6 >e se 6 I 8 ? G AE; R 8 AM;jg2= ;7I r ;

xp = Apxp + Bpu ,

y = Cpxp , e

6 I G Axp ∈ Rn

Ay, u ∈ Rm

n &S<>A G A h A 8 ; r ? 8l ;7~CAEj; 8 uA 8 u; 8 J>?@g r A 8 A 8DG A zE;x< EFAC= ; r ? h76<>I2? l 6 EFuAA<2I>? r E:? 6 ? e A7en

det(CpBp) 6=0 A l ; 8 Au@I>?BuH;;&S<>A 8 6 ; 8F8 <>uH? G ; 8 I 6 )ACu;fe|e

#I r 6 n>g 6G AsR 8 AM;7g>=@? q ;7E 6 A 8Fr ;>?@=B?@~C; G>6 E%; G ;7g r ; r ? hz6DG AMLQE:I>A 8 R ?@=B=@ACu 8%G ; G26 g 6 E

u(t) = −k(t)y(t) , e

k(t) = ‖y(t)‖2 , k(0) ∈ R , e

6 I G Ak : R+ → R

9 <>u z;7I>J 6 ; G ;7g r ; r ? hz6 A 8:q ;x= ;7Ee 8Fr ; =@A? G A q6 I r E 6 =BAmE:A 8 <>= r ; I>;A 8Fr ;>?@=B? G ; G A; 8F8 ?BI r r ? q ; G ; 8 ; G ;

y8 A −Kp

l 6 E-U(<>E ? r ~ LEFI>A 8 ?@=@=ACu 8 "$7 n 6 I G A

Kp =CpBp

9 ;Du; r EF?@~U X N G ;jg>=@;7I r ;2e g>E 6 FA r6G26 (EFA9&S<>AE 6q6 I>J>A q ?BuAI r6G A l ;7? >; 8 g>;7E; 678-h ;7= 6 E:A 8QG>6z8 g>;7E uHA r E 6z8

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q6 I G ? 6 U(<>E ? r ~ 8F6 >E:A ;u; r E:?B~wU X Nag 6G A 8 ACEu;7? 8 E:A= ;2; G ;I 6YqC6 I r A r6 G AA 8Fr ;>?@=B?@~C; 6 ; G ;7g r ; r ? h ;2e &u ;7I 8 AmE:A &S<>ACE;7gZACIK; 8H6 qC6 I>J2A q ?@uHACI r:6 G Am<>uq6 IF<>I r6 3KI>? r6MG A(u; r EF?@~A 8 &S<>A q6 I r ACI2JK;;7= z<>u;u; r EF?@~

Sp∈Rm×m; G A9&S<K; G ;fgK;7E:;5&S<>A

−KpSp

8 A$:;YU(<2E ? r ~7e 8:8 A qC6 IF<>I r:6Y9wG AI 6 uH?@IK; G26 "!+ / 4+(1/ ' ')(+* -!QG A Kp

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yh ACE G ; G A?@E 6

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h AE G ; G AC?BE; G ;Mg2= ;7I r ; 8 A$:;MIS<>= ;2e&u qC6 I r E:; 8Fr A7n 6z8&8 ?BIK;7? 8 I 6z88 ? 8r ACu; 8 gZACEFuw;xI>A q ACu=B?@uH? r ; G>6z8 ;xgZA 8 ;xE G26 E:< G>6G AuHA G ? 6 n G A 8:G A &S<>A ; 7ACE; 6G ; l <>I 6G Au 6G <>= ; 68 A$:;j?@u<>I>A; 6 E:< G>6 e

z

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x1 = Φ11x1 + Φ12yp ,

yp = Φ21x1 + Φ22yp + Γ2up ,]Xj3myl

#+3$,yp∈Rm

sw#hE#$,w;6$,3xp =[xT

1 , yTp ]T ∈Rn

sw#hE#$3w$,#up∈Rm

sw),/76+%2$,u9#+A;#26w $3#?<>26|26 7R$,u9#+A;#26E#742),+,78;78#

up = −Γ−12 ρ(yp, t)

yp

‖yp‖.

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cy ≥ 0

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¢:£

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9269 ),24&a&28#; t<%&|#t<%/? #c1

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S$,# ] 0$ ;$3c <,),#+ OlEm f;9;#,w '*),+,!v"&#$3/?#.$,),24&!v"#$,E;&78'X@

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xp(t)s[),/1p78/[&;78S<,#pQ$,#

xp(t) $%

<>#),/ #,S$,#tcx≥0

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˙y(t) = −γy(t) + c1‖u(t)‖ ,]j3m £ l

#+3$,y(t)≥‖g(t)∗u(t)‖ , ∀t ≥ 0

,$,$, r:),y(0)≥0

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c1/(s+γ)GS qqY.N(~vGSFEJ?BhBAkFN?rOIXBAYABUq

G[D3BF(c1, γ)

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G(s)PON

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GUT>Y.Ng(t)

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Nc1, γ>

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x(t) = Ax(t) + Bu(t) ,]j3ml

y(t) = Cx(t) ,

GUT>Y.Ny ∈ Rp

qu ∈ Rm

Nx ∈ Rn

RN Bγ0

BJ?BF.NEJ Y.NNQPEV`BAkIKUWIXYABAY.NYABJ?BVFZIxw Y.NVXFB[T3P*~vNEFon?TpkEIXB

G(s) :=C(sI − A)−1BNuPON B

γ :=γ0 − δkQGSJ

δ >0PONGT>Y.G1.J?BKkQGUT3PEV`B[T%V N

BF kIKVXFFZIXBRN Bu(t)

.J J?B vGSFB[T%V N I T3PEV B[T%V[TNQGY.Nu(t)

qCIXNxq ‖u(t)‖ ≤ u(t)q ∀t ≥ 0

T%V`x.G q ∃c1, c2 > 0

V`BI6P%NCB[FNQP D%GOPEV`B1IKJdD>zUPEI rSBg(t)

Y.G|PEI6PEV NEJ?BhPBVXI6P*~ BLyB

‖g(t)‖ ≤ c1 exp(−γt) ,]j3m Al

N BSPwPON`.I T%V NQP Y.NQPEI3BUYABAY.NQPdPON BJ PBVXI6P*~vNEIKV`BSP

‖g(t) ∗ u(t)‖ ≤ c1 exp(−γt) ∗ u(t) ,]Xj3mAl

‖y(t)‖ ≤ c1 exp(−γt) ∗ u(t) + c2 exp[−(λ0 − δ)t]‖x(0)‖ ,]j3mOiAl

∀t ≥0qGUT>Y.N

λ0

s B[J?BF.NEJ Y.N|NQPEV`BAkIKUWIXYABAY.N\YAB[J?BVXFZIxwA

FGUrSB! ; ;<>#pQ?78/1<,)3267&h$,#;76p /[]Xj3mlasu$%$%<>#

g(t) =

C exp(At)B

t > 0 ,

0;

t = 0 .

]j3myl

¢

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uh; ;<>#pQ78/1<3),28;7&g(t)

<>#.$3/ ; t<,+:S$%&9#/?#;#/15$,0 ;/?#Zz<>#+, +,97676

Ajltl exp(λjt)

Gjlt

l exp(λjt)&; ;<>9E7&/?+A&

#+3$,j ∈1, . . . , n l∈0, . . . ,mj − 1 mj

sh[/h)32878<,2789 78$%$,\$3#)3#O&28#λj

$,A

Ajl ∈Cn×n

Gjl ∈Cp×m

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r:), ‖ exp(At)‖≤ c3 exp[−(λ0 − δ)t] ‖g(t)‖≤ c1 exp(−γt), ∀t≥0

mt_a#+,; r ,+:/?+Ah),78+:;u$,76),26$%&$,wsw&2876$%

‖g(t)∗u(t)‖ ≤ ‖g(t)‖∗‖u(t)‖ ≤ c1 exp(−γt) ∗ u(t) ,]j3mygl

$,u#+3$,u;),28?|$, ;76)%26$,$,?]jzmAlZmf/1h@\r:),u#?;78+%2R$,u;6$%|$3#?;78 /1]j3mlasu$%$3#?<>#

y(t) = g(t) ∗ u(t) + C exp(At)x(0) , t ≥ 0 ,]Xj3m8vjAl

+:Q"&#[+,#;/11$,#76+%2$,h8$% ‖y(t)‖ <>#.$,h;/1U#$,1<>26[;#/1[$,#/1#&+:;]Xj3ml ‖C‖c3 exp[−(λ0 − δ)t]‖x(0)‖ m. =;9 #26b, +,$,#S ;

c2 =‖C‖c3

%+%&26/?+A\C#zs/]Xj3m8viAlE39#/?<,28 Q&+,$,#& ;u|<,;#O&3m

] dabdc ehgi RN\GuPEI6PEV NEJ?B ~vGSF T>x.GkQGUT%VFGSUx[rNEUGS T>x.G;GkEPONEFGrU[rNEU qoBU T3PBV GUrSBU6GSFQNQP?Y.N

AT>x.G PONEFx.GhD ySU6GOP1Y.N

G(s)quN?T%V x.GBSPJ?BF`.N?T3P?Y.N NQPEV BAkIKUWIXYABAY.N[GkEN;Y.NokQNEFx.G

λ0 ≤ γ0

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vfU sEJ YI6PQPOG qdPONGPEI6PEV NEJ?B B.V UTGSJ[G~vGSF BSPQPEI T%V GSVXI|kBJ[N?T%V NCNQPEV|[rNEU q N?T%V x.G

0<λ0≤γ0

] dabdc ehg RNB N ,BLyx.G YzN1PBXYABY.GPEI6PEV NEJ?B ~vG&Fy(t) = Cx(t) + Du(t)

qN?T%V`x.GBJ?BVXFEIxwY.NCVFB[T3P*~vNEFzn?TpkEIXBY.G?PEI6PEV NEJ?BKs|YABAYABCD%GSF

G(s)=C(sI − A)−1B + DN?G

5NEJ?B ?PON5rNEFZI T7kB BAYI|kEI`GUT>B[T>Y.G`PONCG?V NEFZJ[Gc4u(t)

c4≥‖D‖ B.GOP\J?B vGSFB[T%V NQP N

uBVX.F;BUWJ[N?T%V NCB[FNQP D%GOPEV`B1IKJdD>zUPEI rSBg(t)

Y.N?rNPONEFuJ[GYI T7k;BAYAB[BAY.N 3BAYABJ[N?T%V N

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¢

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U<>#.$,h),/ 76+%&2 9Qb%vv$,# $3# <>#d),/11267$,C9#+A#26 <>#

/?#.$,#h$,2676@+Am 76+%&2Uav

sw# kQGUT%VXFQGSU6NJ s;YI`G $,#|<>28#h%28;;#h<%S %7 ,&\]Xj3m8 £ lEm

W] ehg GUT3PEIXY.NEFQNCGhPEI6PEV NEJ?B

x(t) = Ax(t) + B[U + d(t)] + Bφφ(x, t) ,]Xj3m8 ¢ l

τ Uav(t) = −Uav(t) + U ,]Xj3m8 £ l

GUT>Y.NU, d ∈ Rm

Px.GGOPPEI T>BI6PY.N N?T%VXF;BAYABUqd

s zqτ > 0

s.J?B kZGUT3PEV`B[T%V NY.NV NEJdD%GBF kIKVXFFZIXBUq

x ∈ Rnq

Uav ∈ Rm v ~ZmT Lyx.G

φ : Rn × R+ → Rls8kQGUT%V T,,BD%GSFhD3BFEV NQP

NEJtq U6Gk;BUWJ[N?T%V N ID%P k.IKV wNEJ

xN PBVI6P*~ B w ‖φ(x, t)‖ ≤ kx‖x‖ + ϕ(t)

qGUT>Y.Nkx ≥ 0

s BU zJ NQP k;BUBF Nϕ : R+ → R+ s kQGUT%V T%3BD%G&F|D3BFEV NQPEvPQPE.J[N `PON %N?D3BFB k;BAYAB

kQGUT>YI*Lyx.G I T%I|kEIXBU=BF kIKVXFFZIXB[xT (0), UT

av(0)]T

G PEI6PEV NEJ?BY.N N 3BL&MONQP?YI ~vNEFN?TpkEIXBI6P D%GOPQPE.IdPOGSUW3Lyx.G kQGUT%V T%3BmT%I|kB TG PON?T,VIXY.GY.N*t0IKUID:D%GUr RN B

λ0

BJ1BF.NEJ Y.NNQPEV`BAkIKUWIXYABAY.N YABJ?BVXFZIxw

A RN BJ

cφ > 0N

γ < λ0

V BI6P %N ‖wφ(t)‖ ≤ cφ exp(−γt)q

∀t ≥ 0q[GUT>Y.N

wφ(t)sBFQNQP D%GOPEV`BIKJdD>.UPEI rSB<kQGSFEFNQP D%GUT>Y.N?T%V N J?BVFZIxw Y.NVXFBTzP*~vNEFon?TpkEIXB

(sI − A)−1Bφ

Nγx := γ − cφkx > 0

qN?T%V`x.G sNQPEV|[rNEU TGPON?T%VXIXY.G +:;$,S2876/?7 Q&$%S Q&$,#& 287/?7 Q$3# N ∃ci ≥ 0

i = 0, . . . , 6

V`BI6P%NBPON`.I T%V NY.NQPEI3BUYABAY.N[PONrNEFZI T7k;B ∀t≥0

‖x(t)‖ ≤ c2τ‖Uav(t)‖ + exp(−γxt) ∗ [c3ϕ(t) + (c4 + τc5)‖Uav(t)‖ + c6‖d(t)‖]

+ exp(−γxt) [c0‖x(0)‖ + c1τ‖Uav(0)‖] .]Xj3m8 l

FGUrSB! d]Xj3m £ l0<>#.$3E Z3<3;76/?78U =τUav + Uav

<%;C ;9 \]jzm8 ¢ l09 #/1#

x − Bτ Uav = Ax + B[Uav + d] + Bφφ(x, t) .]Xj3m8Ol

d %+376+,$,#S ;f#\ #a$3d $,#x :=x−BτUav

Cr:)%&!v"#C<>#.$,f;; ;9 ;7Q 9 #/1#

˙x = Ax + (τA + I)BUav + Bd + Bφφ(x + BτUav, t) .]jzm8 l

¢

0+AQ"&#,%$,E%+,78+,$,#& ;u#?76+%&2R;926

r(t) = exp(−γt) ∗ [cφkx‖x‖ + cφϕ(t) + cuav‖Uav‖ + cu‖d‖] + c0 exp(−γt)‖x(0)‖ ,]j3mOAl

\<326789v+,$3#& w#?¡(/1j3m,#3s / ;w#?/1#&+:

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+ c0 exp(−γt)‖x(0)‖ ≤ r(t) , ∀t ≥ 0 ,]j3mgiAl

#+3$,(cu, γ)

(cτ , γ)

;"# FXY%<%; /[S;78@ $, ;+,p'* +,976(sI − A)−1B

(sI − A)−1AB

%; ;<> 9 7&/?+A&

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;76+,2r : R+ → R+ s\;#26),!"#$% r.),!v"&#1$,78'* ; +,9762

r = −γr + [cφkx‖x‖ + cφϕ(t) + cuav‖Uav‖ + cu‖d‖] , r(0) = c0‖x(0)‖ .]Xj3mggl

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r:),f&;76p'X@r(t)≥r(t)≥‖x(t)‖ ∀t≥0

m γx =γ− cφkx >0

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m

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λ0

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γ0 = 1/λmax(P ) ≤ λ0

q|GUT>Y.NP = P T > 0

sB POGSUWzLyx.GYABN ,BLyx.G

AT P + PA = −2Iq0kZGUTv~vGSFZJ[N8s YI6P kE.VIXY.GRT>B RNLyx.G Ev BED>UWI|k;BLyx.G

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£

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$,Q&+, +,976 +: |g2(t)|1Zz<>#+,+39762g]X/

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S(γ, h, kf ) ,]j3m ££ l

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S(γ, h, kf ) = maxk∈T

f(γ, h, k) ,]j3m £ l

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γ3exp(γkh) ,

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i = 1

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γ2i+1 = γ3i

γ4i+1 = γ4i γ3i+1

QT`b dc

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δl =δu =0, 1 s−1 ε1 = 0, 1

ε2 = 10−6

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Ganho DC mínimo

Valor de pico mínimo

Constante de tempo mínima

|1|g (t)

1 2 3 4 50

1

2

3

4

5

6

7

8

t (s)0

Nor

ma

da r

espo

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d .76$3##w&28#; u;9 #26b376$,#\<,#u<%z&/1E;#ε1

ho

5'*#/ 9#/?<,)3$%201

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R

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φ(t)s[$,#

;76<># 9v;$%3(78;#s( r.)378&&26|)3/[ <> ;),;%&!v"#$,| +:;$%d(t)

(+,$3#φ(t) = Bpd(t)

mn #Q+A#3%r:)%&!v"#?$,u$,#?$%<,26+AQ] mjAgla<>#.$,ut; ;9 ;78+%|'*#;/1

xp = Apxp + Bp [u + d(t)] ,

y = Cpxp .] ¢ m8Ol

<%&/? ;;78@v!"# $3#9 #+:;;#28h<%; ;76p /[ / r:),<> ;),;%&!v"#φ(xp, t)

s|+%"#S2876+, u$, ;9$,1s\>#;$%$,+%& ! -.dzm8u zmgzm

<,26+AQ ] ¢ mylw'*#; <> '* 78Q&/1 +:19 #+,b, 978$%[+,+3b.),/1<>p),%!"# ;78 ;;<,+:;1]

d(t) ≡ 0lE3 +:"#|),/[C287 $,9#+A#26dr:),w9v&74hh/1&76@f$3;+,p'* +,976h$%

/128b% '*9Qb%$%9 #/ #/?#.$,28#WM(s)

;76$%$%<>26; ),78+:?<%;/1E76@v&!v"#3r:),1s),)%2(+%|2678;;&),;h#,\9 #+:;;#28u$%<zQ&;78#]Xm ,m] ; c a#.$,#+OAll

u∗ = θ∗T ω ,] ¢ mgl

#+3$,\/1&76@d$,u<%z&/1E;#θ∗

u# ;#t; ; ;#ω(t)

"&#?$%$,#<>#

θ∗T = [θ∗T1 θ∗T2 θ∗T3 θ∗T4 ] ,] ¢ mjl

ω = [ωT1 ωT

2 yT rT ]T ,] ¢ m ¢ l

ω1 = A(s)Λ−1(s)u , ω2 = A(s)Λ−1(s)y ,] ¢ m £ l

A(s) = [Isν−2 Isν−3 . . . Is I]T ,] ¢ m l

Λ(s) = λ(s)I ,] ¢ ml

ω1, ω2 ∈ Rm(ν−1) θ∗1, θ∗2 ∈ Rm(ν−1)×m

θ∗3, θ

∗4 ∈ Rm×m

λ(s)

s ),/ <>#2876+/78#/+,769 #e),t7@u$3\)

ν − 1m

b,#)3 <>p),%!v"&#C$, +:;$%3: <>#.$, \9+,9 26$%\<>28#\;76+,2Wd(s)∗d(t)

$3769 76#+%&$,# |267$,\9 #+A;#26?] ¢ mgla9#/?#; ),

u∗ = θ∗T ω − Wd(s) ∗ d(t) ,] ¢ m Al

Wd(s) = I − θ∗T1 A(s)Λ−1(s) .] ¢ mAl

d Z +3#Q5r:),a/1&76@=$3;+,p'*+,9 74Wd(s)

s Qv2]X)3/[ @ar:),λ(s)

stef),t78;@Olot<,<,743m, =; ;r:), /[u$,t9 #+:;;#28<>#=/1#.$, 26#$,t '*+,9 74ws<3;+AQ$,#+%o76), ¢ m,r:),\7826),p;h#?E'*78;#$,#?76+%2R$3u9v+,9 26/?+:;#$%<>p),%!"#,m

*"%$ + SJA*/24 B.N A*B 3 NO351 B.Sl>O+

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G(s) = NR(s)D−1R (s) , NR(s), DR(s) ∈ Rm×m[s] .

] ¢ m8Oil

d;),/?76+,$,#S ;r:),+%"#ub%u<> )3;%!"#,.w/[&;;78@ $,;+,'* +,976u$,r<%;

yst76),2

#[/?#.$, 26#WM(s)

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+

++

−

−

+

++

++

d(t)

d(t)

e

r

u

y

yM

ω1 ω2

A(s)Λ−1(s) A(s)Λ−1(s)

Wd(s)

W−1d (s)

G(s)

WM(s)

θ∗T1 θ∗T2 θ∗T3

θ∗T4

#.$, 26#

n=24+AQ

O =)3),; <%;/? 76@!v"#$3# 9#+A#26 <>#^/1#.$326#$, ;E'*+,9 743m dpQ9/| ;# ;78+%2$,;E'*+,9 74 ]

rlE?#^76+%&2?$, ;;# ]

el #^76+%&2

Wd(s) ∗ d(t)<%&|#?9+,9 24/? +:#$%|<>;),;%&!v"#?$,u +:;$%

d(t)m

c a#.$3;#+O3,$, Sb, 2678+c a#.$,#+v ¢ l

NR(s)[

Λ(s) − θ∗T1 A(s)]DR(s) −

[θ∗T2 A(s) + θ∗T3 Λ(s)

]NR(s)

−1Λ(s)θ∗T4 = WM(s) .

] ¢ myl

\#19 #+:;;#28w'*#t<,/? ;;78@v$3#<># ] ¢ mgl ] ¢ mla+AQ"# ∃θ∗$3\'*#;/1|r.)3 ] ¢ m8Ol ; ;

;&78'*7Qu] p wc a#.$,#+?O3Sn=;#<>#;76!"# mj3mjAlZmF =Qt9 #+,$,78!v"#t$309v;/?+:;#r:), θ∗T4 = K−1

p

mC# +:+:;#,5 ),+,78978$%$,C$,θ∗

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θ∗s ,+,769 #

u'*#78/1<>#Q||; ),76+Au;;78!v"##26)3!v"#$,1] ¢ m8yl?Z

∂ciD∗(s) ≤ νi − 1 , ∀i ∈ 1, . . . ,m ,

] ¢ m8ygl

#+3$, νi"#h# 8+,$,789$,#3;p&3762678$%$,$,

G(s)

∂ciD∗(s)

"#h#a;),a<>#a9 #28),+%$,

D∗(s) := [θ∗T2 A(s) + θ∗T3 Λ(s)] ∈ Rm×m[s]mno&789)324/?+:;>

νi = ν , ∀i +:Q"&#

θ∗

s ,+3769 #,,),/[C@\r:),ν = maxi(νi)

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z

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θ∗+%"#

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ω1

ω20+,# +AQ+A#,

78# r:),w#9 #+,b, 978/1 +:# B|D>FZI`GSFEI($,h;#.$,##8+,$,769 d$,|#3;p&3762678$%$, $% <,26+:3m d#OS+:;

θ∗$, +,#?[POGSUW3Lyx.G1YABN 3BLyx.GUWI TN;BF\Y.N k;BSPBJ[N?T%V G[GkVIXYABuD%NEU6GJ sEV GY.G1Y.N

J T%IKJ[GOP 3BAYF;BAY.GOPE,h%/$,u;w'*#/|)324a<,9 76&/1 +:;\t26 76 $,u9#+A#26&m

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ω1 = Φω1 + Γu , ω2 = Φω2 + Γy ,] ¢ mOjl

#+3$,Γ∈Rm(ν−1)×m

Φ∈Rm(ν−1)×m(ν−1) 9)#?<>#2876+/?78#C9v&9E6p769 #|s

det(sI − Φ)=

det(Λ(s)) = λm(s)m ; # ;#h$, pQ$3#

X := [xTp , ωT

1 , ωT2 ]T

m 76;/1/ /[&26b%> 9#/1<>#p#<>26|<,24+AQ] ¢ mylau<>28# %28;;#\] ¢ mOjAla<>#.$,ut;<3;+AQ$,#1<>#

X = AoX + Bou + Bodd ,] ¢ m ¢ l

y = CoX ,] ¢ m £ l

#+3$,

Ao =

Ap 0 0

0 Φ 0

ΓCp 0 Φ

, Bo =

Bp

Γ

0

, Bod =

Bp

0

0

, Co =[

Cp 0 0]

.] ¢ m l

g

0+AQ"&#?#E#t;#su$%&$,#?<>#

ω = Ω1X + Ω2r , Ω1 =

0 I 0

0 0 I

Cp 0 0

0 0 0

, Ω2 =

0

0

0

I

.] ¢ myl

),,p78;),76+3$,#& u

<>#u∗ $,$,# <>#h] ¢ m Alt / ] ¢ m ¢ ltC26 /h,+3$,#& ; r:),|[<> ;),p

%&!v"#su9v+,9 26$%|<>#u∗ ,#3;s/| ;u)376+Au;2678@v!"#|+%"&#& /?8+,78/?C$,

WM(s)

XM = AcXM + Bcr − BoWd(s) ∗ d(t) + Bodd ,] ¢ m Al

yM = CoXM ,] ¢ mOAl

#+3$,

Ac = Ao + Boθ∗T Ω1 =

Ap + Bpθ∗T3 Cp Bpθ

∗T1 Bpθ

∗T2

Γθ∗T3 Cp Φ + Γθ∗T1 Γθ∗T2

ΓCp 0 Φ

,] ¢ mgiAl

Bc = Boθ∗T Ω2 = Boθ

∗T4 =

Bpθ∗T4

Γθ∗T4

0

.] ¢ mgzyl

/1&;;78@Ac

s ef)3t78;@),/[@ r:), #/1#.$, 26#f$, '*+,9 74d #o%2# $, pQ$3#w"# Qv78m 76+%&26/+:;zC r.),!v"&#$3pQ$3#|$,#h;#hs#z76$, ;),z78+,$,#S ;|] ¢ m8 Al ] ¢ mOAl$,1] ¢ m8 ¢ l] ¢ m £ lEz#r.)3\; ;),2Q /

Xe = AcXe + Bc(θ∗T4 )−1

[u − θ∗T ω + Wd(s) ∗ d(t)

],

] ¢ mggl

e = CoXe ,] ¢ mgjAl

#+3$,Xe :=X−XM

sf# Q&$,#C$%\r:)%&!v"#C$,# ;#,m0_#/1# Ac, Bc, Cos),/1\v2876@!v"&#

+%"&#& /?8+,78/?u$,WM(s)

(θ∗T4 )−1 =Kp

3 /[S;78@$,f;+,p'*+,9 74C$% r:)%!"#h$,#h;#C$,j

;6$%] ¢ mggl ] ¢ mgjAl<>#.$,u; 978Q|9#/?#

e = WM(s)Kp

[u − θ∗T ω + Wd(s) ∗ d(t)

].

] ¢ mg ¢ l

¢

R &

! $& "

/1&76@dI T%V NEF;BmkEV GSF $,t76;/1=$,&); 24&;78#),+,7'*#;/? ),+,7Q76#sξ(s)=sI

:9 #+3'*#p/?fu$, ,+,76!"# $,)?; 24&;78#u),+,7'*#;/?t<,; +:$% +% !v"&#[mgzmj3mj3mW)zu9#+3$,76!"# r.)378&&26+A <%& r:), # ) ; 24&;78#$% <324+AQ ] ¢ m8Ol?;; ),+,7Q76#s

det(CpBp) 6= 0]a#.$,;#+ c w;#@:76?t769 @?OAlZm d;78/ ; +,$,#,>+, 9<,8;),26#?b,76< &[] ¢ ls\;)3,7 ;),6$%h<>26b376< ;;u/178 ;<> 9%9vZ

Og 8 &); 24S78#|$,<,26+:|su),+,7'*#;/?u),+,7Q&;76#3z7`m mdet(CpBp) 6= 0

m

f \9;#,%/1&76@$3\+,b,#$,\28C'*;r d +,976?$,<,26+:|su$%$,<>#Kp =CpBp

m +,769 7428/? +:;;),/?E r:), <>p),%!v"&#&)% +% +:;$% $%<,26+:39 #+3'*#;/?

#_ &<,8;),26# ¢ mhd !v"# £ mj 9#+,76$, S ;|r.)31 <>;),;,!v"#sY.NQP k;BSPBAYABm5d26s / $376;#,9 #+,;78$, S ;ar:), #d;78 /[sa2676+3v f<>p),%!v"&#sa76+,$, <>+,$3+:; $,#pQ$3#w f; ),78+:;b,78< ;u),,;78;),75tb,78< ; C] la1]wl?Z

hg < ;),;%&!v"#$, +:;$%d(t)

s9 #+:;6+:)%t<>#5<,;0;9 #+,b, 9),/^/1#+Ad(t)

/h>s/ 9 #+:;6+:),#w<>#q<%p A$, '*#;/1dr:), ‖d(t)‖≤ d(t)≤ dsup <+∞ ∀t≥0m

f/1 v2876@v&!v"#$,#/1#.$, 26#$,;E'*+,9 74^] mj ¢ l ]pmj £ l[$,;) ; 24&;78#),+,78;78#]L(s)=1

lasu$%$%|<>#

yM = AMyM + r , AM = −diag γ1, . . . , γm .] £ myl

£

d;76/R<, #9;#$,#&) ;26&7#[),+,78'*#;/?Ch),+,7Q76#, r:)%!"#$,#;#] ¢ mg ¢ l<>#.$,ut; ;9 ;78+%),78+:;w'*#;/1

e = (sI − AM)−1 Kp

[u − θ∗T ω + Wd(s) ∗ d(t)

].

] £ mgl

<%&780$,pQ <%&/1E;78@v!"#\$3#C9#+:#28.&;;),/?fr:),\26 7 $,f9 #+:;;#28<>p+39|; ),76+Au9 24; Z

hg 26 7R$,u9#+A#26w;&78'X@u|$,76),26$%&$,

‖ut‖∞ ≤ Kω‖ωt‖∞ + Krd ,] £ mjl

#+,$,Kω, Krd

"#9#+,pQ+A f<>#;78;78&&m

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<%p78o$,r:)%!"#u$,#;#?] £ mgl $3t&9#$,#w9 #/ n0#<>#;78!v"&#\g.m8287z$, 9#+:#26

E#742),+,78;78#h<,;#<>#Q|s

u = unom + SpU ,] £ m ¢ l

U = −ρe

‖e‖ ,] £ m £ l

unom = θnomT ω ,] £ m l

#+3$,Sp ∈ Rm×m

s ),/[/[S;78@r:), <>#.$3 ;9#28b,76$, <>28#<,;#U ;76pQ #3;p&+,$3#&] £ lt

θnom sC),/ &&26#+,#/?78+%2(<%;θ∗

m 9#+A#26\+,#/?76+,2unom s )3782678@$,#?<%&

<> ;/?7878C$,)3!v"#h$%h/?<,287),$3$,#C ;/?#C9Zb,O$,#|$,#C;76+,2>$,d9#+A#26fw 76+,9 p@<%&/?s ;;789v ‖θ∗ − θnom‖ '*#<>r:), +%3m .f_ sur.)3/1&78@v$3#?+% o78),; £ mm

+

−

+

+

+

+

d(t)unom e

rρ

u

U

y

yM

G(s)

WM(s)

Sp

#.$,28#

n=24&+:Q −ρ e‖e‖

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ρ'*#/

#/?787$,# <,|;78/1<,287%9a#?$,76&/13m

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#z76$,|\#?76+%&2R$,u/1#.$,)324!"#ρ

&;76p%@ f|$, ;76)%26$,$,

ρ ≥ δ + ce‖e‖ + (1 + cd)∥∥∥S−1

p

[

(θnom − θ∗)T ω + Wd(s) ∗ d(t)]∥∥∥ ,

] £ ml

#+3$,ce, cd ≥ 0

;"#9#+3Q&+: 0$3r:)%$%&=r:), ;&78'X@ / o$, ;76)%28$%$, ]`gzmyl5δ ≥ 0

st),/1u9 #+,+Af;378;;76zmo#Q&+,$,#& ;tr:),d(t)

st2678/?7&$%Wd(s)

st<,<,;76w Qv2),/1h'*),+3!v"#?$3\/?#.$,),26!v"&#|r:),\S78'X@?] £ mla1] 3mAlas

ρ = δ + c1‖ω‖ + c2‖e‖ + c3d(t) ,] £ m Al

9 #/

d(t) = d(t) +c4

s + γd

∗ d(t) ≥∥∥[I − θ∗T1 A(s)Λ−1(s)

]∗ d(t)

∥∥ .

