• Nenhum resultado encontrado

On the asymptotic stability of linear discrete time-delay systems: The Lyapunov approach

N/A
N/A
Protected

Academic year: 2017

Share "On the asymptotic stability of linear discrete time-delay systems: The Lyapunov approach"

Copied!
4
0
0

Texto

(1)

8 LQGXVWULMVNLP SURFHVLPD VH SUL SUHQRVX LQIRUPD FLMD LOL PDWHULMH L]PHX UD]OL LWLK GHORYD VLVWHPD HVWR MDYOMD YUHPHQVNR NDåQMHQMH +HPLMVNL SURFHVQL VLVWHPL WUDQVSRUWQL VLVWHPL NRPXQLNDFLRQL VLVWHPL L WHUPRHQHU JHWVNL VLVWHPL VX WLSL QL SULPHUL VLVWHPD VD NDåQMHQMHP ,QWHUHVRYDQMH ]D GLQDPL NR SRQDåDQMH VLVWHPD VD ND åQMHQMHP RNXSLUD SDçQMX YHOLNRJ EURMD DXWRUD YHþYLåH RG SROD YHND ]ERJ LQMHQLFH GD SULVXVWYR YUHPHQVNRJ NDåQMHQMD X VLVWHPX YHRPD HVWR GRYRGL GR R]ELOMQLK SRJRUåDQMD X QMHJRYLP SHUIRUPDQVDPD L VWDELOQRVWL 0H XWLP VD VLJXUQRåþX VH PRçH UHþL GD MH WR LQWHUHVRYDQMH ELOR X GDOHNR YHþRM PHUL ]DVWXSOMHQR NDGD VX X SLWDQMX ELOL NRQWLQXDOQL D GDOHNR PDQMH NDGD MH ELOD UH R YUH PHQVNL GLVNUHWQLP VLVWHPLPD VD NDåQMHQMHP -HGDQ RG JODYQLK UD]ORJD ]D WR MH åWR VX GLVNUHWQL VLVWHPL VD ND åQMHQMHP NRQD QH GLPHQ]LMH SD VH NDR WDNYL PRJX LVND ]DWL SUHNR RGJRYDUDMXþHJ HNYLYDOHQWQRJ VLVWHPD EH] NDåQMHQMD 5HG RYRJ VLVWHPD VUD]PHUDQ MH SURL]YRGX YUHGQRVWL NDåQMHQMD L UHGD PDWULFD VLVWHPD 3UL YHOLNLP L]QRVLPD LVWR YUHPHQVNRJ NDåQMHQMD UHG HNYLYDOHQWQRJ VLVWHPD SRVWDMH L]X]HWQR YHOLNL SD MH WR X SURåORVWL VWYD UDOR ]QDWQH PDWHPDWL NH SRWHåNRþH

8 WRP VPLVOX RVQRYQL GRSULQRVL X RYRP UDGX LåOL VX X SUDYFX GD VH RYDM SUREOHP SUHYD]LH DQDOL]LUDMXþL RVRELQH YUHPHQVNL GLVNUHWQRJ VLVWHPD VD NDåQMHQMHP QD SROD]QLP PDWULFDPD VLVWHPD L EH] QXçQLK SUHYRHQMD LV WRJ QD HNYLYDOHQWQL VLVWHP

3UYL ]QD DMQLML UDG NRML VH EDYLR SUREOHPDWLNRP DVLPSWRWVNH VWDELOQRVWL RYH NODVH VLVWHPD SRMDYLR VH WHN JRGLQH >@ 8 RYRP UDGX GDWL VX MHGQRVWDYQL GR YROMQL XVORYL VWDELOQRVWL ]DVQRYDQL QD NRULåþHQMX QRUPL PDWULFD L QMLKRYLK IXQNFLMD 8 UDGX >@ UD]PDWUDQD MH DVLP SWRWVND L '²VWDELOQRVW GLVNUHWQLK VLVWHPD VD NDåQMHQMHP L SUHGORçHQL VX GRYROMQL XVORYL NRML VX PDQMH NRQ]HUYDWLYQL

RG >@ 8 SRVOHGQMLK GHVHWDN JRGLQD SUREOHP VWDELOQRVWL QHRGUHHQLK VLVWHPD VD NDåQMHQMHP SRVWDR MH MDNR SR SXODUDQ PHX LVWUDçLYD LPD 6DPR PDOL EURM UDGRYD ED YLR VH QD RYX WHPX QHRGUHHQLP GLVNUHWQLP VLVWHPLPD VD NDåQMHQMHP >²@

