Stabilization of a class of Delay Systems using PI methods

HAL Id: inria-00070424

https://hal.inria.fr/inria-00070424

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Stabilization of a class of Delay Systems using PI methods

Catherine Bonnet, Jonathan R. Partington

To cite this version:

Catherine Bonnet, Jonathan R. Partington. Stabilization of a class of Delay Systems using PI methods.

[Research Report] RR-5583, INRIA. 2005, pp.13. �inria-00070424�

ISSN 0249-6399 ISRN INRIA/RR--5583--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème BIO

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Stabilization of a class of Delay Systems using PI methods

Catherine Bonnet — Jonathan R. Partington

N° 5583

mai 2005

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)

Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30

$&./"0/

1!2436587:9<;>=87@?AB=8=87C3EDGFBAH=8243I5J24=LK@MONP249<3I;Q=8RB36AH=

∗

SUTWVX!YZ[]\L^`_badcfe]V<XY6cUgihkjmlkjmnmhpo:qiY<c

rGs]jmtuYe_W\_i_W\v

w%xzyiyBjms]e%{dYs|Y<}~TiY<s|}~TiY€H8‚m‚zƒC„…^†X"xzh‡vmˆmˆC‚^ ‰<„yWxmnmY6c

ŠŒ‹Ž<’‘i“mm”• Ycfe]qW{da)h–/e|Tihkcs]Y<y4jCsfeUe|TiY

H ∞

^—

cfe|xmgih–lkhk˜hkin"yis|jmyBYs]e]hkY<cjz™8rO[}jCCe|s]jCl–lkYs~cš™›jmsyWxms

—

e|hk}qilpxzsO_b[f_W\{iYlpxIacfadcfe]YX"cŽjz™4e]Tis|YYU}lpxmc|c]Y<cšœ{dY6xm{

—

e]hkX!YUc]adcue|YX"c8xmcŽYlklbxCcŽs]Ye|xzs~{dY6{jmsŽiY<qde]s~xzl

c]adcue|YX"c<žŸijCs xŒc]qigH}lpxmc|c%jm™OiYqde|s|xmlc]adcue|YX"c¡4e]TWhkc YWxmgilkY<c%q4c%e|j){dY6{dqW}Yx"yWxms|xmXYe]s|h–˜6x’e|h–jC/jm™

xml–l

H ∞

^—

cue~xzgihklkh–˜<h–in"}jm:e]s|jmlkl–Y<s|c<ž

¢¤£:¥H¦|§©¨

’ªŽm”

{iYlpxIa c]abcfe]Y<X¡

H ∞

^—

cue~xzgihkl–h–euam¡6rO[#}jC:e]s|jmlkl–Y<s<¡}jmyis|hkXY8™«xC}¬e|jms|h–˜6x’e]hkjm‡¡yWxms|xmXYe]s|h–˜6x’e|h–jC

∗

­Ž®u¯’°~±«²Q³®u´²µ~¶4·°|²>¸6®f³%°|²Q¹»ºu¼]½I¾´I¹¿®u±«¼>¹²ÁÀµ~¶WÂb®u®fÃ6¼]½’Âb®u®uÃI¼‡ÂiÄ6ŬƬǬÈJ½z¾GÉÊ

$&% 0) © '8 +-,



”

xmWc}Ys|xmyiy4jCsfe6¡HijmqWce|qW{dhkjmWcPlkY<cPyis|jmyis|he6c{dY6c}jm:e|sCl–Y<qis|c {dY eua:yBYrO[%yBjmqis

lpx cue~xzgihkl–hpc|x’e]hkjm

H ∞

{dY}Y<sfe~xzhkWc!cfadcfe]VX!Y6c…s|Ye|xms|{Wc_b[f_4\ xzyiy4xzs]e]Y4xz:e…e]s|jmhpc}lpxmc|cfY6cŒœ/lkx

}lkxCc]c]Y{dY<ccfadcfe]VX!Y6c%{djm:e%lkYe]s~xz4cu™›Y<sfe%Y6cue{dqeuabyBY

R(s)e ^−sT

^j

R

Y6cue s|xze]hkjmiiY<lQ¡Wlkx"}lkxCc]c]Y{dY<c c]a:e]V<XY6cŽs]Ye|xzs~{6cYeJlkx}lpxmc|c]Yš{dY6cJcfadcfe]VX!Y6cWYqde|s]Y6cžŽr8jmqis8qiiYc]jmqWc

—

}lpxmc|cfY{dYcfadcfe]VX!Y6cWYqde|s]Y6c¡

}Y<cUs|Y<c]qil–e|x’e~cijmqWcUyBYs|X!Ye]e]Y:e%{dY{6{dqihks]YqiWYPy4xzs~xzXe|s]hpc]xze]hkjmŒ{iYPl! YWc]YX gil–Y{dY6c}jCCe|sCl–Y<qis~c

cfe|xmgihkl–hpc]xm:e|c<ž

" ¨ 6 ¦ “#  ” c]abcfe]V<X!Y$ s]Ye|xzs~{ic<¡cfe|xmgihkl–h–e

H ∞

¡%}jm:e]s%mlkYqis!rO[¬¡G™«xm}e]jms|hpc]xze]hkjm}jmyWs]Y<XhkVs|Ym¡

y4xzs~xzXe|s]hpc]xze]hkjm

4

,B 8"0/')G

•

Y}jCWc]hk{dY<sh–/e|Tihkcs]Y<y4jCsfeUe|TiY

H ∞

^—

cfe|xmgih–lkhk˜<x’e|h–jC ¡dhk@xcfe|xmW{ixms|{Œ™›YY6{dgWxm}65/}jCCe|Y7be<¡djm™98’xms]hkjmqWc

euabyBY<cUjm™J_b[f_4\€{dY<lkxIaŒc]adcue|YX"c<ž • Yc]TWxzlkl#gBY}jm4}Ys|iY6{ŒTiY<s]Yh–e]T/e]TWYxzWxml–adc]hkcjz™ŽeujX!Ye]Tijd{ic

e|TWx’eTWx&8CYg4Y<Y qWc]Y<{¤Y7:e|YWc]h:8mYlkah–Ee]TiY ™›YY6{dgWxC}65cfe|xmgihkl–hk˜<xze]hkjm@jm™š{dYlpxIa¤cfadcfe]Y<X!c<¡HWxzX!Y<l–aC¡BrG[

}jm:e]s|jmlklkYs~cŽxmW{!}jCyis]hkX!YU™«xm}e]jms|hk˜<x’e|h–jC}jm:e]s|jmlklkYs~cžŽSUTiY<;Ws~cueOhpcG}jmW}Yyde|qWxzlkl–a X!jms|YUc]hkXyWl–YC¡:xCcJhe

s|Y<o:qihks|Y<ce]TiY%}<xzlp}qilpx’e|h–jC jm™4eujyWxzs~xzX!Ye]Ys~ce]j{dY;WiYUe|TiY%}jCCe|s]jCl–lkYs>=’e|TiY%cfY6}jCW{©WYY<{WcJX!jms|YUxml

