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Nonconservative LMI Criteria for Characterization of Delay-Independent Properties of Delay Systems. Application to Stability and Input-Output Analysis of Systems with Complex Parameter

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(1)INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Nonconservative LMI criteria for characterization of delay-independent properties of delay systems. Application to stability and input-output analysis of systems with complex parameter Pierre-Alexandre Bliman. N° 4278 Octobre 2001. ISSN 0249-6399. ISRN INRIA/RR--4278--FR+ENG. THÈME 4. apport de recherche.

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(219) XU      \[  b b   [!   [F . +j%<  1j% ` 1  b  . {a­ ›[ s† ²'yEvxw„ˆr

(220) ®]]}~zxo2®]rrZ„' } {]vxqLv€Lrzxª½‰Y]‚gE„ˆ} y2o2qLr[]vx2Œ~r)žgq§¨Œ ƒrZsY¸œ2y2¯r)n

(221) Jo2‰)‡Gr[]„ Œ~´r)q6r[r)‚2›Yvx›]2³ žK” †«{]³ n

(222) † o2r/yJ„ˆyJr„ˆyE]§{az2yJqL‚2‚Er}~r[o2Œ/ ™k† ]y2§¨] Œ%† w]y2§ qLr[Œ v|¯J‰){]´-zƒo2®]vxr)‰Œ~o¶v{zEv|‰)„-{ao2} r[vx]Œ gr a)54§:6"‰+/]# } †  v¬}~v€Y ƒ sY‡œ †«³ uw´'r)r[‚2v€Jžgq9{©‡9™kr{©®]Yv † r † ¯wyJ„ v€2žL}~o2r(§ {]‰}

(223) } oJ{a} ­ ›)sw²vx„-rZ¸y2v€®!{]z€r[Y}

(224) }~ .  . RSS SRS

(225) U   + %  

(226) U  3+ %  SST STS ! "C

(227) U   + %       +j%<  +  3#  ! "#&%

(228) U . !

(229) U   + %   + 

(230) U   + %   +     [! #&%

(231) U   +j%< . ! ". !

(232) U   + %  

(233) U   + %      [F U #&%>

(234) U . ! ".  \[   \[  \[   +,%<  1,%' %. . C

(235) U   + %   1

(236) U   + %   1 % 1   . ! ". % ` aa ` aa aa aa 5$# B b b 9 :&%>?. ´-2]v€} }<o rŠƒo2vxr)„ˆŒ~}-r qLr} @oJ † ] §r)Šƒˆ {YŽ ‰ž }-o r)®!ˆ {]ž ½z€yJo {a } x vx]I§¨]Œ- }~o2ˆ Žvx„-Y ˆ „ˆ} iŒ~yJ‰} Y y2Œ~r †i „ˆvx2x žYp y2x zx{]³ Œ-<®!{a!zx´'y2r)r]®Y³ r)ŒZ¯Ey2‚4} ]yJŒ»w2!´-zxr † žYr]¯k} oJr)Œ~r † wr[„ zx‚2r)Œ~qD]™2n

(237) ¯Ezx„ o2r)]qr

(238) zx®!Œ~{ar[­ „ ™2„ r)y2zxrur(z¬}v€„ƒ DÀ2a¯k§:‚kœ]]n

(239) •‡zx†«o2‡¯Er)J{]Y] Œ qLr[† q vx{]„ ‚Ez:›'rZ} !‰vxqLv{kzEr[r]‰)Œm³{]{az€ zxL‡]{!qLƒ œ}ˆ}~‚2”‡Œ~† yƒ{Y}§¨‰]{!}~Œ/} v€®Yvx]‚2rDŒ wY]y2™2§qLzxr)}~q9r)o2Œ~r„-v|‰)„ŽŒ~{a}~r)z2Œ z|yE{]{!} z¬‰}~r } r)†yJŒ~Œ J} r {! † }~v€† „ ®Yr[vxr]2zx¯a{©ž]{Y‡y2„z|„ˆ{aŒ~‡ƒŒ-r[„ˆ„ } ®!]r[{]zxqLyƒz€y2}~„rZv€²Y„-³-7v|n

(240) „-ao2§µ»wr72¹ !¶Œ ´-rZ¼Ž„ˆI„$y2z€v|}~}~„%9„<{˜™Ev€ ‰r1Yn

(241) { ®Yo2/rr[Š‹m]Œ~ª«‚2r)oJŒ~q {a]Œ™2† › ª o2q9r[{aJvx‰Kr/´'§¨y2r[Œ {aJ»wvx2„ rZo2„ r(„[¯]{avx‚29‚2}~Œ~o2©Šƒr/v€‚2q9Œ~{ar[} „ vxr)]L}

(242) a„Ž§}{!„ } ]rYqL¯Yv|r „ <':‹ 8/'S‚J2 .Œ Y)51™2% z€. r[%Iq95 %(„)#&%(¯.ga0 §€%{]2Œ 1™Jv¬v€}~ƒŒ~§¨{]YŒ Œ ‡1q9‚2{aŒ } rZvx]‰v|9„ˆvxv|]„  »w¯Y2™w!‡g´-§ 9{aqLY9vxz€vx} r[o2„'r˜a„ˆ§Nvx‹j·)r/‚2aŒ~§:]} ™2oJzxrr)q9zxr[„[{Y³„Ž)}n