3RVHEQX NODVX VLVWHPD VD NDåQMHQMHP LQH YHOLNL ODUJH²VFDOH GLQDPL NL VLVWHPL VD NDåQMHQMHP NRML VX RNDUDNWHULVDQL YHOLNLP EURMHP SURPHQOMLYLK VWDQMD L NRP SOHNVQLK LQWHUDNFLMD L]PHX SRGVLVWHPD 8 VNRULMH YUH PH SUREOHP VWDELOQRVWL RYH NODVH VLVWHPD UD]PDWUDQ MH X UDGRYLPD >²@

8 RYRP UDGX L]YHGHQL VX QRYL GRYROMQL XVORYL DVLP SWRWVNH VWDELOQRVWL ]D OLQHDUQH GLVNUHWQH VLVWHPH VD ND åQMHQMHP 3RND]DQR MH GD RYL XVORYL GDMX EROMH UH]XOWDWH RG XVORYD GDWLK X >@ D YUOR VOL QH UH]XOWDWH VD XVORYLPD L] >@ 0HXWLP ]D UD]OLNX RG >@ L >@ JGH VX NULWHULMXPL VWDELOQRVWL L]YHGHQL X NRPSOHNVQRP GRPHQX RYGH VX RQL L]YHGHQL X YUHPHQVNRP GRPHQL NRULVWHþL GUXJX PH WRGX /MDSXQRYD

ƒ ² 5HDOQL YHNWRUVNL SURVWRU )% ² 7UDQVSRQRYDQMH PDWULFH ) ! ² 3R]LWLYQR GHILQLWQD PDWULFD ) t ² 3R]LWLYQR VHPLGHILQLWQD PDWULFD O) ² 6RSVWYHQD YUHGQRVW PDWULFH U) ² 6SHNWUDOQL UDGLMXV PDWULFH V) ² 6LQJXODUQD YUHGQRVW PDWULFH __)__ ² (XNOLGVND QRUPD PDWULFH

3RVPDWUDMPR OLQHDUDQ DXWRQRPQL YUHPHQVNL GLV NUHWQL VLVWHP VD NDåQMHQMHP RSLVDQ VOHGHþRP GLIHUHQ FQRP MHGQD LQRP

[ N $[ N $[ N²K

JGH VX [N  ƒQ $

M ƒQ[Q D K FHR EURM NRML SUHGVWDYOMD YUHPHQVNR NDåQMHQMH

6SHNWUDOQL UDGLMXV PDWULFH $ ELþH PDQML RG D D DNR L VDPR DNR ]D SURL]YROMQX SR]LWLYQR RGUHHQX UHDOQX VLPHWUL QX Q [ Q PDWULFX 4 SRVWRML SR]L WLYQR RGUHHQD UHDOQD VLPHWUL QD Q [ Q PDWULFD 3 NRMD MH

! "#

"

! $% & $% &

8 UDGX VX GDWL QRYL NULWHULMXPL DVLPSWRWVNH VWDELOQRVWL VOHGHþH NODVH OLQHDUQLK GLVNUHWQLK VLVWHPD VD NDåQMHQMHP [ N $[ N $[ N²K2YL NULWHULMXPL VX QH]DYLVQL RG NDåQMHQMD L GDWL VX X REOLNX GRYROMQLK XVORYD VWDELOQRVWL'RELMH QL VX NRULåþHQMHP GUXJH /MDSXQRYH PHWRGH L PDQMH VX NRQ]HUYDWLYQL X SRUHH QMX VD QHNLP SRVWRMHþLP NULWHULMXPLPD VWDELOQRVWL5DGL LOXVWUDFLMH GRELMHQLK UH]XOWDWD SUH]HQWRYDQ MH QXPHUL NL SULPHU

5DG MH VDRSåWHQ QD 6NXSX 6DYUHPHQH WHKQRORJLMH L SULYUHGQL UD]YRM /HVNRYDF RNWREDU ²

$GUHVD DXWRUD 6 6WRMDQRYLþ 7HKQRORåNL IDNXOWHW 8QLYHU]LWHW X 1LåX /HVNRYDF (²PDLO VVUHWHQSWW\X

5DG SULPOMHQ 6HSWHPEDU 5DG SULKYDþHQ 0DUW

(2)

D²77

! " D² #

"7"7 $

% " & ' % 7( ) % 7( ) $ * + ( U" UD²U ,- U . D

' ' /

% 7 ( ) '

/ % 7( )

7 7 0

+ &

1 /%

23N 37N 3

¦

M K

[7N M 6[ N M 3 37! 6 67!