] £ ml

/1#+Ah] £ mAlsw#z76$3#h< 24h<,28769v&!v"#C$,#¡(/1hj3m8#+,$,γd

&;76p'X@γ0 >γd >0

+,$,#

γ0|/1; / $3\ Q&,762876$%&$,$,#,28#9 #//1&;;76@d$,w&+,'* +3974

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Rm×m(s)m?_ $,28/?+A#$, P S76p'X@wb,76< ;1] yl

] jAlZ\] ¢ ml] £ l<%Sp

ν

z3#m 76/?<,28/+:!v"&#$%'*),+,!"# $,/?#.$,),24&!v"#] £ m l] £ ml;r:), #?)376+:;[Z

hg "#9#+,b, 976$3#5&28#; 5<%&9 #+,pQ+Aci≥0

]i = 1, . . . , 4

lRγd >0

&$,='*#;/1r.)3|[$3;78)%&26$%$3 ] £ mlt; ;[&;76p'*78 ∀G(s)∈P 9 #/ 28),/

θ∗9 #;<>#+,$, +:

9$%28/?+A#$, P r:),\;&78'X! r.),!v"#?$,76#&'X+:;76+,] ¢ m8ylEm

] dabdchgi v T%I|kEIXYABAY.NY.Nθ∗

T>x.G s.TNokQNQPQP%FEIXBhNEJ v KB 5J?BD3BFBJ[NEVXFEIxwOBLyx.G%N\PBVXI6P*~ BLyBBKkQGUT>YI*LyxzG1Y.NCmT%I|kEIXYABAY.N ?FNQPEzUWV`BNEJ T0UWVFGOPhY.N?GSFY.NEJ J T%IKJ?B\D3BFBB.NEF;BLyx.GY.GOPPEI T>BI6P

ω1

B[T>Yω2

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ν

] dabdchg GSV`B`PON N GOPhPEI T>BI6Pω1

Nω2

D%GYzNEJ PONEF[NH D>FNQPQPOGOP kQGSJ[G PEI T>BI6P T0U VXFBAYzGOP[YABPBXYAB

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T qN D3BF;BGJk;BSPOG j T%V`x.GuD%GY.N `PON\J?B vGSFBF ‖ω1‖N ‖ω2‖

D%NEU6GPEI T>BU

ρy

%NwPBVXI6P*~ B w

ρy =c5s + c6

s + c7

‖y‖ ,] £ m8Oil

GkVXIXY.G BVXFB[r[sQPhYABBED>UWI|k;BLyx.G1Y.G5NEJ?B KkQGUTv~vGSFZJ[NCB8jtkEPONEFGrSBLyx.G vdPQPEIKJGkVsEJ `PONB~ZT Lyx.GY.NJ[GY.UBLyx.G?NEJ J[NEU B[TGUrNEVBU ymqN qNHkQNEV G %NCG vT>x.G[FN %NEF?GKkQGUT %NokEIKJ[N?T,V G[Y.N

Kp =CpBp

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[

;),76+A; ;),2Q&$,#$, Q&,762876$%&$,C'*#7q#z76$3#/ ]Xef;) &2`m(gii3(e;)Ro_a),+3b%3_a#Q?c ¡(76@;;26$3ugiiAg3B _a)3+,b%3 ef)R _a#Q?c ¡(76@;;28$,ugiijl?Z

] c ] hg iG TzPEIXY.NEFQN GPEI6PEV NEJ?B~vGSFEJ?BAY.G D%NEUB N 3BLyx.GY.G NEFEFG qBU6NEI Y.N kQGUT%VXFQGSU6N N B ~ZmT Lyx.G Y.N J[GYzUBLyx.G RN BSP .ID ySV NQPONQP v v q v KB q v q v KB N v KB C~vGSFNEJ PBVI6P*~vNEIKV`BSPoqhN?T%V`x.G B NQPEVXF;BV s`IXB 7v ^PONEFU6GkQBUWJ[N?T%V NNH D%GUTN?TpkEIXBUWJ[N?T,V NNQPEV|[rNEU qIXNxq ∃k, λ > 0

V BI6P N ‖Xe(t)‖ ≤k exp(−λt)‖Xe(0)‖

q ∀t≥0.vfU sEJ YI6PQPOG qqPON

δ>0qN?T%V x.G GNEFZFQG1Y.NwPBXYAB

e(t)PON\V GSFGT>BF

T%.U6G[BED yOP BU .JV NEJdD%GTWT%IKV G

FGUrSB! #\28#+,#$%w<,;#O&w#u;6/h>#28#ki

]i∈N

lo$3+,#0kQGUT3PEV`B[T%V NqD%GOPEIKVXI rSBhBED>FGQD>FEIXBYABmF r.),!v"&#w$3# ;#] ¢ mggl ] ¢ mgjAlR<>#.$,; o;<3;+AQ$,)3782678@+,$,#& hY.NokQGSJdD%GOPEI*Lyx.GY.NCBUWJ?B[T]d;/c 78+3/[OA.,<,<Rm(vi3 >viAgl

˙Xe =

A11 A12 0 0

0 A22 0 0

A31 A32 A33 A34

0 A42 0 A44

Xe +

B1

0

B3

0

[u − u∗] ,

e =[

C1 C2 0 0]

Xe ,] £ myl

#z76$,d&;vso$,&26)3/[;+,'*#;/1!v"&#d2876+, $, r:)%$%TXe =Xe

mF Y.NokQGSJdD%GOPEI*Lyx.GY.NCBUWJ?B[Tf<%p769 76#+%t#f76;/1/ ;),z ;78 /[& #3;p&O 76 +%"#& #,SO 769 #+A;#26O 761+,"#& 9 #+A;#24O76 m f/1 @[r:), /1&76@|$,?;+,'* +,976$, ] £ m8Olws1$,$%< #

e =

C1(sI − A11)−1B1[u − u∗]

=r:), s 78)%2 /1&;;76@?$,[;+3'*+,9 74$% r:)%!"# $3# ;;#] £ mglE / ;[r:),/[S;78@1eY $%<,26+AQs[$%$,<>#

Kp = C1B1

q#+,$3C/1&;;76@ r:)%$3$%&

C1

B1

+%"&#?"#?;78+,),26; m +%"&#& 76+,)32478$,$,$,pQf/1&;;78@<> ;/?78u<326789v!"#|$%h&+,'*#/[&!v"#

e

x2

x3

x4

= TXe , T =

C1 C2 0 0

0 I 0 0

−B3B−11 0 I 0

0 0 0 I

,] £ mygl

;)328Q&+,$,#?+,w~vGSFZJ?B1FQN`.UBFw]K0#),+, f2mRvAz,;:76+vAgl

e

x2

x3

x4

=

A11 A12 0 0

0 A22 0 0

A31 A32 A33 A34

0 A42 0 A44

e

x2

x3

x4

+

Kp

0

0

0

[u − u∗] .] £ mOjAl

_a#/?<%&+,$,#& ;/1&76@$, ;+,p'* +,976w$,u] £ mOjAla]e=(sI−A11)

−1Kp[u−u∗]l 9#/

/1&;;78@h$,;+,p'*+,9 74$%r:)%&!v"#$,# ;#] ¢ mg ¢ lZ59#+,928),7 ;|r:),A11 = AM

m f26s /$,78;#, &t;),z /[S;78@

Aii

]i=1, . . . , 4

l "&#1ef)3t78;@u),/1h @ r:),Ac

s e),78@&md#\<,28769vy \n0#<>#76!v"&#hg.m8f# 76p/1u/ /128b%'*9Qb%&$% 9 #/?<>#;# <>#f] £ m ¢ l

] £ m lEa] £ m Al ] £ ml] £ mOjAl9 #/K = KpSp

π1(t) = K−1A12 exp(A22t)x2(0)

π2(t) ≡ 0

dU(t)=S−1

p [unom − u∗]m0f/1\ @r.)3 −K

sef)3t78;@d / z78;),$,d$% b,76< &|] £ l0dr.)3h'*),+,!"#$,u/1#.$3),24&!v"#] £ m Al&78'X&@u|$,76)%&26$%$3?] £ ml9#+3'*#;/?u|b,78< ;1] 3mAlZ;/| ;wr:),\#;#?$,u&6$%|su2876/?7 Q&$,#h<>#

‖e(t)‖ ≤ (k1‖e(0)‖ + k2‖x2(0)‖) exp(−λ1t) ,] £ m8 ¢ l

∀t≥0mWd26s / $,76;#,%

δ>0%+AQ"#?#/1#.$,#$3;2876@+:;u+,#?<>#+A#

e=09 #/1 !v / &26)3/

;/?<>#,+,78;#ts≥0

m0f/1h @\r.)3

x2(t) = exp(A22t)x2(0) ,] £ m8 £ l

x3(t) = exp(A33t)x3(0) + exp(A33t) ∗ [A31e(t) + A32x2(t) + A34x4(t)] ,] £ m8 l

x4(t) = exp(A44t)x4(0) + exp(A44t) ∗ [A42x2(t)] ,] £ m8Ol

∀t ≥ 0o ),,/1&76@

Aii

"&# ef)3t7@1e(t)

9#+A;[Ez<>#+, +,9 7428/1 +:;1<%;@#,<>#.$,Z \/?#; r:),

‖xi(t)‖ ≤ k3 exp(−λ5t)‖Xe(0)‖ , (i = 2, 3, 4) ,] £ m8 l

∀t ≥ 0#+3$,

0 < λ5 < minλ1, λ2, λ3, λ4

λi = minjRe(−λij) > 0 λij

s# <> 9 #$,

Aii (i = 2, 3, 4)m d#/1#+A ] £ m8 ¢ lC] £ m AlZ09#+,9 26),7 r:), ‖Xe(t)‖ ≤

k4 exp(−λ5t)‖Xe(0)‖ , ∀t ≥ 0=# r:),<,#y&r.)3# 76;/1s 28#%28/? +:;[Zz<>#+, +,9762

i

/?+Au O 2`m ] dabdchg e j kQGUT %NokEIKJ[N?T%V GY.N zJ?B J?BVFZIxwfkQGUT3PEV`B[T%V N

Sp

V`BU %N −KpSp

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Sp

sJ[N?TGOPFQNQPEVXFZIKVI rSBD%GSI6P PONFN %NEFBED%N?T>BSP G kQGUT %NokEIKJ[N?T%V GY.N?.J kQGUT ZT%V GuTWT%IKV G Y.N?J?BVFZIxwSNQP %NJkQGUT%V N?T 3B BU .J?B

Sp

BAY.N ,BAYAB 0PQPON*kQGUT ZT%V Gs Y.N?TGSJ|I T>BAY.G<>9E;),/| ),+,/?7z76+, Y.N

Kp

j BU .GSFZIKVJ[G Y.NNQPEV`BAkIKUWIxwOBLyx.G<kQGSJ|.V BkEI|kEUWI|k;BJ[N?T%V NhGNEU6NEJ[N?T%V G1Y.G\;<>9E;)3/ ),+,/?7z76+3;E1B?PONEF\APBAY.G T>B[U6NEIY.NkQGUT%VXFG&U8N \GN?T%V`B[T%V G qGBU .GSFZIKVJ[G D>FQGQD%GOPEV G|28#.9m 97vm T>x.G s\U6GkQBUWJ[N?T%V N?NH D%GUTN?TpkEIXBUWJ[N?T%V N?NQPEV[rNEUNBU mT3PwPEI T>BI6P %N\V NQGSFZI|k;BJ[N?T%V N D%NEFEJ?B[TNokQNEFZIXBJ TWT%IKV GOP D%GY.NEJV GSFGT>BF `PON|NHkQNQPQPEI rSBJ[NGT%V NFB[T>Y.NQP kQGSJ[G kQGUT3PON n?TpkEIXB[Y.GFZXYzG1Y.N\J[N;YI*Lyx.G

" -0B.1 'J=?+ A*BuC )F+pB7>O+

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Gα(s) =

s+0,6s2−1

−1,2s2−1

0,4s2−1

s+2,2s2−1

Kp .

] £ mOAl

0pQ<324+AQ / < 26#|/s = 1, 1,−1 a)3/ @ #$,;+,/178;;"# /

s = −1, 8

8+,$,789w$,u#3;p&,782678$%$,ν =2

u/[&;;78@$,uA+3b,#?$3 &28QC'*r d +,976

Kp =

1 2α2

−2 α

,] £ mgiAl

#+3$,wh9#+,+:;α∈ [0, 3; 4]

sw)3/ <%/? #h78+,9 #3m=#.$,# # $, /[&76a<,z/? ;;#a;"#9 #+,b, 978$,#m n #pQ+A#,q<>#.$3E &%/[\r:),<,24&+:Q<>;+,9 926; P = Gα(s) :

0, 3 ≤ α ≤ 4 m <> )3;%!"# $,+A;$% s)3+,78'*#/1 /?+:; 2876/?7 Q&$%<>#d(t) ≡ 5

$,9 #$,#?9 #/ |b,76< ;1] mAlZm

09#28b, +,$,#Sp = I

;/| ?r:), −KpSp

s1ef)3t78;@?;1?;#/1 +:?; −1 < α < −0, 25#)

α>0% +:"#?#? ._ <>#.$,u<,26789v&$,#,m 0/ 9#+A&&%; <,z/?E;#|76+,9 p#

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<%&r:),KpSp

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3

/ ]X # c #+,+,#) v3 _ab,78+c(%) OAgzo_ab376 +E|2`m0O lZo# r:),[s 9 24&/?+A),/178/1<>#76!"#|/h),78;#/[&76; ;;78;78S$,#?r.)3C?9 #+,$,78!v"#1ef)3t7@m d28s/$,78;;#3 <>#.$,E ;/?#;r:),

@Sp ∈R2×2 det(Sp) 6=0 2 r:),

KpSp =(KpSp)T ∀α∈ [0, 3; 4]

mwn #pQ+A#,<,26+AQ$39 24;|+%"&#<>#.$, / ;w9 #+:;;#26$%d<>#w26#;7;/?#dr:),Z37 ;/ ;78/1E;76$%|/[S;78@we ur:),\;),/[&/r:),

Sp

; ;?),/1|/1&76@f333m

/?#.$,28#|$,u;E'* +,9769#28b,76$3#s

WM(s) =

1s+2

0

0 1s+2

.

] £ mgzyl

w[%2;# $3 pQ$3#"# ;9#26b,78$,# 9 #/λ(s) = s + 1

m /1&;;78@$3 <%/? #+,#/178+%78as\9289),26$%h<%;

αnom =1,#?r:),\;)328Q|/

θ∗T =

−0, 99 0, 57 −0, 42 −0, 42 −0, 39 1, 41 0, 2 −0, 4

−0, 23 −0, 11 −0, 34 −0, 34 −1, 03 −0, 43 0, 4 0, 2

+

p1

p2

[

1 −3 −2 −2 1 1 0 0]

,] £ mggl

#+3$,p1, p2∈R

;"#19#.E%978+A&;,7&;78#r:), <>;/?78;/ S;f9 #/?<,26EQ/?+Au#19 #+z),+A#$%/[&;;78@ $,|<,z&/1E;#

θ∗r:),|&;76p'X@ / r.),!v"&#$,78#'X+A78+% ] ¢ mOlf<%&

α = 1m =9#28b,)z ;

θnom = θ∗9#/

p1 = p2 = 0#r:), ),/1 ;#28),!v"# $,/18+,78/?#

r:)%$3$,#<%& r.),!v"#?$,76#&'X+:;76+%zm '*),+,!"#d$,/?#.$,),26!v"&#C] £ m Al($3 a; o<,#U Q&$%f#,; &&+,$,#& ;w]zml5<3;#.9)3+,$3#&

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c1

su9289),26$%h<%;;&78'X@ |$, ;76)%26$,$,

c1 ≥ c1 =2

λmin(Q)

∥∥∥PKp (θnom − θ∗)T

∥∥∥ ,

] £ mgjAl

r:), '*#7a$3; +:#2.76$% <%p78C$, ] £ ml\ ]`g.m8Ol\/[&h)z762878@v&+,$,#& ;1# /1#+A1/1 +,#9 #+,; S&7# ‖PKp(θ

nom − θ∗)T‖ ≤ ‖PKpSp‖‖S−1p (θnom − θ∗)T‖ r.)3<>#.$,;?#3;76$,#

Ag

<%&78$, ]`g.mlEm /[S;78@P = P T > 0

&;76p'X@r:)%&!v"#$,¡A<,)3+,#yPKpSp +

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$ 78+,s97C$,#9 ;;# 7#u]M =diagm1, . . . ,mm

l / ;78@ Aij

"&#76+,9 mW/ 76@

S =

1 −1 0 0 · · · 0 0

0 1 −1 0 · · · 0 0

0 0 1 −1 0 0

0 0 0

m m m m m m mmmmmm mmm mmm

1 −1 0

0 0 0 · · · 0 1 −1

∈ Rm×m+1 ] £ m ¢:£ l

s<326789$C+9 #/1<3) !"#$ t ;)32 &+ $ '*#;! t$,# )&$,# m< ),;&!-

d1(t)

d2(t) <, + &/ ;7 # 9#/1<>#+ + $3#< ;#r.)

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),/ r:) / ;78@A11

e),7 @|< 7p' @ b376< $ ' /?6+,7 /] CylZm

hgOg hgi \*] # abdc-, (

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)$,# ;"#?/?#.$ 2$<>#\] #), 2`m%giiiAl

f = F u , F = diagF1, . . . ,Fm+1 ,] £ m ¢_ l

#+3$ #6+,$,789 $ ' 26b Fi =1 #\IsQPEIKJ[G )$3# #< +,$3#f9# / +

0<Fi <1 # )$ /1< +,b,#1su$ $&$,#?#) Fi =0

?9 #/?<,2 / + 76+,#< &+ m3

" !

/ @hr:) #7 / ] £ m ¢ jAl ] £ m ¢_ l <>#;),7m

;76+7$ &6$3 "#[+ 9 76# < + m

76+&7$ 9 #+ ;#2 76+,$ < +,$ + +,$,#& / .7 #?#m 78#$ 9#+ #2 <,#<># #,m

u∈Rm# #$ 76+7d$ 8$ $3#9 #+ ;#2$,#muf/ ;r:) / 7# );)2q<&

u' 28b / )$,# <,28769),// ;76@|$ < ;#]W

lu$ +,#/?76+$/?7 ), $,#] J|I HzNEF;l< h$,7 ;78,),78a# ;78+7a$ 9#+ #2 # )$,# ,7m m8

u=Wum /17 )3$,#

Wst 9#+3%), $,#\$ &9#$,#\9 #/ w' 28bw78$ + 7%9&$w< /+ '*#;! u ;)32 &+

$ $] <7`,_ab+,1c ¡ )gii3ylEmtr:),7 $,# ),/ >#;$ / < ;7w &$u+,#u .f_$ ;97 #u+ !v"# £ mj

)3/u/ ;78@/?7 ),&$,#9#+ &+ $ '*#/wr:) / /26b' 9Zb&$ &,762876$&$ #$ /1< +,b,#[+,#? &/ + # &/ 78/|)3+ 26)3/ ' &26b +3# )$,# m / 76@/?7 ), $,# h<,#<># s

u = ST Mnomu ,] £ m ¢ l

#+3$ Mnom s),/ 28#+,#/?76+2d< / ;76@ $ 78+,s97

Mm; / 76@ $ &+,b,#

$ 2 '* r d +,97$,# 7 /9#/?<># #<>#1] £ m ¢ jAl ] £ m ¢_ l / 9#+),+ #9 #/ / 76@/?7 ), $,# ] £ m ¢ lasu$$|<>#

Kp = M−1SFST Mnom .] £ m ¢ Al

0+ "&#, <326769 +,$,#S / 76@e | r:)!"#$ $3#$9$ 7$ 9;#<>#.$ a 97 + '*#;/[] £ mg £ l] £ mg lEmon &<,267890#C9#+ #2 #;72>)3+,7 ;78#\s +,9723r:) f/ ;78@eY $,# 7 / we),t7 @#r:) s 2 978$,#+d<,#<>#76!v"&# ),76\]`_a)3+,b3>_a# ?c e ;)giij,B%n=;#<>#;78!v"#ylZm

c c abdc hgi RN BKp = M−1SFST Mnom qGUT>Y.N F = diagF1, . . . ,Fm+1

NCB[J?BVXFZIxw

SNQPEV|Y.N*TWT%IXYABNEJ RGkFNB&PJ?BVFZIxwSNQP

M,Mnom ∈ Rm×mBSPQPE.J[N `PON %N

(MnomM−1) > 0 RN Fj ≥ 0

D3BFB BU zJj ∈ 1, . . . ,m + 1 N Fi > 0

q ∀i ∈ i ∈1, . . . ,m + 1 : i 6=j qN?T%V`x.G

(SFST )>0N −Kp

PONEF dzF =IKV w

FGUrSB! / ;78@(SFST )

<>#.$ w <, + $< 2[9 #/h,76+!v"#[2876+ f$ /yAg

;76@ < # ;7 78 /?76$ %+,76$ \]Si

l

SFST =m+1∑

i=1

FiSi ,] £ m ¢ Al

#+3$ S1 = Blockdiag1, 0m−1

S2 = BlockdiagJ, 0m−2

Sj = Blockdiag0j−2, J, 0m−j]

j =3, . . . ,m − 1lE

Sm =Blockdiag0m−2, JSm+1 =Blockdiag0m−1, 1

0l ∈Rl×l

s / 76@d+.),2

J =

1 −1

−1 1

.] £ m £ iAl

f/\ @r:) Fi≥0 ∀i∈1, . . . ,m + 1 z<>#b,78<

Si≥0z<>#.$ 9#+,9 26)376r:)

(SFST )≥ 0mf26s / $,7 #,+,# +,$,#S r:) db,78< ;)3/178$ #, #d9# ,97 +

Fi

78/1<32679&/r:) ?/ ;78@(SFST )

<>#;),7 <># #[9 #/?<,2 #, + "#[<>#.$ 9#+3926)376r:) (SFST )>0

mn <,;#Or:) \/ 76@ −Kp

sfe),t7 @&.)y &+'*#/&!v"#u$ ;78/1782&;7$$ Kp = MKpM

−1 = SFSTMnomM−1 m\ +,$,#S P = (SFST )−1 > 0

+ r:)&!v"#$ ¡ <,)z+,#O

PKp + KTp P = MnomM−1 +(MnomM−1)T > 0

9 #+,928),7 r.) \/ ;78@ −Kp

9 #+ r + / + −Kp

"&#1ef),t7 @m <,28769 !v"&# $Cn0#<>#;78!v"&# £ m8 / 9 #+U)3+ #h9 #/ #C # / £ mg\ ),2 $,76 &/ +

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] c ] hg e G TzPEIXY.NEFQN GPEI6PEV NEJ?B ~vGSFZJ?BAY.G=D%NEUB0D>UB[T%V B q:D%NEUBfJ1BVXFZIxwaJ|I6P VXzFBAY.GSFB q:D NEU6GtJ[GY.NEU6GfY.N0FQN*~vNEFzn?TpkEIXB q:D%NEUBdU6NEI Y.NkQGUT%VFGSU6N kQGSJ

Sp =I q

q q N D%NEU6G T0UVFG RNG T zJ[NEFG Y.NBV3BAY.GSFNQP kQG&JdD>U6NEV`BJ[N?T%V NI TGQD%NEFB[T%V NQP ~vGSFJ[N?TGSFuY.G %N

2N\BSP .ID ySV NQPONQP v q v Kk [N v Kk ~vGSFNEJ PBVXI6P*~vNEIKV`BSPoq

N?T%V`x.G B|NQPEVFBVs`IXB 7v PONEFU6GkQBUWJ[N?T%V NfNH D%G T N?TpkEIXBUWJ[N?T%V NfNQPEV|[rNEU qIXNxq ∃k, λ>0V`BI6P N ‖z(t)‖≤k exp(−λt)‖z(0)‖ q ∀t≥0

.vfU sEJ YI6PQPOG qqPONδ>0

qN?T%V`x.GGNEFZFG[Y.NwPBXYABe(t)

PONuV GSFGT>BF#T%zU6G1BED yOP\BU zJV NEJdD GTWT%IKV G

j

" -0B.1 'J=?+ A*B & 'J=?/ 7354765+ 35+ + Sl>@),+5=2B R+5=2B)35Sl>B 3

:039= J35N

n & 7628) <326769 !"#[$,# f .f_ # 9#+ #2 #2 + [' 26b 9#+;78$ S 9 $ 7[9#/1<># [<># 9 ;# ;r:) / 76@ $ +*o78), £ m 3m9 ;76+7f$ 6$"# 26#.9 76$$ a$,# 9;# 7#

y=[v1, v2]T #h # $ $,#|<&;972>s

x1 =[l31, v3]T

#+3$ l31

s <>#;78!v"# $,# 9 ;# < 78#]*+,# 9 + #:l / 2!"# # <,;78/ 76#9&;;# v3s 28#.978$$ $,#9;#< ;78#3mK 9 ;# "&#9#+ 9 $3#C<>#\),/ /?# 9 $3#\9#/

9 # %9 7 + $ /?# 9 76/ + #b23 = 1 Ns/m

),/[/?#2?9 #/ 9 # %9 7 + 26 769 #k31 =

1 N/mm

m

v v vF F

Atuador 1k

m

3

2

2

b23

3

31

1

m

Atuador 2

Atuador 3

31l

2FF

2

10

31

d7&/$9 $ 7$ 9 ;#\9#+ ;76$ &$+,# /?<,26#$ <,28769&!v"#$,# .f_ #9#+ #2 #2 &+ |' 28b m

d/ $$ 76d$,#f9;# 78# "#m1 =1 kg

m2 =2 kg

76/ 1/ ;76@$ 78+,s97 s

M = diag1, 2 m / $3#9&;;#< ;7#s|76+39 3R+,# + + # r:) ?+h' 7

0, 5≤m3≤1, 5 (kg)m

$,78+ /?789a$9$ 7$ 9;#(<>#.$ 5 <3 + $f< 2 r:)!v"&#f$ $,#C] £ m ¢ jAl] £ m ¢¢ la9 #/

A11 =

0 1

−k31/m3 −b23/m3

, A12 =

−1 0

0 b23/m3

,] £ m £ yl

A21 =

k31/m1 0

0 b23/m2

, A22 =

0 0

0 −b23/m2

.] £ m £ gl

f# S r:) / ;76@d9#/?<+,b 78A11

s /?<, ef),t7 @ ;78/ b376< $ ' /?6+376/ C+ 9 ;;7 <,28769&!v"#h$,#1 ._ s 7' 7 3m

¢

/?#.$ 28#|$ ' +,97 9#28b,76$3#s

WM(s) =

1s+2

0

0 1s+2

,] £ m £ jAl

#r:) < /17 ;78/1<,287%9 !v"&# $ '*),+3!v"#$ /?#.$,),2!v"&#] £ mj[AlC),/ @r:) <>#.$ ;9#26b

c1 =0 / .76 ),$ $,#[_a#;#26;78#gzmm

_a#/?# a/ a$3#a9;# 78#;"#h9#+,b 9 76$ < p' 7 / + .) 782678@ S C/ 76@$ 78+,s97t+,#/?76+7

Mnom =M+,#19v&269)326#$?/ 76@/17 )3$,#] £ m ¢ lEm f/ ;78@

+,#/178+7Knom

p

Anom

22

<,26789&$ +C !v"&#h$,#h9#+ ;#2 +3#/?76+2] £ mj lZ"&#h9#/?<,) S$t<|#?9 # /r.) #.$,# t# )$,# t#< &/< p' 7 / + ] F1 =F2 =F3 =1

lEmf/h @\r:) +%"#?b%< ),;&!- $ + $]

d1(t)≡0

d2(t)≡0lZ%#t9# %97 +

$['*)3+,!v"#$ /1#.$,)32!"#[9# ;<>#+,$ + ;"# ;9 #28b,76$,# 9#/1#c4 = c6 = 0

m w$ /7<&z/ ;#$h'*),+,!v"&#1$ /?#.$,),2!v"&#h'*#/ ;9 #26b376$,# t$ '*#;/hr.) #?$ /?< +3b,# //28b[' 9Qb$ /+ 76$,# / ;7 )! - u+ ur:)7&#/ +,#w$,#7 )$,# u#< /< ' 7 / + u/ u76+,9

m3

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c2 =3, 9c3 =2, 2

c5 =3, 4

γ1 =0, 32 s−1

δ=0, 1

m

$ /1< +,b,#$3#;7 /?$ 9 #+ ;#2 #2 + ?' 28b r:),7<,;#<># #1<>#.$ <, 9 7$3#[+,#d ),2 $,#f$ ;78/h),2&!v"#<, + $,# w+ o78), £ m £ mygzm /?<,26#3# ;76+2$ ' +,9 7s

r = [16 sen(12t) , −8 sen(6t)]Tm / $,# 9 ;# < ;7# s

m3 =0, 5 kgm

0+3r.)+ #0≤ t < 0, 67 s

#.$,## )$,# #< / < ' 7 / + ] F1 =F2 =

F3 = 1lEm /?#.$,# $ 2678@+ s|78+,769 7&$,#+ w$

0, 1 s(r:)+3$,## ;;#$ / + #

#++:),26#9 #+3'*#/ # + o78), £ m8Oizm n t ≥ 0, 67 s

# )$,#1 9 #/?<,2 / + 78+,#< + ] F1 =0

lZ39 #+ r + / + F1 =0 N

'*#!C$3# )$3#dsw $37 ;78,),6$h< 28##) ;# )$,# %9#/1# < 9 +o78), £ m3m $ /1< +,b,#$,# f X_ s78+ +; 2 ' 28b 3;),< 3a#r:) s .76$ + + o78), £ m8 £ myg.m

£

|&,9#$ |'*#;! + o76), £ m "&##9 #+ ;#2 r:),72 + 76/$,# |<>#%2 #5< S 7 $ <,;78/ 76 #$ / 9 #/ 9#+ + $ /?<>#f< r.) +|]

1 mslE&r:) "&#

+ 9 76#),/u @dr:) #;78+7$ 9#+ #2 "#C$ ;9 #+ 6+:),#a$,)3+ #C$ ;2876@&/ + #,m

%#;! &$ < 28# )y$,# C]

F1

F2

F3

lEm

=;;# $ 6$3m

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l #;76+2$ 6$1$,#[/1#S$ 28#?$ ' +397]

vM1

lZm

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vM2

lZm

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#72),+,7 ;78#,md"# ),/ #t9 #+,b 978/ + #$/ 76@o$ &+,b,#t$ &2 '* rm +,9 7t$<,2&+ u]

Kp

lEm |<,76+,9 76<&7C ),2 $,# 2 9 / r:) # 7 / s28#28/ + z<>#+ +,9726/ + v 2 <>#.$ q<,#U &$,#< dr.) #/?#.$,#d$ 2678@+ +;)3< '*897

e=0 76+,78$,#

/ /?<>#%+,7 #, $ ;$ r.) −KpSp

),// ;76@?e),t7 @&q#+,$ Sp

s1),// 76@9 #+ + ;< 978,9$h< 26# <3;#U 7 3m, #Cs/ +,# ;7 7# $3#Cr:) \b,78< $ r:) KpSp

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Kp

md28s/$ 76+,9 @ < /?s 769 +<,2+ 3$,/?7 /| < ),;&!- $ + $

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A

&

! $& "

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ξ(s)= sn∗

I B - n∗≥ 1 C D- E GF # 2 H +JI 2 K , 2ML n∗=1 C 5*/ N O +

P Q 0- 3 F R &AS T! PVU + W( X Y/ N O FZ .[\ 5 2 2 H]7 $ ^ 2 +

I - 3$- F 3 Q `_ \ 2 \ 2 a 0D- GF b , # - 2 2 L c 2 +ed 3 f ] L D 2 V B2 ]7 >g ? ?- 3 \- # - 2 ihj D- ,+Gk ,+ml !on B /p , & B /p qMr % ls-sWl < B n B /p & B /p qMr % l s-sWlt ( B # - 2 2 & 3$ 3 , u wvxg P /y!z <:|4< t~ 8*|6o:|;6.:`8 P / (R 2 vZv I !on +k B n +k-W ( +

2 ^ C \ W D \ % '\- 3- C ? $ !n +k- (

e = WM(s)L(s)Knom[U0 − L−1(s)UN

],

! +k (

- Knom C 2 F -? 2

K :=KpSp

+ P ]7 * -L(s) B

-*! k-+ ( BC # % b 2L G(s)L(s)

WM(s)L(s) & 2 D- \ F # - 2

`_ +eI -L(s) C g 2 p 2 \0]7 3$ F

-

+

−

+

+

−

+

+

−

+

−

1 2

d(t)unome

e

ε0

L

L−1

rρ0

u

U0

U0

UN

y

yM

G(s)

WM(s)

WMLKnom

Sp

ρ0ε0

‖ε0‖

"!$#&%f(''g*),+ P / $ 2 'D- GF G# - 2 n∗ ≥ 2+ I ]7 3 e

N ρ0

# 2 2 $ 2 ]7 j D- 2 +P % R \ L C Rhj D- + l.+ d g \ 2 ]7 3 >g X 3 b a - .- k0/ - l1/ F.% a $ - 3$- B - # - 2 v * + O +,+

# L ! ~ 8<3254 ~ 6 8;:$( L 2 % 5 5hj D- + l B L C X 5\- - F 3- `_ +I D- \ F Y `_ # % \ g

L(s) = L = I+

6 B e ≡ 0\- L37 2 2 & " $ C 2 B

, , L 2 5hj D O +k-+

I 84 ~ 6:`=92.8;:=<>24<F−1

i (τs) hj D- +ml 3 )g X 2 2 3 %w

# $[

F−1i (τs) = [favi(τs)I]−1 ,

! + l (

- favi(τs)

3 &? 2 n' A@ % 2 τs - & # - 2 L u] 3$ \ &

D & =Bw/ `_ !favi(0)=1

( B + D + B favi(τs)= τs + 1+ v \- 2

τ >0# - ,]7 2 \ L B ]7 2 C "# \ 2 2 2

kss

−

+

−

+ε1εN

F−11F−1

N

L−11L−1

N

ρ1ρN

U0U1UN−1UN ρ1ε1

‖ε1‖ρN

εN

‖εN‖

"! %6 3$ 3 c a] 3$ F # X Vc\- - F 3 x 3_ ! \ - L ( +

`=76:`= ~ 808.4 z < ~ 876 8 ! k-Wl (

(Ui−1)eq ≈ F−1i (τs)Ui−1 .

! +m (

6 2 # ]7 F X ' vxg ),+ P / $ 2 0vZv I 2 !n k-s ( +YI X & \ !on +ek- B n +kW ( 2 $ F 2 p X ' vWg*),+ P / F 2 # 3 4 ~ 6o:3= 2.8 :=<>24< +

! #"

d 2 2 >g w 5'ag ),+ P / 2 2 ]7 R Z K 2 g \ 3 # 2 E!n B /p , & B /p q r % $ ls-s-l <( +%$ Q\ 2 C# 3 2 !on k-s ( B 2 g L j# @ e 2

ρi

L e L 2 % 2 3 j\ 2 2 D 2 2 ]7 3

tsi 2 2 & 2

\- - F 3 `_ + B $ % g \- 2 τ 4 ~ 6o:3= 2.8 :=<>24<

2 g (Ui−1)eq =F−1

i (τs) (Ui−1) B - L37 2 B (UN)eq =L(s)(U0)eq

+ 6 B $ e 3 Nw X .

WMLKnom \ 2 F. $ S ^]7 2 3_ F B - 3 B 2 ; -\ 2 \ + 2 F % L T 2 % 2 3E.\- 2 2 & T 3$- F 3- `_ B 2 g

ε0 = 0 B L7 2 B \ N

e(t) 2 L \

% ; -\ 2 \ 2 & N* X .\WMLKnom + T HD-

GF 2 - L T 2 B \- F \D-[\ \ N C S .\- 2 2 H]7 3 F. & (' 2 R X .\

WMLKnom +ksZk

6 5 F 5D T F D- X R, L \ 3 2 L b \- 3 2 C 2 \- - L gF \ -$ $ - \ 2 L 2

τ → +0+

P # 2 & $ u!e( *]7 F. H 7_ 2 g

]7 + 3 B 2 2 &T \ , &F $X X ^ 2 % 2 \ 3 R 2 2 &0 H- 3$- F 3- 3_ L V\- $ -

U0

!n k-- ( F. p L 3 p \ f 4 ~ 6o:3= 2.8:=<>24< + P $ 2 # ? 2 \ ,<6o6 8;:|4 T \F. _ F +0I 2 & c \ C f 2 c % 2 A _ H 0 2 2 & \- - VF - _ +

"! "$##%'&(&(# )+* ,&

P L - .+./ @ / 0 Z HT# - 2 1 L " $ \- - b + 7_ 2X !on 3 + k-W B n B /p & B/p q r % ls-sWl <( +

B ! +ml (" ! +k ( B $ Z ε0 =e − e 4 ' 2

ε0 = WM(s)Kp

[u−θ∗T ω + Wd(s) ∗ d(t)

]− WM(s)L(s)Knom

[U0−L−1(s)UN

].