—

nCYgis~xzhp}%X!Ye]Tijd{ic<¡icfhkW}Ye|TiYyWxzs~xzX!Ye|s]hk˜<xze]hkjm"jm™Žcfe|xzgWh–lkh–˜<h–Wn}jm:e]s|jmlkl–Y<s|cGhkcU}jmX!X!jmilka"Y7dyis|Y<c|cfY6{

hk…e]Ys|X"cjz™UcfjCl–qde|h–jCWc e]j¤x/Z <˜jCqdeY<o:qWx’e|h–jCEj8CYs e]TiY"xzlknmY<gis~x

H ^∞ ( C ₊ )

jz™šgBjmqWW{dY<{ xmWxzlka:e]hp}

e|s|xmWcf™›Ys™›qiW}e]hkjmWc%{iY;WiY6{/jm)e]TiYs|h–nCT:eUT4xzl–™

—

yilpxzWY

C ₊

^ž

? @hkXyBjms]e|xm:e s|Y<}Y<:es]Y™›Ys|YW}Y™›jmse]TiYrO[ cfe|xzgWh–lkh–˜6x’e|h–jC¤jz™0;Ws~cue jms~{dYsP{dY<xC{

—

e|h–X!Ycfadcfe]YX"chkc

@

‰m‰A>ž8SUTihpc8nmh:8mY<c‡e]TWY%}jmX!yilkYe|Yc]YeJjm™Bcfe|xzgWh–lkh–˜<h–Wn rO[ yWxzs~xzX!Ye|Ys~c ™›jCs8g4jme]T!jmyBY

—

lkj:jCy cfe|xmgil–YUxz4{

qWWcue~xzgilkYyilkxm:e|c<žB¤jms|Yj8CYs%hk@e|TiY"}xmc]Y©jm™šxz…jmyBY

—

lkj:jCy@qiWcfe|xmgilkY yilpxz:e6¡#henmh:8mY6c x)iY<}Y<c|c]xms]a

xmW{/cfqDC }h–Y<Ce}jmW{ihe|h–jC)jCŒe|TiYe]hkX!Y{dYlpxIa"™›jmsUe|TiYY7dhpcue|YW}Yjz™Jcue~xzgihkl–hk˜hkin!rO[

}jm:e]s|jmlklkYs~cž

• Y%c]TWxzlklWxz4xzlkabc]YUTiYs|YcfhkX!h–lpxzsšoCqWY<cfe]hkjmWc<¡zgiqdešhe|T e|jbjmlpc8e]TWxzeGcfY<YX e]jgBY%cfhkX!yil–Y<sJe]T4xze]Tij:cfYjm™

@‰m‰AWTihp}~Txms]YUs|Ylpx’e]Y6{e|je]TiYFEYs|X!he|Y'GbZhkYTilkYsŽe|TiYjCs]Y<X/žHB¤jCs]Y<j8mYsŽYhklkliY7ixzX!hkiY%x{dhIBY<s]Y<Ce

}lkxCc]c%jz™G{dY6xm{

—

e]hkX!Y©cfadcfe]Y<X!c xmcšY<l–lŽxmcc]jmX!Ys|Ye~xzs~{dY<{xmW{iYqie]s~xzlc]adcue|YX"c™›jCs%e|TiY;Ws~cfe%e]hkX!Ymž

Jjze|Ye]T4x’ee|TiY

H ∞

^—

cfe|xzgWh–lkheua/jz™Gx iYqde|s|xmlŽ{iYlpxIa/c]adcue|YX hkcPx"X!jms|Ys|Y<cfe]s|hk}e]h:8mYWjze]hkjm@e]TWxm¤e]TiY

xmgWc]YW}YPjm™qi4cue~xzgilkYy4jCl–Y6c

@K

A>ž

? ijze|TiYs)s]Y™›Ys|YW}Y¤jm e|TihpcŒo:qiY<cfe]hkjm hpcŒe]TiY…g4jbjL5 jz™Jhk}qil–Y6c]}q ž ŸWjmsŒo:qiY6cue|h–jCWc jm™™›YY<{igWxm}65

cfe|xmgihkl–h–euam¡TiY }jCWc]hk{dY<s]hkin/e]TWY rO[

—

cue~xzgihkl–hk˜<xze]hkjm jz™x¤cfadcfe]Y<X h–e]T e]s~xzWcf™›Ys™›qiW}¬e|h–jC

G(s) = e ^−sh

s

¡’TWYGhpc‡X"xzhkilka}jm4}Ys|iY6{he|T cfe|xmgih–lkh–euaPjm™de]TiYU}lkjCc]Y<{

—

lkjbjmye]s~xz4cu™›Y<s ™›qiW}e]hkjm

G(s)K(s)(1+

G(s)K(s)) ⁻¹

^¡TiYs|Y

K

{dY<ijze|Y<cx¤}jC:e]s|jmlkl–Y<s=šY c]TWxzlklOxz4xzlkabc]Ye]TiY)x)yWs]hkjms|hOcue|s]jCinmY<sijze|h–jC

jm™h–iyWqde

—

jmqde|yiqde¤cue~xzgihklkh–˜6xzgihkl–h–euaNMÁe]TWY@™›jmqWs)cfe|xz4{ixzs~{ e]s~xzWcf™›Ys ™›qiW}¬e|h–jCWc)TWx&8mY¤e|j gBYEhk

H ∞

^O

¡

xmle|TijmqWnmT/jmqis}jCW}lkqWc]h–jCWcxzs|YcfhkX!h–lpxzshk/e]TiYc]y4Y6}hpxzl }<xmc]Y<cxz4xzlkabc]Y<{Œh–

@QP

A ž

• Y!X!Y:e]hkjm xzlpc]j

@R

¡O‰Ivb¡J‰<„&AxmcPyWxzyBYs~cPe]T4x’e}jCWcfhp{dY<sPrG[ }jC:e]s|jmlŽjz™{dYlpxIa@cfadcfe]Y<X!chk x)s|j

—

gWqWcue|iY<c|cjms

H ∞

™›s|xmX!Yjms5B¡#xmle|TijmqinCTEe|TiYhksxzyWyis]j:xm}~TiY6cxzs|Y©e|jze|xml–lkaE{dhIHYs|Y:e™›s]jCXjmqWs|cxz4{

e|TiYhkss|Y<c]qile~cUjmilka lkjbjCc]Ylka s]Y<lkxze]Y6{#ž

• Y)}jmWc]hp{dYs h– c]Y<}e]hkjm vE{dY<xC{

—

e|h–X!YŒc]adcue|YX"c jz™e]TWY euaby4Y

G(s) = e ^−sT p(s) q(s)

xmW{ nCh8CYŒcfjCX!Y

s|Y<c]qil–e|cšjm"e|TiYPY7dhkcfe]Y<W}YPjz™ rG[šcfe|xzgWh–lkh–˜<h–Wn }jC:e]s|jmlkl–Y<s|c<ž

?

lkc]jW¡bhk e]TWYP}<xmc]YTiY<s]Y

deg p

^xzW{

deg q

xms]YPlkY<c|ce]TWxm/jCsY<o:qWxmlHe]j!jmiYC¡be]TiYY7dhkcfe]Y<W}Yjm™Žcfe|xzgWh–lkheuaŒh–W{ij’%cUhkcY7ixzX!h–WY<{#ž