3RWRQMD UD]OLND RYH IXQNFLMH GXç WUDMHNWRULMH VLVWHPD MH

'9[N [7N $73$36 [N

[7NK $73$ 6 [NK [7N $73$[NK [7NK $73$[N

8NROLNR ]D PDWULFX 6 XVYRMLPR YUHGQRVW

6 $73$

RQGD MH

'9[N [7N$73$ $73$ 3 [ N [7NK $73$[NK

[7N $ 73$

[NK [7NK $73$[N

[7N$73$[N[7N$73$[N '9[N [7N$73$ $73$ 3 [ N

>$[N $[NK@73>$[N $[NK@ 2 LJOHGQR MH

'9[Nd[7N$73$ $73$3 [ N 3UHPD WRPH DNR ]D VYDNX PDWULFX 4 47 ! SRVWRML PDWULFD 3 37! NDR UHåHQMH PDWUL QH MHG RQGD YDçL

9[N ! '9[N [Nz SDMHVLVWHPDVLPSWRWVNLVWDELODQ

$NR ]D VYDNX VLPHWUL QX SR]LWLYQR GHILQL WQX PDWULFX 4 4%! SRVWRML VLPHWUL QD SR]LWLYQR GHIL QLWQD PDWULFD 3 3%! NDR UHåHQMH VOHGHþH PDWUL QH MHGQD LQH

$73$ 3 4

RQGD MH PDWULFD 4 ² 3 SR]LWLYQR GHILQLWQD DNR L VDPR DNR MH ]DGRYROMHQ XVORY

U$ —

CC 3UHWSRVWDYLPR GD MH PDWULFD 4 ² 3 SR]LWLY QR GHILQLWQD åWR MH HNYLYDOHQWQR 3 ² 4 1D RVQRYX YDçL

!3 4 3 $73$ 3 $73$ 3 WM

$73$ 3 5

JGH MH 5 57! ELOR NRMD VLPHWUL QD SR]LWLYQR GHILQL WQD PDWULFD 3RVOHGQMD QHMHGQDNRVW þH ELWL LVSXQMHQD DNR L VDPR DNR MH ]DGRYROMHQ XVORY /HPD ,]PHX NRUDND X RYRP GRND]X YDçL UHODFLMD HNYLYDOHQFLMH SD YDçL L REUQXWR WYUHQMHXNROLNR MH ]DGRYROMHQ XVORY RQGD MH PDWULFD 4 ² 3 SR]LWLYQR GHILQLWQD

8VORY JDUDQWXMH GLVNUHWQX VWDELO QRVW PDWULFH $

6LVWHP DVLPSWRWVNL MH VWDELODQ DNR VX LVSXQMHQL VOHGHþL XVORYL

U $ —

CC __$__OPLQO4 3

PD[3

JGH MH PDWULFD 3 37 ! SR]LWLYQR GHILQLWQR UHåHQMH PDWUL QH MHGQD LQH ]D SURL]YROMQR L]DEUDQX SR]LWLY QR GHILQLWQX PDWULFX 4 47!

1D RVQRYX 7HRUHPH VLVWHP MH DVLP SWRWVNL VWDELODQ XNROLNR YDçL

$73$$73$3

2YDM XVORY VH PRçH UD]ORçLWL QD VOHGHþD GYD XVORYD =D GDWX PDWULFX 4 4%! SRVWRML PDWULFD 3 3%! NDR UHåHQMH GLVNUHWQH /MDSXQRYH MHGQD LQH

0DWULFH 3 L 4 ]DGRYROMDYDMX VOHGHþL XVORY

$73$4 3

3UYL XVORY ]DKWHYD GD PDWULFD $ EXGH GLVNUHWQR VWDELOQD D XVORY ]DKWHYD GD PDWULFD 4 ² 3 EXGH SR]LWLYQR GHILQLWQD 1D RVQRYX /HPH WR þH ELWL LVSXQMH QR DNR MH ]DGRYROMHQ XVORY