! + (

5 g6u=θnomT ω − SpUN B K =KpSp

U := (Knom)−1Kp

[

(θ∗ − θnom)T ω − Wd(s) ∗ d(t)]

−[I − (Knom)−1K

]UN ,

! + O (

$ . ε0

2 !o + (p 4 ' 2

ε0 = WM(s)L(s)Knom[−U0 − L−1(s)U

],

! + (

- ] \ L R R 2 Q!θ∗=θnom K =Knom ( $ $-

U0

;F \#7\ 5 X

d(t)+

I - $ 8 Z K ]7 ^ F. w#943 ? -

εi = F−1i (τs)Ui−1 − L−1

i (s)Ui .! + (

ks-l

P g6U0

! X3 !o + ($( 2 ! +m ( B $ Z 2 \ $E]7 F. w#9 C -

ε1 = L−11 (s)

−F−1

1 (τs)L1(s)L−1(s)

[(Knom)−1W−1

M (s)ε0 + U]− U1

.

! + U (

P F C V F ' $. \ 2 B $ Z \ \ 2

εi = L−1i (s)

[−Ui − F−1

1,i (τs)L−1i+1,N (s)U

]− πei − π0i , (i = 1, . . . , N − 1) ,

! + (

εN = −L−1N (s)(Knom)−1K

[UN +F−1

1,N (τs)Ud

]−

[I−(Knom)−1K

]βuN − πeN − π0N ,

!o +Gks (

- Li,j(s)=

∏j

k=i Lk(s)!Li,j(s)=I

j <i

( B Fi,j(τs) C ]7 w# - 2 _-D B

Ud = S−1p

[

(θ∗ − θnom)T ω − Wd(s)∗d(t)]

,!o +Gk-k (

βuN = [F1,N (τs) − I] F−11,N (τs)L−1

N (s)UN ,!o +Gk)l (

πei =i−1∑

j=1

Lj,i−1(s)F−1j+1,i(τs) εj = Li−1(s)F

−1i (τs)[πe,i−1+εi−1] ,

!o +Gk (

π0i = [WM(s)F1,i(τs)Li,N(s)Knom]−1 ε0 ,!o +Gk (

πe1≡0 B -?\- F +

# & # "$# & & ## +* 4&

/p >g $ 2 2 L @Y $ ! +ml-l ( ! +ml ( B !o + ( B !o + ( B !o +Gks ( + d \ $3S ]7 $N # \@*2 B $ % + r / - 2 \ πρi(t)

!i = 0, . . . , N

( B + + B |πρi(t)| ≤ R exp(−λt) B ∀t ≥ 0 B 2 - R ≥ 0

λ>0 L . + I F !o + ( C 2

Xε

L L x0FL

2 2 $S D- -

L−1 2 !o + ( B F−11,i L−1

i+1,N

ks

2 !o +m (30 b 2 - 2 . 2 βuN , πei, π0i

2 !o +Gk)l ( !o +k ( B C 2 ]7 N w .K# \@^ 2 + 2 0F\% L 3 - 8 ]7 $ 3 _ F\ B ∃KFL, aFL >0

# 2 L

∥∥x0

FL(t)∥∥ ≤ KFL exp(−aFLt)

∥∥x0

FL(0)∥∥ .

!o +Gk O (

P ] 2 \F 2 \- $ 0# 2 \ 2 + \- @c B 3 %)g F - 4

z 2 ]7

zT :=[(z0)T , εT

N , XTe

],

(z0)T :=[XT

ε , εT1 , εT

2 , . . . , εTN−1, (x

0FL)T

].

!o +Gk (

L D- B K 8 a 8 2 X - G% 2 a $3GF. D C . - g2 Y 2 -*! ‖·‖ ( C %Y 2 L∞e

! FZ*v \ k-+mlZ+Gk ( -Π/ 9-

Π0 /$ 2 X - % 2 L L 2 5# - 2 K‖z(0)‖ exp(−at)

K‖z0(0)‖ exp(−at) B $ g

F. 2 +I D $ 2 X 4 F $D-[, H $ Z J# $ F. 2 !n B

/p & B /p q r % $ ls-sWl < B $ - 2 ( 2 ;F. 2 2 2 2 g - 2 " \

πρi(t)L L 2 Y# \@

2 +

<:$<N = n∗−1 ≥ 1 `=,;4 2.8;:`8a=c8;:|:3='<-4 ~ 4<:38` !"$# !&%'#b8( !&)*$#!

+ 8 −Knom 8 −(Knom)−1K , =>:380:.-V.:0/e4 62180<> , 43658`;2.8 :*=2 ~ <73 8.=R<64$491>8;:380:

ρ0 ≥ (1 + cd0)∥∥L−1 ∗ U

∥∥ + cε0‖ε0‖ + πρ0(t) ,

!o +k) (

ρi ≥ (1 + cdi)∥∥(F−1

1,i L−1i+1,N ) ∗ U

∥∥ + πρi(t) , (i = 1, . . . , N − 1) ,

! +k U (

ρN ≥ (1 + cdN)∥∥F−1

1,N ∗ Ud

∥∥ + πρN(t) ,

!o +k (

∀t ≥ 0 `= : `=,;6 < 698` <2.8 <2W<>cε0 ≥ 0

8cdi ≥ 0 (i = 0, . . . , N) 876:8.= =u8;:;:3=

<6-4 ~ 4<:38`εi (i = 0, . . . , N − 1)

6 8 2.8;:;8.= <<1>8;:3= <.= :*8= 8 '==8 ;4< ~ :*876 8!?> ~ <0:ks

24`=

‖εi(t)‖ , ‖Xε(t)‖ ≤ Π0 ,!o +mls (

‖εN(t)‖ ≤ τ∥∥I−(Knom)−1K

∥∥KeNC(t)+Π ,

! +mlZk (

8

‖πei(t)‖ , ‖π0i(t)‖ ≤ Π0 , (i = 1, . . . , N) ,!o +ml-l (

‖βuN(t)‖ ≤ τKβNC(t) + Π0 ,!o +ml (

= 2.8

C(t) = Mθ‖ωt‖∞ + Mred ,!o +ml (

`= : `=,;6 < 7698` ==;4 64 z <>Mθ

8Mred

<2.8 ,<2W<>!

:3= z <

v D w /f--_ l.+ l B ε0

Xε

2 !o + ( - F $D 2 c% 2 , 2 B # , 2 ,

ρ0. ] % D ! +k ( +

L L 2 $ L−1 C -

π1(t)

π2(t) ≡ πρ0(t) BL + 2 - - Π0 + /p- L37 2 B ;g6 -\ L ‖Xε(t)‖ ≤ Π0

‖ε0(t)‖≤Π0 B ∀t≥0+

B !o +Gk ( |g \F π0i = Hi(s)ε0 = hi(t) ∗ ε0(t) + π0

0i(t)+ 2 F% L

‖π00i‖≤Π0 ‖hi(t) ∗ ε0(t)‖≤Π0 B ` ‖π0i‖≤Π0 B ∀t≥ 0

+ P D-- B N > 1

i = 1 B 2 F% L πe1 ≡ 0 B L !o + ( 2 ‖ε1(t)‖ ≤ Π0 B ∀t ≥ 0 B u r 2 lZ+ \ 2 β(t)≡0

+83 G% g 2 D- 2 $ 2 B -\ g F 2 u 3 !o + ( B

! +k (0E r 2 lZ+ \ 2 β(t) ≡ 0 L ‖πei(t)‖ ≤ Π0 ‖εi(t)‖ ≤ Π0 B ∀t ≥ 0 (i =

2, . . . , N − 1)+

P D- \- $ >g6εN

2 !o +ks ( + L B &AS. ! PcU + W( ! FZ / N O ( BMθ

Mred

2 - & ? # - 2 L ‖UN(t)t‖∞≤C(t)+ 2 HF% L πe,N−1

εN−1

2 2 - T -Π0 B 2 2 2 - 1 πeN B + + B

ks O

‖πeN(t)‖≤Π0 B ∀t≥0+ B ! +Gk)l ( 2 g

‖(βuN − β0uN)t‖∞ ≤

∥∥(F1,N (τs)−I)F−1

1,N (τs)L−1N (s)

∥∥

︸ ︷︷ ︸

O(τ)

C(t) = τKβNC(t) ,! + l O (

- β0

uN(t) C 2 - -Π0 + 6 ` B g r 2 lZ+m & L !o +Gks ( B L

4 ? \ 2

εN = −L−1N (Knom)−1K

[UN + F−1

1,NUd

]−

[I − (Knom)−1K

] [βuN − β0

uN

]−

−[(

I − (Knom)−1K)β0

uN + πeN + π0N

],

!o +ml (

|g \-\ L 2 - !o +mlZk ( C F _ + 2 F% L ‖(βuN − β0uN)t‖∞ ≥

‖βuN−β0uN‖≥‖βuN‖ − Π0 B 3 2 - ! + l ( C X !o + l O ( +

" = A8`=>:380:< !&) < `= 24 3 8.= -V.:0/e4 621Y= t :`8 −Knom 8 −(Knom)−1K=7=2.8;:|4<8;:w<6o4 , 8;4 6 <8``= ~ 78 2.= 8

Knom = knomI `= :knom ∈ R knom > 0

! <:|6o4 ;~ <: :*876 8 8

Knom = knomI<u`= 24 3 8.== t :38*< :<6:|4 1 2.82W< 7= 2.8< ~ 6 < , :`8 ;4< <

;4: = ~ 8`0:*876 8 −K2.8 z 8c8;: -V.:0/e4 621 =>T8\< <0.4 =>6 8`8 >#!

" ! " #"

I $ "# 2 %$ l B'& #(")F *)" 2 + n -, B/. ,01 B/. " q r3254 6#-#*75) lssWl < B $ 8"#89+ O: B X78; & #-" & # 2 8) *)<)5 X 2 7 2 ) *)=)" 2 9;8"9 >") .". L , *?@@)" 2 0 2 ) 8#-#-"A)*) & "# + %$B: B + %$ : + %$ ks :C$ d 6# N =0 + n∗ =1

: 0 X 2 7 2 ) 6), & "00; 2 *7C X78; 2 ) 0" $ 8"#89+ OD$ k $

FE HG*I(GN =n∗ − 1≥1 G `; :KJC J LJ G : G*MNIPO 1J` −Knom J −(Knom)−1K

J G : -V I / OQM 1RJLJ G , 4365J` 2J :KS2 ~TG 3 8S G*MNO , G 3 G : !&)U #6V !&) %#6!AW M 8S = G*I(Gτ > 0

; 4K O J M J0:KJ M J =J JXS SE O M J0: G YS : =S M S =J ~TG ZJ G 365J` 2SZJ IPI S\[!F]B]# !"$# !&%#^J !&) *$#cYS :_J` M`G 2S

z2J$4 O 2S^J0: !&)$#KJ I(a ~ S tYG~ :KJ M J@J '=SXJ ObG~ :AJ M J

J` Mca z J ~ J0: I J ~TG 3 8S G :yYSd| M S I J` O 2- G~ 2JeS I 2J0:τ O M S < ∃a,Kz >0

M`G*O #LJ

‖z(t)‖ ≤ Kz exp(−at)‖z(0)‖ + O(τ) , + %$ l :

ks

∀z(0) ∀t≥0!

<I S z G

d "#=;8"080 2 [ 0; 2 #(;#;g + %$ ks : ;8"9A"

εN = −L−1N

[UN +U

]+βuN−πeN−π0N , + %$ l U :

"0%)βuN =(F1,N−I)F−1

1,NL−1N U & ")5#<9 ("#*)%" & "*# C(t) + 2 ) + D$ l :-: 2 9 2 7*#-9/0 -

βuN

89 + %$ l O: 2 $ $ ‖(βuN−β0uN)t‖∞≤τKβNC(t)

$ +88789HX%#*0)%" g L , u=unom−SpUN 0".>g6 L , L−1

N

9 + %$ l U : " & 8# 0"R90-9 " 2 0 *7UN

L ,0 L ,7/89 + $ l-l :C$ P & 6# 2 #) + $ ll : + D$ ks : "0#(#-" )#-#6980 -" )%" 9A")7F" & ")4#=#-;8# 2 -" ;"9 "

Xe = AcXe + BcKnom

[

˙eN + αN eN

]

, + %$ l :

"0%)eN := εN−(βuN − πeN − π0N)

$ d *#7 2 9 2 0 *#- )# 2 .*) ˙eN 89 & # )g 0-#*0g "#(9+6?*") *# 2 _ 2

Xe :=Xe − BcKnomeN

#-,%760)"/9

˙Xe = AcXe + (Ac + αNI)BcKnomeN . + %$ -s :

9 4 L , Ac n ,# @ 2 4 & 6# 2 # ) + %$ lZk : 9Hg L , ‖Xe(t)‖≤τKC(t)+Π ∀t≥0

$ P 7 9 ) 2 (" ;"0 "*#-90 -;876#; 2 )"0X 2 "

‖Xe(t)‖ ‖e(t)‖ ≤ τK0C(t) + Π , + %$ ,k :

‖ωt‖∞ ≤ τK1C(t) + K2‖z(0)‖ + Km , + %$ Wl :

C(t) ≤ K ′red + K4‖z(0)‖

1 − τK3

. + %$ - :

B .>" ) 2 , 67F) *) 9 + %$ ,k : [ 9 )Xe = Xe −BcK

nomeN

$ B #76?*"+ $ k : 895g6 L , ‖ω‖ ≤ KM +KΩ‖Xe‖ 80 6" "-X 9 + %$ Wl : & *# 2 #) + %$ ,k : "0%)

K2‖z(0)‖ 89 )%" .*75"# 2 0 2 ; 2 67 )"/8#-9 "Π89 + %$ ,k :P$ P *"# & 6# 2 # ) + %$ l :

+ %$ -l : C(t)*9RX 9 & ")3#H9 ("#*)%" & "# C(t)≤τK3C(t)+K4‖z(0)‖+K ′

red )" L , 67

"X 9 & "# 2 0 8#-9 ) 2 "A)e9+60 2 & ,%76?367 X# 2 ; " 9 ("#*0 + %$ - : L , _ 7 2 )%"ks-

& 6# τ <K−13

$ . "0 "*#-90 & 7 2 ;6)"0/, 2 # & ");g64-;8# #869HX 9

‖z0(t)‖ ≤ Kz exp(−azt)‖z0(0)‖ , + D$ :

‖ze(t)‖ ≤ τK5

(‖ze(0)‖ + ‖z0(0)‖

)+ O(τ) + Π . + %$ O:

B .>" .*# 2 _ 2 Xε εi+ i=0, . . . , N − 1

: x0

FL

*"9 ("#6) & "*# Π0 0%" $ 8"#|g9 %$ k $ P 2 9 & *# 2 ; 2 "0*0)" g6 zT =[(z0)T , zT

e ] "*0) zTe := [εT

N , XTe ]

"-X 9Hg + %$ : "0%) ] ,%# & 0 . ;"*0) 2 ?*" 2 0 2 ; 2 67B) z0

$ P "# ) + %$ lZk : + %$ ,k : + %$ - : 9 + %$ O: "0%) " #(9A" # 2 ), *7

O(τ) 2 0) & 80)80 ) !;"0) 2 ? 2 0 2 ; 2 2 $ ".*0)%" g L ,A"2 0*0 2 0 2 ; 2 67 2 #-#7.60 0)8# 2 *?6" ) .8 & # *; 2 9+ & ")|g 4(;# #

‖ze(t)‖ ≤ τK5 + K6 exp[−a(t − tk)]‖ze(tk)‖ + ‖z0(tk)‖

+ O(τ) , + D$ B:

‖z0(t)‖ ≤ Kz exp[−az(t − tk)]‖z0(tk)‖ , + D$ W :

‖ze(tk+1)‖ ≤ λ(‖ze(tk)‖ + ‖z0(tk)‖

)+ O(τ) , + D$ U :

‖z0(tk+1)‖ ≤ λ‖z0(tk)‖ , + D$ - :

& 6# t ≥ tk ≥ 0 + k = 0, 1, . . .: *#X 2 ># _ # 2 "$+*7 ,9 & 8#N5")" T1 = tk+1 − tk > 0

*) L , *)"& 6#!"-X8# λ<1

$P L , 6? + D$ U : + %$ - : *"^"-X 2 ) . ) + %$ B: + %$ W : ;"*9A" *, & *# τ <K−1

5

2 T1 > 0

67 L , λ = maxτK5 + K6 exp(−aT1), Kz exp(−azT1) < 1$ 6 0 *" *"

_ 7 2 ) ) 2 , 67F) 6) 7 2 06#H#8;,# 2 + %$ U : + %$ - : L , & 8#-9 2 9 ;"0%;7F, 2 # L , & 6# τ > 0

-, ] ; 2 0 900 & L ,80" " 2 89+R) L , 6?e)%"28#-#(" 75"-X *759/0 & "*00%; 2 67F900 4 _ 73;8"9 #76?*"/,9 ;8"0(,%0 >" # 2 ), 67 )!"#()9

τ$

## " #

83?*"4 X "#-) )g " 2 9 & 79080 *?*"') ,0? )@9A"),%7*?*" & *# "V(''g*),+ P . #),?*"e)" G*MbM J IPO $ P & #0 >g6 *9RX 9 ,9+ 8# *")"R('ag ),+ P . X *0 2 95g& 7 2 ] ;*) , & ") # ,6) ^0 & # _ 2 ; , 60)" "# (, ] ; 2 0 ^"X80?*"^) X 2 7 2 ) *)75";*7 $

ks U

6#!" ;-" 89 ,n∗=1 " & #(" >") ,%0?<)'9 "),7*?*" )0 "7 2 )" )*751 ! ) 6980 0 Jv 8?#" $%$ " $'&D$%$$)( , 2 *"!)80"7 2 ) ,0? )=9 "),7*?*"3 2 9 2 76#

& 6#e";-" n∗>1$*( & 7 2 ;! 6/" r 89+ $+$ & ) 2 , *75) *) + %$%$ : + %$%$ : ,!)9 #

# & 2 *) & 7 ,0?')!9A")%,7*?*" ρi+ i=0, . . . , N

: "-,80)". 6 & *# ρ0

ρ0 = δ0 + cε0‖ε0‖ + cω0‖L−1(s)ω‖ + cU0‖L−1(s)UN‖ + d0(t) , + D$ Ws :

d0(t) = cd0d(t) +cd0

s + λd0

∗ d(t) , + D$ $ :

;8"9 ;8"0 *0 & #-" & # 2 *) cε0, cω0, cU0, cd0, cd0 ≥ 0 δ0 ≥ 06#/, 2 >#106# 2 "

λd0 > λd0 > 0 "0%)

λd0 9*# 89 ) ., 2 7 2 ) 6)^)"327>#("L−1(s)Wd(s)

$B "*#-9/6040*75" e"-, 93 6

ρi = δi + cωi

∥∥F−1

1,i (τs)L−1i+1,N (s)ω

∥∥ + cUi

∥∥F−1

1,i (τs)L−1i+1,N (s)UN

∥∥ + di(t) , + D$ &*:

di(t) = cdid(t) +cdi

s + λdi

∗ d(t) , + D$ - :

& 6# i∈1, . . . , N − 1 ;"9 δi≥0*#5, 2 -#10*# 2 " ;"*0*0 & #-" & # 2 6) cωi, cUi, cdi, cdi≥0

λdi >λdi >0 "0) λdi 9+6# 89 )4., 2 7 2 ) 6)^)"327>#("

F−11,i (τs)L−1

i+1,N (s)Wd(s)$

*#ρN

96

ρN = δN + cωN

∥∥F−1

1,N (τs)ω∥∥ + dN(t) , + %$ :

;8"9δN ≥0

*#/, 2 >#/0*# 2 " /;"*0*0 & #(" & # 2 *) cωN , cdN , cdN ≥0 dN(t))*)" & "# + D$ W :

;8"9λdN >λdN >0 "0) λdN 9*# 9 )4-, 2 7 2 )*)^)"627-#-"

F−11,N (τs)Wd(s)

$( ;"9 & ,*?6") ) 2 #=;8"0*0 '0;106# 2 . & 2 9 & 79080 6?*"e). ,%0?

)/9 "),7*?6" & *#2 2 9@;8"9 n∗>1 & ") , 2 # & #(";8) 2 900 >" 8987F1 60 & ,7 , )67F1 6)"0 v 8?6"7" $'&D$%$ & *# 2 9=;8"9 *#*, #87. 2 ",0 2 "#(93!,0 2 806# 2 " $

( 2 9 & 79080 6?*") . ,0%? )9A")%,7*?*" & ") #8 2 9 & 7 2 2;*) /.-# ) & 7 2 ;*?6" )" r 89+ $+$ & 6# 8#*#9("*#*0 2 9 & 7 & 6# ‖ω‖ A",-#-" 2 0 2 927>#*)" ;8"0 "#(9 ) *751 *)" 9 + n (, . ,0%1 . " q r3254 *#(#*75) & s-s & G v 8?*" $'&: + . ,0%1 . " q n (, & ss-7t v ?*"$ :C$

$ s

( 2 9 & 78900 *?*" )"w(' *),+ ( . & ") 8# 2 0 2 2;. 2 .6900 2 9 & 7 2 2;6) 89 & 7 2 ;6? , # ,#89 & 0 -, 2 7 2 ) 6) 75";*7 $ 6 2 9 & 78900 *?*" ;"0 2 0" ,(" )*9 & 7 2 >,%)=);"0 >#-"*7!;"*060 ρi≡ ci >0 + i=0, . . . , N

:P$ 6 0 *" "27-#-"8 & 7 2 ;*)"0 8#*?*" ) . ,0?@)/9 "),76?*" 6"e)-0;10*# 2 " $ # >#*900 >" ;"*9 # 2 ?*") & #-,#/, 6?*" & "),8#="-, 2 )"e, 2 7 254 *0)"! .*75"#-6) , 6)" & *# *9 & 7 2 -,) ci

$ ;"0 >#("7*)%"# #-,%760 )0"*9 2 0 *)"=J ~ < > & "# ,/0%" ;-"RvZv " 2 0 2 )!;8"0 -#-"7, *" #*)" & "*#=#87 - 2 9 & 7 Ui =cisgn(εi)

+ n (, 5*7 $$ - :C$

! #"$"%'&)(+*

( & #80? ) 2 >" 0 *" 2 ) 2 ;8"9 " & ,0"0.>#-" K1 2 8# 0"e) 2 & " 2 2 ") ;P1 6980 -" # 2 "*,Z) 2 0 ' 9 2 ;0 *"! 9 ")7*) 0 & 7*0 #-,7A0"+-,# 2 900 >"A)" 80 ? 900"A)80"9 2 0 6)" G*MNM J ICO + 8 2 0 $ W-, v ?*" " )" . &/. >,75" :C$ 9 "#(9+) 2 *#" G*MNM J IPO -, , 2 >, 2 #" ;"0 >#("7 >"# 2 *7 ,0 2 806# 2 " )(;"0 . 0," + &D$%$ : & "# & #-" 2 9+6?H;"0 . 0, ;"*9A"0 4 "07 2 0*# + 6 ) @ *#() q v & ,%# 8"0 $ -0, & $ $:

U =

−ρ(x, t) v(x)‖v(x)‖

‖v(x)‖ ≥ δv ,

−δ−1v ρ(x, t)v(x)

‖v(x)‖ < δv ,

+ %$ " :

", 0 & #-" 2 9+6?*" 2 *9A" 2 )*7 + 6 ) @ 6#-) q v & ,%# 8"0 $ -0, && $$ " $ & :

U = −ρ(x, t)v(x)

‖v(x)‖ + δv

, + %$ B:

"0%)δv >0 ,9 -;*7*# ) .67F"# & ,0" ,@) 20 '7*# ,#=) '# 2 6"') & #-" 2 9*?*";8"0 . 0 , $ ( 96,. & #-" 2 9*? 2 0 >#("), 4 89 ,%9+ 4 "0 Z;8"0 . 0 , 0 2F482 01*0? )

)-;8"0 2 0 , 2 ) 6) "*# 2 2 0 *7 $ #(,7*)" #8),?*" )" *0%1" )" ;"0 >#("76)"# 0 2F482 0%1 *0? $ & #?" # & " " -,# 2 900 >" ),9 #(#-"R# 2 ),*7 + $ $ S21KJ M : ,*,%980 0 90) 2 ) 389 ,

δv

;8#-; $ "# ", >#-" 7*)" 4 & #-" 2 9+6?*" & ") # (,*) & 6#*#8*#5, 2 -#*# 2 69/0 & #43 2 9+e)7 2 )!;"0 >#("7 2 )67 (;"7510%)". δv & ,80" $

( 7 9 ) 2 (" & 7 2 ;*?*"$89 2 0 6" *)9 2 2 #^"(; 2 7*?!)*7 # 7/5 0; 2 & "# , ) "#()9 & #10 2 ; & ") 6*27>#6# "- 2 0 *7 8#*)" & 87H7 2 ) ;"0 >#("7 )-;"*0 . 0, 90(9A" ,0 6"$89 & #, 4 "0 ) & #(" 2 9+*?*"A;"0 . 0, $ 576 2 ,# %$%$ & ")8#

$ $$8

2 2 0;87F, 2 # ,%9 27-#-" & . , 2 80 ># " & "0 >" - $ / - & / 89 ) >#-, 2 #R" 9 ")")-7 2F4 *0 -9 ;*) !9*7F1 /)';8"0 >#-"*7 -"# 2 *7,0 2 806# 2 "3 *7 9 ) 2 -" " 2 0 *7)=;"*0 >#("7# 2 -, 254 6)" *0 )08#2 & 7 2 ;*)" & & 7*0 $ 6 & #-";) 2 980 -" /0 " 2 & 7 2 ;6)" 0"'wv *),+ ( . & ")8# 2 28#- & 7 2 ;6)" 0"5' ),+ ( . , 2 0)". 64., "#-) 9 ) ;"*0 >#("7)-7 2F4 *0 4(, & #(" & " /89 + (, $ W :C$

* # " ! #

( & #0 ,%9 9 & 7F" & *#Z9 " >#6# 1 & 7 2 ;-, 2 7 2 ) *)A)""(' *),+ ( . & #(9 2 2 #2.*7 2 *?*" )%" 8, )89 & 801" .># !)/ 2 9e,%7*?*" + (, . ,01 . " 2F4 6#-#67F)&-8-8 & G v 8?6" , :C$ ( 6 2 ,# D$ & #0 " ) 2 #*9 ) ,9 2 89 )1(, & 80 *"*# 2 ;,7*)" $ " , 2 " 4 #0 & " 2 ?6" ) ;6# + y2

: R" ' 0 *,7F" )R"# 2 0 *?*" ) ;*# + φ : # >#*#9 . ) )" 9A")7F" ) # # 5 0; 2 .># )%" ;8"9*0)" ) )" 2 .-, *)"*#!))-7F";*900 >" 7 2 06# $ 4 2 0 2 ^);8"9+60)" *" . "#(? )" >, *)%"#u1

u2

$ ( & 7*0 7 2 0*# 2F4 *) 0 2F482 01 60? )φ = 0 rad #-,7*0%)" 0 , 2 0

# & #80 *?6"+80 >#*) ! 6 . )

y =

1s2 0

0 1s2

Kp [u + d] , + D$ W :

Kp =

l1/J −l2/J

1/m 1/m

, d = K−1

p

0

g

, + %$ , :

"0%)e" -"#^) . ) y = [φ, y2]T " -"#/)e;8"0 -#-"7 u= [u1, u2]

T =u − dnom$

;8"0 -#-"7 0%"9 2 0 *7dnom =[50, 50]T N , 2 7 254 6)" & *#A;"9 & 0*?6" )" & (" ) A;*#

" , & 8#-9 2 #8), 482 # *9 & 7 2 >,)Z)%" 2 0 *7/) ;"0 >#-"*7u$ ( (,-", 6

d0(t) ≡ 250

d1(t) ≡ 500 ,0?K) 9 "),7*?*" 7 .*0)%". 9 ;8"0 & #-,#/,*?*" # 2 ), 67

+ d− dnom:P$ *"!,0e9+.-# 254 ) & *# ' 9 >#-"H0"9 2 0 2 + θnom =0

: & *# ,!" & #(" >")" ;"*0 >#("7*)"# 0; 2 ) & ",;" ;"01; 2 900 -" (" ,# " .67F"#e)" & *# ' 9 >#-"e) & 7*0 $

& 6# ' 90-#-" ) & 7*0 ,*)%"0 2 9e,76?2 6" A9Z) Z;*# m = 10 kg

9 "900 >" ) 2 0 #(; 2 ) ;*# J = 1 kg m2 *;7#6?*" ) # 2 )*)

g = 9, 81 m/s2 $ $-$

( -, *)"*# $ ( >, *)%"# &

g

l

l1 l2

. 6#

7 "#(9

u1 u2y2

φ

"!"% B 2 #*9)"+ 2 9A)-, & 0 6" $

0 5 10 15−0.2

−0.1

0

0.1

0.2

0.3

g

ll1l2

u1

u2

y2

φ

t (s)

e 1(r

ad),

e 2(m

)

"! %6 #-#-" ) . ) )%" 2 9+e)4-, & 0 *" $

0 5 10 15−0.1

0

0.1

0.2

0.3

t (s)

φ,φ

m(r

ad)

"! %/0 ,7F" )^"# 2 0 *?6"e) ;6# " 2 0*7<)$ . ))" 9 ")87F" + φ, φM

:C$

0 5 10 15−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

t (s)φ, φm (rad)

t (s)

y 2,y

m2(m

)

"!"%#" 2 ?6" ) ;*# A" 2 0 *7 ) . ) )" 9 ")75"+ y2, yM2

:P$

;8"9 & # 2 980 -" ) & 7. "*#-9 l = 4 m & " 2 ?6" ) ;*#

l2 = 3 m$ 6 & *# ' 9 >#-"

0 6" "#69 ,*)" 0" & #-" >" )"E(' *),+ ( . $ ( 9 ) & 7 "#(9 9+.. )".-, *)"*#8!"8.># 2 -" "*#*9 ) & # 4 *)%" $

9A")87F")!# # 5 0; 2 /-;8"7F1 2 )"

WM(s) =

1(s+1)(s+5)

0

0 1(s+1)(s+5)

. + %$ W :

& 6# ' 90-#-" )" ;8"0 -#-"7*)"#8 6" cω0 = cU0 =1 cω1 =0, 2 δ0 = δ1 =0, 1 θnom =0 L(s)=λ(s)=s + 5

Knom =0, 1 I

$( ;8"0) 2 ?6" ,# @ 2 4 # ,8# 2 ) & 6# & 7 2 ;*?*" )" (' ),+ ( . .

2 2 & 6#$ $ &

Sp = I/-"900

l1 >−J/m " , 89 & #8#-) 6) 2 #-" ,9+ 4 ,0 ;*# 9*0 2 )R)0 -#-" ) & 7 "#(9 $ " 80 *0 -" " & 7 2 ;*) *7 ,9 7 2 )K;"*0 >#("7 ,3 2 ,4 9.># 2F4

KpSp

& " 2 2 . ) 20 2 ) ;8"9 " & "#" 89 & 7F" + $ *" "*00",$ -,- . 1 2 80 367 $ $ - : 0 *"# 2 0;/0*# 2 "+ -, 2; 2 80 ,2.8*, 2 0 ';"0%) 2 ? "89 . 2 2 2 9,%760*900

l1 > − J

m, + %$ " 8B:

e l22 +2J

ml2 +

J2

m2− 4Jl

m< 0 . + %$ " $:

5 89 & 7F" ;6# )8# 2 # & " 2 ; 2 "*0 *) # & 2 *0%)". # ># 2 ?6"l2 <

1, 164 m ;(" ;"*0 >#/0*# 2 " )8# 2 8# (;"751 2 )",-#9.-# 2F4 Sp & *#,>"#(0 *# KpSp & " 2 2 e) 20 2 ) $8B ,8#H#*7*)%" ,,e) 20 2 ?6" & " 2 2 . ) KpSp

2 9 & 7 2 ; , (−KpSp) ,%# @ 2 4 " , -, # ,e"(' *),+ ( . & ")2#3 & 7 2 ;6)"+A;7^9 2 5*9 & 7.) & 7*0 =)" , ",>#("5*7"*# 2 -9 " ;"*9A" & "#5 9 & 7F" + $ 6" " 600", $ 0,0, $ *" " 600", $ 0,- . 1 2 80 5*7 $$ - :C$

( ;"*0*0 0 ,0? )H9 "),7*?6" + cω0 cω1cU0

: "#69 -;"*7F1 2 ) & *#/9+60 # *9 & 7 2 >,) )" 2 0*7)';8"0 -#-"7 u

*" & ,0 , *0 >" & " . 7 9+ " ,80)"! 5#(" ,, 4 )/ -, 2 7 2 ) *)3!)")9 & 01"A)" ;8"0 -#-"7*)"# 2 0;# 4 '0" & *# ' 90-#-") & 760 $

5

5 2 9,%7*? & #80 6) 0 6 2 ,#. %$ %$ "4 2 0 67%)=# 8# 5 0%; 2 r=[0, 1 sen(t);

0, 1 sqw(2t)]T "0) sqw(·) :=sgn[sen(·)] ,9"0) ,*)#6) $( ;8"0# 5 0; 2 )" 2 0 2 )" #(#-" ) . ) & "00; 2 *7 ;8"9A" "-,8# .*)%" 0

6 2 ,# %$ $ " 0 60 >" ),#*0 ^"2>#60 2 3# 2 " 2 0 2 ; 2 67 6"" ,# .*)" #*0) & 2 ;8"@;"9*9 & 7 2 >,%) 0, 25 rad (≈ 14)

9e1

0, 11 m

89e2 ,+ *" ;*,*)" & "# ;"0%) 2 ?