STS

´U2V'V'W'X

12!' ! %!D 1 . '+,

w%Ye|xms|{iY<{ xmW{ iYqie]s~xzlcfadcfe]Y<X!c!jm™e|TiYeuaby4Y

G(s) = 1

p(s) + q(s)e ^−sT

xzs|YŒe|TiY xmWxzlkadcfY6{ hk

c]Y<}e]hkjm©„iž9B@xmbaPY<inmhkiY<Ys|h–injmsŽgih–jCl–jCnmhp}xmlmX!jd{dYlpc‡lkY<xC{dhkinW¡’yBjCc|cfhkgilkax’™Áe]Y<slkhkiY<xms]hk˜<xze]hkjm ¡6e]jPcfq4}~T

e|s|xmWcf™›YsP™›qiW}e]hkjmWc}<xz g4Y!™›jCqiW{…h– M

@QP

A ¡‡}~TWxmy v O ž •

YnCh8CY TWYs|Y!c]jmX!Y"}jCW{dh–e]hkjmWc M«iY<}Y<c|c]xms]a

xmW{!cfq C"}h–Y<:eGh–!e|TiY }<xmc]Yjz™#xiY<qde]s~xzlWc]adcue|YX xzW{"c]qDC }hkY:eOjme]TiY<s]hpcfY

O

™›jmsOe]TiY%Y7dhkcfe]Y<W}Y%jm™BrG[

cfe|xmgihkl–hk˜hkin }jm:e]s|jmlklkYs~cGxz4{ qWc]Ye|TiY<c]YPs]Y6cfqil–e|cše|j{dYe]Y<s]X!hkiYPx yWxms|xmXYe]s|h–˜6x’e|h–jC"jz™xml–l#cfe|xzgWh–lkh–˜<h–Wn

}jm:e]s|jmlklkYs~cGhk/e]TiYyWxmsfe|hk}qilpxzs}xCcfYTiY<s]Y

deg p = deg q = 1

^ž

0d$& "#%$&

• Y}jCWc]hk{dY<sUh–/e]TWhkc%c]Y<}e]hkjm{dYlpxIaŒc]adcue|YX"cUhe|T/e]s~xzWcf™›Ys™›qiW}e]hkjm

G(s) = e ^−sT p(s)

q(s) ,

^Mu‰ O

TWYs|Y

T > 0

^xzW{

p

^xmW{

q

xzs|Y s|Y<xml4yBjmlkabijmX!hpxzlpcc|x’e|hkcf™›abh–Wn

deg p ≤ deg q

hk)jCs|{iYsše]j!{dY<xmlBh–e]T yWs]jCy4Y<sUc]adcue|YX"c M›h>žYmžOjCiY<cc]qW}~T)e]TWxze

sup _{Re s>0, |s|≥R} | G(s) | < ∞

¡d™›jms%c]jmX!Y

R > 0

O ž

•

Y©Uxz:e%e|j Y8’xml–qWxze]Y™›jCs c]qW}~T@c]abcfe]Y<X"ce]TWY

H ∞

^—

cue~xzgihkl–hk˜hkin"yis|jmyBYs]e]hkY<cjz™OrO[%}jm:e]s|jmlklkYs~c¡de|TWx’e

hpc<¡djz™J}jm:e|s]jCl–lkYs~cGh–e]T/e|s|xmWcu™›Y<sU™›qiW}e]hkjm

K(s) = k p + k i

s ,

^{MQv O}

TWYs|Y

k p

xmW{

k i

xms]Ys|Y<xzl#}j:YC }hkY:e|c<ž

G

K r - e ₁

+ e

+

-

? e ^{− d}

+

6

e ₂

ŸŽh–nCqis]Y!‰CœG_:e|xmW{ixzs~{ŒŸWYY6{dgWxm}65šjmD;WnCqis~x’e]hkjm

• Y }jm4cfhp{dYsŒe]TiY cfe|xz4{ixzs~{ ™›YY6{dgWxm}65 c|}~TiYX!Y…jz™©ŸŽh–nCqis|Y ‰m¡ xzW{ s]Y6}xzlkl%e|TWx’e

H ∞

^—

cfe|xmgihkl–h–euahkc

Y6o:qih8’xml–Y<Cee|j!e]TiYe|Tis|YYe]s~xz4cu™›Y<s™›qiW}e]hkjmWc

(1 + KG) ⁻¹

^¡

K(1 + GK ) ⁻¹

^¡

G(1 + KG) ⁻¹

g4Y<h–in cfe|xmgilkYmž

9S

• YTWx&8CY

sq(s)

sq(s) + e ^−sT p(s)(k p s + k i ) , sp(s)e ^−sT

sq(s) + e ^−sT p(s)(k p s + k i ) , k p sq(s) + k i q(s) sq(s) + e ^−sT p(s)(k p s + k i ) ,

xmW{e|TijCc]Ye|s|xmWcu™›Y<s™›qWW}¬e|h–jCWcPs]Y<yis]Y6cfY<:e%e]TiYgBYTWx&8bhkjmqis jz™Gx)s]Ye|xzs~{dY6{¤{dYlpxIa¤cfadcfe]YX h–™

deg p <

deg q

xzW{)jz™JxiYqde|s|xml {dYlpxIaŒc]adcue|YXh–™

deg p = deg q

^ž

• Y}jmWc]hk{iYsH;Ws~cueGe]TiY}<xmc]YUjz™ xiYqie]s~xzl

—

euabyBY%}lkjCc]Y<{l–jbjCy =me|TiY%iY7beOyis]jCy4j:cfh–e]hkjm‡¡mh–e]TijCqdeGnmh:8bh–Wn

x }jCXyWl–Ye]Y}~TWxzs~xm}e]Ys|hk˜<x’e|h–jC!jz™ e|TiY 8’xml–qiY6cjz™

k p

xzW{

k i

c]jml:8:hkine|TiYyis|jmgilkYX¡bxzlkl–j’%cšjmiYe]j"; 7

e|TiYlkjb}<x’e|h–jC)jm™

k p

xzW{)e|j!{iYe]Y<s]X!hkiYm¡i™›jms%x"; 7dY<{

k p

¡iT4x’ehkc%jmgde~xzhkWxzgilkYmž

 ¨G¨ « ¨

2'3L! 12!$'#&'!(

G(s) = e ^−sT p(s) q(s)

12!!

deg p = deg q

T > 0

³

1 ! 6% !

K(s) = k p + k i

s

/

| lim |s|→∞ q(s)

p(s)k p | ≤ 1

^{1 !}

K(s)

^{3L>!' ' L Q>!}

G(s)

/

| lim _{|s|→∞ p(s)k} ^q(s)

p | > 1

^{12! 12! ! 6)D 12!} !3" > 12! H($' 2! #

( %#)D 'L ! ! 3 *-!63

k p

HF1 ! ! !'

k i

6)261 12H1 ! !*F")D ' L ! !

1 !