8VORY HNYLYDOHQWDQ MH VOHGHþHP XVORYX

OL>$73$4 3@ O d LdQ 3RåWR VX VYH PDWULFH X SRVOHGQMRM QHMHGQDNRVWL UH DOQH L VLPHWUL QH NRULVWHþL :H\OORYX QHMHGQDNRVW VOHGL GD MH

!"#$ %&"&' ($

(3)

&" " &,&" " &,

&" " &, &"

&, &

" &, &

&, & "

&,

& " &, &

& " &, && - 1/ 1 " " && '/

/ % $ - 1 /

/ - 1 - 3

"& ,

&5

+ - .

3

&" " &0 - 1 - , , /

/ 3 2 ( 3

/ < -

, , - - 2

. &0

&/ = , 3 &" ", & - &5 -1/

/ % "

2

. &0 -2 & - 1 /

/ 8 3 .

@ " @ " 4#/ "

/& /5/

"

/5 /&

+ 4 /

; & 1 - 1 -+ - - 1 / < 2 , ++ /

: 1 ;2 -1 12 - &/ < - + + - - + 1 2 - / # 2 - 1 - ; 1 /

A - 1 1 ;2 & . 1 2 / 71 - / =$./-8ÿ$.

8 UDGX VX SUH]HQWRYDQL QRYL GRYROMQL XVORYL DVLP SWRWVNH VWDELOQRVWL X YLGX NULWHULMXPD NRML VX QH]DYLVQL RG NDåQMHQMD 3RND]DQR MH GD GDWL XVORYL SRVHGXMX PDQMX NRQ]HUYDWLYQRVW RG QHNLK XVORYD L] SRVWRMHþH OLWHUDWXUH

/,7(5$785$

>@ !""" # $ %& '()*%+ ),-)-->@ . # / #

0 / ! 1 $ -( '())2+ ,)3242

>@ " 5 # !66 7 8 0 !"""" $ $ 9 :# '())2+ 3*-3)(

>@ 5 ; . .B # '())*+ ((,(((,,

J

J

D

>@

¦

M 1

__#M__ _J_ _J_

>@ 7ULQK $OGHQ U

§ ¨ © ¨ ¨/

¦

M

1 _$M_

· ¸ ¹ ¸

¸ _J_ _J_ 7HRUHPD $73$$73$3 4 _J_ _J_

3RVOHGLFD

$73$ 3 4 __$__ OPLQ43

OPD[3 U$

—

CC

_J_ _J_

3RVOHGLFD

$73$ 3 4 __$__

OPLQ4 OPD[3

_J_ _J_

*UDQLFD

VWDELOQRVWL _J_ _J_

6672-$129,ý L VDU $6,037276.$ 67$%,/1267 /,1($51,+ +HP LQG

(4)

*+ , - B.*/*0 *1 ./2/% 0 3/ 4/5 /263+ 03 // 7/*% /. #888' 9)(298

5*7 *:7/ B .*/*0 *1 ./ 3/ 4/5 7./ ;/ 7<*4 03 7/*/% . #"""' !(2!$9

- = 4* >032 ; / *:7/ /:0?/* *1 7./ ./2/ //2 03 3/ *. 9 @, ,*+ 7/*/* ,*/*0 #""' .0*

@ A 5< >032>; / B ,*/*0 *1 A./ >./ >03 3/ 7*; *70 *1 ,*/*0 #""9' 82$(

B* , =+ - =+ >032 ; / *7/;7/21 :.< /:0?/* *1 ./2/ 3/ 4/5 /263+ // 03 *. 2,*/*0 5*3 ;; *0 9 * #""&' !82!8)

5 +40 2 *7< 2 57 *:7/ // 1 :.< .*/*0 *1 ./ /2 03 0 3/ 4/5 *2:*7 7.//3 // 3/ . #"""' &"82&9

A *6. CB0 ./0? /:0% ?/* *1 0+2.0 0 .*/7*7 ./ /2 03 3/C / ,*/ 6*0 &! * #8)(' (2&"

A *6. C>./0? /:0?/* *1 0 .*/7*7 ./ /2 03 3/ 4/5

03 /.*./*C 7/*/ ,*/ 6*0 ,2$$ #8))' (9(2(!