2 0 2 ; 2 2 ) "#10 2 $ -" (, 2 2; & #/0. 2 ; ) 1 (,*# K;"*0) 2 ? 2 0 2 ; 2 2 )"9 ")87F")!# # 5 0; 2 & *# ,!"08#-#-" 2 0 2 ; 2 *7 & ,0" $

B , ;6#*; 8# . 2 ;H)%"5' ),+ ( .& *# & 760 H;8"9 #*, #7. 2 "e,%0 2 "#(9n∗>1

*"/ 2 9 & " 2 , 2 7 2 ) 6)K) ;8"08# 5 0; 2 )"1#-#(" )$ . ) 89 9 & " 20 2 >" "8#-#-" & ") 0 *" 0%)#0 4 #-" 9 & *# ,9 ;"*0(,0 -" # 2 ), 67 $ >" ;"*0 ># ;"9 "

$ $

5' ),+ ( . & *# & 760 .;8"9 n∗ = 1 , & ")9 # & #(" 6)" & *# , " #(#-" )#-#6980 >";"*0 2 # & *# 4 #("289 9 & " 20 2 -" + -, # /;8"9 & *#6# 6 2 ,# D$ ;"9 6 2 *,#3" $ :P$

( ;8"0*0 H) 9 & "/)" 4 MNI S52J :=<>2 ObG + τ =0, 003 s: " 2 (;"751 2 )5-, 2; 2 0 900

& ,0 & *# 9*0 #" 2 8919 9*7F1 ;Y1 6) 10 87- " 8#-#-" ) # >#*900 >"& ,0"+ & 3=">#60 2 3# 2 " 2 0 2 ; 2 *7 ;"9 "/" ,# . 0 +6 2 ,# %$ %$m & *# t>5 s

$

0# & ) " # $# #

$#

'! *),+ ( . + !: "5' *) + ( . + ) ) !: & *# & 760 .') #6, #87 2 " ,%0 2 "*#-90n∗ > 1

, 2 7 2T4 *9 4 MNI S 2J :=< 2 ObG ;8"9 ;"060 K)89 & " τ-, 2; 2 0 900 & ,80 $

6 ;"0 *0 K)9 & " 2 9 2 76# *" & *# ' 90>#(" ε) & ,80" *7F"# , ;*#6;8# 254

"" ,# *)"#A) *7>" 601" + O G*O S#"$J I&% J I(' *) !: 89 & # 6)"/9 67,0;8"0 -#-"7*)"# & "# 9A")" )-7 2F4 *0 1 #*7 2 900 *?*" ) . ) $ 0 ? 900%" =LJ G,+O )-;8# 2 -"/89 + ,(9+*0%0 .- " ">" 20/ $121 $ - 1 *7 2 7 $3121-B: 04 8?*" 1%$%$'& & ") ";8"#-##0" 2 0 2 )" *) ;8"0 7 0 980 0" 2 0 2 ) ;"0 >#-"*7K0Z90) 2 ) ,

ε →+0

$ $ 60 >" 0"/;"0 >#("7*)%"#, *)%"39 "-,8# .6)"# )*7>" *01" , 60 >" 0" '! ),+ ( . /0"R(' ),+ ( . " #-#("@)/#.>#*900 >" 80)89% 4 #("0 90) 2 ) 89 ,^"& 6# ' 90-#-" & ,80" 0)9 4 #(" $ "-0 *0 >" 9 *7,%0 2 9, *)%" 89 5) + $ $ + 6 *0) 2 *# 2 6-!1*7 2 7 $1#1 & 6 90787 60" +8*7 $$121 & G 6 90797 *0" + 5*7 $$1#1 & " :(: " 2 0 2 & ")9 2 , 2 # & 2 ;8"H0" >#60 2 3# 2 " 2 0 2 ; 2 67 ;, -*9 & 7 2 -,) 6,980 *9 090) 2 ) 89 ,

ε → +0$ 6 9 ;"0 ># " '! *),+ ( . K"E5' *) + ( . & "(,9 & #-" & # 2 ) *)

) ., 2 7 2 ) 6), & "00; 2 *7 7F"-, *7 ,0 2 "#(9-89 #87*?*"τ ∈ (0, τ ∗] & *#0*7 ,9 .67F"#

(, 2; 2 0 8980 & ,80" τ ∗ > 0$ >" 2 9 & 7 2 ; ,+ -, 2 7 2 ) *) 75" , *73)"T(' ),+ ( .

+ ", '! *),+ ( . : & ## *) 3 7 2 # )" 80 ? 900" =J G,+O <0 /908) 2 ) ,τ → +0

$ " . &/. -,7F" 1 # 2 ! 6 " *) )0 "7 ,9 ),+ ( . , .6)" 89 ;"*0 >#("7

& >#(,>,%# .*# 2 07 0,%9 5) & *# 2 97 2 06#: e$ 2 9R)A;"*0 >#("7#-,%760 7 2 D# ) =J G,+O 3 & "(, 2 & #(" & # 2 )*) 75" , 2 @)4-, 2 7 2 )*) $

$ $<;

!#"%$ &'

( ) &#*+ , -. , /'0

8 . &/. -,7F" "21(' ),+ ( . )80"7 2 )" 0%" . &/. -,7F" " 80) 2 )%" & *#A;7. ) 2 9e0 *". `7 2 0*#e# & #80 6)" & 71 , *?6" ) *)" + $$43 &: , . 2 4 9

1 2 & 3 + ( $ : + (53B: + (5;$ : + 2 -" *#*, #87. 2 ",0 2 "#(93!,%0 2 10*# 2 " : + ( " : + (76:P$

9A")87F")!# # 5 0; 2 ) *#*, #87. 2 ",0 2 "#(93!,0 2 806# 2 " ) 6)" & "# + " $%$ :C$

6 . &/. >,75" ;"*96, 2 0 1. 2 ) 2 . 2 0 -#-"), 42 ) 9 + ) 2 0 (, &-8-808B: & *# ># *#& 8#>,%#/, *?R0 6". 7 2 06# 89 2 89+. ;"9 "81(' ),+ ( . & *# 2 89+. 9e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`7 2 0*# 2 ) *) ) & 80)80 $ & 80 )" 9 & " A) . ) ) & 7*0 0 , 60 >" , , 2 & ")#) & 0)0 69 , 9 )%"0 *)%"A0 6" 90) 2 )" ;"*0 "#-9041 2 & 3 + (76*:C$

$ $ "

,& & % 4# "$# ## & #

( & *#*90># 2F4 *?6" )%";8"0 -#-"7- , 2 & #0 *) 2 9 2 76# & ,7 )-;# 2 !0" . & . >,%7F" ;$'"40 *0 >" " -"# #*#-"# - 9.># 2F4 ) & 6# ' 90-#-" *" & 6# 2 ; 2 "0 *)" ) ,9+ "*#-99 2 ;"0 0 2 0 & *# " )0 "7 2 980 -"!) 7 2 ) ;"0 >#("7 & *#3 2 89 0 *". 7 2 0*# $ 6#";("09 & # >,#5, *?6" + φ(xp, t)≡0

: 7 2 )!;"*0 >#("7

u∗ = θ∗T ω + θ∗T4 r , + 6D$+$:

"0%)

θ∗T = [θ∗T1 θ∗T2 θ∗T3 ] , + 6D$ &*:

ω = [ωT1 ωT

2 yT ]T , + 6D$ 3:

0)" "=)9 2 8#-9 " ) 20 2 )"-9 + ;%$ " : + ;%$ 6:P$

"! "$# '& & #

)0"*7 2 900 >") , *?@)%" 8#-#-" & *#4 2 89+. 0 *". `7 2 0*# )*7F1 6)" 89 + -, ,*7 $ &-80893: , ?*" ;$'& ) $ $ 2 99 9+*751 0-,# A;"9 & "-" & 7& 7*0 + $$43 &: & 87F" 27-#-" + ;$%$<3B: & ") #=# & #0 *)%" & "#

X = AcX + BcKp

[u − θ∗T ω

]+ Bφφ ,

y = CoX , + 6D$ ;B:

"0%)!" >"#=)4 *)%" X :=[xTp , ωT

1 , ωT2 ]T

Bφ =

I

0

0

. + 6D$ " :

$ $ :

9A")87F")!# # 5 0; 2 & ") #=# & #0 *)" & "#

XM = AcXM + BcKp

[θ∗T4 r − Wφ(s) ∗ φ

]+ Bφφ ,

yM = CoXM , + 6D$ : :

;8"9

Wφ(s) = K−1p [WM(s)]−1 Co (sI − Ac)

−1 Bφ , + 6D$ 6*:

, ,9!9.># 2F4 )->#60 8# 5 0%; 2 4 80 87 $ 5 & 760 & "-, 2 # #*, #87. 2 "!,%0 2 "#(9 ,0 2 806# 2 " 0 *",/9.># 2F4 )">#*0 # 5 0; 2 Wφ(s)#/0 & # 3 & # 2 $ #(9A" Wφ(s)∗φ

;*0;7"/ 2 >") & # >,#5, *?6" φ

0 . ) yM ;"0 "*#-90 " 2 & #-" & "-" & "# + ) 2 0 -, &-8-808B:P$

( "*# (, ,># 2 0%)". + 6D$ : : ) + 6D$ ; : ;8"0;75, 2 , " -"#H)36)")" #-#(" + Xe :=

X − XM

: !"/#-#(" )4 . ) . 2 4 9

Xe = AcXe + BcKp [u − u] ,

e = CoXe , + 6D$ , :

"0%)

u = θ∗T ω + θ∗T4 r − Wφ(s) ∗ φ . + 6D$ 1B:

1 9 4 , Ac, Bc, Co ,%9+#*7 2F4 6?*") WM(s) 0 , 6?*"K)%"#-#(" & ") ##(;# 2 *,0)"+# & #0 *?*"0 -#*)! . )

e = WM(s)Kp [u − u] . + 6$+$ 8:

2 9 & "# *0 =#*7*# ,3 , *?H)" #-#(" + 6D$ , : + 6$+$ 8: *" !0*7 2 ) & *# 2 9=;"9 9.># 2F4 O M J I(G ; M S I *#/, 2 >#/0*# 2 )-) , "A9 ")75" ) # # 5 0; 2 0 (;"751 2 )%"*) , *)*980 $ '" 0 *0 >" 0"e;.-"#67 9.># 2F4 )5>#60 8# 5 0%; 2 Wφ(s) & ")8#102#2 9 & #43 & # 2 ;8"0 "#(9 ) 2 -;8, 2 )"0" . &/. -,7F" , $

$ $<6

&(# # ,& #! ,& "

& % # # & ,& #

( 7 2 ) ;8"0 -#-"7 >"# 2 *7 ,0 2 806# 2 " 89 e90(9+ "#(9+ ) e7 2 ) ;"*0 >#("7 & 6# 2 97 2 0*#<) *#*, #7. 2 " ,0 2 10*# 2 " + 2 ) + " $ ; : + " $ : : + -, . " . ,01 &-808 & G (, . " . ,01 &-808 & " :-:

u = unom + SpU , + 6$+$ $:U = −ρ

e

‖e‖ , + 6D$%$ &:

unom = θnomT ω + θnomT4 r , + 6D$%$<3B:

"0%)θnom θnom

4

*" .*75"#=0%"9 2 0 2 & *# θ∗

θ∗4 # & 8; 2 .6980 $

B #-" & " 2 ?*" &D$+$ 96 ,-^;"0 # 5 0; 2 , & "*00; 2 67)%"48#-#("!) . ) & *# 4 #("#10" , 2 ) ,0%?*" )9 "),7*?6" 2 2 4 8#5) 2 *, *75) *)

ρ ≥ δ+ce‖e‖+(1+cd)∥∥∥S−1

p

[

(θnom − θ∗)T ω + (θnom4 − θ∗4)

T r + Wφ(s) ∗ φ]∥∥∥ , + 6D$%$3; :

"0%)ce, cd≥0

*";"060 8*) , *) . ,4. 2 4 9%H) 2 *, *7F)*) + &D$%$ $ : δ≥0

,9/;"060 -6#/, 2 >#106# 2 $ 1 9 ,0?*" )H9A"),%7*?*" , 2 4 + 6D$%$3; : %;87F, 2 0)"! #-9 " & "00; 2 *75980 !)8;#-;0 #*)" & 87F"-#*0 2 43# 2 ") 27>#(" 80 2

ρ = δ + c1‖ω‖ + c2‖r‖ + c3‖e‖ + φ(t) , + 6$+$ " :

;8"9

φ(t) =

[

c4 +c5

s + γφ

]

‖φ‖ ≥ (1 + cd)∥∥S−1

p Wφ(s) ∗ φ∥∥ . + 6D$%$ : :

1 9 4 ,/K9+.-# 254 )2>#*0 # 5 0; 2 Wφ(s) & #43 & # 2 807 " , 96 e" 9! ("#*0 + 6D$%$ : : .-# H) & 7 2 ;*?*" )%" 9 3%$%$ "0) γφ

"-,)8;γφ >γφ >0 0)%" γφ

9+6# 89 )4-, 2 7 2 ) *)^)Wφ(s)

$ #(9A" ‖φ‖ 89 + 6D$%$ : : 0 6" & ")"# 98) 2 )%" $ "# 9 ) /1 2 & 3 + ( 6: H79 ,#*0)".

$ $ ,

, ‖X‖2 = ‖xp‖2 + ‖ω1‖2 + ‖ω2‖2 "-, 93 64/ 2 9+. 2

‖φ‖ ≤ kx

√

‖X‖2 − ‖ω1‖2 − ‖ω2‖2 + ϕ(y, t) . + 6D$+$<6:

4 2 0 2 ω1 ω2

ϕ 6" ) 2 & "*0 . 2 $ ( 0"#(9+ )" 6)" ‖X‖ & ")08# 2 9*)

.-# =) 0 & 7 2 ;*?*")" 89+ 3D$ & & + 6D$ ;B: ,#(,70"9+ ("#*0

‖X‖ ≤ c6τ‖Uav‖ +1

s + γx

[c7ϕ + (c8 + τc9)‖Uav‖ + c10‖ω‖ + c11‖r‖] , + 6$+$ , :

Uav =I

τs + 1Ueq =

I

τs + 1U , + 6D$%$1B:

"0%)Ueq " ;YS MNI S,FJ J O %G FJ M J γx := γ − cφkx > 0 λ0 > γ > 0 λ0 R9*# 89 )

., 2 7 2 ) 6)^) 9.-# 254Ac

cφ >0

10 ) 20 2 )" 0%" 9 3%$ &$ ;YS MNI S,FJ J O %G FJ M J

Ueq

0 *" ) 2 & "0 . 7 & "*# 9 & ") 8# (, , 2 -, . )" & 7F";PS MbI S,FJ:=<>2 O S

Uav

#*)" & 75" 27>#-" & . , 2 + 6D$%$1B: ;"*9 ;8"0 *0 )889 & " τ >0) .67F"#

*#5, 2 -#10*# 2 " $ ,(" )" ;YS MbI S,FJ=:=<>2 O S0 6" ,9+ & #-" 2 9+6?*" )%" ;YS MNI S,FJRJ O %G FJ M J& " 2 " 2 2 0%;75, . )"e# 2 "#-"*9080 ^0" 9 3%$ & !0 *"/!# ,8# , τ

& ,0" $

" > J McG " O O 2 G 2J 2S O M J0: G 2J ;YS MbI S,FJ I J J I G O : OQM`G 3 8S 2S G 2S2 OQI J OQM S 2 G 2J O G 2 G 2JYU !&)'# S LJ I J J I

γx > 0! S M`G JJ

γJ

cφ

2J =J 2J0:2S :KS2J$FS,2J I J , J I ; ObG J 2S(4 MNI S 2J J MbI-G 2 G J 2J4 G 2 G J <;YS, O 2S! >5 O : 2J % JCJ M J McG*I J <;YS, J I S :KS 2J$FS 2J I J , J I ; ObG = G*I(G : O O : O 1 G*IG I J$ G 3 8S

cφ/γJ J M 8S J

G 2: OQMbOQI-a LJ G O ;YJ IPM J;1 G J G : G*O S I O M S < S %G FS I 2Jkx

=S2J I(a J I : G*O S I !: G% J;1 J J McG ;YS McG M J 2J % J0: J I J <;YS, O 2 G 2J , S I : G LJ ‖wφ(t)‖ ≤ cφ exp(−γt) ∀t ≥ 0 S 2J

wφ(t)< G I J =S McG O : = O %G ;YS ICI J =S 2J M J : G*MNIPO 1 2J MbI(G , J I ; ObG

(sI − Ac)−1Bφ

%O 2J J0: G !F]# S G S 2S >cφ/(s + γ)

2 G 2S =LS Icφ/γ

#2J % J!J I : O O : O 1 G 2S S J =S2J!J I!I J G O 1 G 2S =J$FS :=< M S 2S 2JS MbO : O 1 G 3 8S = I S$=S M S J0:) G <S McG - ] * * , # %O 2J + J 3 8S ! ! !F] 2J McG J J #!

5E W: % J;1 2J*J : O O : O 1 G*I S2 G S 2J: > ;YS , S I :KJ G "J I&%G 3 8S U !&) G G M J I G*MbO %G = I S$=S M`G J0: - <S McG G ] * *] , - <S McG G ] * *] # < G : G O : O 1 G 3 8 S 2 G : G*I J0: 2J!J McG " O O 2 G 2J

γ0

# 2 G : G*MNIPO 1Ac

J MNO : G 2 GG*MbI(G2% < 2J , S I : G G 2 I-a*MbO ; G 2J G = XS % %O 2J "$J I&%G 3 8S ! #! HG*I-GAG = G*IPM J* O XJ G*I2 G = G McG J0: =J ICM I " G 3 8SJ G :KJ McG 2 G =LJ$FS4 MbI S 2J!J MNI(G 2 G J 2J G 2 G J M J = I S#"<FJC

$ $31

: G =S2J5J I I J S, %O 2S =J$FS = I S dJ M S 2 G FJ O 2J ;PS MbI S,FJu=θnomT ω

LJ S MNO : O 1 GKG : G*I J0:2J/J McG " O O 2 G 2J J MbO : G 2 G 2S5 O M J0: G J0: : G G , J ; G 2 G !@W J < :?= I S#"<FJ0: G 2J5;PS MbI S,FJI S#" M S"CJ0:;YSJ ; O 2S J =S 2J J I I J S, %O 2S =S I S MNO : O 1 G 3 8S ;YS % J G G 2SeJ O G 2 G 2J O XJ G*I J G*MNIPO ; ObG*O 2 0*# ) .># 2 0 , *7 2 2 #<^S 2 G G,+IPO G W G S O J I S ) % %[#! SAJ M`G M S S S 2J , S I : G G 2 I-a*MbO ; G 2J G = XS % = G*I(GG ;YS : = M`G 3 8S 2S;YSJ$4 ; O J M J 92S > , S O J %OQMcG 2S XJ McG J J =S O =S2J I J McG*I J0:J MbO : G*MNO %G : OQM S ;YS J I&%G 2S I-G ;YS , S I :KJ5J ;YS ;$ O G + J 3 8S ! [!

HG*I-G LJ G G = O ; G 3 8S 2DS 3J0: G !F] I J M J : O M J0: G J M`a#% J$@J0:: G G , J ; G 2 G < XJ ;YJ a*IPO SLJ G : G*MbIPO 1

Ac

J G - I / OQM 1 ;YS 2 O 3 8SJ < G*MNO , J OQMcG=S O J G < G : G*MNIPO 1 2 G J G 3 8S 2SRJ ICI S! S 2 G J G 3 8S 2SRJ IPI S O : = O 4 ; G O O 4 ; G*MbO %G :KJ M J G J MbO : G 3 8DS 2 G XS I : G 2S J M`G 2S ‖xp‖

LJ=< G 2 G XS : G dS I(G M J YU !&)U #!W: O - ] * * *$# J MbO O 1 G#%G :;YS MNI S, G 2S I O XJ G*IG 2 O ; O S G = G*I(G J M`G " O O 1 G*IG= G McG J J M 8S =S O " O OQM`G*IG J MbO : G 3 8S 2 G XS I : G ‖xp‖

!

( 9.-# 2F4 ) -#*0 # 5 0; 2 ) & 7*0 G(s) 2 0;# 2 & 8#0; ,9 ) *) ;7 P , ,9 -, , ;8"0(,0 >" )

Rm×m(s)$ . *) 7900 >" ) P 2 4 1 2 & 3 + ( $ :

+ ( 3B: + ( ;$ : + ( " : & *# Sp

ν2 %" $ ( 2 9 & 79/0 *?6" ) ,0?*" ) 9 "),76?*"

& 7 2 ;*0)%". + 6D$%$ " : + 6D$%$1B: # ,#5/, 2 0 !1 2 & 3. *) 2 ; 2 "0 *7 *"A;8"018; 2 )" .*75"# & 6#0';8"0*0 ci≥0 + i = 1, . . . , 11

: γφ, γx >0

) "#-9 ,+ ) 2 , 67F) *) + 6D$%$3; : . 2 2 + D;75, 2 0)%". /8#-9 "4 & "080; 2 *7F900 e);8#-;0 : ∀G(s)∈P ;"9 *7*,9 & *# (θ∗, θ∗4)

;8"#-# & "*0)0 + ;*)7900 -") P , 2 *?/ , *?*"A) 2 " *0 2 0 + ;$%$ $ :P$

B *7F15-" ,%# "K; 0*7F;8,7F" )^;8"0 *0 *" & #0 *)"/0 8?*" 6D$ ;$ *7*". # 2 >9 " 1(' *),+ ( . , 2 & #(" & "-" #-,9 2 )%"0 $ -,7 6D$+$*$

*#/80,0; 2 6#'" $ "*#9) /-, 2 7 2 )*)^) ;8"0 -#-"7*)"# ;8"0 2 )#! "0 2 9 "#(9+6)" & 7 , 6?*" )" 8#-#(" + 6$ , : " >"*#)$ *)" 6,980 *)"

z = [XTe , x0T

F ]T "0%) " -"# ) 6)"

x0F

# & #80 " 8#-9 "$>#*0 2 43# 2 " )%" 27>#(" 9 + 6D$+$ : : + 6D$+$ , : + 6$+$31B: & 7 2 ;*)"!0 #*?*" )" 2 0 67

φ$ 6 0 6" "-, 5 +9 + -, . "

. ,0%1 &-8-8 & G (, 0*7 $ & 80893B: " #(,7*)" )$ -, 2 7 2 ) *)0,%0; 2 *)" 0" $ "#9 , 2 # $

$ & 8

3 !$#&% ( 7*"# 2 >9" & #(" & "-" & *#" ;8"0 -#-"7 >"# 2 *7,0 2 80*# 2 " & "*#<9 ")87F") # # 5 0; 2 & *# 2 9 0 *"! 7 2 0*#H;8"9 #*, #7. 2 ",0 2 "#(9 !,0 2 806# 2 " $

yM = AMyM + r

! "#$%&e(t) = y(t) − yM (t)

'("")&+*,-.*&)/0ω1 = Φω1 + Γu , ω2 = Φω2 + Γy

12)034+.**0ω = [ωT

1 ωT2 yT ]T

5%.06)&0u = unom + SpU , U = −ρ e

‖e‖

7 0 )&0#089:;$unom = θnomT ω + θnomT

4 r

7 0 )&08<.=Uav = I

τs+1U

'>?A@0BC8 => 0?A@0ρ = δ + c1‖ω‖ + c2‖r‖ + c3‖e‖ + φ(t)

DFE$Gφ(t) =

[

c4 + c5s+γφ

] [

kx

√

max

X2 − ‖ω1‖2 − ‖ω2‖2, 0

+ ϕ(y, t)

]

H2*&)&"8B$)&"I0 ‖X‖ X(t) = c6τ‖Uav‖ + 1s+γx

[c7ϕ + (c8 + τc9)‖Uav‖ + c10‖ω‖ + c11‖r‖]DFE$GJ K>?A@0max·, 0 L0M" )& =>N.PO#$/PI6")/$,Q/$R"N-ST>#0=/09U V8+*34T$)&"IW+*3S >

OX & .0&,I6B09)&/$*&")Y0&B+*[Z")&+*3>)&"""N\0+*,#P4W/0?A@0*&"#$X(t) ]

S O 2J I J@S O M J0: G ;YS : =S M S =J$ G = G McG ) ! ]#^JHS > 2DJ ; ICOQM S G G "CJ$ G U !&)! + J G / O = M J J >()#6V > # > [! G # > #6V >U # J >(% ! ; # , S I J0: G*MNO C, J OQMcG J M 8S S > J I(a #FS#" G :KJ M J J '=LSXJ ; ObG :KJ M J J M`a#% J$ O M S < ∃k, λ>0McG*O LJ ‖z(t)‖≤k exp(−λt)‖z(0)‖ ∀t≥0

! > <0: 2 O S J δ>0 J M 8S S J IPI S=2J! G 2 Ge(t)

J M S I G*I(a FS G =L G : M J0: =LS.4 OQM S!

<I S %G( & #-" 8 & #0 *)889 + -, (*7 $ &-8-893B: 6040*7F" & & #-" . )" $ 8"#89+" $%$

& 6#2 2 9@7 2 0*#<) #*, #87. 2 " ,%0 2 "*#-905^,0 2 806# 2 "2 & #0 *)0 $ $ ( ) 2 #0?=9+ 2 H#87 .*0 8 & *#;9 0 0 & 7 2 ;*?6") #(" & " 2 ?*" &D$%$

( , 2 2 0 -#-"), 4 " 2 0 *7 & "*00; 2 *75900 )8;#-;0 π2(t)

+ ‖π2(t)‖ ≤ k4

exp(−λ5t) ‖x0F (0)‖ ;"9

0 < λ5 < min(γφ, γx, τ−1)

: & *#Z# & #0 *#" >#*0 2 3! # 2 ")"27>#(" , #69 ,0?*" )!9A")%,7*?*"

#(9A") & 8#-,#/, 6?*" ) *)%" & "# dU(t)=S−1p [unom − u]

$

$ & $

&(# " # " &(# #

& #-"-" )"51(' *),+ ( . & *#8 2 9 0 6". 7 2 06#3) #6, #87. 2 "=,0 2 "*#-90 ,0 2 10*# 2 "& ")4#=)0"*7 2 )"0, 2 0)". 6!" & ("')-;# 2 -"8/, 2 # $

$$ 6 & ; 2 2;! "9A")7F")# 8# 5 0%; 2 ;8"9 "/9 + " $+$:C$

&D$ 6 (;"751 ,9 & "*7 2 0 ? 9 2 "9 ? 0 2 ;" ,# @ 2 4 λ(s))#6,

ν − 1 & *#"27-#-" )0 >#6) @) . ) + ;$ " :C$ 6 0 *" 6"3(;"751 2 ) 9.-# 254 Φ Γ & *# /#*7 2F4 *?6")27>#-" + ;$%$<3B:P$

3%$ 6 2 9+! 6,%9 7 2 9 2 2 0 8# 2 "# & 6# 9+*# 9 )-, 2 7 2 ) 6)γx = γ − cφkx

;"0*0 cφ, γ>0

.># ) '9 2 0 2 9 2T4 6?*"=)%" *01" B . + cφ/γ: )%"6 ( 6

cφ/(s+

γ) & *# 9.-# 254 ) -#*0 # 5 0; 2 (sI−Ac)−1Bφ

+ 2 ) 9 3%$'& ,8# .6?*" 6D$%$ :P$ >" & ") 8#3#*7 254 *)%"- ># )"=9 -")"=) ". 2 9 254 *?6" & #-" & "-"=0 8?*" 3%$43%$ 3D$ &D$

;$ 49*# 9 ),-, 2 7 2 )*)γx

"#H9e, 2 >" & ,0 ", 0 . 2 . 0 *"#>"*#-0 *" & (" $$9( 3 & ; 2 2;6?)" 9 ")87F"K)e# # 5 0; 2 e)"27>#-"/) 0 -#*) )8 . ) !)89 # # 2 . , γx

8>"*#-0 & " 2 2 "48-, 2; 2 80 8900 #*0%) $

" $ . "9 & , /!9.># 2F4 ) & *# ' 90>#(" 0"9 2 0 2 θnom θnom4 & *# & 7*0 0"!

9 2 0 *7 $ ( A9.># 254 ) & *# ' 9 >#-" 0"9 2 0 2 )9 . 2 4 #A;8"0) 2 ?A);*900 -"R)-;8# 2 489 + -# 7 @")("0 $1 , 1:C$9B A0"*# ," .*75"#^)θnom & ")/0 *"28#%0 2 ;8" ;8"0 "#(9 ) 2 -;8, 2 )"289 + ) ) >187 2 0 @")("0 $121#;B:C$( 2 9 " & #-" 2 7 2 # & *#,(;"*7F18# ,*7 ,8# .*75"# )=;*900 -" , "# ;8"0 0 2 0 $ ( , 2 8-, #54-;"*7F1 !) *7F"*# ,-01 69 0"*#-9 (, 2; 2 80 8980 & ,0 29 )49*0 # & ,0 0*9 & 7 2 >,)^)%" ;"0 >#("7 0%"9 2 0 *7 $

: $ 6 (;"751 *7F"*# & *# ;8"0 *0 <0 ,%0?*") 9 "),7*?6" + 6D$%$ " : , & " 2 , 2 7 2 9 ,8 ) 2 , *75) *) + 6D$%$3; : , 2 2 $ 6 ;"*0*0 )9 . 2 4 8#

c1 ≥ (1 + cd)∥∥∥S−1

p (θnom − θ∗)T∥∥∥ , + 6D$'&-8B:

c2 ≥ (1 + cd)∥∥∥S−1

p (θnom4 − θ∗4)

T∥∥∥ , + 6D$'& $ :

c3 ≥ ce , + 6D$'& &:

$ &-&

"0)ce, cd≥0

2 4 9%e) 2 *, *7F)*) + &D$%$ $ : & *#e9.># 2F4 " 2 9 ># 2 ; P,Q>0,+. 2 *?*9 , 6?*" ) 7 & ,0" + &D$ 6*:C$ B K79 ,#*# ,+R9.># 2F4) #67 2F4 *?6"R)" 9 ")75"R)A# 8# 5 0%; 2 (;"751 2 )%" "#!) "*#-9

AM = −γMI;"9

γM > 0 0 *"1 ,0?*"Z)K9 "),7*?*" & ")#/0 #2 2 9 & 7 2 2;*) ;"9 c3 = ce = 0 ;"0 "#-90$.,78; 2 )" 0" . "#-"7 0*# 2 " &$+$$ ( ;8"0 *0

δ ≥ 0 & ") #/-;"*7F1 2 ) *#/, 2 >#*# 2 69080 & ,0 ,9+ 4 , 08;10*# 2 $ & 80 & *# *#60 2 # K;8"0 # 5 0; 2 )"08#-#(" )4 . ) & 6# 4 #("89%89 & "620 2 >" $

6D$ 6 (;"751 .67F"# & 6# .!;8"0*0 c4, c5 ≥ 0

γφ > 0 2 , ) 2 , 67F) 6)

+ 6D$%$ : : 4 2 2 ^)-;8"0 *0)"! -#(9A" & "080; 2 *7F900 )8;#-;0 $ *#@" ,# .*75"# & *#5 ;8"0 *0 #-;# - 9+.-# 254 ) >#*0 8# 5 0; 2 /;8"9A"(1+cd)S

−1p Wφ(s)=W (s)+(1+cd)S

−1p K−1

p Cp "*0) W (s)=(1+cd)S−1p K−1

p (CoAc−AMCo)(sI − Ac)

−1Bφ

$ 6 0 *" & ") 6 -;8"7F1#

c4 ≥ (1 + cd)∥∥S−1

p K−1p Cp

∥∥ . + 6D$'& 3B:

( <;"0 *0 c5

γφ

)89 # -;8"7F1 2 ) .<;"*0 "#-90'" 9 3%$%$$ *# /9*0 8#& ,0/2*9 & 7 2 -,)') ,0?*"e)/9 "),7*?6" @;"060 H)89 8#H" 2 9 2 4 *) & *# 9 2 0 2 9 2T4 *#/" *01" B . )" 6 ( 6

c5/(s + γφ) ;8"0 "#(9 )-;#0 8?*" 3%$43%$43%$

, $+ .*75"# & *# ;8"0 *0 γ, γx

cφ

/0 "#*9 " , 2 )"0" & -" 3%$ & #43. 2 9 " & -" R" ,80?6" ) ;"*060

cu, cτ ≥ 0 2 ,A" 27>#-"

cu/(s + γ)

cτ/(s + γ) *9 6 ( 6 & 6# 9+ ># 254 ) >#60 8# 5 0%; 2 (sI − Ac)

−1Bo

(sI − Ac)−1AcBoSp # & ; 2 .*900 $ ( 9 2 0 2 9 2T4 6?*"). ;"0 *0 ) 2 96

& "# ' 0; 2 & #/0. 2 ; $ , *0 >" 980"#^" .67F"#/) ^;"*0*0 *0 -" 900%"# 8#100*9 & 7 2 >,) )%"0 2 0 *7 )!;8"0 -#-"7 $

1%$( ;"0 *0 c6, . . . , c11≥0

) 9)#5 2 , A) 2 , *75) *) + 6D$+$ , : . 2 2 & *# , *7 ,# *7F"#"-;"*7F1 2 )" & 6#;8"0 *0 )389 & ")" 27-#-"e) 9 ) 2 + τ > 0

: )-;8"0 *0)"! 8#(9A" & "00%; 2 *7F900 =);8#-;0 $( & *# 2 # ) + 6D$ ; :$ & 3

4*, 2 0)%". !" & -"') & #-" .)" 9 3D$ & "-, 93 6

c6 ≥ ‖BoSp‖ ,

c7 ≥ cφ ,

c8 ≥ cu‖Sp‖ ,

c9 ≥ cτ + cφkx‖BoSp‖ , + 6D$'& ; :c10 ≥ cu‖θ∗ − θnom‖ ,

c11 ≥ cu‖θnom4 ‖ ,

"0)4!,69Bo = BcKp

) 20 2 ?*" )%" ;"0 >#("7 0%"9 2 0 *7 9 + 6D$+$ 3B:C$

* # " ! # # ## & # #"&

6# 2 75,-#*# & 7 2 ;6?*"!)" 1 ( . ;8"0 2 )8#! 6 0 ?6" " ;"0 >#-"*7 >"7#60 8 *751 )H,9+/;*) 2 /;"9 & " & "*# ># 5 ;*#(#-" ;8"0 "#(9 6 2 *,# 6D$%$ + (,/ 67 $-&-80893:C$ 2 0 *7)@;"*0 >#("7 " >"*# ) "#(? )*) )" .-, *)"*# u ∈ R3 " 2 0*7)" . ) " >"*# ) 75"; 2 ) *)

y=[v1, v2]T<" >"#)"*)" xp =[v1, v2, v3]

T$ 0 ,9 .-, *)"*#

#),0%) *0 & *# ,A" 2 9 >"7#*0 & & #() >"*7 ) ,9 )" .-, *)"# ) "#(9+-6040*7F" -*" , 2 >"=0 5 8?6"9" $ ;$ ( 9+.. )" ;6#-#-" *" m1 =1 kg m2 =2 kg

m3 =0, 5 kg$

Atuador 1 Atuador 3

Atuador 2

F1

F2 F2

F3v1 v2v3

m1 m2m3

B31 B23

Fd(v3 − v1) Fd(v2 − v3)

2 #*9 ) R;*) 2 R)A;*#-#(";"0 2 )#6) 0" 9 & 7F" )+ & 7 2 ;6?*" )"1 ( . *" ;8"0 -#-"7 -"78#*0 2 *751 $ ,*9 "#;)"#3(, & # 2 "# *"0 *". `7 2 0*# !" 2 0 8# 2 "*# & "(,9 *9 "# 8; 2 9080 -"7 2 0*# $

;*#(#-"2 *"Z;"*0; *)%" & "# 69A"*#8;8)"# )" 2 )" , 2 & "-,89 ;8" 2; 2 80 $ &,;