K(s)

^{' !}

G(s)

 ¨ ¨ ‰ O

Ÿis|jmX rOs]jCy4j:cfh–e]hkjm viž–‰Uhk

@K

A>¡CšYTWx&8CYe]TWxzeGh™

| lim _{|s|→∞ p(s)k} ^q(s)

p | < 1

^e]TWY

K(s)

^{ij:Y6c

Wjze%cue~xzgihklkh–˜<Y

G(s)

žOSUTiYY6oCq4xzlkheua y4xzs]e™›jCl–lkj’%cš™›s]jCX SUTiY<jms|YX(viž–‰hk

@K A ž

v O

ŸWs]jCX rOs|jmyBjCc]h–e]hkjmLviž–‰@hk

@K A ¡%YETWx&8CYe]TWxze/h–™

| lim |s|→∞ q(s)

p(s)k p | > 1

^e|TiY e]TiYEe]s~xzWcf™›Ys

™›qWW}¬e|h–jCWcGjm™He]TiY}lkjCc]Y<{!lkjbjmy"TWx&8CY%x’ešX!j:cue ;Wih–e]Ylka!X"xzba qWWcue~xzgilkYyBjmlkY<cšxzW{ h™#e|TiYa!TWx&8CYijmiY

e|TiY/e|TiYa)l–hkYh–

H ∞

žš_bj!e|TWx’e6¡de]TWY}jbYC }hkY:e

k p

gBYhkin$; 7dY<{#¡Wh™Že]TiY<s]YY7dhkcfe|cx

k i

nmh:8bh–in!yBjmlkY<c

jCilka h–/e|TiYl–Y™Áe%TWxzl–™

—

yilkxmiYPe]TiY</e]TiY}l–j:cfY6{Œlkjbjmy/hkccfe|xmgil–YCž

)£ ‘d ! " /'$#&%('*) +2,-! ,-, 12! ($!*L! & ( ($3L! 1 !.2 ' )D 6&!

p

³

q

^{3L!,!! !}

• YY7dxmX!h–iYWj’Le|TiYWxze]qis~xzloCqWY<cfe]hkjm¤jz™Jcfe|xzgWh–lkh–˜<h–Wn

G

^he|T@x"rO[}jm:e|s]jCl–lkYsTihp}~T¤TWxCc%gBYY

{iY<c]h–nCiY<{Œe]j"cfe|xzgWh–lkh–˜<Y

G

^TiY<

T = 0

žJSUTiYiY7:e%yWs]jCy4j:cfh–e]hkjm)yWxzs]e]hpxzlkl–aŒxmWc]šY<s|cše]TWhkco:qiY6cue|h–jC ž

ŸŽhks|cfešYs|Y<}<xzlklHe]TiYcfe|xmW{ixzs~{"™«xC}¬ee|TWx’e

)£ ‘d ! /1032 )23 $! 2 )2

ax ² + bx + c

¹

a > 0

³

b

c

^{! 1}

" !42&'&&! > 3 ! !,&!

c < 0

⁵

!6

b ² − 4ac > 0

^{5 3}

STS

´U2V'V'W'X

!6

b ² − 4ac < 0

 ¨G¨ « ¨ & ) 2&! 1 ! .0/F6% !

1 2! 6)D

K(s) = k p + k i

s

' Q>!

p(s) q(s)

#1 12! '#&'!( #

(

1 ! !

T = 0

^12!

/

deg p < deg q

K

^{' !}

p(s)

q(s) e ^−sT

^{') !% #$6(}

T

/

deg p = deg q

³

| lim

|s|→∞

q(s)

p(s)k p | > 1

K

^{' Q>!}

p(s)

q(s) e ^−sT

^{') !2 Q#$'($}

T

/ 12! 2 ' )D 6&! '#&'!(" 1 !#2!

G(s) = e ^−sT s − σ

01 ! 6'.& Q)2!

T 1

12!3L! #

1D1

93!''L Q>! 12!" !63 ( !

cos(ωT 1 ) = k i ω ² + k p σω ² k _i ² + k _p ² ω ²

3

sin(ωT 1 ) = ω ³ k p − σωk i

k _i ² + k ² _p ω ² ,

12!!

ω = r

1 2

k _p ² − σ ² + q

(σ ² − k _p ² ) ² + 4k ² _i

T > T 1

12!" &!63 ( )D26L ! 12 12! ! 'L Q#

3

21 !( !

-,

/' 12! 2 ')D ! '#&'!( 1 ! #2!

G(s) = s − α

s − β e ^−sT

K(s)

^{F1 ! 1 !}

+, ( ! '!'

| k p | < 1

¹

K

^{6L Q>!}

p(s)

q(s) e ^−sT

^{*') !2 Q# '($}

T

^3$1 !.6'.& Q)2!

T 1

1+61

3!'' L Q>!'$1 !"'#&'!( ')21 1 12! !3 (

[G, K]

^{!($ )D2' !<}

T > T 1

12 01 ! !", 1 ! ! 'L Q#

3

12! ( !

<

| k p | > 1

^{1 ! :&!63 >}

[G, K]

^{)D2' ! "}

T > 0

)£ ‘d ! 12! >$ . ( 6

! !' !! '&! # 1 ! Q 61 0!61

%

2

)2! #!6! % )

1+61

! '! # !66 12!!

1D($!1 >3

!"36,&!

12!1 ! 1 ! >"

A(s) + C(s)e ^−sT

^{1 ! !}

A

³

C

^{! !6 2 Q# ( Q!$ 1 ! !612 :! 3 3!! '(" ! 12! -,-! # Q)2!}

h

^{1D1 1+}

$! !6 120 626' 1 !6! '!% 12!.6'9'! $!+( ! ' #<

T = 0

^12!

!66 3 6 ! '3L! 6 1 ! ! 2!''( Q#*'( 2&'&!

T

^{3* '! 3 1 ! !6L6)D}

12!6( :! !1 ! %! 2)D( ! $ ! >12! 2!6 !63 12! 1+ 3 '!

3 2&'&! )2!'

h

^{2# 1+61 1 ! ! ! >!}

A(s) + C(s)e ^−sh

^{Q#+, 1 !}

( ,L'# 3 3! !'( !

1 !1 ! 12! >! & )2611 ! &6 ( !$1

! 12! 12!

1 !6&'-,

h

^{Q 3! 6 1 ( L3! 1 ! '! & 1}

A(iω) + C(iω)e ^−iωh

12&>!

ω 0

12!

W (ω ₀ ² ) := A(iω 0 )A( − iω 0 ) − C(iω 0 )C( − iω 0 ) = 0

9S

"! #$%#&'!("*)+'-,$.0/($!123

. > 0. 2&' ‰ O xmW{©v

O

™›jmlklkj’ Y6xmc]h–lkaP™›s]jCX

@

AHœ‡h™

K

hkc8c]qW}~Te|TWx’e

| ^p(∞) _q(∞) K( ∞ ) | <

1

^e]TiY<

[ ^p _q , K]

^hkc) —

cue~xzgilkYL=Pe]T4x’e)hpc<¡%e]TWYs|YEY7dhpcue~c

T 0

c]qW}~T e|TWx’e

[ ^p _q e ^−sT , K]

hkc/cue~xzgilkY@™›jCs)xml–l

T ∈ (0, T 0 )

^ž

Jjze|Y e|TWx’e%TWY

deg p < deg q

e]TWYxzgBj8mY}jm4{dhe|h–jChkcxml–UxIadcc|x’e|hkc;WY6{#ž

„

O Ye

A(s) = sq(s) = s(s − σ)

^xzW{

C(s) = p(s)(k p s + k i ) = k p s + k i

ž • YP™›jms|X

W (ω ² ) = A(iω)A( − iω) − C(iω)C( − iω) = ω ² q(iω)q( − iω) − (k _p ² ω ² + k _i ² )p(iω)p( − iω).