D 7 C>./0? /:0?/* *1 0+ .0 /.*% ./ 3/ 4/5 03C 7/*/ ,*/ 6*0 ,2$8 #88&' )"2)

5 0 C .*/ * E>./0? /% :0?/* *1 0+ .0 /.*./ 3/ 4/5 % 03C 7/*/ ,*/ 6*0 ,2&" #889' 8&28!

F7 C 032 ; / /:0/3 *1 +2.% 0 3/ 4/5 >03C 7/*/ ,*/ 6*0 ,2&" #889' 8$"28$$

7+ 5* D D5+ C/:0/3 03 *1 0% +2.0 3/ 4/5 03C 3/ G ,*/*0 // 6*0 #889' (92()

, C>032 ; / /:0/3 ,/ 1* >./ 7./ +2.0 3/ 4/5 >% 03C @<0 / 6*0 $$& * #88(' 992!! = =+ B 7 C/:0?/* //* 1*

/7: >./ 2>03 +2.0 3/C 7/*/ ,*/ 6*0 & * #;/ 88(' ((2)

< C*:7/ >./0? /:0?/* *1 A./ +2.0 >./2 3/ 4/5 >03C ;% / 5*3 ;;0. 6*0 $ * #""' "928

6800$5<

! "#

" $ %&!%'( 'UDJXWLQ /M 'HEHOMNRYLþ ,OLMD 0ODGHQRYLþ

)DFXOW\ RI 7HFKQRORJ\ 8QLYHUVLW\ RI 1Lå /HVNRYDF)DFXOW\ RI 0HFKDQLFDO (QJLQHHULQJ 8QLYHUVLW\ RI %HOJUDGH %HOJUDGH

1HZ FRQGLWLRQV IRU WKH VWDELOLW\ RI GLVFUHWH GHOD\ V\VWHPV RI WKH IRUP [ N $[ N $[ N²K DUH SUHVHQWHG LQ WKH SDSHU 7KHVH QHZ GHOD\²LQGHSHQ GHQW FRQGLWLRQV ZHUH GHULYHG XVLQJ DQ DSSURDFK EDVHG RQ WKH VHFRQG /\DSX QRY·V PHWKRG 7KHVH UHVXOWV DUH OHVV FRQVHUYDWLYH WKDQ VRPH LQ WKH H[LVWLQJ OLWHUDWXUH $ QXPHULFDO H[DPSOH ZDV ZRUNHG RXW WR VKRZ WKH DSSOLFDELOLW\ RI WKH GHULYHG UHVXOWV

.H\ ZRUGV /LQHDU GLVFUHWH WLPH²GHOD\ V\VWHPV N $V\PSWRWLF VWDELOLW\ N 6XIILFLHQW GHOD\²LQGHSHQ GHQW FRQGLWLRQV N

.OMX QH UH L $VLPSWRWVND VWDELO QRVW N 'LVNUHWQL VLVWHPL N 9UH PHQVNR NDåQMHQMH N 6LVWHPL VD NDåQMHQMHP N /MDSXQRYD VWDELOQRVW N !"#$ %&"&' ($

Referências

Documentos relacionados

Os objetivos específicos buscarão:  Observar o cotidiano escolar tendo em vista capturar os movimentos da sala de aula comum, dos espaços de planejamento e de formação, das

A porcentagem da polpa foi determinada de acordo com os resultados obtidos nos experimentos, de modo que durante este experimento inicial foi utilizado 0,4L de

Pensando nos diversos obstáculos que impedem o uso frequente de um planetário, por parte de professores e alunos, com finalidade de ensino, a tecnologia VR (virtual reality)

The probability of attending school four our group of interest in this region increased by 6.5 percentage points after the expansion of the Bolsa Família program in 2007 and

For time domain signal interpretation, several methods have been proposed, including recommendations related to the excita- tion shape and frequency of the input wave, which effect

CCM-205 Sistema de Bancos de Dados - 2° quadrimestre de 2011 70 Sintaxe simplificada para atribuir privilégios de objeto (qualquer usuário) GRANT ALL SELECT INSERT DELETE INDEX

Deste modo, a piramidação dos locos de resistência Run1+Ren3, não acrescentou na resistência ao oídio, porém, mesmo que o Ren3 confere apenas resistência parcial, a

E esta expansão do agronegócio canavieiro, principalmente nos três municípios em que este processo se deu de forma mais intensa, se deu no âmbito de uma