) *9 "#; 2 900 >"A7 2 0*#B31 =B23 =1 Ns/m

"/",>#("/)" 2 3*9 "# ;)"*# *" 0 *"! 7 2 0*# $ ( "#(? ) ;*)+*9 "#;8)"#'0 6". 7 2 06# ,9 ,0?*"K) 75"; 2 ) *)#87. 2 .80 ># "5>#9 " )" *9 "# ;)"*# + Fd(v)=0, 1875v + π−1 tan−1(v)

: # & #80 6) *#! 2;*900 0 6 2 ,%# 6D$'&D$ 69A"# 8;)%"# 0 *". `7 2 0*# " 2=2 0 & 2 #*)" 0" >#-, *751" -" ,%#(, & 80 *"+9 2 . 2 e) + ) "#-)%"0 " $1#1#; :P$

d

−2.0 −1.6 −1.2 −0.8 −0.4 0 0.4 0.8 1.2 1.6 2.0−1.0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.0

−2.0 −1.6 −1.2 −0.8 −0.4 0 0.4 0.8 1.2 1.6 2.0−1.0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.0

−2.0 −1.6 −1.2 −0.8 −0.4 0 0.4 0.8 1.2 1.6 2.0−1.0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.0

F (

N)

v (m/s) 6"#-? )"+*9 "#;8)"#'0 *". 7 2 0*# 9 ,%0?*"A) 75"; 2 )*) + 7 2 0%1 ;Y1 2 :P$

( =7 2 01 & "0 2 7F1 *)H# & #0 *9 "=9 ("#60 ±(0, 5 + 0, 1875|v|) $

>"*#=) "#-?=#(,7*0 =)"8.-, *)"#Fr∈R2 ) *)" & "#

Fr = SF u , + 6$ & " :

"0%) 9.># 2F4

S =

1 −1 0

0 1 −1

+ 6D$'& : :

& 7 2 ;*) & 8#*?*" ) "#-?=#-,%760 -

F = diag F1,F2,F3 , + 6D$'&96:

"0%) " . 0) 2 ; ) *7F1 Fi =1 " O O S >, *)%"# 80'" & 8#*0)%" ;"#(# *900 0<Fi <1

$ & "

"/, )89 & 801" )*#*) 6)" ", Fi =0410 ;8"9 & 7 *900 2 0" & #60 $

1 9 4 , "3 2 89+ & "-, 2 )" 2 2 0 2 ) . ) *" 08;10*# 2 " & 0 )" 2 2 0 2 );8"0 -#-"7,80)". 6 89 2 " " , 2 " ) ,2 7F"; 2 ) 6)

v1

v2

2 69 8 . );8"#-# & "*0)0 =)"9 ")87F" )^# # 5 0; 2 $ 6# , " 2 9 #-,%760 e#-"-,,-"/ *751 H)"8.-, *)"*# , 2 7 254 ! 6,9.-# 254 9 2 -,#*)%"# + " $ ; 6: & #-" & " 0 8?*" " $ ;$ 3

u = ST Mnomu , + 6D$'& , :

"0%)Mnom ,9 *7F"# 0"9 2 0 *7 & *#- 9.># 254 ) 2 0 #-; 2 M :=diagm1,m2

$ ( ) 2 0 ' 9 2 ;) ;6) 2 )=;6#-#-" 2 0%;75, 2 0)%"4 9.># 2F4 9 2 -,#*)%"# & ")-8# # & #0 *) & 673 , 6?*")46)" + $$43 &: ;8"9

Ap =

−B31m−11 0 B31m

−11

0 −B23m−12 B23m

−12

B31m−13 B23m

−13 −(B31 + B23)m

−13

, Bp =

m−11 0

0 m−12

0 0

SFST Mnom ,

Cp =

1 0 0

0 1 0

, φ(xp) =

Fd(v3 − v1)m−11

−Fd(v2 − v3)m−12

[Fd(v2 − v3) − Fd(v3 − v1)] m−13

. + 6D$'&#1B:

18 2 7 2T4 *0)%". Mnom =I 9.-# 2F4 ) *01%" ) *7 # 5 0; 2 /8#10) *) & "#

Kp =

m−1

1 (F1 + F2) −m−11 F2

−m−12 F2 m−1

2 (F2 + F3)

. + 6D$4308B:

( ;8"0) 2 ?6" + ( " : ) , 9+ ># 254 −KpSp

,# 2 4 . 2 2 1-;8"7F10)"! Sp =I

89 + 6D$+$-$ : ;"0 "#-90,6 #-" & " 2 ?*" " $%$$ ) 2 0"e) 0" , e9.># 2F4 2 0;# Kp

+ 6D$ 3-8B: & ")48#8 2 0 *7 2 0) 20 2 )"0 & ")0 6"8#5 2 9 -# 2 ; " , 2 9 & ) 0 & 7 2 ;*?*")*7 "# 2 >9" ,!# ,8#89)) 20 2 ?*" )4 2 0 *7 2 9 ># 2 $

"*0)"! ," 2 89 + $$ 3 &: ;"9 + 6$ &#1: 10 0 "#-9K#*,7*# ;"0%;7F, 2 , & "-, 2 ,%9 4 #(" ) >#*0-9 2 *"A75";*7 254 *)"089

s=−(B31 + B23)m−13 ;8"0 0

900 !"/ 2 89 ) .!9 . 0 2 90 " , . 2 4 1 2 & 3. + ( $ :C$ 6 2 *,# 6D$'& <0" , "#-? )"(*9A"*#8;8)"#0 *"! 7 2 0*# 6"/7 2 9 2 6) & "#|Fd(v)|≤0, 1875|v| + 0, 5

$ 0 6" "8#-9 "0 *". `7 2 0*# & ") #=9+ ("#*)" & "*# ‖φ(xp)‖≤$ & :

0, 96‖xp‖ + 2, 1 ;"0 "#-9041 2 & 3 + (76:P$

9 ")75")!# # 5 0; 2 /(;"751 2 )%"

WM(s) =

1s+4

0

0 1s+4

, + 6D$43 $ :

" , & #(9 2 2 9 & 7 2 2;*?6" ) ,0%?*" )R9 "),7*?6" + 6D$+$ " : ,9+ 4 , & ")-;"*7F18#

c3 =09 2 # >,))" . "#-"7 0*# 2 " &D$%$$ & "7 2 0*9 2 ";*#6;8# . 2 ;"/-;8"7F1 2 )" & *#

" 27-#-" ) *)" λ(s)=s+10 " , #-,7 0 9+6# 89 ) ., 2 7 2 ) 6)γx =1, 36 s−1

$( <9.># 2F4 ) & *# ' 9 >#-" 0"9 2 0 2 θnom θnom

4

"#*9 ;"9 & ,*) & *# " ;-"289 ,4-")"="8.>,*)"#=" & 8#*9 & # 2 *980 + F1 =F2 =F3 =1

:P$ & 6# ' 90-#-" )" ;8"0 -#-"7*)"# "#69 -;8"751 2 )"/) "#-9 ,e" 2 99 9+67F1

8;P1*) 9*0 0%1 "0, , "9 )9 & 0%1"K)-) , *"980"=)" 2 -.>, 6)"#'" & #89;8"#-#6980 $ !)9 2 & 6# ' 90-#-"^)"R;8"0 >#-"7*)"*# "#69 ;*7F;8,7*)"3, 2 0)". 6"& .-")-;8# 2 >"!0 ?*" 6D$ ; c1 = 110 c2 = 3, 2 c4 = 4, 8 c5 = 21 c6 = 2, 7 c7 = 2, 5 c8 =4, 5 c9 =23 c10 =477 c11 =7, 2 γφ =3, 98 s−1 τ =0, 01 s

δ=0, 1

$

5

'" #-,7*)" ) 2 9,%7*?*" & #80 6)" 0 76 2 ,# 6D$ 3 6D$ : " 2 0 67 )Z# 8# 5 0; 2

)*)" & "*# r = [10 sqw(60t) , 10 sen(60t)]T$ 6# 9A" >#*#2 #-"-,, 4 )"Z)89 & 801"

)" 1 ( . #60) 2 0;# 4 & 6#*9 -# 2 ; " .-, *)"*# $ 10 ;"*9 & 7*900 2 0" & 8#*0 + F1≡0 F2≡F3≡1

:C$ 5 6 2 *,# 6D$ ; !" ,# .0;8"0# 5 0; 2 )%" #-#(" ) . ) & *# 4 8#-"09%9 & "320 2 -" $

$ & 6

0 0.02 0.04 0.06 0.08 0.1

−200

−100

0

100

200

t (s)

u1,u

2(N

)

2 0 2 H)!;"*0 >#("7 $

0 0.02 0.04 0.06 0.08 0.1−1

−0.5

0

0.5

1

t (s)u1, u2 (N)

t (s)

e 1,e 2

(m/s

)

#(#-"=) . ) $

0 0.02 0.04 0.06 0.08 0.1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t (s)

v 1,v

M1

(m/s

)

87F"; 2 ) 6) )" ;*#(#-" $ + v1

: "4 2 0 *7%) . ) ^)"!9 ") 75")!# # 5 0; 2 + vM1

:C$

0 0.02 0.04 0.06 0.08 0.1−0.2

0

0.2

0.4

0.6

0.8

1

t (s)v1, vM1 (m/s)

t (s)

v 2,v

M2

(m/s

)

87F"; 2 ) 6) )" ;*#(#-" & + v2

: "4 2 0 *7%) . ) ^)"!9 ") 75")!# # 5 0; 2 + vM2

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!#"%$ &'

( ) &#*+ , -. /

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n∗ = 2,

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-#.*980 -" )" #-9 " 0 *". `7 2 0*# )-;6)" " 22 0 & 2 #*)"K0" >#-, *751"K) + 2 0 -, &-808-8B: + 2 0 &-808 $ . & . >,%7F" " :C$ # 2 2 0*759/0 " #(9A" 0 *"! 7 2 0*# & ")# 2 #) & 0)0 2 & 0 ) . ) e) & 7*0 + 2 $ $ φ(y, t)

89 + $$43 &:(: , ,9% 2 0 *7X90) 2 )" $'" . &/. >,75" 6 ) $ & #-9 2 6 , " #(9A"e0 *". 7 2 0*# e) & 80)0 !) >") *# 2 0 2 ) *)" + 2 $ $ φ(xp, t)

89 + $$ 3 &:-: 2 0) , 09 >") 2*9 90) 2 ) )-) ,"#(9A"

φ(xp, t). 2 6?/=1 2 & 3 + ( : : + ( 6:C$

! ?*" 6D$'& "3 2 0 *7 Wφ(s)∗φ 2 0 >#("), 482 )" & 6# & #(9 2 2 # ,- & # >,#5, *?*" )(;! *)

φ,># *) ^;"9 " & # >,#5, *?*")-0 >#6) $ '";-")- 2 9 ) #6, #7. 2 "

$ & 1

,0 2 "#(9 ,0 2 80*# 2 " & ") 6 6#*0 2 # , 9.># 2F4 ) >#*0 # 5 0; 2 ) & # >,#5, *?*"Wφ(s)

+ 6D$ 6: & # 3 & # 2 $ "$80 60 >" ,9K) 2 2;,75) *) ,-,%# e0" ;.-" ) 2 9) *#*, #87 2 " ,0 2 "*#-900*#5, 2 -#10*# 2 " ,+ 9.-# 2F4 )0-#*0 # 5 0; 2 ) & # >,#5, *?*"Wφ(s) & ") # 2 9 & #43 & # 2 $

*# ," 8#-9 "0 *". 7 2 0*# & "/8# -#.*)";"*9A" & # >,#5, *?*" )40 >#*) 0; &/. >,75"+ 2 9 & ,K0"#(9+ ) & #-,#/,*?6" )20 >#6) + , 2 .*780 + Wφ(s) ∗ φ

9+ 6D$ , : + 6D$ 1:-: & "#H9 ("#*) & "*# ,0? ) & 0)0 =)3 2 0 2 <900(,#10 2 + $ $ . ) y

: !9+ ("#*0 #6)" & "# 6 ( 6 ",

‖Wφ(s)∗φ‖ ≤ c1‖xp‖ + c2ϕ(y, t) +c3

s + γφ

∗ [kx‖xp‖ + ϕ(y, t)] , + , $+$ :

"0%)c1, c2, c3≥0

6"^;8"0*0 *) ,*) γφ

" ,)8;γφ >γφ >0 80)" γφ

9*# 9) ., 2 7 2 ) 6)!)

Wφ(s)$( ,0?6"

ϕ(y, t)≥02 ;"060

kx ≥06"K) 20 2 ) 0

1 2 & 3 + ( 6:C$ ( ,0%?*"ϕ : Rm × R+ → R+ # & #80 9 ("#60 /) 2 0 2 3 8#9

) 20 2 )"=" & "# >,0*980 $

4# "$# #4& " & & 4#

#! ,&

8 ?*"2 *" )0 "7 2 )"<9+ ("#*0 & *#20"*#-9) & 8#>,%#/, *?*"2 , 2 .670 ')80 >#*) + ‖Wφ(s) ∗ φ‖ :P$ $ 80)" 2 -"/89 900 ) 79 ,#*# , 0"#(9+e) & # >,# , 6?*" + ‖φ(xp, t)‖

: & ") 8#9 ("#*) , 2 7 2T4 *0)%". + 6D$%$ 6: + 6$+$1: , *" & # !0*7 2 ) & *#0 2 9 ;8"9 #*, #7. 2 " ,0 2 "#-90 *#/, 2 >#/0*# 2 " $ "*# ",-#-"A7*)" A9.># 2F4),-#*0 # 5 0; 2 Wφ(s)∈Rm×n(s) )8) 2 0 2 ) 89 + 6D$ 6: & ")4#=)8;"9 & ";"9 "

Wφ(s) = WNsN + · · · + W1s + W0 + Wφ(s) , + , $'&:

"0%)Wi ∈ Rm×n + i = 0, 1, . . . , N

: Wφ(s) ∈ Rm×n(s) ># 2 *900 & #43 & # 2 1 807 N :=n∗−1

n∗ " #6,#7 2 "/,0 2 "*#-90@) & 6# 7 2 0*# )%"4 2 89+ ;8"0 "*#-90

1 2 & 3. + (5;B:C$ *#+ 2 9/) #6,Z#87. 2 "A,0 2 "#-90 ,0 2 80*# 2 " Wφ(s) & # 3 & # 2 ;8"0 0

900 & 80 3 9+ ># 254 W0

2 9+ ># 254 #*; 2 "0 *7Wφ(s) & ")89 8#/0 *"A0,%7 $( 2 9

$<3-8

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*#+ 2 9/) #6,Z#7. 2 "A,0 2 "*#-90n∗≥ 2 & ")89 #^08;10*# 2 "^9 ("*#*0

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& # & *# 2 9H;"9 #*, #7 2 ",0 2 "#(9n∗=2

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$

# & " & & # 4#( % !

& " & 2# #4& #

" #*, #87. 2 "H) & 7*0 "# ,0 2 "#-90 n∗=2 80 6" N =1 H9+ ># 254 ) >#*0 # 5 0; 2

) & #-,#/,*?6" + 6D$ 6: #(,7*0

Wφ(s) = K−1p

[s2I + s(α1I − AM) − α1AM

]Co(sI − Ac)

−1Bφ , + , $43B:

"0%)AM = −diagγ1, . . . , γm

"/-;67*#α1, γj > 0

*" ) 20 2 )" & 6# " 9 ")75")# # 5 0; 2 WM(s)

9 + $$43 " : + $*$ 3 6*:C$#( 9.># 2F4Wφ(s) & ") 8# );8"9 & "0,0%)"

+ , $ &*: ;"9

Wφ(s) = K−1p

[CoA

2c + (α1I − AM)CoAc − α1AMCo

](sI − Ac)

−1Bφ , + , $ ;B:

Wo = K−1p [CoAcBφ + (α1I − AM)Cp] , + , $ " :

W1 = K−1p Cp , + , $ : :

"0%)3^, 2 7 254 ",K" >"e) ,Cp =CoBφ

$ + , $ : : ;"0;87F, 2 , W1 6=0 ;8"0 0

900 " #(9A" d(W1φ)/dt & ") 0 *"1# 2 )80 2 ;6980 0 ,7F" $ *, 2 0)%". + 2 0 -, & 80808 ?*" &: 96

d[W1φ(xp, t)]

dt=

∂W1φ

∂xp

xp +∂W1φ

∂t, + , $ 6*:

$<3 $

;8"9xp

" , 2 )%" ) 2 # *900 ) , *?*" ) *)" ) & 7*0 + $$43 &: " , & #(9 2 ;8"0;87F, 2 # ,

d[W1φ(xp, t)]

dt= K−1

p

[∂Cpφ

∂xp

(Apxp + φ + Bpu) +∂Cpφ

∂t

]

. + , $ , :

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( ,0%?*"Cpφ(xp, t)

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$$( ,0?6"Cpφ(xp, t)

)28# ;"0 . 0, 089 #76?*"*"88,3*# ,900 -"8 -,')8# 2 .*) . & *#(; 2 2 H) & # 2 90 2 # "#-)9 ) 9 2 2 # ∀xp, t

$&D$( 0"#-9 ) )# 2 .*) & *#-; 2 *7 ∂Cpφ/∂xp

) 8# ,0 2 "*#-90900 7 2 9 2 6) & "#

∥∥∥

∂Cpφ

∂xp

∥∥∥≤kφx ∀xp, t "0%) kφx≥0 ,%9 (;*7*# ;"0%1; 2 )" $

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) . 2 4 8# ∂Cpφ

∂xpBp≡0

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) # 7 2 9 2 *) & "#∥∥∥

∂Cpφ

∂t

∥∥∥ ≤

kφt‖xp‖ + ϕφt(y, t) ∀xp, t "*0) kφt ≥ 0 ,9 -;*76# ;"01; 2 )" ϕφt :

Rm × R+ → R+ ,%9+ ,%0?*" ;"01; 2 ) ;8"0 . 0 , & "# & *#89 #7*?6"t!;"*0 . 0, 89 #87*?*"0

y$

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n∗ = 2 ) *) & 75"& 6# ' 90-#-"R) 6#*"

Kp = CpApBp ; $ $ + $*$ & 3:C$ A" #(9A"R) & 8#>,%#/, *?*" φ "

7 2 0*# 2 0 .*# 2 *0 0" 9 & " 2 $ $ φ(xp) = Aφxp 80 *" ∂Cpφ

∂xpBp = CpAφBp ",

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2 0 *7

d(t) = c1Xp + c2ϕφt(y, t) +

[

c3 +c4

s + γφ

]

φ (≥ ‖Wφ(s) ∗ φ‖) , + , $ 1B:

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√

max

X2 − ‖ω1‖2 − ‖ω2‖2, 0

(≥ ‖xp(t)‖) , + , $%$ $ :

X(t) = c6τ‖Uav‖ +1

s + γx

[c7ϕ + (c8 + τc9)‖Uav‖ + c10‖ω‖ + c11‖r‖] (≥ ‖X(t)‖) ,

+ , $%$ &:

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τs + 1UNeq =

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ρ0 ≥ (1 + cd0)∥∥L−1 ∗ Unl

∥∥ + cε0‖ε0‖ + πρ0(t) , + , $+$3;B:

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1,i L−1i+1,N ) ∗ Unl

∥∥ + πρi(t) , (i = 1, . . . , N − 1) , + , $+$ " :

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1,N ∗ Udnl

∥∥ + πρN(t) , + , $+$ : :

∀t≥0 ;8"9 ;8"0*0 5*) , *) cε0≥0

cdi≥0 (i=0, . . . , N)

!"- 2 0 2

Unl := (Knom)−1Kp

[

(θ∗ − θnom)T ω − Wφ(s) ∗ φ]

−[I − (Knom)−1K

]UN , + , $%$ 6:

Udnl := S−1p

[

(θ∗ − θnom)T ω − Wφ(s)∗φ]

, + , $%$ , :

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OOQMcG") S =J$ G ;YS-, McG , M Jkz0 > 0

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M`G*O 80J

‖z(t)‖ ≤ kz exp(−λzt)‖z(0)‖ + O(τ) , + , $%$1:

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xp = Apxp + Bp[u + d(t)] ,

y = Cpxp ,8 [ $ YCN

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y = G(s)[u + d(t)] ,8 [ $$# N

%)

G(s) = Cp(sI − Ap)−1Bp = Kp

Np(s)

Dp(s),

8 [ $ N

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@&AIHCFEHGG(s) P% -# ( * " & #` & # ( $

@&AJHCFEHG o #() )5 ( _8nN PO 7 22( ) $

@&ALKMCFEHG M .#R# #- ( ,* n∗ ) G(s) PO *7 '2( )5 $

@&ANHCFEHG M ( * )".!*7) *C: # # '^4O %2 '( Kp PO 7 '2( ) $YC u

@&AHCFEHG o & #&#&0v1* d(t) Pa " & # & 6# 7 2 *#*" d(t)*

^ |d(t)|≤ d(t)≤ dsup <+∞ , ∀t≥0$

49:7 ( & ` *#-# & %) ^ /$ ( ) & & *# * ># & #6 ") m) # '#3 #%2 '( ) ( 9 6 ( *# w5 ( ,-*# (x , '( H) .#* # / ( ,s5 (# *#& *# ( -# x # ( $

o 7 ( & ` 8 o $ N P * x m.! 7 ( & ` 8 o =N ^ $ *7 '2( ") %) (m ) #-,* ( ( )*) ν

) ( s 9 ( ,*# (mx , 2( _8 , ( ) g 0v1TY $ $ # N #, & 6# : 2 # _#() ) C>#&) 9:*) 8h[ $ Y # N $/o 7 ( & ` 8 o $ N & ) # ( ## # *)/) t# #& # 2^ # #4 & b5 #*" & *# n

p " PQ#32( 8 !*V giYC[[ipg 021* $ $ YnN $ * &p ( 9% *#-# :6# ( Z &!)L#-) )! -#&!) E 9:*) ( () *)!Z 9>#&&# )5 ">#& & *#X ) m ) # '#3 #&%2 2( &)s # ) & *# %* -#&!H) * >##6)# $

9>4D<E4<, B!<4A A /BUACB 0eBKp > 0

c<*D<A$:Zc : 1&<*D< , >U<;>" ) =,, ( ) ">#&/*)*# $ g Kp < 0

p " 1* ( ') * ># ) , # ( , # ( ) $ o ( _p 7 ( & ` 8 oXqD$ N P ( #32( : $

o 7 ( & ` 8 o $ N & ) # ( > # # # 6)5p : 5p L '( ) -#& # &5 : # ( ( & F _8 "*5V g; YC[*[5peg 021*! $ \*N $

#" $6%:036&%:0+9;0(' 0H9*)3 + *

M ")*,2)*,!#-, # ,#&%' 2(/.sP )*, ( )5 & #a8 . 5V gi YC[[05pHg ,2021*! $ $ YnN

yM = WM(s) r , r, yM ∈ R ,8 [ $1 N

%)*,

WM(s) = KM

NM(s)

DM(s)= KM

spm−n∗

+ bpm−n∗−1spm−n∗−1 + · · · + b1s + b0

spm + apm−1spm−1 + · · · + a1s + a0

,8 [ $rq N

%)*,KM > 0 P . . 7 )*, . 32 .?# #-, ^4O %2 '(4. ,5 & , 2(~e6. ) & . # . ")*,'mR)*,A#7, # ,#%2 '(4. $

M ")*,2 )*, #7, # ,8#&%2 2(4. )*,$,,857,#9,572 x ,,',K)*, ##. 57, ( . $ M:59.*# . *5 )5 & ( *; ( 5Ik#7< ( 2 ) ,s =; (m 5 NM(s)

,DM(s)

51pm − n∗ , pm

p3#-,5 & , 2 ( , . >,2?2-, $o #()*,2 ) ")*,'m

pm

)*,$,,#-,5 & , ( 2 . # pm≤n$

YC*[

o 5-5 >,$3 57, ^ *, =5 ( . 3)*, #7, # ,#%2 '(4. r(t)57, . *?2 ;5 & *# & . #72 , 5 ,a (# *#& >,2 >,2 2 ,

( ( 2 . ) $z . # . & . 2 . 5 * .*# . #7,2 . 2 ( , ( 2 x # ( 5p .# 021* ) ,!2># . *5 # ,#&%' 2(/. )5 ")*,'m )*,

#7, # ,#%2 '(4. 8 WM(s)N )*,',, 57,# ,5 2># ( 2 . =,'?2 , #-, . & 05 ( 2 ( , . 8hg5z G AhDF: #wce>CA$: : E*B9D B&<0 NFp

& ( 59*, 572 , 6. 5 )*, 5-,',~, ( >,2?2& ) *?2>#/,!5-,'.*, . . #() . .,' )*, 8hI 5& V 572 .YC[ u [=NFp ^ *, P #7,', ( 5 2 . . g ,2021*[ $1$

z . # . & . 2 . 5 * .*# . #7,2 . 2 ( , n∗>1p ,2 8hI 5&>,2 . $ YC[*[ pI 5&>, 2 . $ YC[[\N ,5 7*,$3

57," & ( *; ( ? >; ( 8,Ik#7< ( 2 ) L(s))*,s.# . N := n∗ − 1

) , # # .^ , .?# 0v1)*, 2># . *5 # ,8#&%' 2(/. WM(s)L(s)

5-, . g5z $o ^ ( . , 5 *m7 . )/ ")*,'m )*, #7, # ,8#&%2 2(4.P .( 5#7,572-# ( 2 ( , . & ( 5 57, #-, ^ *,8# ^ , WM(s)L(s) = KM/(s + γ)

p ,2 , ( # 2) , ) . ,572-#& 2&# . )IZKaM 8 , ( )*,dML*5-,8#, . 0v1* [ $$# . g ,'0v1*a[ $ $ YCN,=) .X# # .L " 5-, ,' & #7,2. . ,' . /$$# .& #n, . )*, ,5 2 . ( ( ) . )*, )5 ( 572-,2 . )*, ?2-#& , ,2 . 7 .# , 7 . ) . $ao 575 ( _p & . # . & . 2 . 5 n∗>1

p .9# 021* )*, 2># . *5 # ,#&%' 2(/. )" ")*,'m)*,!#-, # ,#%2 '(4.sP

WM(s) =KM

L(s)(s + γ)=

KM

spm + apm−1spm−1 + · · · + a1s + a0

,8 [ $ =N

%)*,KM , γ>0

ppm =n∗ , L(s) P 5 & ( *; ( a =;* (m ,XIk#7< ( 2 ) )*,L.# . N :=n∗ − 1

$ ,$,,$3 5-,02 . # ^ , .*^ ( . 5 # . ) , 5 )*, L(s)

e1* & #-, 2( 5 . 5-,8#A#-, .*( 5 $ ,5 2 , . 5 & , 2 7 x .*( 5b ( ,#() . )*, & # 7,2) ^ *,kOg3 o )*, 5-,',~, ( ) & *#d8hI 5=, 2 . $ YC[[ pI 5&,2 . $ YC[*[=\N$p ^ *,A#7, ^ *,8# ^ , L(s)

2-,27 . . & ,' . 5# . ) ,5!#-, .*( 5'p . ,, )^ *,_ & ,# . )#L(s) P #7, . (m). ) m> $: & #/ . , 1* ,2 6. 5 6. 2 . )*,

N 32>#&05K)*, . , . 0' )*,##. 57,a . 5-, . )5 ,' , 572># 2%# . , . # (yx ,,2 ) ,#&05 5 ( & ,5 ,#-, .*( 5'p 5 ( ( . #-, 5 ^ *,'/, 5 2-#&5. & #-,57,2 2 . )55k . ( .# . $ # & . # . 5 ( 5 2 ,2 . 5 9 2 ( , . # (mx ,, ( 5 $

7 0 1*6 %:0 + 649;60

M 7, 2 ( , P & #&7, 2 . #d5 . , ( )*, 2>#&*/, u)*, # #& .s^ , . 5 . ) . ) . & . ?2 . y

# . 572-#-, ( ,. 575 ( ?2 2 (m. >,2 2 , . 5 . ) . yM

)_ ")*,2 )*, #-, # ,#%2 '(4. p . ) ( 2 ( )3 57,w & , ^ ,2 ,8#-#&#7,5 ( ) . $ .( 5 & #-, 2( 5 . >,2 2 ,*p/* 7,2 ( ,*K) , ?2-#& , P 2 ,8# . ,,8#&.*%2 '(4. . 5-5 ( ?2&`02 (6.) ,#(#& )*, 5 . ) .

e(t) := y(t) − yM(t)8 [ $ \N

Y

& . # .w) ,8#&" & . # . . & , ^ *,2 . , (m)2( 7 . 0 . #-, 5 ( ) . )*, ) ,# $

3 3

o & . # . >,2># ()6. 021*Z) 2>#&*/, & #9 )*,'mZ)*, #-, # ,#%2 '(4. )*,85 ( 572 ,' . 5"gi-g5M P. x . . ^ *,' . )*, 5 # ( 2 . . & 2&m & . # . 5 ( 572-,2 . 5 M $ z . # . 5 ( 5 2 ,' . 5wgi-g5Mwp *?2>#/,^ *, . 5 .>. & . ?2 .9 " ")*,'m)*,!#-, # ,#%2 '(4.sP ) . )5 & #

u∗ = θ∗T ω − Wd(s) ∗ d(t) ,8h[ $ru N

Wd(s) = 1 − θ∗T1

A(s)

Λ(s),

8 [ $ [N

%)*,Os,,2#=)*, & . #&%* >, 2>#5 P ) . ) & #

θ∗T =[θ∗T1 θ∗T2 θ∗3 θ∗4

],

8h[ $ Y N

θ∗1, θ∗2∈R(n−1) p θ∗3, θ∗4∈R

,Os,,2#=#7,2.#7,5-5# P

ω = [ωT1 ωT

2 y r]T .8h[ $ YYnN

o ^ ( 5 32>#5 )*, ,5 2 . )5=5&1* ) . )55 & #

ω1 =A(s)

Λ(s)u , ω2 =

A(s)

Λ(s)y ,

8h[ $ Y # N

A(s) =[sn−2 sn−3 . . . s 1

]T,

8h[ $ YC=N

%)*,Λ(s) P 5 & ( *;* ( . #& ( 2># x # ( 5pR >; (m ,Id%#7< ( 2 ) ) ,_.# . n − 1 ^ *, 2 P

NM(s) * ##. 2&#2p ( $ , $ p

Λ(s) = Λ0(s)NM(s) ,8h[ $ Y N

^ , ( & (.^ *, Λ0(s)57, . =; (m p Id# < ( 2 ) ,A)*,/.*# . n0 = n − 1 − pm + n∗ p i3

# # =, 8 . 5 V+gi YC[[ipRg ,'0v1 $ $$# N $ ] . #-, . ()6. 021* & . # . 5 32>#&05 )*,8, 572 . )& )*,/ 2 ( (). # . 5 . 2-# () , 5 Φ

,Γ)*, # # . 5 ( ( . # . 5 32>#5 ) , ,5 2 . ) & . # . 5 ( 572-,2 . 5

Y Y

M 8 $ YC=N $o 5 ) ( 02S , 5 )*, 6. 5 . >,2 2 #-, ^ *,#7,2 ^ *, θ∗4 = K−1

p KM

8 . * VJgi YC[*[5p, ^ $ 8 $ $ Y # NN $

o , ^ . 0v1*e)9,#(#& P . & . #72 ( . # ()6. 0v1*) ^ *, x!# ( )*, 5-,',~, ( )9 . g ,'0v1* %$ # & . # .5 ( 572-,2 . 5 M9p ( 572& P

Xe = AcXe + Bc(θ∗4)

−1[u − θ∗T ω + Wd(s) ∗ d(t)

],

8h[ $ Y q N

e = CoXe ,8h[ $ Y =N

%)*,Xe := X−XM ∈ R3n−2 P ,5 2 . )5Z) . , ^ . 0v1 ) ,#(#& $ ,',,F3 5-,02 . # ^ *, .^ (

(θ∗4)−1 =K−1

M Kp

p ,' ^ . ?2& ^ *, & . # . 5 ( 572 ,' . 5 M 2-,2 w3 57, (θ∗T4 )−1 =Kp

,2 , ( #72%)*,H)*,57, 2 ,8# , 5 *m7 ( )" ")*,'m)*,#-, # ,#%2 '(4. 8Y $ q N . . 2># () ) ,a. . 7 )*, . 2 .9# #-, ^4O %2 '(4.P9."( )*,'?2 ( ) . ) , $ " Ac, Bc, Co P . #-, . ()6. 021*"e13 ( . ) , WM(s)

p!,8#-#K)*,5 . ) . & )*, 57,# ,5 # ( 2 . & . # 2 ( # ) ,/8 [ $ Y q N8h[ $ Y =N "

e = WM(s)Kp

KM

[u − θ∗T ω + Wd(s) ∗ d(t)

].

8h[ $ Yn\N

o & . # 2 ( # ) ,572 . & . # . >, 2-# (m). 021*!) 2>#&*/,*p . 5-5 =,F3 57, ^ *, . /, ( )*, 2>#/, & ,8#72-,20 . . 575-,/)*, ( ) . . 7 ( & `02-,5-,98 oZu%$ . N & . # . 5-, ,', ( 2 . #OB A &<$ceBOB BZce> :>=)5 5 ( .*( 5 )5 ( 572-,2 . p # # =, ) ( 5 2 ( )s . & 2&m q$

3 .

z . # . n∗ =1p2-,2 w3 57,

N =n∗ − 1=0pL(s)=1

,a/ ")*,2A)*, #-, # ,#%2 '(4. WM(s)) ,',,95-,#

, 5 *m7 ( )Lg5z $4o 5-5 ( _p . & . #72 ( # ) . , ^ . 021*^) ,#-#w8h[ $ Yn\*N(,<)*, .* #() ,' . /$ Yp . , ( ) , 2>#&*/, . ,5 2>#& 2# . , . # (yx ,,2 & # ")*,2^)*, #-, # ,#&%' 2(/. 8Xg3 o N # ( . ( . ,2?2-,

Y "#

& # & 5 2 . ,' 8hI 5V 5 2 . YC[ u [=N P ) . ) . & #

u = unom + U ,8h[ $ Y u N

U = −ρ sgn(e) ,8 [ $ YC[=N

unom = θnomT ω ,8h[ $$# N

%)*,θnom P , . m#9* ( . <)*, θ∗

$ M 2>#&*/, ( . unom & ,8#& ( 2 , . #7,)50v1* ) .. & ( 2%)*,^) .w# 50v1* )*,O ") . 0v1 ρ

57, .w( ,# 2 , ). ‖θ∗ − θnom‖ # # & , ^ ,2 . $o & . #72 ( # ) (,2 . /$ Yp; 2 P w3 57, , 572 . ( ( ) . )*, , & **,2 '(4. 5-, .X# 0v1 )*,k ") . 0v1*

ρ∈R+ 5 . 2 ( 5 e) ,# . )*,5 ( . . F) . ) ,

ρ ≥ δ +∥∥∥(θnom − θ∗)T ω + Wd(s) ∗ d(t)

∥∥∥ ,

8h[ $$# YnN

%)*,δ ≥ 0 P 5 . **572 . ?2-, . #& ( 2># x # (/. $ ] . # 50v1* )*, )5 . 021* ^ *, 5 . 2 ( 5 ##.*) .