^{M«„ O}

SUTWY{iYijCXhkWxze]jmsjm™Y6xm}~T}lkjCc]Y<{

—

l–jbjCy e]s~xz4cu™›Y<sU™›qiW}¬e|h–jChkc

sq(s) + e ^−sT p(s)(k p s + k i ) = s(s − σ) + e ^−sT (k p s + k i ).

?

ce|TiY©}lkjCc]Y<{lkjbjmy¤hpccfe|xmgil–Yxze

T = 0

¡4šY©jmgde~xzhk™›s]jCX(e|TiY©wjCqde]TE}s]h–e]Y<s]hkjm/e|TWx’e

k p > σ

^xzW{

k i > 0

^ž

• YTWx&8CYe|TWx’e

W (ω ² ) = ω ⁴ + (σ ² − k _p ² )ω ² − k ² _i

¡dTihp}~TTWxmcxqiihpo:qiYyBjCc]he|h8CYP˜<Ys|jW¡bnmh:8mY<)gba

ω = r 1

2

k ² _p − σ ² + q

(σ ² − k ² _p ) ² + 4k _i ²

.

^MR O

• Y!}xz@e]TiY<E}jCW}lkqW{dY©e]TWxze e|TiY;4s|cfeP{dY<lkxIa

T 1

TWhk}~T@hklkl8{iY<cfe|xzgWh–lkh–˜<Ye|TiY }l–j:cfY6{¤l–jbjCy@h–lkl8g4Y

{iY;WiY6{/g:a

cos(ωT 1 ) = Re {− iω(iω − σ)

k p iω + k i } = k i ω ² + k p σω ² k ² _i + k ² _p ω ²

xmW{

sin(ωT 1 ) = Im { iω(iω − σ)

k p iω + k i } = ω ³ k p − σωk i

k _i ² + k _p ² ω ²

MQ‚

O

xmW{#¡<e]TiYGs]jbjze

ω

gBYhkin%qiihpo:qiYm¡IYJTWx&8mYŽe|TWx’e ™›jCs

T > T 1

e|TiYG}lkjCc]Y<{Plkjbjmyh–lklmgBYOqi4cue~xzgilkYmž‡SUTWhkc‡hkc

gBY<}<xzqWc]Y M«c]YY

@

‚>Aiy4xznmYUvm‚

O

h–™

W (ω ² ) = 0

TWxCcWj s|YyBY<xze]Y<{©s]jbjme|c‡e]TiYcfe|xzgWh–lkh–˜<h–Wn xz4{©{dY6cue~xzgihkl–hk˜hkin s|jbjze~cxmle|Ys|Wx’e|YGxzW{ xmc

W (ω ² ) > 0

™›jmsŽlkxms]nCY

ω

e]TWYšTihknmTWY<cfeŽs|jbjzehpc8xzlkxIadc{dY<cfe|xmgih–lkhk˜hkinWž[W{dYY6{#¡

e|TiYs|Yhkc%ijcfe|xmgih–lkh–eua h–4{dj’%c dyWTiYijCX!YijC ž

R O [ )e|Tihpc%}xCcfYC¡

A(s) = s(s − β )

^xzW{

C(s) = (s + α)(k p s + k i )

^c]je]TWxze

W (ω ² ) = (1 − k _p ² )ω ⁴ + (β ² − α ² k ² _p − k _i ² )ω ² − α ² k _i ² .

^M O Jj’¡be]TiY{iYijCXhkWxze]jmsUjm™e|TiY}lkjCc]Y<{)l–jbjCy)hpcUY<o:qWxmlHe]j

(1 + k p )s ² + (β + k i + k p α)s + αk i

STS

´U2V'V'W'X

SUTWY}l–j:cfY6{Œlkjbjmy/g4Y<h–Wn cue~xzgilkYx’e

T = 0

¡de|TiYs|YY7dhpcue~c

k p

xzW{

k i

c]xze]hpcu™›abhkin







1 + k p > 0 β + k i + k p α > 0 αk i > 0

jms







1 + k p < 0 β + k i + k p α < 0 αk i < 0

[™

− 1 < k p < 1

^e]TiY<

1 − k _p ² > 0

^xmW{

1 k p

> 1

^{c]j e]TWxze}

W (ω ² ) > 0

™›jms"lpxzs|nmY

ω

^xmW{e]TiY

X!jd{dqilkqWcjm™Ge]TWY }jbYC }hkY:ejz™še]TiY"TWh–nCTiY<cfe{dYnCs]Y<Y©e|Ys|Xjz™

A(s)

hpcnms|Y<x’e|YsPe]T4xz…e]TiY"X!jd{dqilkqWc jm™Je]TiY}jbYC }h–Y<Ce jz™Je]TWY©TihknmTiY6cue{dYnCs]Y<Ye]Y<s]X jz™

C(s)

Tihp}~T@hkXyWl–hkY<ce]TWxze

[G, K ]

hkcPcfe|xzgWl–Y™›jCs c]qDC }hkY:e|l–aŒc]X"xzlkl

T

^{MQcfY<Y M@‚>A}¡iyWxznCYvm‚

O O ž

Jj’ ™›s]jCX w%YX"xzs5vižvb¡bY}xmŒ}jCW}lkqW{dY%e]TWxzešY<o:qWxze]hkjm M O

T4xmcšjmilka jCiYyBjCc]h–e]h:8mY%s|jbjzešxmW{™›jms

e|TiY"c]xmX!Y s|Y<xCcfjC@xmcxzgBj8mY©e]Tihpcqiihpo:qiY!s]jbjzehpcP{iY<cfe|xzgWh–lkh–˜<h–Wn/xmW{¤™›jCs

T > T 1

e]TiY"}l–j:cfY6{¤l–jbjCy

hklkl g4YqWWcue~xzgilkY M›TiY<s]YxmnCxzhk)e]TiY<s]Y}xmiijmeUgBYxzba hkW{dj’ yWTiYijCX!YijC dTiYs|Y

O ž

[™

k p < − 1

^jms

k p > 1

^e]TiY<

W (ω ² ) < 0

™›jmslpxzs|nmY

ω

^xmW{

[G, K]

hkcUqWWcue~xzgilkY™›jms%xml–l

T

^ž

)£ ‘d !

# 0 !! ' '

'#&'!("

G(s) = e ^−sT λ ^s−α _s−β

^{3 2 ' -}

'#&'!("

G(s) = ^λe _s−σ ^{− sT}

¹

λ > 0

⁶ !6&' Q# 6!3!3)26!3 # ! +,

G(s)

¹

λ ⁻¹ G(s)

³

K(s)

^#

λK (s)

^{1D1 3L>!'} <12-,-!"12!%! ' ' # 2! '! 1 ! :&!63 >

# 1 !$6&!

σ = 0

^{L6!" !6 !63$ %' 0 ! - )}

# /

G(s) = e ^−sT q(s)

1

deg q(s) = 2

^{! 6 12! ' # 3L 21 !($!1#!6!}

. 2&' ' 3 !( '

%(' )

#&

. '

". ( '

! 3. / 6% ! 6 12 ! L)+' ! &!