(243) !:o2r[¯v€v€Œ § {} aw‡Yž]¯wr)§¨}(]Œ

(244) {]f´-o2r)Šƒv|‰{YoI‰}}~o2‰rYJ¹ ‰¶zxyJ¼Ž„ „-v€Y{]fŒ r(]„ f]zx} ®©o2{]rg™2zx®]r r[Œ~­ {Yv€D‰v€}Ža‡D}~o2ar)§%Œ-}~´'o2]rLŒ ‚2† Œ „[Y¯w‚Ear[§ Œˆ}~}Žo2‡Dr(y2zx{]Œ † žYr)r[Œ„ˆ}'„ˆ} !Iy † ´-‡2o2²v|³1‰oGn

(245) v|o2„

(246) v|„˜2rZvx„‰r[{]„~f„~{avxŒ~qg‡L‚k} g]Œ } }~rZ{]„Ž}-}(w‰)y2]qLqLr)‚2Œ~zxvxr‰[Šƒ{av¬zx}Žz€‡‡ v|­ „ „ „ r)y2r9r]™k¯ r)™Ezx]!} ´ o uw§¨Œ r[Y‰q} vx]} –oJr)„]]²(Œ~vxr }~†vx‰[v|{a‰)zm{a} {arL †}~oJ‚2{!}7Œ{]„ ‰q9} v|{a‰){]zxz%zm®!‚E{aYzxv€y2r[}„a]§'§'®! vxr)´­ }Ž‡w³I‚2–vx‰[4{azxz€} ‡ o2r9„ˆq9]} {]o2z€zxr[r)ŒŒ(oJ}~{aoJ {a† ¯ } „ oJ]r qL† rLvxqgyJr[qgJ„ r[v€YŒ v|¸‰){]e zmraŠƒ§

(247) ‚k}~r)o2Œ~rKv€qL„Žr)}{!}~}r„ ®!{aŒ~vx{]™2zxr]§}~o2r„ˆ‡ƒ„ˆ} r[q9²']§ } r[G‡wvxr)z † ‰)]Œ~Œ rZ‰}-{aJ„ ´'r)Œ„)³ } ˆEr[¯:q ‰ ˆ:y c`‰

(248) rŠd ³2‹]!¼«³}$´ ¼½}~ r)y2Šƒ† Œ E‚2r[rŒ „$†rZ]„ ¯Yyƒ„N} }$} o2o2} roEr'„ˆ{!Œ~r)}%r[}%„ {aay2zx§µqLz¬}} „Yo2„ˆ{ r}† „ˆ2{a‡ƒ‚2(„ˆ} } žYrr)r)†q92} r[„$Œ~§¨{]yJ} z€o2z9v€zJ}Žr'‡(zxzxvxv€„~vx2„„ ž1y2zxYr {(„ˆ]}$‚2§E´-Œ ‰YoJoJ‚E{ar)r[gŒŒˆ{]}ŽŒ ‰‡1r[} ‚2r[aŒ zx§k{Yvx·[} ‰o2{avx2} r/vxž/]‚J(vx{agŒa{a} § o2qL† r'r)r}~z|}~{©Œ~Œ {]‡Yvx·)Jª«r vx„Ž † §¨†r[„ Œr)‡ƒ‚k} „Žo2r)}~r)r'q9† } r)r[„ Œ q9c }— ‚J%„ ¯Œ Œw‰Y=]‚EEŽ r[d „ŽŒˆ¯}~}~v¬v€Qn}~rZyƒ„N|~} arZ§k}„m„ˆ¯!} ‡ƒ™woJ„ˆ‡ r ª %('". ) ,0%I8/, a§} o2r7„ r}a§} o2r7„ ‡ƒ„Ž}~r)q9„

(249) §¨y2{z zJzxz€vx2žL}~o2r7‰YŒ Œ~r[„ ‚k] † vx2ž † r)z|{©‡ªWvx † r[‚Er[ † r[Y}<‚JŒ Y‚Er[Œˆ}Ž‡Ga§ c`d ³'n

(250) o2r}~]‚k]z€ª ]žY‡9´-o2v|‰ov|„

(251) qLr[{]Y}Z¯2v|„'} o2r(}~]‚k]zx]ž]‡Iv€ † yJ‰r † ™w‡9}~o2r2YŒ q6]a§ §J}~}~o2o2rr<‰zx{]wŒ rVžY¤9r[„ˆ‰)}

(252) v€r[„ vx}~2„ ž]y2z|{aŒ-®!{]z€y2r(YI}~o2r‚2Œ~ † yJ‰}/„ ‚J{]‰)r ³ '. .

(253)  . . . . . . . . . 2. m oSpqo rWm oSpqo r#m oSp

(254) v rWm oSpwv rWmx po rWmx po rWmx p

(255) v rWm%x pwv à ÃÅÄÇÆqȘɘÊNË. FIB D Y<B M Y;T D Y;T M YN\ D Y<\ M Y;a D Y<a M J.

(256) ¼½y2J‰)‚Jr){aŒ Œ }~} {]v|v€‰y2} z|vx{ar[ŒZ„

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