)*, 5 ( . . 5) . )*,"8 [ $$# YnN , . 7 ( & ` 2 ,57,8 oZu%$ . N P

ρ = δ + c1‖ω‖ + d(t) ,8 [ $ # # N

d(t) = d(t) +c2

s + γd

∗ d(t)(≥

∥∥(1 − θ∗T1 A(s)Λ−1(s)

)∗ d(t)

∥∥)

,8h[ $$# =N

%)*,c1 ≥ ‖θ∗ − θnom‖ $ M . # . ?2-,8h[ $$# =N P 2 ( ) & ,2 . . & (6. 021*R)(,2 . $ Yp)*,

%)*, 57,:2 ,' ^ *, c2≥0 P . *572 . ?2-, . & # & # (/. ) . , γd

5 . 2 ( 5 ##.) γ0 >γd >0p 5-,')

γ0 . . #&.0,2 )*, ,572 . ( ( ) . )*,^) 2-#& . 2-# () )*, 2># . *5 # ,8#&%' 2(/. A(s)Λ−1(s)$

M 57,2. ( ?2-,K#7,5&5 2 . ) )*,=,572 . ( ( ) . )*, # ( & #&C, . )# ( . ( . ,2?2-,>,' 8 I 5& V 572 .YC[ u [=N $

E C B <*D<n∗ = 1

> 5A$: ;B$DBUACB >TA$:mA B"<v>4D "< ;> ceB~< B5<;> ;>jB$DFD > aBbceB~<B$:! ;B: > hD>mB #"$ &%(' *)Bd<4A,+;:rc.- B ACB A 0/ > 0/ > 1v>4DBA2<0 :mA0vB$: <4AdBk<2 ;>3 ;B />4 0 ~<;>dA2<0 :mA 654B$D6<7+;:rc.-B ACB 0/ " <aBd<8 ;B A$::905<0: =< ;B &% B ;> < B AhD<0<;=9*: <?>@) =ACB /ED ;F90m>*@ <0 /B BBHG'ce> B $: <0 B BB A=I*E*B X: B 3 ∃k, λ > 0U<*:mAJeB ‖Xe(t)‖ ≤ k exp(−λt)‖Xe(0)‖

∀t≥ 0 / &;K *:mA AC> ACB

δ > 0!B * ;> > B$DFD>L ;B

Y

A2< =<e(t)

ACB >4D <*DHI m><$c.-CAa<0 90 BZce> e:>

D>4E4< o n, .?P .8. (6. 021* ( , 2 . ,' . Y . 85 ( 572 ,' ./ 572& ,2 . , . (m). 0v1*e1*43 ( . . , ^ . 0v1* 8h[ Yn\NFp . . ,2 . , ^ . 0v1* , 8h[ Y q N 8 [ Y =N$p , ,' . /, ( *, *?2 /,s8h[ Y u N 8h[ $# N

E C B : &<B$EC: ;B B eB > =AC> ;Ba>@$AvB$D$E4< ;>4DB A3; ;B A BFB A A$IDF:U>6c5<*D&<>= > D>mB8 ;BLA$:mA B"<4A3 ;B '<4ACB e:"<?B 9*D<0_DB~<0h: E*> e:=I*DF:U>?B HBLACB9BZcDB=9=<?B AhD ;D<E4<*DF:I*E*B6BDB&<0 :/B <;> iB/A2< =< a> B U< > s> : ECD > R<*DH 4A"! )c;D=9;B > "$BZcDB=9=< >*@$ACB$DFE4< ;>4DB AXce>4D=/>4 ;> ;B A ::5C< B > > hD>mB ;BsA$:mAB"<4A > 9*D<0 D B~<0 : E*> e: I*DF:U> >J< <0 : =< ;B ;B 9;B$D<*D_< ;> ;B8/>4 0 ~<;> LACB B (+ "< E4< < 9;B<$c5<*DB BwB DB~<;>#_<=@$>4DH < 9;B ;B A <%$B ACB &!B A </ACB7;>?B > D <$c' m> kB$EC: <0UACBs>=AC> ;Bk>*@$ACB$D$E4< ;>4DB A <0 D&<*E ; A, =<!<$c : &<;> ;B D>CA ;BdB A < ;>!B ;B )( /* A c5<*D&<!< 9;B$D&<;> =<4A ,+CB A iB />4 0 |< ;> A <9<=@$>4DH =< 9;B c5<*D B B"<*:mAQA$:ZcmB AZB <0#;D&<0c5<*D<XA$:mA B<4A: B$D >CA ;> eBa> Av> ;B!>*@$ACB$D$E4< ;>4DB AHcH>4:mA ;> D BeB$D!>= > (+eB$:/B >E =</B AhD ;D&<J ;>A$:mA B"<

.- 3 0/

z . . . ?2 . 5 * . . ,' . 2 ( , n∗ > 1p 2 /, ,572 2& . , . (mx ,,'d . 5-, . ,'

, . ( >,'?2 . 0v1**, 5 . . e1* 1*, 57, . (m. ( , 2 . =,'?2-,*p5 . ,, )s^ *,w? 1*,'m*, , # , %2 '(4. 1* P g5z 5o . . .,' 2 ,s8hI 5,2 . Yv[[ piI 5, 2 . YC[*[=\N P . 57, . . , 3 , ,4 ( 0v1, ,2 2 55 , . , . 0'6 , ##. 57, ,5 2 2 . , . (yx ,,2 8 , ( *, . 2&m =N

o ^ ( 5-, 7 S ,!5 . , 572 . 2 P . (4. *, *?2 , . 2-, . 2 ( , . . 57, . . *57, , . *,. 32w. . 78 +;::9 +J9=<*: >*@$ACB$D$E*B$DbG ILK!MON e] .s( P2(/. . 2& . 5-, (4. , 572 ( . >,572 . 8 . . 2 . " 5 ?2 . 3 . . , 572 . ( (m). 0v1* *5 2 . ,' 8UML7 V+WX7 . ( kYC[*[ q N ,'?2 . ?2&p .*^ ( 57," 2 ( (). _ "1*,20*, , # , %2 2(/. 8#57,2 ( , 2 , )6. N:,2 ,, ) 5 "1*,20 . . 2 . 8 ( , 2&N . . , **572 ( :,5 2 . 59 . , ^ . 0v19:, 5p . 502 . 73 57,Q . .5 =,'?2*,"8 7 . 5.V (,,sYC[*[=N:*, ^ *, P .*( 5 # x*2( *5 2 ( *57, , . ; . . s 1*,'m<*, , # , %2 '(4. 5 ^ , . . . . 2 . b] . . . .,2 5 ( ( . . , 572 .wP 572 . 8hMX7VWX7 . ( YC[[=\*N . . . 572 , . =,'?2&*,:2 . 7, 2` (4. 5 ,2 5 ( 572 ,' . 5 e1*43 ( , . ,5 ,2 2 . 2p

Y 0

2 . ?2&8hMX7_V WX7 . ( Yv[[ q N ^ . ?2& 8hML7V WO7 . ( Yv[[=\Nt 2 ( (). 5 . , ( Og ,2 ^ *, .# 50v1**,O 15 . 021* P .m . >,'?2 ,L ( ( 2 . . .w *, 57, ,' ( ( . # ,2*;* =,'9cHB&<4f4: 9*p . 5 . . . 43 5-,X ,20'8*, ^ *, . 5 7 ( , . *, 5;*, ,5 2 . ( ( . *,:5&1*?ACBw:90m>*@ <*:mA

o ^ ( p?5-,k 2 ( ()6. *57, , . : . . .0, . . /, ( *, 7 . ,, . =,'?2&,*p . ! >,5 2 ,' 5p 2 ( ()6. 3 57, 5 .L# 50v1* *,6 "1 . 0v1X., . . 32 05 *, ,572 . , M o 5 . . 57,k 2 , . ,5 2 . ( ( . ,L. . 5-,' ^ *,O # ,2*;* =,'9ceB<4f4: 9:57,! . (# ,5 2 ,L . . 2 . ,O 5 ( . *, . p . ,5 . *, ^ ,!wceB&<4f4: 9 1*,O , " 5-, , . 5 *, . 32w. . 57

*5 ( *, . 3 5-, . , . ()6. 021**,! *,' ( . .w# .96. ; (m. "**5-, , . . .s 1 ,2m , , # , %2 2(4. WM(s)

. . ,2 . 5 . 2 () , 5

AM =

−apm−1 1 0 · · · 0

−apm−2 0 1 0

−a1 0 0 1

−a0 0 0 · · · 0

, BM =

0

bpm−2

b1

b0

KM , CM =[

1 0 · · · 0]

.

8h[ $# N

b?21*p . 5 , ^ . 02S , 5 >, 8h[ Y q N 8h[ Y N 1*,' 57, ; , ,5 ( 2 . 5 "

xe = AMxe + BM

Kp

KM

[u − θ∗T ω + Wd(s) ∗ d(t) + πe(t)

],

8h[ $#q N

e = CMxe ,8h[ $# =N

7*, ./ * ( 021* ( (2(/. xe(0),9!5 ( . ,5 6. . , *,' 2(/. m >,2 2 ,6 , ,5 ,2?2-, πe(t)

51. , ^ . 55 . . , , 5-,2 2 . 5 , # , ( 25 . 5 * ( 0'S ,5 ( (2(/.*( 5 5 "15b*57, , x ,, ( 5R . 5e13 ?2 x ,, ( 5 . , ^ . 0v1*85>, 8 [ Y q N8h[ Y =N M 5 ( . πe(t)

5 . 2 ( 5 ##.)

|πe(t)| ≤ ke exp(−λet)‖Xe(0)‖ , ∀t ≥ 0 ,8 [ $# \N

. . . *572 . ?2-, . (/. . ke ≥ 0,

λe > λe > 0pX**,

λe P . . .,2 *,, 572 . ( ( . *, . . 2 (m) Ac

o ML*57, , . 0v1 [ 1 . g ,2021* [ Y *5 2 (~e6. . ( m 51* 5 ( . πe(t)

. , ^ . 021*0 ( D%* (m. ,2 ,, ) *, 5-, . , 5 ,2?2 . 5 ( . , ,2 '(4. =,'?2-,*, ,5 ,2?2 ,w . , ^ . 021* *, 5 . . . . , ,5-,'?2 . 5 , # , ( 2&5 . 5 * ( 0'S ,5 ( (2(/.*( 5 05 "15 5-, , x ,, ( 5k . 5 e13 ?2 x ,, ( 52p " 1*, (4. . , , .( 5 . 2 .

Y =q

5-5 ,,' 7, 2 . 5 . . 2 (m) S ∈R1×pm ^ *,*, *, . 5& , # '( ,< , 7 . ,, . >,2 2( *, . σ(xe)=0

p7*,σ(xe) :=Sxe

p . .Z^ *, AM , BM , S 5-, . . , . (m). 021* .d# 0v1*,!2 . 5 # , %2 '(4. giz WM(s)L(s)

z . . 6. 5& ,2 ^ *, 5-, ,5 7*, NM(s) = 1pb ^ *,

( (~6.6^ *, . *,' Z "1*,2 *, , # , %2 '(4.dP . . pm =n∗ , BM =[0, . . . , 0, KM ]Tp

2-,2 w3 5-,"8hW .*( . 2&7 YC[ u p 7 [ =q N

ST =

0 · · · 0 0 1

0 · · · 0 1 apm−1 1 apm−1 apm−2

0

1 apm−1 apm−2 · · · a1

−1

1

lN−1

l1

l0

,8h[ $#*u N

7*,li51*L5 , e2( ,2 2 ,5 5 ( *;* ( L(s)=sN +lN−1s

N−1+· · ·+l1s+l0,N =n∗−1

o 5-5 ( 43 5-, ^ , . ,572 ( . 2 ( , . xe

*,xe

,572-, . ( 5 * ,,'Up ,'?21* . /, ( *, ?2 , P

u = unom + U , U = −ρ sgn(Sxe) ,8h[ # [N

7*,d 2 */,6 ( . unom P . 5 8h[ # N g , xe(t)≡xe(t), .O# 021*6*,d "1 . 3

021*ρ5 . 2 ( 5 e) , 8h[ $# YnNFp,2?21*d ,572 . xe

,t , *, 5 . . e =,, .,2 , *,' 2(/. m >,'?2 , . .w) , p 57, 1*, m ( . 2 . , P 5; .>. (m. 0v1 (,' . Y

M

] . ,, )X^ ,O ,5 2 . xe

e1* P ( 5 * ,,'Up , 572 , 57, x 5& 572 ( 2 9 . /, ( *, 2 */, ,2, 572 . =, 572 ( . xe

2 ( ,2"**5-, , . , . 2w. . 7

˙xe = AM xe + BM

Knomp

KM

U −[α(ε−1) − aM

]e ,

8h[ N

e = CM xe − e ,8h[ 5YCN

Y

7*,e P , %*, 5 . . *57, , . p Knom

pP , . ( . . . 57 , . 2 .

# , ^ O %2 2(/. . . 2 . Kp

,!w,, 2&

aM =

apm−1

apm−2

a1

a0

.8h[ # N

M 5 , e'( ,'?2-,5 αi

5s,, 2& *, , . ( >,2 2 . 021*s*5-, , . 8 V g .,' Yv[[ u N

α(ε−1) =

αpm−1

ε

αpm−2

ε2

α1

εpm−1

α0

εpm

8h[ =N

*,$,,2 5-, , 5 m7 ( 5) , # .a^ ,O ( *;* ( 6. .* 2-, 572 (m 5s**5-, , . 5 ,2 . m7 .# , 7 . . 5-, . Ik < ( 2 ) p ^ *, 57,!* 2 P 5-,O ( *;* (

Nα(s) = spm + αpm−1spm−1 + · · · + α1s + α0

8h[ N

# Id < ( 2 ) , ε>0

M . %* >, 2 ε ,',,57, . *5 2 . 5 . .w^ , . 5 ( , 2-, ). 5 , , 2& . 02S , 5d>5 ( 572-,2 .

.# ,2 ,' " ,5 2 . , 572 ( . xe

z . .w( 5-5&5p . . 5s. . 57 . , . ( ,2 2 . 021*< 5-, , . 5 8 ‖α(ε−1)‖ N *,',, 5-, . 2 . p5 ^ *, , ^ *, ^ , ε

5-, . , ^ *,25

$ 689;10 %:6 0H996 %:0<032 1* 6

M ,' ,2 . [ # . , 5-,2 2 . 5 . 57,2. ( ,5 2 . ,'/, ,5 . . ?2 , . . . . , *, , 572 ( . 0v1*=, 572 . xe(t) := xe(t) − xe(t)

, . . "`1=, 8*, 5 . . e(t)

5-, , . 5 *, . 2w. . 57 5 2 , ,2 ,2 . 5-, x 2 ( . n, . *, ,572 . ( ( . *,*>5 ( 572-,2 .*, ?2 , ,2 . 7 . # , 7 . .

z . . 5 ( (|e6. *,57,2,*, ( >,2 2 *, 572 , ,2 ,2 . p . 5-5 >,$3 5-, ^ *, . . .,' *,

Y \

, 572 . ( ( . *, 5 * ( ; ( Nα(s)8λα

N ,d9,5 . . λα

5 . 2 ( 5 ##. 0 . . *, 5 ( .* . . *, 0<λe <

λα < λα

o P ( 575&p ( S ,$3 5-, ^ *, 0 < ε ≤ 1 . .^ *, . . .,' *,=,5 2 . ( ( . *,5

5-, , . 5 e1>57, .s( # , ( w . .,' *, , 572 . ( ( . , λα

z . . *,57,2=,,, ,' ,' . . 57,2. ( p*, *,$3 5-,!>5 ( .

U :=[1 − (Knom)−1K

]U + (Knom)−1K

[

(θ∗ − θnom)T ω − Wd(s) ∗ d(t)]

,8h[ q N

7*,K :=K−1

M Kp

pKnom :=K−1

M Knomp

,O>5 ( . *, ?2 , U P . 5 8h[ $# [=N

E C I D > iA$: ;B$D BQ>L>@$AvB$D$E4< ;>4D ' dBQ<!B5<;> ;>LB$D$D> )B"< +;:rc.- B ACB 0/ " < v>4D A2<0 :mA0vB$:U<X>CA A$: <*:mA

r(t)B

d(t)v>4DB e: v>4D /BB *B :w:U<0

;>CAk>ace> : w:U>Nα(s)

v>4D iD b:&5NM(s)=1

B>!c5<*D0/BhD>εv>4D9"< : ;> < '<*: G<

0<ε≤1 B ;> ∃k1, . . . , k6 >0

<*:mA8eBw>?B$DFD > ;BwB A :"<;> ;>B A < ;> xe

?Bw>?B$DFD>J ;BA2< =<J i>>*@$ACB$DFE4< ;>4D

ewA2<0h:mA'< 5nB <4A ;B A$::90<0: =< ;B A

‖xe(t)‖ ≤ k1

εpm−1‖xe(0)‖ exp

(

−λα

εt

)

+ k2ε‖Xe(0)‖ exp(−λet)

+ exp

(

−λα

εt

)

∗ [k3‖ω(t)‖ + k4] ,8h[ N

‖xe(t)‖ ≤ k1

εpm−1‖xe(0)‖ exp

(

−λα

εt

)

+ k2ε‖Xe(0)‖ exp(−λet) + εC(t, 0) ,8h[ =\N

|e(t)| ≤ k1‖xe(0)‖ exp

(

−λα

εt

)

+ k2εpm‖Xe(0)‖ exp(−λet)

+ εpm−1 exp

(

−λα

εt

)

∗ [k3‖ω(t)‖ + k4] ,8h[ u N

|e(t)| ≤ k1‖xe(0)‖ exp

(

−λα

εt

)

+ k2εpm‖Xe(0)‖ exp(−λet) + εpmC(t, 0) ,

8 [ [=N

∀t≥0 > ;B

C(t, t0) = k5‖ωt,t0‖∞ + k6 .8h[ 1 N

D>4E4< ] 2 ( ()6. 3 5-, . *, ( 0v1* , *,!, 572 ( . 0v1 , 572 . ,, 5 ,',,$3 57, .Y u

, ^ . 0v18>, *, 5 . . "**5-, , . 8 [ 5YnN "

e = CM xe .8h[ 1 YnN

gi 2 .*( 3 5-,L8h[ $#q N *,X8h[ N(5-, * 2 P . , ^ . 021*9 . ( D% (. :, **, ,5 2 ( . 021* ,5 2 .

˙xe = AM xe + BM

[KnomU − Kπe(t)

]−

[α(ε−1) − aM

]e ,

8h[ # N

^ *, 1*,:5-, , ,5 ( 2 . 5& 572 ( 2 ( 3 5-,L , *, 5 . . 5s**5-, , . 8 [ YnNR . , ^ . 0v1* . ( ;%* (. , *, ,5 2 ( . 021*_8h[ 1 # NFp , 5&32 . 7>,'

˙xe = Ae(ε−1)xe + BM

[KnomU − Kπe(t)

],

8h[ 1 =N

7*, 5-,! 2 ( (). . . 2 (m)

Ae(ε−1) = AM −

[α(ε−1) − aM

]CM =

−αpm−1

ε1 0 · · · 0

−αpm−2

ε2 0 1 0

− α1

εpm−1 0 0 1

− α0

εpm0 0 · · · 0

.8h[ 1 N

2 1p . (m. 53 57, . 2 . *5 # . 0v1

xe = T (ε)xe , T (ε) = diag1, ε, ε2, . . . , εpm−1

,

8h[ 1q N

,,5 ,',,$3 57, . , ^ . 021*85>, 8 [ =NQ . n, . . 5-, "

ε ˙xe = Aexe + εBM(ε)[KnomU − Kπe(t)

],

8h[ 1 =N

Y [

Ae =

−αpm−1 1 0 · · · 0

−αpm−2 0 1 0

−α1 0 0 1

−α0 0 0 · · · 0

, BM(ε) =

0

bpm−2ε

b1εpm−2

b0εpm−1

KM .8h[ 1 \N

"NM(s) = 1

p 2 ,' s3 5-, ^ , BM(ε) = [0, . . . , 0, εpm−1]T KM

,p 5-, ^ O ,'?2 ,2 >,2 2 ,*p‖BM(ε)‖=εpm−1KM

bo 575 ( 5-,27p . . >, 572 . xe

1*, 5-, . . .

‖xe(t)‖ ≤ k1‖xe(0)‖ exp

(

−λα

εt

)

+ k2εpm‖Xe(0)‖ exp(−λet)

+ k7εpm

εs + λα

∗ ‖U(t)‖ , ∀t ≥ 0 .8h[ 1=u N

] . ,, )Z^ *, . 7 ( `02 , 5-,98 oLu7 . N P 5 . 2 ( 5 # , ( 2 . ,d5 5 ( .( 5 r(t),

d(t)5&1* (~# =,' =,'?2-,

( ( 2 . 5'p m ( 3 57, . . 2 ( *,X8h[ q N ^ , ∃kUω, kU ≥02 .*( 5 ^ *, ‖Ut,0‖∞≤kUω‖ωt,0‖∞+

kU

p ∀t≥ 0 z 2 . p . . 2 ( . *, ( 021* . 2 . *5 # . 0v1* 8 [ =q N$p2 ,' s3 5-, ^ ,

‖T (ε)‖=1, ‖T−1(ε)‖=ε1−pm

pi5 . ,, )L^ *, ε∈(0, 1] ] 2 ( (). 43 57, ,572-,5 ##. 2&52p; 2 P w3

57, . . 2 ( *,8h[ 1=u N 5k . . ?2-,5 8 [ =N,8h[ =\N . .=. . 5!, 8*,,5 2 ( . 021* ,5 2 .

M 5a . . ?2 , 5/8 [ u N , 8h[ [N . . `15m 5 , *,=5 . . 5*57, , . 51* 2 ( 55 2 ( (~)6. 53 57, . *, 5 ( . . . *, |e|= |CMT−1(ε)xe|= |CM xe|≤‖xe‖

E C I ( $B >4DB"< &%:Zc +nB3eBw>>/>4 ;Bm> ;B!DB0vB$D $: <wACB'< < v>4D "<

WM(s) =KM

spm + apm−1spm−1 + · · · + a1s + a0

,8h[ 1 [=N

> >4D ;Bpm = n∗ dc5<*D&< HB?ACBwce>CA A2< 9=<*D&< : D eB > B$DFD> ;B B Ah:"<;> ;> B AU< ;>

DB A$: 05<05ACB'<F =<s>4D ;B ;> c5<*D0/BhD>ε / < I0 :mACB ;>9>*@$ACB$D$E4< ;>4D ;Bd<0 > 9=< (+e> c5<*D&<EeB

ACBX< 0w:U</> />4 ;Bm>E ;BZD B0vB$D $: < "<*:mA,9;B$D<0 kc5<*DB BkACB$DL@$B "<*:mA >ZcmBHG<Bce>4D:mA > ;>,v>4: ;B ACB eE*> EC: =< B A < $B ACB

E C J <*D&<ACB E*B$DF: &<*Dw>F>Zce>4D U<0/B >J ;>>*@$ACB$DFE4< ;>4D8 ;B9<0 >E9=< (+e>B > *: ,+CB A >w: <*:mA

K = Knom Bθ∗ = θnom B9ACB cHB$D #iD'@ <;>C ;B/B D< <

d(t)≡ 0

Y q

mB"@2D<0UACB eB B AB &<4AC>9 B UACBU(t)≡0

t> eBdACB F> i: <Zc5<D : D7 =< ;B e: ;> / A A$:9 <tc5<*D h: D* =<X< I0 :mACB* =<aB5<;> =< *: 0w: &< ;>OB$DFD> ;B6B A :"<;> > ;:ACB eB!>sB$DFD>F ;BOB A :"<;>

xe(t) → 0BHG'ce> B $: <0 B B "<9E*B5 eB!>OA$: <0

πe(t) → 0BHG'ce> B $: <0 B B"B"< "<0 D$::5

Ae(ε−1)

; :;D b: 5J ;B A4 ;B HBε > 0

A> ; ;B ACB/B A cHB$D<*Dce>4:mA/> ( ; >*@$ACB$DFE4< ;>4D : B&<*D HB<$cDB ACB < B A B8 >Zce>4D <0/B >B > *: ,+nB A >w: <*:mA BLACB ceB$D#;D2@ <,+CB A ;> /B *: =<4A E CFK >4 ;B$D$: <c5<*DB B$D "<*:mA <0 ;D<0dD B cDB ACB U<*D >CA B0vB$: >CA =<4A > *: ,+nB A: e: $: <*:mA3 ;>CA /> ;>CA >*@$ACB$DFE I*E*B$:mA "<4A ;>- > D>:I*E*B$:mA3 =<B<;> ;>B$DFD > <0 D<E ; A =<X: =A;>7 ;B A$: <0eBHG'ce> B $: <0 /B B* ;B$D B A B B

πe(t) < B5<;> ;BRA2< =<

&%( 2HB c5<4A A2<*D$: </<"ACB$D

e = CMxe + πe(t) ,8h[ q N

B ceB$Dw: : DF: <<_B :w: <;> ;>9A$: <0πe(t)

=<B5<;> &% a>?B U< > t</: A;> ;> B$D />

πe(t)D B A U<*D$: < >>"< v>4D< B

‖xe(t)‖ ≤ k1

εpm−1‖xe(0)‖ exp

(

−λα

εt

)

+k2

εpm−1‖Xe(0)‖ exp(−λet)

+ exp

(

−λα

εt

)

∗ [k3‖ω(t)‖ + k4] ,8h[ rq YnN

> ;BZACB >U< HB9>> B$D /> k2

εpm−1‖Xe(0)‖ exp(−λet); i: >>"<*:mA9 > 5ACB$D$E4< ;>4D7 ;>JeB9>

B$D />k2ε‖Xe(0)‖ exp(−λet)

cDB ACB B > "< v>4D&< B eBJ;>*@h: ;> h: ::5C< ;>UACB<B5<;>J ;>?B$D$D> &% &%( X<0 ;D&<0 /B B ;B$E*BUACB > 5A$: ;B$D<*D3HBs>B AU< ;>

xe(t) =<B 5<;> ;>_B$DFD> &% ;9/>4 *: &< ;> > &<4AC>B eBaACBsB :w: <>"A$: <0

πe(t) ;B A <

B5<;> E C N ( =AC>3 =<4A hD< 5A0v>4D "<,+nB A

T (ε)BT−1(ε)

<9>*@ B ;>7 ;>CA"< v>4D< B A aB tc5<*D< < >4D "< ;> B$DFD>3 ;BZB A :"<;> ;> B AU< ;> ‖xe‖

ce>4 ;B29;B$D<*D DB A < ;>CA ;:> > 5ACB$DFE4< ;>4DB A > />sACBdce>4 ;B> > i: DE ;> '<0 > ;BFHB ‖T (ε)T−1(ε)‖=1

ZB"@$>4D<> "< v>4D< Bac<*D< B A < >4D "<sB AU<=@$BmB$: ;>/ce>4D ‖T (ε)‖ ‖T−1(ε)‖ = 1/εpm−1 !ACB >4D <BHG< 9;B$D< =<0/B B 9*D< ;B <=/B *: =< eB

ε → +0 "<E*B5FeB

pm≥2

( AC>J =< >4D "< ‖T−1(ε)‖ <"A B ACB8 =< ;>J ;B />4 0 ~<;> >wA$: <0 ;B > hD>mBB /B*< >4ELB <0 %C B #"$ cH>4 ;BtDB A <*DkB , .( . !eB cH>4 ;B ACB$D6>*@$ACB$DFE4< ;>

Y q Y

> > hD>~< ;>4DQcD> ce>CA >Om 2( 2 ACB=90 ;> ( + !7+5<0 :

(' 03 03:6

M # ,'*; >,29ceB&<4f4: 9X5"*57, , . *, . 2&w. . 7 PO6. .* 2 , (m). ,2 2 ,

k1

εpm−1‖xe(0)‖ exp

(

−λα

εt

)

,8h[ rq # N

,57,2?2-,w55Z . . 2 ,5 , *,9,5 2 ( . 021*8 , 572 . 8h[ =N ,?8h[ =\*N z . *,$,,$3 5-, ,5 7*, . % =,2 ε

5 e'( ,2 2 ,' =,'?2-, , ^ *,' . . ,45 )2( , , ,5 2 ( . 021* ,5 ( . , . . 2 . .*( 5 x ( 2 . *5 ( 2` ( **5-, , . 4 z 2 . 5p ( 572 1*, ,5 2 . 5 . 5 =,'?2&3 (m 3, 3*, ,572 ( . 0v1 ,572 . 65 . 2 ,d 2 . 5 ( 2&` ( ( (m'(4. 5 . 021* *, 1/εpm−1

b] ##. 2 ( 5 Pk^ *,d1*!7 x cHB&<4f4: 9 5, *, 5 . .s*57, , . p # =, 57, 1*, **572 . 2 . 56 . . 2 ,5O8h[ u N ,8h[ [=N

z . .w( *572 . # ,2 ; =,'9cHB&<4f4: 9LIZKaM9p *5 ( *, . 3 5-, . , ^ . 021*8 ( D%* (6.

˙xe(t) =

−2

ε1

− 1ε2 0

xe(t) , xe(0) =

1

0

, t ≥ 0 ,8h[ q N

^ *, , 7 ,57,2 2 . . ( D% (. , ;*, ,5 2 ( . 021* , 572 . 8h[ N *, 5 ILK!M *, 5-,'. . *,2 ^ . 5 U(t)≡0

,πe(t)≡0

g ,' ,5 2 . 5 ( , 2 , )6. 5'pi! . . 2 ,!8 [ 0=N ,5 2 . ,2

‖xe(t)‖ ≤ k1

εpm−1‖xe(0)‖ exp

(

−λα

εt

)

, ∀t ≥ 0 .8h[ q N

02 . 43 5-, ^ *, . . .,2 *, ,5 2 . ( ( . ," * ( ; ( Nα(s) = s2 + 2s + 1 Pλα =1 s−1 p , 5 m7 ,2i3 5-, λα =0, 3 s−1 <λα

. .^ *, (m / . . ?2-, 57, . , ^ *,2 ,=5-,2 2 . *5 ( 2` ( >57, . .*) . ,,'m >,2 2 , x ( o *5 2 . 2 , k1 =1, 2 # ( 2 . . . . ( ( (). s, . m *, (m " . . 2 ,s8h[ q N . . w, . , 572 ( . *, λα

o ( . . [ Y . ,5-,'?2 .:. 5 . , *, ,5 2 ( . 021* , 572 . . ?2-,t 2 . 5 ( 2&` (

*,!8 [ rq =N M # ,'*; >,2ZceB&<4f4: 9 6. ,', ( *,2 2 , ^ . O . %* >, 2 ε P , )'( 9*, 1 . .

0, 1 02 . 3 5-, ^ *,5w . . 2 , 5 ,5 2 . ,'/, '( 05 ,' . *, 5 ( .* . . *, 8 [ q N51 . 5 2 . 2 ,

*5-, , . ,52p ,' w . , ( 2 . "ceB&<4f4: 9 ( 5 .=. ( 2&*,&5/ . . 2 , , . 5& . 2 .n*.*,*, .*( >,2 2 . >,2 2 . . =, ( . ,' ^ *,L . %* =,2 ε P ,45 )2( p 2 . ^ . , . . >,5 2 . 5

Y q #

Majorante para = 0,1

ε

Norma do estado para = 1

Majorante para = 1

ε

ε

ε

Norma do estado para = 0,1

0,5 1,0 1,5 2,00

2

4

6

8

10

12

t (s)

Nor

mas

e M

ajor

ante

s

0

1

,2*;* =,' ceB&<4f4: 9 *57, , . *, . 2& . . 7 o . , 572 . 8 ‖xe(t)‖N5 . , ^ . 021*5 , 6*, , 572 ( . 0v1*35 ,5 2 . 58h[ rq =N PZ . . .

>5-,' . . 2 ,s8h[ rq N . . ( 5 , . , 5;5 . %* =,2 ε

C HC IHC B * E L E E ! E

] , ( 2& ( 2 . ?2-,>5&* ,/ 5-, , . ,5<*, . 2_. . 7 P BZce> ;B_BHG : ;> ;>c: >4p ^ , P *, ( . .*(| # >, . ( . . [ $#

" #$ E C B D > 5A$: ;B$D BUACB/< B5<;> =< *: 0w: &< ;> B$D$D> ;BB A :"<;> ;> B AU< ;>xe(t)

J ;B ( >U(t)≡0

Bπe(t)≡0

=< =<Oce>4D

˙xe(t) = Ae(ε−1)xe(t) , t ≥ 0 ,

8 [ rqq N

> ;BZ< "<0 D$::5 :;D b: 5Ae(ε

−1); ;B e: =<"B < ( 2 ,' *, , 2 ( 50v1*35 (

te

;> ( ;9>>/B >4DXE4<0m>4D3HBXA2<0 :mA0'< 59< ;B A$::905<0: =< ;B

‖xe(t)‖ ≤ ‖xe(0)‖ , ∀t ≥ te ≥ 0 , ∀xe(0) ∈ Rpm ,8h[ rq =N

c5<*D<= E4<0m>4D !Gi> ;>Lc5<*D 0/B D>ε∈(0, 1]

M BZce> ;BBHG : ;> ;>/c: >8*, ILK!M 1*,85-, (# 2( *,85-, 6. . z ( 5-5&5p ,2 ,, ) *, . x 3 , . 2 . >,2?2-,*p 1 ,5-, .( 5 =,,2 ( ,2 2 ,,5 2 ( x 3 . 2 . , P 5

Y q

Majorante da norma do estado

Norma do estado

te

et 0,5 1,0012

4

6

8

10

12

Nor

ma

e M

ajor

ante

0 t (s) $BZce> ;B BHG : ;> ;>c: > 8

teN *, ( . . 2 ( . . , 572 . 8 ‖xe(t)‖

N!, "< v>4D< B ;> BZce> ;B BHGh: ;> ;>c: > 8teN , ( . . 2 ( . . ?2-, 8 [ rq N o ^ ( 57,? 2 ( (~)6. " , ,2 # ,2 ; =,'ceB&<4f4: 9a5*57, , . *, . 32. . 7 , ^ . 0v1* , *,9,5 2 ( . 021*8h[ rq =N . . ε=0, 1

. . 2 ,s8h[ q N ^ *, ,5&5 2 . . ,5 ( . . . ,

k1

εpm−1exp

(

−λα

εt

)

≤ 1 , ∀t ≥ te ≥ 0 ,8h[ q \*N

7*,te≥ te

8 , ( *, ( . . [ $# N$p .w^ . (5-, m ( ^ *,

te =ε

λα

[ln(k1) + (1 − pm)ln(ε)] =pm − 1

λα

ε[(pm − 1)−1ln(k1) − ln(ε)

].