6( '( ! ! ') 6

'Œ8 /0 8 80)0 0) % "#Ž%$&

• Yc]TWxzlkl }jCWcfhp{dY<shk/e]Tihpc%c]Y<}e]hkjmcfadcfe]YX"cUh–e]T/e]s~xz4cu™›Y<sU™›qiW}¬e|h–jC

G(s) = 1

p(s) + q(s)e ^−sT ,

h–e]T

T > 0

^xz4{

deg p ≥ deg q

^žJ[™

deg p = deg q

e|TiYcfadcfe]Y<Xhkc%jz™WYqde|s|xmlHeua:yBYm¡Wjze]TWYs|hkc]Yhe%hpc jm™s|Ye~xzs~{dY<{Œeuaby4YCž

9S

"! #$%#&'!("*)+'-,$.0/($!123

? nCxmh–/šYPe~x5mY

K(s) = k p + k i /s

=de|TiY/e]TWYe]Tis|YY}lkjCc]Y<{

—

lkjbjmy e|s|xmWcf™›YsU™›qiW}e]hkjmWcxms]YnCh8CY/g:a

(I + K(s)G(s)) ⁻¹ = s(p(s) + q(s)e ^−sT ) s(p(s) + q(s)e ^−sT ) + (k i + k p s) , G(s)(I + K(s)G(s)) ⁻¹ = sr(s)

s(p(s) + q(s)e ^−sT ) + (k i + k p s) , K(s)(I + G(s)K(s)) ⁻¹ = (p(s) + q(s)e ^−sT )(k i + k p s)

s(p(s) + q(s)e ^−sT ) + (k i + k p s) .

• TWY }jCWcfhp{dY<s]hkin e]TiY@cfe|xmgihkl–h–eua jm™

G

^gbax }jCCe|s]jCl–lkYs

K

Tihp}~TTWxmc!gBYY< {dY<c]h–nCiY<{ ™›jms e]TiY qWW{dYlpxIamY6{cfadcfe]YX¡Ih–e‡hkcY8bhp{dY:e ™›s|jmXLe]TWY

•

xmle|jm+GDB@xzs~c]TWxzlkl’s]Y6cfqWle~c#e]T4x’e e|TiYšcfe|xmgih–lkh–eua yis|jmyBYs]e]hkY<c

jm™

(G, K)

xms]Y}jC:e]s|jmlkl–Y6{ gba"e]TWYs]Y6xzl#yBjCc]he|h8CY*8’xml–qiY6cUjz™

ω ²

M›h–™Oxzba

O

e|TWx’e%c|x’e|hkcf™›a

W (ω ² ) := A(iω)A( − iω) − C(iω)C( − iω) = 0,

TWYs|Y

A(s) = sp(s) + (k i + k p s)

^xzW{

C(s) = sq(s)

^ž

 ¨G¨ « ¨! !

G(s) = 1

p(s) + q(s)e ^−sT

1 !6 96! !2 3

K(s) = k p + k i /s

1

k p

k i ∈ R

#

/

deg p = deg q

³

| lim

|s|→∞ p(s)/q(s) | ≤ 1

^{12! 12!!" 2! 3( !26 H6% !}

' L Q-,

G

^{3"12! !}

[G, K]

H ∞

' !

#

/

deg p = deg q ≥ 1

³

| lim

|s|→∞ p(s)/q(s) | > 1

^{12! ! !'# . /F62 !}

K

^{1D1 ' !}

G

^12!

T = 0

^{9 $' L Q>!}

G

^12!

T

^{*') !2 Q#$'(}

!6! 1 ! 2 ' )D : 6&!

1 ! !

p(s) = αs + β

³

q(s) = γs + δ

^*

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k _i ² (α ² − γ ² ) < 0

^12!

K

^{' L Q>!}

G

T

#

/

p(s) = β

³

q(s) = δ

s

³

| (β + k p )/δ | > 1

^{12! !!'#.0/ 2 :!}

K

1D1 6L Q>!

G

^{1 !}

T = 0

^{' Q>!}

G

T

^/

| (β + k p )/δ | < 1

^{1 !}

[G, K]

)D26L ! "

T > 0

#&

/

deg p > deg q

^{12! ! !'# .0/ 2 :!}

K

^{1+61 6L Q>!}

G

^{1 !}

T = 0

^Q

' L Q>!

G

^{1 !}

T

^{6) !% Q#$'(}

!6! F 1 ! 2 ')D 6&!

1 ! !

p(s) = αs + β

³

q(s) = δ

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k ² _i α ² < 0

^{1 !}

K

^{' L Q>!}

G

^"

T

 ¨ ¨ M«h

O

SUTWhkcUhpc%x!cfyBY<}hkxml }xCcfYjz™8SUTiYjCs]Y<X

R

žk‰h–

@K A ž

M«h–h

O

_bqiyiyBjCc]Y

K

cfe|xmgihkl–hk˜Y6c

G

^xze

T = 0

^{ž Ye}

α 0

xmW{

γ 0

gBY e|TiY}jbYC }hkY:e~c©jm™e]TiYŒe]Ys|X"c©jm™

TWh–nCTiY<cfe{dY<nms|YY jm™

p(s)

^xzW{

q(s)

s|Y<c]y4Y6}¬e|h8CYlkamž…SUTWY ¡8e|TiY/}jbYC }hkY:e jm™e]TiY)e]Y<s]X jm™%Tih–nCTiY<cfe

STS

´U2V'V'W'X

{iYnms|YY jz™

W (ω ² )

hpcY<o:qWxmlHe]j

(α ² ₀ − γ ₀ ² )

^{cfj e]T4x’e}

W (ω ² )

hpcUyBjCc]he|h8CY ™›jCsUlkxms]nCY

ω

^žJ[ W{dY<Y<{)šYxzlpc]j T4x&8mY!e]T4x’e©e|TiY)}jbYC }hkY:e©jm™e|TiYŒTWh–nCTiY<cfe {dY<nms|YY"e|Ys|X hk

A

^¡

α 0

¡8hpcjm™%Xjd{dqWl–qWc©nms|Y<xze]Yse]TWxm

e|TiY!}jbYC }hkY:ejz™Oe|TiY!TihknmTiY6cue{dYnCs]Y<Ye|Ys|X h–

C

^¡

γ 0

¡‡xz4{@Y }xm…{dY6{dqW}Y©™›s|jmX M@

‚A>¡#yWxmnmY"vm‚

O

e|TWx’e

[G, K]

hkccfe|xzgWl–Y™›jmsUcfqDC }h–Y<Ce|l–a c]X"xzlkl

T

^ž[ Œe]TiY yWxzs]e]hp}qilpxzs}xCcfY TiYs|Y

p(s) = αs + β

^xmW{

q(s) = γs + δ

¡Wx!cfe]s~xzhknmT:ef™›jCs]Uxzs~{"}xmlk}qilpx’e]hkjm/nmh:8mY6c

W (ω ² ) = (α ² − γ ² )ω ⁴ + ( − δ ² + (β + k p ) ² − 2αk i )ω ² + k _i ² .