8h[ rq*u N

z . . 57, ,,27*, . *, ,25%2 2(/. BZce> ;B_BHG : ;> ;>"c: > ,2 ,2 . 021* . . %* >, 2 ε

*, *,$3 5-,! "< v>4D< B >4D "<0 ::5C< ;> ;> BZce> ;B9BHGh: ;> ;>Lc: >

Te :=λα

pm − 1te ,

8h[ rq [=N

Y q

^ *, P .

Te(ε,Kpm) = ε [Kpm − ln(ε)] ,8 [ N

7*,Kpm :=(pm − 1)−1ln(k1)

bo ##. ( . *, , . 5 . ( . . [ , , 5-,'?2 . Te(ε,Kpm) . .=. m. 56, . m ,5 ." *5 2 . 2 , Kpm

z 1 ,$3 57, m ( ^ *,a "< v>4D< B ;>= BZcH>J ;BBHG : ;> ;>!c: > P (~# >,' =,'?2-,a ( ( 2 . =,' ,2 . 0v1 . . %* >, 2 ε∈ (0, 1]

,2 ,'*,.) , . >,4 ( ./^ *, ε → +0

p . ,, )s^ *, 5-, . 5 . 55X5L, . m ,5*05 . % =,2 5k1≥1

pλα >0

,pm≥2

K

e pm0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,2 0,4 0,6 0,8 1,00,0

T

ε . (/. , , . 5 "< v>4D< B >4D"<0 ::5C< ;> ;> BZce> ;B BHGh: ;> ;>

c: >w8Te(ε,Kpm)

N

E C <*D&<!< B :w: <;> ;> , .( 5. D +;:>;D*%('' %4 cD> c +CB >*@$ACB$DFE4< ;>4D ;Ba<0 > 9=< (+e> eB ce>4 ;BLACB$DODB cDB ACB < ;>Zce>4D

˙xe = AM xe −[H(t, ε−1) − aM

]e ,

8h[ 5YCN

e = CM xe − e ,

> ;BH : R+×R+ → Rpm

; E*B >4DQE4<DF: < B >: BZce> e: $: <0 /B B6>LE*B >4DH(t, ε−1)

B >4D "<kceBeB <kc5<*D<FeBO> , .( .ACB'<"B$EC:U< ;> ;> >aE*B >4D H(t, ε−1) > eE*B$D 9;B

BHG'ce> B $: <0 B Bdc5<*D< >?E*B >4Dα(ε−1)

< ;B HB/>B$D$D>C ;B/B A :"<;>ACB'<wceBeB >Y q*q

<$c.-CA > hD< 5A$:-4DF:U> : e: $: <0 !>B < > D +;: >;D %(''%4 DBF> (+eB BeB >>*@$ACB$DFE4< ;>4D ;> B$DF: < A B A AC> B AF: 5<,+CB ADB&<*:mA <4AL<*:mAsceB$D #iD'@ <,+CB Awce> ;B$DF: <0 &<0A2<*D> , .( .5< ;> < >4D "< ;> E*B>4D

H(t, ε−1)v>4D 9*D< ;B"B"@$>4D< > ACB >*@$ACB$DFE4< ;>4D

B :w: B > , .( . B > *: ,+nB AL: ;B&<*:mA 0;D&< Bw> hD< 5A$:-4DF:U>s: e: $: <0 "< <0=A $: < *::9 < ;B > < > >*@$ACB$DFE4< ;>4D ;>A$: <0 ;B > D >mB

U A ACB

B$D />F;6: *:mA ceB 5A4I*E*Bnc5<*D&<2v>4D "<*D ~<> ;B6DB<0 :/B U<;>9 > 9*D&<0"DB~<0h: E*> e:=I*D$:U>< iB eBw>8 > D>mBa<_B A D #iD<E4<*DF:I*E*B 9;B$DB9>>/>4 ;> ;B A ::5C< B

3 / .

/

M ( =, ( 2 . 5 572& . . 5 ( 572-,2 . 5 ( *, . ,5 n∗ ≥ 2 2 ( (~)6. . , ( *,

?2 , . ,572 2& . , . (mx ,,' 8h[ # [N 5. . .L OILK!M 8h[ N 8 [ iYnN z . . 5-, ,$, ( 2 . ?ceB&<f4: 9s 5 ( . *, 2 */, u

,"055 ( .( 5& . . ?2 . p 2 ( (). 3 5-,w5 2 5& ,=, 572 . 8 [ Y # NZ . 5 ?2-,5-, .?# 021* *," "1 . 0v1 ρ

> ,'?2 . ?2&p ,s?ceB&<4f4: 9w , 572 . , 572 ( . p ( 5 ,572-, # ,2 ; =,' P?( ?2 5-, IZKaM 8h[ N 8 [ iYnN 572 , 2 * . P*, 5 ( 2&" . . ,' . [ Y

*240 % 0032"187>** % %:0

z . . . . ( 5 . . , 572 . ( ( . *, 5 5 ( 5 2 ,2 . *, *?2 , . . . ?2 . 5 . . ,2 . 2 ( ,*n∗ ≥ 2

p /,', . 43 5-,9,2 *5 ( *, . 021* *, # ." /,2 .!. 5 ( 02S , 5 ( (2(/.*( 52p 2 ( (). 3 5-,w,, 2& *, ,572 . z . 575 ( , ( ;?

z :=[XT

e , xTe

]T.

8h[ # N

^ *, 5-,'.*,p*2155 5 k 5 , λ 55 ( 9 (). *5 2 . 2 ,5 5 ( 2 ( , . 5Q.,' P (m. 52p K 5 51# 502S , 5 .w . 5-57, K , . . *,X , . 8 ‖·‖ N PX( )2( . ,2 . 5 . L∞e

8 , ( *,g ,'0v1Y $# YnN

M ,' ,' . [ ,572 . ,2 , , . 5 7 ( , . ,5 *, , 572 . ( ( . *, 5 ( 572-,2 .9 2&1 . 5 . 5, ^ . 02S , 55595 ( .( 5 *,!, " . . 5 8h[ Y q N 8h[ Y N:, 8h[ =N z . . N = 0

8n∗ = 1

N ., 572 . ( ( . *,L. . , *,2 2(4. P , 572 . ,2/, 2( . ,' ,' . [ Y

Y q

o m.* ( 2 s 5 2 . . 2 /, "1*,29 , , # , %2 '(4. *, 5 ( 572-,2 . 5 ( , . ,5 . . ,' . 2 ( , 5& , ( . 57,2 ceB&<4f4: 9d55 5 ( .*( 5: . . i32 . 572-, . . ( 2 s P . 5-, . ,2 , 572 2& . , . (yx ,,2 ,; *5-, , . , . 32. . 75 o # 50v1* *," 15 . 021* . ,5-,'?2 . . *, 572 . 2 . ,2 . 1*,57, 5&5*572 ( 2 . ^ . ^ , 2 .w# 021*8 ,! 15 . 021* ^ *, 5 . 2 ( 5 ##. 0 .>. ,5 ( . . . ,"8h[ $# YnN , . 7 ( `02 , 5-,8 oZu7 . N

xM = AMxM + BMr , yM = CMxM

!"# $e(t) = y(t) − yM (t)

% &$(') *+'$&,- ω1 = Φω1 + Γu , ω2 = Φω2 + Γy

./&-01(+''-ω = [ωT

1 ωT2 y r]T

233 +-&$-u = unom + U ,

U = −ρ sgn(σ(xe)) ,

σ(xe) = Sxe

4 -&--56 !"unom = θnomT ω

%7 8:9-; <5=7 -8:9-ρ = δ + c1‖ω‖ + d(t)

d(t) = d(t) + c2s+γd

∗ d(t)

>@? ',$A-- -@ *" &61B" C ˙xe = AM xe + BMKnom

p

KMU −

[α(ε−1) − aM

]e ,

e = CM xe − e

Y q \

E C J <*D<N =n∗−1≥1

> 5A$: ;B$DBUACB6<kc~< U< BL> > hD>~< ;>4D ;B A $DF:> < $e<=@$B~< )B6<4A2+;:rc.-B ACB A 0/ > 0/ > 6v>4D BA2<0 :mA0vB$:U<4AdB < ;>7 ;B/>4 0 ~<;>ρA2<0 :mA 654B$Dd< ;B A$::905<0: =< ;B &%9BZ< +;:rc.- B ACB 0/ " < B ;> Hc5<*D<

ε>0A$:UB B/B B

ceBeB > L>A$:mAB"< >Zce>CA > ceB~<4A/B5<,+nB A ;>CAB$DFD>CA _B > B AU< ;>

z ;B e: ;>B %49ACB$DHIE90m>*@ <0 /B B BHG'ce> B $: <0 /B B B A I*E*B B D B|< ;>"<!

> >"DB A$: 05<0! ;Bw>4D ;BεR:mA> ; ∃kz, λz >0

B "< ;> KX(ε) =< ~<4A ACB K U<*:mA

eB

‖z(t)‖ ≤[

kz

εpm−1‖z(0)‖ + KX(ε)

]

exp(−λzt) + O(ε) ,8 [ =N

|e(t)| e ‖Xe(t)‖ ≤ [kz‖z(0)‖ + KX(ε)] exp(−λzt) + O(ε) ,8 [ N

∀z(0),∀t≥0

D>4E4< ] . ,, )/^ , . 7 ( `02 , 5-, 8 oLu7 . N 5& , 5 ( . 5 , ?2 , u P 5 . 2 ( 5 # , ( 2 . p5 5 ( .*( 5;=5 ( 572-,2 . 5&1*DB=90 ~<*DB A ,p 2 . 2p 1*,2 ,5 , 5/ x4i( , *,' 2(/. |3 >,2 2 , 8hg . 572 VQ1*5&* YC[ u [=N . ,2 . 0v1* 8 YC\N ,/,' w . 53 57, ^ *,/ 5 ( . r(t) P (# =,' >,2?2-,L ( ( 2 . 5p 2-,2 w3 57, ^ *, ‖ω‖≤kM +kΩ‖Xe‖

2 1*5p 1*,F3 57, * ( ^ *,

‖xe(t)‖ e ‖Xe(t)‖ ≤ ke1 exp(λe1t)‖Xe(0)‖ + ke2 [exp(λe1t) − 1] , ∀t ≥ 0 .8 [ q N

g ,2.* , ( 2 BZce> ;B BHG : ;> ;>_c: > 8teN 5 2 . g ,'0v1* [ # p

m ( 3 5-, ^ , te≤ te . 8 [ rq*u N 1 , 5-, . . .X# 0v16*, ε

.a . 5-57,K o 575 ( ps . . 2 ,s8h[ q N 1*, 5-, . .

‖xe(t)‖ e ‖Xe(t)‖ ≤ (ke3 + ke4ε)‖Xe(0)‖ + Ke6(ε) , ∀t ∈ [0, te] ,8h[ N

^ *,p ,' 5?2 J . . 2 , , *, ,572 ( . 0v1 , 572 . 8h[ \N$p , ( 2 , m ( ^ *,

‖xe(t)‖ ≤ k1

εpm−1‖xe(0)‖ exp

(

−λα

εt

)

+ ε [ke9‖Xe(0)‖ + ke10] , ∀t ∈ [0, te] .8h[ =\N

o . . <*, 5-,2=,, ( =,'?2&*,L5 . . 2 , . . ,572 . z 5-5-,'.*, . . t>t1 :=

te . 43 5-, t1 "! C, ( *572 . ?2-, ( (2(/. . `5 . , 2 ( 0v1* ( !5 2 . 5 ( 2&` (

Y qu

"ILK!M9p 2 ,2 w3 57,*p ∀t≥ t1p

‖xe(t)‖ ≤ [‖xe(0)‖+ε (ke11‖Xe(0)‖+ke12)] exp

[

−λα

ε(t−t1)

]

+k2ε‖Xe(0)‖ exp(−λet)

+ εC(t, t1) ,8h[ u N

≤ [‖xe(0)‖+ε (ke13‖Xe(0)‖+ke12)] exp [−λe(t−t1)] + εC(t, t1) .8h[ [=N

*,,' ( ,'?2 , 57, ,,5 ,',, . , ^ . 0v18>, 8h[ #q N "

xe = AMxe + BMK [U + dU + πe(t)] ,8 [ \ N

7*,

dU := (θnom − θ∗)T ω + Wd(s) ∗ d(t) .8h[ \iYnN

] . ,, )a^ *, . /, ( *, 2 /,O 2 ( ()6. =, 572 . !, 572 ( . ,2mIZKaM 8xe(t)=xe(t) +

xe(t)NFp 1 ,$3 57, ,,5 ,',, . , ( , ?2 ,"8h[ $# [N "

U = −ρ sgn (σ(xe)) ,8h[ \ # N

σ(xe) = S(xe + xe) = Sxe + Sxe .8h[ \*=N

,' 9 . 3 5-, ^ *, AM , BM , S P 5 . , . ()6. 021* 1*3 ( . , ?2 x ,,'6*,WM(s)L(s) = KM

(s+γ)

, ^ *, K = K−1M Kp

pk5 ( . , 7 . ,, . =,'?2& σ(xe) 1*, 5-, 0 ,$3

,57,2?2 . . . 2 ( . , ^ . 021*8 ( D%* (6. 8h[ \ N ,& . , ^ . 0v1* . m. P (6. 8 [ \*=N "

σ(xe) =Kp

(s + γ)[U + dU(t)] + π1(t) + β(t) ,

8h[ \ N

π1(t)+β(t)= Kp

(s+γ)∗πe(t)+Sxe(t)

b] 2 ( (). 3 57,ka . . ?2-, . . ‖xe‖ .

8 [ *[=N$p 2 ,2 w3 57,*p ∀t≥ t1p

|π1(t)| ≤ ke14 [‖xe(0)‖ + ‖Xe(0)‖+εke12] exp [−λe(t−t1)] ,8h[ \ q N

‖βt,t1‖∞ ≤ εkβC(t, t1) .8h[ \0=N

Y q [

2 1p 5 . ,, )s^ *, 5-, . 575& ( ^ , Kp > 0, .?^ *, # 0v1 *,w 15 . 021* 5 . 2 ( 5 ##.*)

ρ(t)≥|dU(t)| 8 ∀t≥0N$p 1*,$3 5-, . (6. (,2 . # . 5 ( 5 2 ,' .X# . 5 8h[ \ N, ,2 .

, ( *, ?2 ,"8h[ \ # NFp ^ *, , 5&32 . " . . 2 ,

|σ(xe)| e |σ(xe)| ≤ ke15 [‖z(0)‖+εke12] exp [−λe(t−t1)] + 2εkβC(t, t1) ,8h[ \\N

∀t≥ t1pi7*,

σ :=σ − π1(t)− β(t) ,2 ,2 9 . 3 5-, ^ *, u=unom + U

pi02 . 3 57, ^ *, Kp

(s+γ),' 8h[ \ Nd , . _ >,5 85 ( . U + dU ^ *, .*^ *,'/, ,' 8h[ Y q N wo . 2 ( *,8h[ \ N 5-, m ( ^ *, U + dU = K−1

p [ ˙σ + γσ] b?21*p . . 2 ( *,8h[ Y q NFp , *, . 572 , . >,2 2

s 1 ,2m 1 , 57, ; ,, 5 ( 2& "

Xe = AcXe + BcK[

˙σ + γσ]

.8h[ \ u N

z . . ,' ( ( . . *, ( , . . ˙σp ,' ,2. . 3 57, . 2 . *5 # . 0v1*3*,d, . (yx ,, ( 5 Xe :=Xe −BcKσ

,5&5 2 . >,'

˙Xe = AcXe + (Ac + γI)BcKσ .8h[ \*[=N

] . ,, )!^ *, Ac P Ik < ( 2 ) , ^ *,! 5 ( . σ 5 . 2 ( 5 ##.*) s . . ?2-,s8h[ \\NFp 2 ,' s3 5-, ^ ,

‖Xe(t)‖ ≤ ke16 [‖z(0)‖+εke12] exp [−λe(t−t1)] + εkC(t, t1) , ∀t ≥ t1 .8h[ u N

o P ( 5-5&p * # >, P , 5 . , 2( . .*(~ p ∀t≥ t1p

‖Xe(t)‖, ‖e(t)‖ ≤ ke17 [‖z(0)‖+εke12] exp [−λe(t−t1)] + εke18C(t, t1) ,

8h[ u YnN

‖ωt,t1‖∞ ≤ εke19C(t, t1) + ke20‖z(0)‖ + km ,8h[ u"# N

C(t, t1) ≤k′

red + ke21‖z(0)‖1 − εke22

.8h[ u =N

, ##. 2&p . 5"*, 5 ( . . . *, 5 ,' 8h[ u YnN?,%2 *,Xe = Xe + BcKσ

] . ,, ) ^ *,‖ω‖ ≤ kM +kΩ‖Xe‖

p,2?21*p5-, 2 P 8h[ ru"# N . . 2 ( *, 8h[ ru YCN o . . p . . 2 ( *,8 [ N ,s8h[ ru"# N$p 1 ,$3 57, m ( ^ *, C(t, t1)

2 . 9 P 1 ,:5-, . . C(t, t1)≤εke22C(t, t1)+ke21‖z(0)‖+k′

red

p7 ^ . 5-,L 2 P ( ?2-, P ( 6*,L . ( . 02S , 5 . . P 3Y

(m. 5d/ . . 2 ,/8 [ ru =N$p ^ *, P , x ( . . ε<k−1e22

M:5d . . ?2-,5 8h[ [=N . . ‖xe‖,8 [ ru YnN . . ‖Xe‖

1*,2 5-, . . 565" . . ?2-,

‖z(t)‖ ≤ [ke23‖z(0)‖+εke24] exp [−λe(t−t1)] + εke25C(t, t1) , ∀t ≥ t1 ,8h[ u0 N

^ . 1*,F3 5-,O 2 ( (). " . . 2 ,s8h[ u =N . . C(t, t1) ^ , , 5&32 . ,2

‖z(t)‖ ≤ [ke23‖z(0)‖+εke24] exp [−λe(t−t1)] + εke26 + ke27‖z(0)‖

1 − εke22

, ∀t ≥ t1 ,8h[ u=q N

^ *, . . 0<ε≤kε <min(1, k−1e22)

1*, 5-, ,,5 ( 2 "

‖z(t)‖ ≤ [ke23‖z(0)‖+εke24] exp [−λe(t−t1)] + ε [ke28 + ke29‖z(0)‖] , ∀t≥ t1 ,8h[ u =N

*,O7*, 1*,$3 5-, .4 . ^ *,

‖z(t)‖ ≤ ke23 exp [−λe(t−t1)] + εke29 ‖z(0)‖ + O(ε) , ∀t ≥ t1 ,8h[ u \N

7*," 2 , " , 5 ( . O(ε) P"( *, ,2 ,2?2-, . 5 ( 02S , 5 ( (2(/.*( 5 = 02 . 3 5-, ^ *,( *572 . ?2-, ( (m'(4. PO( ,2 ,', . ?2 ,X . *, ( , . 0v1 . 5 , , 5-5&S ,5 .*2( . p 1*,F3 5-, ,5 ,$,,

‖z(t)‖ ≤ ke30 exp [−λe(t−ti)] + εke31 ‖z(ti)‖ + O(ε) ,8 [ uu N

. . t ≥ ti ≥ t18i = 1, 2, 3, . . .

N . ( 2 x ( 572 . *, 5 ( .* . . *, * ) *, 5 ( . . . *, ( *, . ) , 5 ( , .

‖z(ti+1)‖ ≤ λ‖z(ti)‖ + O(ε) ,8 [ u [=N

λ=ke30 exp(−λeT1)+ εke31

, . m.* , 1 T1 = ti+1 − ti >0 z . . 0<ε≤ε∗<k−1

e31

,:, 5 *m7*,'3 5-,X , 1 T1 >0 . *, ^ . . >,'?2 ,Lm5.p; 2 P s3 5-, λ<1 o 5-5 ( m ( 3

57, ^ *, . . ε > 05& 2( ,'?2-,2 >,2 2 , , ^ *,'p . , 5&1* 8h[ ru [N *,, .,>,2 . ,5-5&1*

.,' P 2 (m. . . 5?2 ,5 ( . *,O *,2 ε

. . 3 5-, .d ,, .*%2 '(4. , **,2 2(4. , 572 . z(t) . . 5?2 ,5 ( .

*,L *,2 ε . . 2 ( ( 572 . ?2 , t1

p5 ^ *, 5-, m ( . , 518h[ u [=NFp . 5 . . 2 ,58 [ 0=N . . ‖Xe‖

, 8h[ =\*N . . ‖xe‖p , x ( 55 . . 0 ≤ t ≤ t1

p * 2 P s3 5-, . >,2?2-,5Y iY

. . 2 , 5k8h[ =N ,a8h[ N . . . 5 . 5 5b,, 2& , 5: , ,5 2 . 53 . 5 , ^ . 02S ,5:55 , 05^ *, 5&1*s, x ( 5 ∀t≥0

E C2( > hD>~< ;>4DL iBD<4A D B&<0/B > iB D< vB-4DF: <ce>4DDB&<0 :/B U<;> ;BA2< =<9cD> ce>CA>ace>4D ( +%! 7+5<0 :1 C;"@F<4ACB&< ;>_B B A D #iD<E4<*DF:I*E*B B ( A A2<B AhD<0<;=9*: < ;B > D >mB ;>9<$cDB ACB U<>vB /B > , .( 5. <dc~< U< HB ;!B$EC:U< ;><0 D<E ; A =<?A2<0#iD<;>C ;>/A$: <01 ;B! > D >mB eB ;390m>*@ <0 /B B :w:U< ;> a>B U< > L>CADB A < ;>CA_>*@ : ;>CA '( 2 9=<*D< B <$ceB <4A< B AU<=@2: : =< iBwACBw:90m>*@ <0 : B 3XDB AhDF:U<< "<DB=9*:#;>?<*D'@2:hD<*DF: <0/B B 9*D&< ;B<4A9 :w:U< =<_B>_B A < ;> =< B5<;> ;>B$DFD> ;BD<4A D B&<0/B >> > eE*B$D 9;B!B <0 90 BZce> e: >dc5<*D< > *> D B A$: 05<0 ;Ba>4DH ;B √

ε

!B A ACB<4A ceB>>CAsDB A < ;>CA/>*@h: ;>CAZce>4D ( + ! 7+<0 : ;>/A;> ;>@$> iA 5< >B A B AB A <=@$BmB$: ;>CA > $B >4DB"< ;B A < $B ACBXB eBwACB>*@; < B A <=@2: : =< ;B390m>*@ <0B >CA B A < ;>CAL =<4A B<,+CB AL ;>CA B$DFD>CA > eE*B$D=9;B BHG'ce> B $: <0 /B *B9c5<*D&< > >DB A$: 05<0 ;B >4DH ;B

ε > hD<$c5<*D : =<<;:9<c~< < ;B$E*B_ACB$D : B&<*D B : eE4<*DF: < B

>8 BZce> : > D&<4AU< ;> > <<=@$>4DH =< 9;B ;B ( +%! 7+5<0 : eB,v>4:k>4DF:UB U< =<<c~< U<4A ;> : B&<*D B A

3 / .

/

3

M 57,2.7 *?2 . 3 572& . . 5 ( 5 2 ,2 . 5 ( *, . ,5 n∗ ≥ 2 2 ( ()6. IZKaM

. . >, 572 . 8 . , ^ . 0v1*85>, 8 [ $#q N , 5 . m . ,' . 2 . *5 # . 021*

ζ = T (ε)xe , T (ε) = diag1, ε, ε2, . . . , εpm−1

.

8h[ [ N

M IZKaMj ( , ,;OceB&<4f4: 965,572 . ,572 ( . P * 2 ( . 2 . , P 5: .. (6. 021*& . 2 . 5-3# . 0v1* ζ =T (ε)xe .

"IZKaM 8h[ N 8h[ 5YnNFp ,5&5 2 . "5"*57, , . ( D%* (m

ε˙ζ = AM(ε)ζ + εpmBM

Knomp

KM

U − [α − aM(ε)] e ,8 [ [iYnN

e = CM ζ − e ,8h[ [ # N

Y #

7*,

AM(ε) =

−apm−1ε 1 0 · · · 0

−apm−2ε2 0 1 0

−a1εpm−1 0 0 1

−a0εpm 0 0 · · · 0

, BM =

0

0

0

KM

, CM =[

1 0 · · · 0]

,

α =

αpm−1

αpm−2

α1

α0

, aM(ε) =

apm−1ε

apm−2ε2

a1εpm−1

a0εpm

.8h[ [=N

z 1*,$3 5-, * ( ^ , , 572-,X**5-, , . P ( , , aceB&<4f4: 9 ( 5 . , ^ . 021* . ( D%* (6. , *,=, 572 ( . 0v1 , 572 . xe(t) := ζ(t) − ζ(t) P . . 8h[ N$p ^ *,e1* , i( ,ceB&<4f4: 9 # >,X" . . 2 , . . . 5>, , , 572 ( . 0v1* ,572 . 8h[ 1=u N

o /, ( , *?2 /, . ,572 2& . , . (mx ,,' *, 572 , 6. 5& P *, ( . ,2 . 5 , # 2( ,*, 7 . 3,, . >,2?2& σ(xe) = 0

p *,σ(xe) := Sxe = ST−1(ε)ζ

,'?2 . ?2&p . ,, )"^ *,s5 ( . σ(T−1(ε)ζ)

1 , (/. . , 5-,'?2 . ceB&<4f4: 95 ,', ( 5 . d 5& . 2 . *5 # . 0v1 T−1(ε) . 3 .9P ε1−pm

p ,572-, P ,5 6. . . *, ^ . . >,2 2 , ,5 2 . w 5 ( . σ(ζ) :=εpm−1ST−1(ε)ζ

^ *,!1*>, i( ,LceB&<4f4: 9 o 575 ( . , ( *, *?2 , P . .

u = unom + U , U = −ρ sgn(

S(ε)ζ)

,8h[ [ N

7*,S(ε) := εpm−1ST−1(ε)

,a ?2 ,a* ( . unom P . 8 [ $# N 5 2 , 2 43 . P *, 5 ( 2&s . . ,' . [ $# ,X5 (4. . . . *,X 05 . ( .* . [ ,5 2 ,& (/. . . .*,L 5 # ( . 5-5 ( . . 8~<> ;> />4 ;> ;B A ::5C< B : ;B&<0-pi5 ^ . .a# 0v1 ,:2 . *5 # ,$3 %2 '(4. 5 ( . U . . 95 ( . σ P *, ##. 57,Z ( . , 575& ( . . ,' . 2 ( ,a ( 2 x ( p ^ , 575 ( ( ( 2 . . %2 2(4. 1 *,5 ()6. 2 , 8 Q* . ,', , 2 . YC[ u=q N M ,' P^ *,., . ?2 , 7 . ,, . ##.*) . 2 ,<*,9 . . m7 . *, , . ( >,'?2 . 0v1* ^ *, . 5-5 . ( , 2 . =,'?2 ,. 2 . , P 5:5a*5-, , . p ^ *, , . ()6. @ c5<4A A . ( D%* (6. . . 2 . o 5-5 ( _p +5<0#B$DF: 9^ *, 1*, (4. 5-, . *5 . ( ;%* (. 5 . . 5 ( 2 . 5 . . ?2 . p 5-,'*5& ,5 , . 2& . ,5 1*,

Y *

57, ,$, ( 2 . 5 ,2a*56a*57, , . . . 7 .X * 2 , 15 *,5 (m). 2 , 8R.,2 . YC[*[[=N$p 8 ] 2 ( , 2 . YC[[*[5pHg ,20v1 u =N

o m.* ( 2 s 5 2 . . 2 /, "1*,29 , , # , %2 '(4. *, 5 ( 572-,2 . 5 ( , . ,5 +. . ,' . 2 ( ,!5 , ( . + ( , ,3*,aceB&<4f4: 9 5 2 , . . ( 23 " P . 5-, . ,2 , 572 2 . , . (mx ,,' , 5-, , . *, . 32 . . 7 ( , ,**,LceB&<4f4: 9 o # 50v1**,X 15 . 021* . ,57,2?2 . . *, 572 . 2 . ,2 . 1*,57, 5&5*572 ( 2 . ^ . ^ , 2 .w# 021*8 ,! 15 . 021* ^ *, 5 . 2 ( 5 ##. 0 .>. ,5 ( . . . ,"8h[ $# YnN , . 7 ( `02 , 5-,8 oZu7 . N

xM = AMxM + BMr , yM = CMxM

!"3$e(t) = y(t) − yM (t)

% &$(') *+'$&,- ω1 = Φω1 + Γu , ω2 = Φω2 + Γy

./&-01(+''-ω = [ωT

1 ωT2 y r]T

233 +-&$-u = unom + U ,

U = −ρ sgn(

σ(ζ))

σ(ζ) = S(ε)ζ , S(ε) = εpm−1ST−1(ε)4 -&$-*-56 !"

unom = θnomT ω

%7 8:9- 5=7 -8:9-ρ = δ + c1‖ω‖ + d(t)

d(t) = d(t) + c2s+γd

∗ d(t)

>@? ',A-- -) " &61B" Cε˙ζ = AM (ε)ζ + εpmBM

Knomp

KMU − [α − aM (ε)] e ,

e = CM ζ − e

*240 % 0032"187>** % %:0

z . . . . ( 5 . . , 572 . ( ( . *, 5 ( 572 ,' . *, 2 */, ^ *,X 2 ( (). sILK!M ( , , ceB&<4f4: 9 . . . ?2 . 5 . . ,' . 2 ( , n∗≥2

p ,', . 3 5-,,' *5 ( *, . 0v1**, # .s /,2 .. 5 ( 0'S ,5 ( (2(/.*( 52p5 2 ( ()6. 3 57,Xw,, 2& *, ,5 2 . z . 5-5 ( *, ( ?

z :=[XT

e , xTe

]T.

8h[ [ q N

M ,' ,' . [ ,5 2 . ,'/, , . 5 7 ( , . ,5*, , 572 . ( ( . *, 5 ( 5 2 ,2 . 21 . 5. 5 , ^ . 0'S ,585>5 ( .( 5*, , . . 5 8h[ Y q N 8h[ Y =N, 8 [ =N ,', ( *,2 2(4. ^ *, 5 ( 5 2 ,' . *, ?2 , P .m . >,'?2 , , 572 x ,,2hpZ ( , , ceB&<4f4: 9 , 58, 5 *,, .,'

Y

+

−

+

+

+

++

−d(t)unom

e

e = CM ζ

e

ζ

S(ε)

r

ρ

u

U σ

y

yM

G(s)

WM(s)

ML*5-, , . 5

1*,'m

zb . ?2 .

!"

−ρ sgn(σ)

$# o . ( 2& " 5 2 . . 2 */, 1 ,2m9*, , # , %2 '(4. *, 5 ( 572-,2 . 5 ( *, . ,5 . . ,2 . 2 ( , n∗≥2 ( , ,&*,OceB&<4f4: 9 M:5 32 05 *,, 572 . !,

. 5 ?2-,57, 5 ( . ρ # . ( 2 ( 5 . . 5 ( (~e. (/. . . .

, **,2 2(4. m >,2 2 , . . 2 , 5 ( . *,O *,' ε

E C K <*D<N =n∗−1≥1

> 5A$: ;B$DBUACB6<kc~< U< BL> > hD>~< ;>4D ;B A $DF:> < $e<=@$B~< &% )B6<4A2+;:rc.-B ACB A 0/ > 0/ > 6v>4D BA2<0 :mA0vB$:U<4AdB < ;>7 ;B/>4 0 ~<;>ρA2<0 :mA 654B$Dd< ;B A$::905<0: =< ;B &%9BZ< +;:rc.- B ACB 0/ " < B ;> Hc5<*D<

ε>0A$:UB B/B B

ceBeB > L>A$:mAB"< >Zce>CA > ceB~<4A/B5<,+nB A ;>CAB$DFD>CA _B > B AU< ;>

z ;B e: ;>B 9ACB$DHIE90m>*@ <0 /B B BHG'ce> B $: <0 /B B B A I*E*B B DB~<i>"<!

> >"DB A$: 05<0! ;Bw>4D ;BεR:mA> ; ∃kz, λz >0

B "< ;> KX(ε) =< ~<4A ACB K U<*:mA

eB

|e(t)| e ‖z(t)‖ ≤ [kz‖z(0)‖ + KX(ε)] exp(−λzt) + O(ε) ,8 [ [=N

∀z(0),∀t≥0

Y q

D>4E4< o C, . *,5 2 , ,2 ,2 . 5-,'.*, . n, . ,2 ,2 . [ 5p *5 ( , . 3 5-,^ *,! , , , 572 ( . 0v1*>, 572 . xe P . 5 ,2 . 2 . *5 # . 0v1* xe(t)=T (ε)xe(t)

E C ( A $B >4D B"<4A B B AU<=@$BmB B <_B AU<=@2: : =< ;B 90m>*@ <0 BHG'ce> B $: <0 ;>CA! > hD>~< ;>4DB AZcD> ce>CA>CA"@ <4ACB&< ;>CAB B A D #iD<E4<*DF:I*E*BRBB ( A?B D B~<;>_< > >9D B A$: 05<0 / A1 ,+CB A1v>4D n< B A

r(t)B

d(t)eB BHG $: <0 >aA$:mAB"<sDB A <0

> B$D/> BHG'ce> B $: <0 /B *B ;B$D B A B B KX(ε) exp(−λzt)cDB ACB B >CA "< v>4D< B A ;>

B AU< ;> B a> &<4AC> <0 >/> !: B 3r(t) ≡ 0

Bd(t) ≡ 0

6ce>4 ;BUACBB A > +eB$D KX(ε)≡0

> D&<4A B >> ) =A B /8D c5<*D&<sA$:mA B"<4A ) ) ( kA BQ<0 B > > =ACB /ED c5<*D&< A$:mA B"<4A A A ( dA ? D (+5< D >CAU< ! ::5C<*DFD<0: ;BC%('' % . A;>90m>*@ <0 /B B9BHG'cH> B $: <0 /B B9B A=I*E*B$:mAwB DB~<;>/< > >"D B A$: 05<0! ;Bs>4DH ;B

τ57,2 2 , " KX(τ) exp(−at)

_ . . 2 ,< . . , 572 . 8# $# \N$pQ<*: =<LHB<4A ,+CB A v>4D n< B A

r(t)B

d(t)B A B'<0 <0 : E4<4A EC: ;B $B >4D B"< &%4

/ /

z . . ( *572 . . . (m6. 0v1*" *?2 . . , 572 2 . , . (yx ,,2 . 5-, . *5-, , . *, . 2& . . 7 ( , ,*,ceB&<4f4: 9*p *5 ( *, . 3 5-, 5-, , >, 6. ( 5 ,' ^ *,857, 2 * . 5 ( 021* *, . ( 7 . .* , 2 ( m75 . 2 . 3 5-," , 5 5 ( 572 ,' . ,' ^ *, . ( D%* (6. ( . ?2 , P *, 57,2.7 . ,2 ,Os. . ,2 . 2 ( , P ( 5

032 + 9;* 6 %:6 210H9 6 0(+ *2 6

] 2 ( () i3 5-,t:57, ,* =, 6. ( 5& "**, 5 (2( . >,2 2d ( *, . ##. (6. 5 ,2 . < 5ACB$D D > 5A : 9 *, 5 ( 2&OO . ; . 8 o . (4. Yv[[ q pg ,202S , 5t Y , q #1 YnN , ( 5 * ,,' . . 2&` ( *, 5.,257 . (/. b P 2 (6. . ] ( ,, 5 ( . *, 572 . 9 ( *, . *, ( M 5 ( 572-,2 .ZPQ w3 5 2 5 . ( 7 .*2( . 5 "02 5&* ,6 2 ( m75 o 5 ( 021*& 6. 8yN P >,4 ( . 02 ,' 2( ; >, 2 932 ( ,32 . 5 ^ *,"., . . 2 ,'*51 '( .