[™

| lim |s|→∞ p(s)/q(s) | >

^{‰¤¡ŽY 5:Wj’ ™›s|jmX} rOs]jCy4j:cfh–e]hkjm viž–‰"hk

@K

Aše]TWxzeh™e|TiY „@}lkjCc]Y<{

—

lkjbjmy

e|s|xmWcf™›Ys™›qiW}e]hkjmWcT4x&8mYij!y4jCl–Y6cUh–/e|TiYs]hknmT:e

—

TWxml™yilkxmiYe]TWY

[G, K ]

^hpc

H ∞

^—

cfe|xmgilkYmž

Jj’¡Oh–™

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k _i ² (α ² − γ ² ) < 0

¡Oe|TiYs|Y)hpc!ij y4j:cfh–e]h:8mYc]jmlkqde]hkjm ™›jms

W (ω ² ) = 0

MQcfY<YwYX"xms5Œvižv O

xzW{)e]TWY}lkjCc]Y<{ lkjbjmyhpccue~xzgilkYP™›jms%xzlkl

T

^ž

M«h–hkh

O [Œe]Tihpcš}<xmc]Y

A(s) = (β + k p )s + k i

¡

C(s) = δs

^¡

W (ω ² ) = ((β + k p ) ² − δ ² )ω ² + k ² _i

^{xzW{ e]TiY}

s|Y<c]qil–eU™›jmlkl–j’%cxCcšhkh–h

O ž

M«h8

O ? c

H ∞

^—

cue~xzgihklkheuajz™ s|Ye~xzs~{dY<{!c]abcfe]Y<X"cOhpcOY<o:qih:8’xzlkY:eOe|je]TiYP}jCW{dh–e]hkjm ijyBjmlkY<cOhk"e|TiYs|h–nCT:e

T4xzl–™

—

yilpxzWY k¡We|TiY • xzl–e]jC

—

B@xzs~cfT4xzlkl#e]Y6}~TiihpoCqWY<c%hkX!yilkae]TiY s]Y6cfqil–e<žPŸŽh–s~cue6¡Be]TiY h–D;4ihe|Y bqiX g4Y<s jm™

WY s|jbjze~cPxmyiyBY<xzs|hkin)TWY

T

hpcijze˜<Ys|j/xms]Yl–jd}<x’e]Y6{Ehk…e]TiY"lkY™ÁeTWxml™—

yilpxziYCž!_dY<}jCW{#¡ hk e]TiY

y4xzs]e]hp}qilpxzs}xmc]YPjz™

p

^xzW{

q

jz™{iYnms|YY jmiYC¡bheUhkcY6xmc]a e|jc]YY e]T4x’eUh™

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k _i ² α ² < 0

e|TiYs|YhkcPij/y4j:cfh–e]h:8mY!c]jmlkqde]hkjmE™›jms

W (ω ² ) = 0

xmW{¤e|TiY"}lkjCc]Y<{¤lkjbjmyEs|YX"xmh–WcPcfe|xzgWl–Y

™›jCs xml–l

T

ž!ŸŽh–Wxml–lkam¡BY xzyWyil–a/e|TiY xmg4j8CYs]Y6cfqWle~c%hkEjms~{dY<se]j)yis]j8bhp{dY©x)}~TWxzs~xm}e]Ys|hk˜<x’e|h–jC)jm™

e|TiY©c]YePjz™Ocue~xzgilkY©}jm:e]s|jmlkl–Y<s|cUhkEc]jmX!YhkX!y4jCsfe~xz:eY7ixmXyWl–Y6che|Tijmqie iY<Y<{dhkin"e|jŒ{dYe|Ys|X!h–iY©e]TiY

qWWcue~xzgilkYyBjmlkY<cOY7dyil–hp}h–e]lka M›hk"nmYWYs~xzlbe]TWYašjCqilk{!gBY%nCh8CY"xmc8e|TiYc]jmlkqde]hkjm4cJjz™ xPe]s~xz4c]}YW{dY<:e|xzl

Y6o:qWx’e|h–jC

O ž

£b¨

 £ "

!

G(s) = 1

αs + β + (γs + δ)e ^−sT

1

α

β

γ

δ ∈ R

) 2&!"12

| α | >

| γ |

^{5 12! 1 ! !<}

H ∞

'L Q-, 62 !6 F,&! #

V + M Q

U − N Q

1 ! !

N (s) = 1 s + 1

M(s) = (αs + β) + (γs + δ)e ^−sT s + 1

U (s) = s(s + 1)

((αs + β ) + (γs + δ)e ^−sT )s + k p s + k i

V (s) = (s + 1)(k i + k p s)

((αs + β) + (γs + δ)e ^−sT )s + k p s + k i

,

Q

^{$ !! ( ! ! "}

H ∞

3

k p

k i

6#1 ! 632

k p + β + δ > 0

k i > 0

³

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k _i ² (α ² − γ ² ) < 0

 ¨ ¨†_dqiyiyBjCc]Y ;4s|cfeUe]TWxze

γ 6 = 0

^xz4{

α γ

> 1

^ž

9S

[™

α > 0

^¡‡xCc

| ^α _γ | > 1

šY!TWx&8CY©e|TWx’e

α + γ > 0

xzW{E™›s|jmX xmg4j8CY Y"}xz }jCW}lkqW{dY!e|TWx’ee|TiYs|Y Y7dhkcfe|cxrO[}jC:e]s|jmlkl–Y<s

k p + ^k _s ⁱ

^Tihp}~T cue~xzgihklkh–˜<Y<c

G(s)

^TiY<

T = 0

¡‡yis]j8bhp{dY<{Ee]T4x’e

k p

xmW{

k i

c|x’e|hkcf™›a"e|TiY}jCW{dh–e]hkjmWc

k p + β + δ > 0

^xz4{

k i > 0

^ž

Jj’¡Be|x5bhkin

k p = − β + √

δ ² + 2αk i

M›Tihp}~T c]xze]hpc ;WY6c

k p + β + δ > 0

^O šYTWx&8mY©e]TWxze

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k ² _i (α ² − γ ² ) = − 4k _i ² (α ² − γ ² ) < 0

^xmW{ cfje|TiYs|Y xzlkUxIabcPY7bhpcfe|cxrO[

}jm:e]s|jmlklkYs

K 0

Tihp}~T/cfe|xmgih–lkhk˜Y<c

G(s)

™›jms%xzlkl

T > 0

^ž

[™

α < 0

¡ŽšY!TWx&8CY

α + γ < 0

xmW{…e|x5bhkin

k i < 0

^xzW{

k p = − β − √

δ ² + 2αk i

xzlklkj’%cqWce]j

jCgde|xmh–

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k ² _i (α ² − γ ² ) < 0

^ž

[ Ee]TiY!}xCcfY TiYs|Y

γ = 0

¡Hc]hkXhklpxzs}xmlk}qilkxze]hkjmWcyis|j8mYe|TWx’ee]TWYs|Y xzlkUxIabcY7dhpcue

k p

xzW{

k i

cfq4}~T

e|TWx’e

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k i α ² < 0

^ž

Jj’¡‡xmc

G

^hkc

H ∞

^—

cue~xzgihklkh–˜6xzgilkY h–eiY6}Y<c|c|xzs|h–lka¤xm{dX!h–e|cx}jCyis]hkX!Y©™«xC}¬e|jms|h–˜6x’e]hkjm…j8mY<s