5 ( 021* 8 ep = 10, 7yN M , (| ? 02& ,w8, (| 02 ,' 2( ; >, 2 51* **, 2 . 55

,'. ,' . .0,2*5 . 5 .w ,' . 7*, ( . ( *5 2 . . . 5 2 ( m75 # =, . ( .* . [ q o : B$D0'< B * 2 . 2 ( (). 5 *,, 5& o ! ,L =,, 5& ! o

,5&*m021**,sY # ( 2-5 , 2 ,2 5&S ,5 *, ,'?2 . . ,& , 5 . . .w##.(~ . , −5 V . +5 V M 5 ( .

*, ?2 ,"8 u N . 2& . . 2 ,'*51=,', (4. .>. s "02 6. Y 0

Sistema

de aquisição

de dados

Amplificador depotência

de sinaisCondicionador

e p =10,7 y

u

Tensão do potenciômetro

o motorTensão para

Cremalheira

Carro

Trilho

A/D D/A

0 y

(4. . . . 5-, , >, 6. ( 5& " ,572 .*6. 3 5-, . 5 *, S , 53 5 ( 572-,2 . *, 2 /, * 2 . 4

M . . ( 2& " , 2 */,a . 5-, . ILK!M ( , , *,!ceB&<4f4: 9/8 . ,' . [ $# N P 1 (6. 3? (4. . . . ,w 5 . 2 . , P 595AC> R<*DBLg ( 9 ( ! . 2 . q $# . . ., . . . . , , 2 x ,,2 o 59, . (mx ,, ( 5 51* . 5 2 . . 5 , ( 1 (6. >,'?2 , ( ?2 , , . m"*,. "5 2 . .,2 *,

1 msp ^ *, P ? >,2 ( ?2 , , . m*, . "572 . .,' ^ *, 57, ;1*,9 2 ,

, 572 , 5 ( 572 ,' . *, .*^ ( 5 ( 0v1* *, . 5 M . . ( 2& "*, *?2 /, Ps( ,' >,2 2 . . 2 . , P 5 .w( 2 ,2. . 021*s P (6. ,'ms P 2&1*, b/, 4

Y \

(' $6%:036&%: % * * + %:6 240H9 6 0(+ *2 6

o # 50v1* *, 2 . *5 # , %2 '(4. 5-, , >, . ( 5 2 ( . . 2 . , P 58 . "1*,2 . .,' . 5 . ( D%* (6. P

Gmod(s) =2, 93

s(s + 13, 1),

8h[ [=\N

7*, # . 2 ( (). 5k5d, . , 5Z ( .*( 5 ( 5 ,, ( 5Z5? . . Q8 o . (4. YC[[ q pg ,$3021* q $# YnN M:5L, . m ,55 . %* =,2 5*, Gmod(s)

51*? . 572 . ?2-, ` i( "5 .*^ *,2 ,59 .# 50v1*"*,=2 . *5 # , %' 2(/. 2 ( . . 2 . , P 5< . 5 , . 5<*, ,5 0572 . ,' # , ^ O %' 2(/. >,4 ( ., , ( =,'?2 . m >,2 2 ,

Gnom(s) =2, 41

s(s + 13, 4),

8h[ [ u N

^ *, # ( . 02 . . " .# 021*0*, 2 . *5 # , %' 2(/. . c~< < >w: <0 . . 7, 2&0 ?2 .

o # 021**, 2 . *5 # , %' 2(/. !57, , >, 6. ( 5 5-5 ( .*( . ( 5 `*m5 ,572 x ,, ( 5 ^ *,# . *, 5 , ). 5 57, ,' *5 ( *, . 5 x ( 5 ?t8YnN `m ,2 −14, 4 ks−1

,2 . 2 ( ,* ( D%* (6. ,2 P 2 (m. 2( ( 2*, . . . 5 "02 ,/8 # N `>,' −165 s−1

( ?2 1 )'( 5 32 . 5-5 . 3 .(~ . 5Q (m2( . 5 *,:5 ( .( 5 ^ , . 2& . * 2 < : <0 : <4A$: 9s8 5 2 V ( 2-2 ,' . YC[[\N

M . 2 ( 26e1* ,572 xZ( k5*, 5-,2=,, ( =,'?2 . "1*,' . .,' ^ *, ,5&5 2* .k# 50v1*,=2 . *5 # , %2 '(4. 8 [ [\N ,2?2 . ?2&p . , 2& . 0v1 .*( 5 5 ( .* (6. 2 ( , .P . 2 ( 2p ^ *,P( . . , 2& . 0v1* *,8,'?2 . . 8 2-,2*5&1*N o , 2& . 0v1* P 5 (# =,' >,2?2-, ( ( 2 . . dsup =0, 3 V

p # =, 57,O,, ( , , ( >,2 2 . m >,2?2-,

#" 96 0 6

M "1*,2*, , # , %' 2(/. ,5 7 ( 5 P

WM(s) =50

(s + 5)(s + 10),

8h[ [[=N

^ *,=2 ,' . . 7 ( 2 x ( 5 m7 ,2i3 5-,/ ( *; ( L(s) = s + 10 . .. ,2 .

` ,' −10 s−1 " "1*,2*, , # , %' 2(/. Y u

M 5 2 55*, ,572 . 9519,5 7 ( 5 λ(s)=s+10b] . ,, )Z^ *, WM(s)

,λ(s)

x# . ,5 , '(e. 5'p ?,, 2 *, . %* =,2 5X 2 ( (). ? ?2 ,w ( . P". . . .w6. 5 . . . 2 . ( . Gnom(s) " "1*,2*, , # , %' 2(/. pe ^ *, , 5&32 . ,2

θnomT =[

−1, 6 −22, 56124 −18, 48031 20, 736434]

.8 [ Y N

o # 0v1 ,O "1 . 0v1*8 [ $# # N:*,$,, 5-, ; 7,2 . . *, # .9^ *,Os . . 2 ,w8h[ $# YnN57, . 5 . 2 ( 5 # , ( 2 **5 ( *, . 3 5-, ^ ,Z05 . %* . 2 5 . . ?2 . 51* ( , 205 ? `m ,572 x ,,' , 2 ,' , ##.(~ . −15, 0 s−1 . −9, 5 s−1 ,. . 7 *, . 2 .?# , ^ O %2 2(4. , 2-,2 , ##.*(~ .1, 59 . 2, 99

wo P ( 5-5&0*,',,F3 57, . ?2-, 5 ( . *, *?2 , , ^ *,' . . ( 2 ,*p ^ *, 1*, 5-, **5-,'. ( 57, . 5 **572 . ?2-,5& ./# 0v1 *,w "1 . 0v1 # ,' , ^ *,' . 5 o 5-5 ( ,5 m7 ,2i3 5-, c1 = 44

w] 2 ( () *i3 57,92 . w P 82-, "d(t)≡ 0, 4

. . 575 ( ( ( 2 . . ,2 . =,'?2& 5>, # , ( 25 . 2 ( 2&

o 2 ( (). 0v1*8 . % =,2 δ ./# 021* *,w "1 . 0v1 P9( 5` . . . 5 ( 572 ,' . 5

. . ,2 . 2 ( , n∗ > 1p ( 5we1 57, 1*,. . . ?2 ( . =,, .%2 '(4. , *,5 . . ,'

2-,2 ( 2& =, 5& " ^ *,>5-, ,5 7 . δ > 0so 5-5 ( 57,25p 2 ( ()6. 3 57, δ = 0

.?# 021* *, "1 . 0v1_8h[ $# # N

032 1 %:62 %:0<24* :6

. 5 5 ( 9 . 02S , 5 ,Z55 , , ( >,2 25X8hg ,2021*w[ [ rq N 95 ( . '*, , # , %2 '(4.aP . .a^ . 3 . .a * . ( 2*, 20 mm

, # , ^ O %' 2(/. 3 rad/s . 5 5 ( 9 . 0'S ,5Re1* 7 x , 2& . 0v1*

. (6. . . ,'?2 . . . . ?2 . p?,2 9 . 95 ( 572-,2 . , . 7 . .. 2 ( 2 57, ^ , 1*, (4. 57, ( 5 " , 2 . 0v16*, ,2?2 . . ,'?2 . ?25p . . 5-, . , . (4. 5 , # , ( 205 . 5 ( , 32-, )6. 5 . . P 2 (m6. 52p4 . 5 5 ( 9 . 02S , 5 5 , . m ,5 05 . %* >, 2 5 .6# 50v1* *, 2 . *5 # , %2 2(/. 5-, , >, . ( 5& ">5&1*" . 5 2 . 2 ,& ( 572 ( ?2&5 05 , . m , 56* ( .*( 52p ( 5 2 P p

Gdist(s) =1, 59

s(s + 15).

8 [ Y YnN

1 . 5 . 5 ( 02S , 5 ( (m'(4.( 5 5&1*w;5 . 5'p , i , 2 . 5 ( 021* ( (2(/. . ^ *, P y(0)=

100 mm

o 5 ( . . 5 [ ,O[ \ . 7 ,57,2 2 . 5 , 5&32 . 05 , 5 ( w5 . 021* 2 ( 55 . . ( 5Q, . m43 ,5 . %* >, 2 ε .s *,5-, . , . (4. ."( %2 '(4. *, ε

8*,57,2 ,275 =5 ( 5 2 ,' . *,Y *[

?2 , M *,57,2 ,257 5 *?2 . P ,' ,', ( ,2 '(4. 5 ,2 . x ( . *,, 3.%' '(4. , 0*, 5 . .^ *, 1*, 5-, . , '(4. . . ( . . [ # >, ,5 , . p , ,5 ( . . `5 2 . *5 ( 2&` ( ( (2(/. P 5 ( . (6. 2 ( , . >,2 2 , , )'( w . =, ( . ,2 ^ *,ε . 5-5 . *, 0, 1

8 ( . . [ . N . . 0, 018 ( .* . [ HN

0 0.5 1 1.5 2 2.5 3−0.02

0

0.02

0.04

0.06

0.08

0.1

t (s)

e(m

)

@ G0 0.5 1 1.5 2 2.5 3

−0.02

0

0.02

0.04

0.06

0.08

0.1

t (s)e

(m)

@ G , 5&32 . 55& . 5 5 ( 9 . 02S , 59 , 0*,>5 . . . . ?8 . N ε = 0, 1

,8 HNε =

0, 01

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

t (s)

ζ 1

ζ 2

@ G0 0.05 0.1 0.15 0.2

0

0.1

0.2

0.3

0.4

0.5

t (s)

ζ 1

ζ 2

@ G , 5&32 . 55 . 5 5 ( 95 . 0'S ,5) . 5 , . (yx ,, ( 5 , ,5 2 . 5>,5 2 ( . . 5 ,2mw*57, 3, . *, . 2&w. . 75 . . ?k8 . N ε=0, 1

,8 HNε=0, 01

Yn\

M # ,2*;* =,'aceB&<f4: 9Z1*w ,:,' *,'7; 255 ( .( 5 5 ( 9 . 052p; . ,, )X^ *, , 572 , ?2 . P ( , ,3*,!ceB&<4f4: 9 z . 2 (m . >,2 2 ,*pH02 . 3 57, ^ *,9"ceB&<4f4: 9!e1*/ , . 5, . (mx ,, ( 5*, , 572 . , 572 ( . . 5 ζ1

,ζ2

p ( 5 ^ . 5, . 5 . % =,2 ε P ,4 )'( 5*,

0, 1 . . 0, 01 . 5 . ( 2 ,595 ( 5&*, 572 . 5!, . (yx ,, ( 5 e. . 2 (6. >,'?2 , ( . 2-,$3

. . 5'pH,' " . 5 *572 . ?2-,5 *, 2 ,' 56 1056 .*( 5 ,2?2&52p # >, 57, , ,2, . . 3 5-, . ( . . [ \ .s . ( .* . [ \

M . 575& *, ( 2 ,2. . 021*w 2 ( (). 9*,5 2 . 5 5 ( 9 . 02S , 5 P . 5 2 . 2 , , ^ *,' 8 10 µsN . .

575 ( ( ( 2 . ^ *, , . ( ?2-,2. . 0v1 P (6. 5-, . 2 , x ,,2 . . 6. 5& ,2 ^ , . ( D%* (6. 5 ILK!M P . .( 5 x ( . p ( , p ^ . ε=0, 01

, ##. 2&p 2i3 5-, ^ *,d . 5-5&*, ( ?2 ,'. . 0v1* , . m ( . . . ( 2 , , . 8*, . 0572 . .,' 05:, , ( =,'?2&5w8 1 ms

.g ,'0v1"[ [ rq N 5&` 5-, (/. 5 . 2 ( 5 ##. 2&` ( . . ε=0, 1

z ( 5 2pi05 , , ( =,'?2&5 . , 5-,'?2 . 5 . 5-,'0v1* . 5-,'. ( 2 ( () *i3 57, ε=0, 1

p ^ *, ,5&5 2 . ,' , 5 *, . 5 2 , . >,2 2/ . 5 2 . 2 ,5 ( . (~e. 2 ( ,5

032 1 %:62 0 ,.0H9;* 031*2

,572 . 5-,2021*=51* . ,5-,'?2 . 5 ,5 2 . 5 , , ( >,2 2 .*( 56 2 ( 05 * 2 %56, . (/. 02S , 5 . 5 , ( 5 *, *?2 ,X . 5-, . . 5kIZKaM 5-,2 ceB&<4f4: 9 ?

Y M 2 */, P ( *, . ,O ( . hp ( , p u=θnomT ω . ( . . [ u7

# M 2 */, . ,5 2 2 . , . (yx ,,2e1*9 2 ( (~)6. *?2 /,d ( . hp ( , p u=−ρ sgn(σ) . 5 ( . . 56[ [ ,![ Y 7

M ?2 , . , 572 2 . , . (mx ,,' ( m ( 2 */, ( . hp ( , p u=θnomT ω−ρ sgn(σ) . 5 ( . . 56[ YY ,![ Y #

o . 5-5 . ( . . P mnom = 0, 455 kg M:5=, , ( >,2 205!5&1* , . ()6. 552p

2 . 9 P _p ^ . . . 575 . 6. P . =,'?2 . . . . mdist =0, 665 kgp ^ *, Pa 5-,$3

. ( ,2 .* P 5 2( "*,X . 7 . 572 , . 5-5 . , 0, 210 kg ,$,,$3 5-, , 5-5 . 2 . ^ *, , 572 .

. 32 , . 021* P . 572 . ?2-, 5 ( .* (e. 2 ( , . ,' * . . 021* . . 575 . ( . b8 sN . ( . . [ u ,, (e. 3 57, ^ *,a 2 */,O ( . ( , . Pa( 6. .*) *, , . (m). . 53

2 , . =,'?2& , 2( 59 . 5 2 . 7, 2&` (/. 52p .*( .d^ , . . 5-5 . 6. :5-, . . * ( . M , **, . 572 , . =,'?2 P .( . .*( ^ . . . 575 . . P . >,2?2 . . 8 , ( *, ( . . [ ru HN o . ,'?2-,2 >,2 2 ,*p . ( '( . 6. *5 . *, 5-57,5 , 5 , . .0, . 5 P . 2 ( 2& 5-, p >, # , ( 2& P

Yn\;Y

.*( 5 ,$, ( *,2 2 , ^ . 7 . ,,2 2( . *, 6. P , ^ *,2 . p ^ , . *5 . , (##. >,2 205 ( . *, 5 ( 0v1* 6. . ( .* . [ u7 M . 2 ( 2&!5-, P .*( . * . 575 .. 5 =,'?2 . . *,$, ( . . >,2 2 , 5&5pH ^ *, .* ,'?2& . , " . 5 2 , . >,2 2

m

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

0 2 3 4−20

−15

−10

−5

0

5

10

15

20

t (s)

y, y

(m

m)

1

@ Gm

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

t (s)

y, y

(m

m)

@ G , 5&32 . 55 , , ( >,2 2 .*( 5 . 572 , . =,'?2*, 2 . 7, 2&` (/. 5 *?2 /, ( *, . ( . ?T8 . N 6. . 575 . ( . , 8 HN 6. . 575 .

. (2( * . 56. x 5 ? yG ( 57 .9 *?2 ; . yM

G ( 7 . 2 .* , . .

M ?2 , . , 572 2 . , . (yx ,,2i . 5-, . XXILK!M , 5&32 . ;5 . 572 , . =,'?2Z . 572 . ?2-, , '( 5&5p .*( .a^ *, . . 5-5 . . 5-, . . >,2 2 . . 8 , ( , ( . . 5Q[ [ . [ Y # N 2 . 3 5-,^ *,Ow*5 2 /,L5 ( . . (/. 5 . ?2 , . , 572 2 . , . (mx ,,' ,4 ) 5 ( . (~e6. 2 ( 3, . =,'?2-,d, 3*, 5 . . p ^ *, 5-, * ( .! * . . 021* 5 , 5t . ( . . [ Y 8#5-,' 2 /,X5 ( . yN 5 , 56 . ( . . [ Y # 8 ( ( 5w 2 */,X* ( . yN

Yn\ #

m

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

t (s)

y, y

(m

m)

@ Gm

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

0 1 2 3 4−20

−15

−10

−5

0

5

10

15

20

t (s)

y, y

(m

m)

@ G , 5&32 . 55 , , ( >,2 2 .*( 5 . 572 , . =,'?2*, 2 . 7, 2&` (/. 5 *?2 /,

. , 572 2 . , . (mx ,,'?57,2 2 */, * ( . ? 8 . N 6. * . 5-5 . * ( . ,8 HN 6. . 575 . . (m2( . 05L. xe 5 ? yG ( 7 ." *?2 ; . yMG ( 7 . 2 .* , . .

0 1 2 3 4−3,0

−1,5

1,5

3,0

t (s)

e (m

m)

0,0

@ G

0 1 2 3 4−4

−2

0

2

4

t (s)

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t1 :=1

λ1

ln

[γ(r(0) + δ/γ + cR)

δ

]

< +∞ ,XZ]9 k

STO OlU'O2WTO@TK?"K O[a)"oK U6eW6)FX]9 k6Q¡S Kxp O ∀t>t19 I .oKnV\ UTV p"c;OlU

r(t)Q"O S SlUTOR

r(t)≥ 0\"O S"WdUTK OlU

r(t) OoS6 K?"S K4LV0"S BSlUTWdO*6

t≥ t19

oG 781:9<;=(FA >9BA #$'& (FA >( R$'CED29 GH®G

I . UPK ®uLO2*0S P *oOp TO G"V oO J ©O0OLS [O W p OoK^eW UTSTK [KdO *.0S TO©S6 F0t0uLOfM9 fM9;VW ]*oOp!OLVOoK KnS6LWnV20S a".0SgpgOLWnpmK]0V XZY;oU\]_U"ST`P\6]_O b GKdc4eWHfII fThL\ I *,+0SPK#"B]9 f-5]9 kJ9 I OlUTK$O K KdS"WdU TOOCoKnS6LW%0O V

w±I

4eKdV00S OA/MM*sO S"S0KLWπ2(t)

S6!UTSTt 5 "]?VjOFPUTW^Lt0uLO0O K`."pmK6A*)oOqpSATOG VARfM9 i)9

]_OLSoK?".FUTVA7"LWdKnc4Lt0uLO7 J KdWnKdc p"W a p0W`jXfM9 fLigk

x = Ax + B (U + d + π1) ,XZ]9 gk

ε = Cx ,

O S)M(s)=C(sI−A)−1B

9 I VA [Kdce LSP`TO]) LW LO+0S"K^<QK =CB

9 [K 0VAX]9 k *sOFa."P"oK ORS6 vª hj hL©XpmKRX! O UPS LW 9^w± a\#" %$ KdS w± ± f kok

x1 = A11x1 + A12ε ,XZ]9 k

ε = A21x1 + A22ε + K (U + d + π1) .XZ]9 °gk

p ORN TO 4eWdKdc"Lt0uLO Qxe := [xT

1 , εT ]T9 I PKdS#&LVRK4POc"oO-

X]9 Lk'^X]9 ° k_Qa62*sO

x1 =A11x1 .XZ]9 ±gk

]_O VRO(.R.oUTVRKdUOlUM(s)

Q Z.,V STKnVAa\ 0S uLOA11

Q,UTVAV oKncjY<U('K c9M \

λ0

VA "V / J KdWnK6C6V oKncA11

0 < γ0 < λ0

9 V S ueOT\ (0OTO"O VO(G"ViM9nw*PWdK#"TOCLO1oU J oK? "V~XZ]9 k "O V VA oKdc7 LS)L".+0S"K^C66C*sOx1 =(sI − A11)

−1A12ε\O J Q0V .2OHVAS\[OLS

‖x1(t)‖ ≤ c1

[|εf (t)| + |ε0

f (t)|]+ ‖x0

1(t)‖ ,XZ]9xw Igk

"O V

εf (t) = exp(−γ0t) ∗ ‖ε(t)‖ ,XZU9nw wk

O S)ε0

f

x0

1

®uLO "[VjO@ MM*sO S0S 0K^eWdV00S 7) ".""0S "TOlU2ouLO-/pmKTOXb 0O S)TKdt 5 "KnSTK#"K^eK?96r:O O U oO=W^TOT\LUTV<p0cPOlU −K

Q'Y U)('K c e\ 0O S."O0S 0V00S L\mSuLO oKnS UPW^"\w±aw

7"OlU6Lt0uLO X]9 °gk *sOF)X."P"oK 0OLVjO

ε = A22ε + K[U1 + d + K−1A21(sI−A11)

−1A12 ∗ ε + π3(t)]

,XZU9nwqfLk

O S)

U1 = −ρ1ε

‖ε‖ ,XZU9nwi k

ρ1 ≥ δ + (1 + cd)‖d(t)‖ + cε‖ε‖ + cf exp(−γ0t) ∗ ‖ε‖ + π4(t) .X]9xw k

"[VjOπ3

KnS0WnUTK O "[VjOπ1

x0

1

9 "[VjOπ4

KnS0WnUTKO "[VjOπ2

ε0

f

9 I V J OO "[VjOP K?ZLc0V

‖π3(t)‖ e ‖π4(t)‖ ≤ [c1‖xea(0)‖ + R] exp(−λ1t) ,X]9nw k

O S)xea := [xT

1 , εT , εf ]TQ Op O[3" PO 0OLVA*PW OPORoK "V"V VALWn`6 L `6

λ1 <min(γ0, λ)9 I .oKnV\)6Hre[O*sO[KdtvueOjfM9xw0OLS0WnUTK [OmU)T7oK U6LW?6HXfM9 fLk_Q@ K

L"K X*6 ‖ε‖ \M".@OlU@ "O S LS " cd, cε, cf ≥0.Q\LV "0O Wn`TK?6 "OlU66LV00S 9

" V=p"c]OlU ‖x1‖ ‖εf‖

K?ZLt"LV UTVAX)"oK U6eW6);[KdVRKdW?"\ "S uLO ‖xea(t)‖ eV J Q0V

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δ > 00V XfM9 f k \;"S uLO

ε OoS6 =K)0S K#"LV00S =SlUTWdO07*6 K? )TeW UTV 0V0*sOTSTK O

ts≥09

rX[KdV0*TWdK 4ea*oOpP\F[UTVA .UOlU]a"LWdKnc4Lt0uLOaM(s)

Q 0O S oOLW p0W 9U 4[OT\.j` " TOBKnSTO J ." p p"K"\6O1"W0V0"S O

A21

6 O[VR UTW^jXZ]9 k' XZ]9 °gk©QjK?"S K4V00S SmUPWdOT\mK 9 9x\

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O 4[O )<UTVAa"LWdKnc4etvuLO2SueO V STKnV0@OlUU.Q\FSuLO 0O S [O W p0WªO UjSuLO O J [p p0WZ\M*[OqpS[*sOF ."W)""Slp O WxpmK6[EUTVA OoVAUoKdVRKnW¯U KdWnKnc4LS)TO m / m m h ¨ h XpaK?jX] V b K . 0SPV $ w± ± m\** 9w IPw 'sw4Igf k[k 6OoVA7. UPW^2XZ]9 k'^XZ]9 °gk 9M

γj = γ > 0\(∀j ∈ 1, . . . ,m) \B0S uLO&/*TWdK4etvuLO1TO ]_OoO W oKnO fM9xwS(*)oOqpS/TO

G"VRfM9 f7*,oVRK "O S0WnUTK OlUX.X*sOFa"0O Wn`"cε =0

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I ORWnO S O767*oOpHO2 V J O WnOki

Xi∈Nk 0SPO h / hh/s® s h h9

"<VAp"c&OlU −KQ@Y U)('K ce\; MaK 0V VA [Kdc"

P = P T > 0

Q = QT > 0OmU)

K?ZLc0V 2"OlU6etvuLO0XG "*TUTSTOp

KT P + PK = Q .XZ]9xw gk

VS ueOT\ 7"[Knp626 OoVA3OlU6 K4

V (ε) = εT P εX]9nw k

Qa662*sO

V = −2αεT P ε + [εT PKU + UT KT P ε] + 2εT PKdU .XZU9nw° k

" KnWdKxc4LSPO .@HW?0K 0OLS [O W?RXfM9 igf k 0V .aOlU

V = −2αV − ρεT PKε + εT KT P ε

‖ε‖ + 2εT PKdU .XZU9nw± k

I OP\LK?oO WLSTO O "[VjO]/MM*sO S"S0KLWπ2(t)S UPSTtvuLO[EVROFTUTWLtvueOXZfM9 iLigk O J Q"V

.

V ≤ −2αV + 2‖PK‖ |π2| ‖ε‖ − ρεT PKε + εT KT P ε

‖ε‖ + 2εT PKdU ,XZU9«fI k

OlUX*sOF)X."'V \[OTO7*sO

V ≤ −2αV + 2‖PK‖ |π2| ‖ε‖ − ρεT PKε + εT KT Pε

‖ε‖

+ ρ(π1 + β)T PKε + εT KT P (π1 + β)

‖ε‖ + 2εT PKdU − 2(π1 + β)T PKdU

≤ −2αV + 2‖PK‖ |π2| ‖ε‖ − ρλmin(Q)‖ε‖

+ 2ρ‖PK‖‖π1 + β‖ + 2‖PK‖‖ε‖‖dU‖ + 2‖PK‖‖π1 + β‖‖dU‖ .X]9«fmwqk

w±Li

MQ\HUPV UTSPtvuLO0)FVROFTUTWLtvueOρ LW OlU

(1 + cd)(1 + cd)dU(t) ≤ ρ(t) ≤ (1 + cd)dU(t) ,X]9 f f k

∀t≥0\ "O V

cd ≥ 2‖PK‖

λmin(Q)− 1 ≥ 0 , cd = (1 + cd)(1 + cd) − 1 ,

X]9 fLigk

AOlU6LK.OlU"cd > 0

\cd ≥ cd

XpaK? XfM9 i ka XfM9 i LkokJ9 " KnWdKnc"LSTO HO V \[OLS dU(t) ≥

‖dU(t)‖ \ ∀t≥0\TO J Q0V a2* K jXZ]9 fawqk

V ≤ −2αV + 2‖PK‖ |π2| ‖ε‖ − 2(1 + cd)‖PK‖‖ε‖dU

+ 2(1 + cd)‖PK‖‖π1 + β‖dU + 2‖PK‖‖ε‖dU + 2‖PK‖‖π1 + β‖dU

= −2αV + 2‖PK‖ |π2| ‖ε‖ − 2cd‖PK‖‖ε‖dU + 2(2 + cd)‖PK‖‖π1 + β‖dU

= −2αV + 2‖PK‖ |π2| ‖ε‖ + 2‖PK‖ [(2 + cd)‖π1 + β‖ − cd‖ε‖] dU .XZU9«f k

]_O VROε= ε+π1+β

0V .aOlU ‖ε‖≥‖ε‖−‖π1+β‖ L\ .oKnV .0STOP\

V ≤ −2αV + 2‖PK‖ |π2| ‖ε‖ + 2‖PK‖ [(2 + cd + cd)‖π1 + β‖ − cd‖ε‖] dU .XZU9«fLk

VS ueOT\6"SOlU6LS OjO2oKdSLWε(t)

Kxp" O36 J O W7 STK?62*sO

‖ε(t)‖ ≤ 2 + cd + cd

cd

‖π1(t) + β(t)‖ ,XZU9«f k

7 UTKdS aoK U6LW?6@." p WdK

V ≤ −2αV + 2‖PK‖ |π2| ‖ε‖ ≤ −2αV + 2‖PK‖‖ε‖R exp(−λt) .X]9«f k

I OP\ O VALSTO @2oK U6LW?6@ > "mW"K `" > K c

λmin(P ) ‖ε‖2 ≤ V (ε) ≤ λmax(P ) ‖ε‖2 ,XZU9«fL° k

w±

S r :=

√V

cD :=‖PK‖/√

λmin(P )\*62.FO J "

r ≤ −αr + cDR exp(−λt) .XZ]9 fL±gk

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r(t) ≤ r(0) exp(−αt) + k1R exp(−λ1t) ,X]9 iIgk

O S)λ1 <min(λ, α)

9 I *TWnK#"LSTO TA)"oK U6LW?6)T > "mW"K ` > K cjXZ]9 fL°gk 0V X]9 iIgk/\O J Q0V a2oK U6LW

‖ε(t)‖ ≤ k2‖ε(0)‖ exp(−αt) + k3R exp(−λ1t) ,X]9 iawqk

OlUHQ2 KL"K R"SOlU6LS OO-oKnS6LWε(t)

" Knp OR6 J O W^" J "W0K-*sORXZ]9 fgk 9 VA \[OLS

‖ε(t)‖ ≤ k4 [R exp(−λ1t) + ‖βt‖∞] ,X]9 i f k

S?UTVA J O W^7OmU) 0O S Q0V J OLW^X]9«f gk/\O262OmULW%0"[KnpS6VQFV \[O67*sO

X]9«f kJ9 I .oKdV5\ 7 0OLSTKntvuLOKdSTK0KLWε(0)

" Knp" O6 J OLW^@X]9 igf k/\ \. EoKε(t)

." WnKdVRK 60* OHX]9 iPwqkU Q3OlU2 J O W^ XZU9 igfLk;Q\R KdS K6M9©r:O©OLU [OWTOT\dOlU6eSTO"O STKntvuLOHKnSTK#"K^eWd Kxp"P"S oO06 J O WXZ]9 igf k \TO UC7*6 K PORKnS LS

t1"V OlU

ε(t)"S S6 J O W^XZ]9 igf k \ 0S uLOX S\. E[K^

ε(t)*,"[VLS) " )0S oOa" J O W^ ∀t≥ t1≥0

\*sO K? O2"W^

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*6'O KdS "[KdO 6 J O W^FXZ]9 igf k S0WP*,"[VeS P*6 K)t19 I oKnV .0SPOT\LO V \[OLS

‖ε(t)‖ ≤ max k2‖ε(0)‖ exp(−αt) + k3R exp(−λ1t) , k4 [R exp(−λ1t) + ‖βt‖∞] ,XZU9 i i kQ p WdKPO ∀t ≥ 0

9 KnS6LWnV0"S L\¡W"V J LSPO ./OlU ‖ε‖ ≤ ‖ε‖+‖π1‖+‖β‖ O J Q"V (oK U6LW?6RXZfM9 igk OlU2QFUPV VAS\[OLS [*6X]9 iLigkJ9

w±

w±

5 /

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<OAOmU) . Ue\ O "[VjOπ1(t)

Q?UTV oKnS6LW`"LW^U MM*sO S0S 0KLWdV00S X "0S L\^K 9 9x\|π1(t)|≤R exp(−λt)

\ ∀t≥0\6*2LW UTVA 0OLS LS ";"LK

R≥0

λ>09

yzm|~j / m /d h

e(t) = M(s) [U + dU(t) + π1(t)] ,X]]9xwqk

mM(s)∈R1×1(s) § h m m! [h / h ,/2 / sh

dU(t)

§ / sh π1(t) § / h j m

U = −ρ(t) sgn(e) ,XZ]]9«fLk

ρ(t) ≥ δ + |dU(t)| , ∀t ≥ 0 ,X]U9 igk

mρ § δ ≥ 0 § h h hL" L hª¢ m ∃ c1, c2, λ1 > 0

h dj h m/w±

Ph h m

|e(t)| ‖xe(t)‖ ≤ [c1‖xe(0)‖ + c2R] exp(−λ1t)XZ]]9 k

§ h Zd / h ∀t ≥ 0¢ m

xe § h m Th x , h ¨ h m hg ¨ L _ m L m /L , m hL § Ld ¢

δ > 0¢ m

e(t) ªsh

m h , ,h/ h ts≥0

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σ(t) =Kp

(s + γ)[U + dU(t)] + π1(t) + β(t) ,

X]]9 k

/ sh

σ(t) = σ(t) − π1(t) − β(t) ,X]]9 gk

mKp, γ >0

m h h hL ¢σ, σ, U, dU , β, π1∈R

¢dU(t) § ¢ π1(t) §

/ h , m B =/ sh dπ1(t)

β(t) m,hg h , Z ¢ ∀t≥0

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ρ(t) ≥ |dU(t)| , ∀t ≥ 0 ,X]U9 °gk

mρ § ¢ m =/ sh d σ(t)

σ(t) m ¨ h ª

|σ(t)| |σ(t)| ≤ |σ(0)| exp(−γt) + 2 [R exp(−λ1t) + ‖βt‖∞] ,XZ]]9 ± k

mλ1 <min(γ, λ)

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