H ∞

¡ e|TWx’e

hpc<¡Ce|TiYs|YY7dhpcue

N

^¡

M

^¡

X

^xmW{

Y ∈ H ∞

c]qW}~T e|TWx’e

G = N M

xzW{

M X + N Y = 1

žJSUTiYPc]Yejm™xzlkl cfe|xmgihkl–hk˜hkinŒ}jm:e]s|jmlkl–Y<s|cUjm™

G

hkc%e]TiY<@nmh:8mY<¤gbaŒe|TiYOjCqilkx yWxzs~xzX!Ye|s]hk˜<xze]hkjm

Y + M Q X − N Q

¡

Q ∈ H ∞

xmW{¤e|TiYs|Y!Y7dhpcue~c

Q 0 ∈ H ∞

c]qW}~TEe]TWxze

Y +M Q 0

X−N Q 0 = ^V _U = K 0

ž J%jze|Ye]TWxze

U

^xzW{

V

^{xzs|Y Z ˜<jmqde}

™«xC}¬e|jms~cxCc]c]jd}hpx’e|Y<{"e|j!e]TiY}jmyis|hkXYP™«xC}¬e]jCs]hk˜<xze]hkjm

(N, M)

xCcšY<l–l>ž

[ehkcšY<xCcfa e]j"8mY<s]h–™›a©e|TWx’e

(N, M ) = 1

s + 1 , (αs + β) + (γs + δ)e ^−sT s + 1

hpcšx }jCyis]hkX!Y™«xm}e]jCs]hk˜<x

—

e|h–jC©jm™

G

^j8CYs

H ∞

xmc

inf

{Re s>0} ( | N (s) | + | M(s) | ) > 0

^{xmW{hk @}‰<ˆAbYU}xzcfY<YGe|TWx’eŽe]TWYU}<xzlp}qilpx’e|h–jC

jm™ŽZ ˜<jmqdeU™«xC}¬e]jCs|cUgBY<}jmX!Y<cc]hkXyWl–Y<sjmW}YPjCiYxzlks]Y6xm{da 5:Wj’%cx!cfe|xmgih–lkhk˜hkin"}jm:e|s]jCl–lkYs6ž

• Y/TWx&8mYŒe]TWxze

1 1 + GK 0

= M U

c]j@e|TWx’e

U = s(s + 1)

((αs + β) + (γs + δ)e ^−sT )s + k p s + k i

=Uxzlpc]j

K 0

1 + GK 0 = M V

c]j©e|TWx’e

V = (s + 1)(k i + k p s)

((αs + β) + (γs + δ)e ^−sT )s + k p s + k i

ž • YTWx&8CY

( − δ ² + (β + k p ) ² − 2αk i ) ² − 4k _i ² (α ² − γ ² ) < 0

cfj)e]TWxze"¡‡xmchkEe|TiY!yis|j:jm™šjm™šrOs|jmyBjCc]he|h–jC…„Wž–‰m¡#e]TiY"Z ˜<jmqde

™«xC}¬e|jms~c

U

^xzW{

V

^l–hkY%hk

H ∞

žŽSUTiYOjmqilpxyWxms|xmXYe]s|h–˜6x’e|h–jC jm™#xml–lBcue~xzgihklkh–˜<h–in }jm:e|s]jCl–lkYs~c8jz™

G

^}xz

Wj’ gBYnmh:8mY)gba

V + M Q U − N Q

¡

Q ∈ H ∞

)£ ‘d "

/' 1 !$6&!

γ = 0

^{2 ($! '>}

H ∞

' +, 2 :!6*12&

!3#6!6! L!63 1 )L,>1 3!96 )D : F >)D 3L!2# #!6!% ) 1 !*L3,&2,-!

12!"6! !'')D Q ,&3 66 ! %! 6!63) !

/š '/‡

• Y%TWx&8CY%xzWxml–adc]Y<{ e]TWY}lkjCc]Y<{

—

lkj:jCycfe|xmgihkl–h–eua jm™ xhk{dY }jmlklkY<}¬e|h–jC!jz™ {dYlpxIa c]adcue|YX"cOh–e]T"s|Y<c]yBY<}¬e

e|jErO[}jm:e]s|jmlklkYs~cž¤ŸiqWsfe|TiYsY7be]Y<Wc]h–jCWcjm™e]TiY6cfY)hk{dY6xmc }jmqWlk{ gBYŒg4xmc]Y<{ jm XjbY/cfjCyiTihpcue|hk}<x’e]Y6{

STS

´U2V'V'W'X

e|Y<}~TiWhko:qiY6c™›jms{dYe|Ys|Xhkihkine]TWY y4j:cfh–e]h:8mYŒs]jbjze~cjz™U™›qiW}e]hkjmWc cfq4}~T xmc

W (ω ² )

xmg4j8CYm¡xmW{ e]TWhkc

jmqWlk{)TiYlkyqWcUqiW{iYs~cue~xzW{)g4Yefe|Yse|TiYyiTiY<ijmX!YWjm/jz™Jcue~xzgihkl–h–eua hkW{dj’%c<ž

\ WYs|YX"xzhkihkinhpc]c]qiYhpcGe]j©y4xzs~xzX!Ye|s]hk˜YUe|TiYPcfe|xmgihkl–hk˜hkin©}jm:e]s|jmlkl–Y<s|cOjz™‡x©{iYlpxIa!c]adcue|YX ™›jmsTihp}~T

jCiY¤{djbY<c!ijme"TWx&8mYxz Y7ixm}e 5:Wj’l–Y6{dnmY)jz™Pe]TiYqiWcfe|xmgilkY/yBjmlkY<c<ž M›[™Pe]TiY6cfY¤xms]Y 5bij’ ¡Oe|TiY

e|Y<}~TiWhko:qiY6cJcfqW}~T!xmce|TijCc]YUh–

@

v>AWxzW{

@

‰'Aiyis|j8bhk{dYxPcfjCl–qie]hkjm žO

SUTiY<jms|YX „Wž–‰Uxm}~TWh–Y8mY<ce|Tihkc8hk"cfjCX!Y

c]hkXyWl–YP}xmc]Y<c<¡mgbaqWc]h–in©e]TiY rO[

—

gWxmc]Y<{e|Y<}~Tiihpo:qiY<cš{dY8CYlkjmyBY<{!hk"e]TWhkcšyWxzyBYs6¡:giqdee]TiY nmYWYs~xzlW}<xmc]Y

hpc}lkY<xms]lkaŒx e]jCyihp}™›jCsU™›qis]e]TiY<s%s]Y6cfY6xzs~}~T ž

"/ 0/*%$&

SUTWY…xzqie]TijCs|c"jmqWlk{ lkh:5mY¤e|j e]T4xz 5 ?

lkgWxm qWxm{is|xze ™›jms)TiYlkyihkin e|TiYX&hkXyWs]j8bhkin xz Y6xzs|l–hkYs

8CYs~cfhkjm)jz™8SUTiYjCs]Y<X(„ižk‰zž

’ 8)

@

‰A  žŽZjmiiYe©xzW{ 4ž‡wžr8xmsfe|h–Wnze]jC ¡‡Z ˜<jmqde™«xC}¬e|jms~cxzW{

L ¹

^—jCyde]hkX"xzlG}jm:e]s|jmlklkYs~c™›jCs©{dY<lkxIa c]adcue|YX"cqWc]h–in x¤euj

—

yWxms|xmXYe]Y<s©}jCXyBY4c]xze]jms!c|}~TiYX!Ymž / 0 )D % RR

Mf‰ KKLK

O

¡#‰6‚i‰6v>GH‰6‚Cvd‰mž

@

vAwžmŸž šqis]e|xzhk ¡ž • Y<hkc|cJxzW{ B ž • Yhpc|c¡ šjmyis|h–X!Yš™«xm}e]jms|hk˜<x’e|h–jC™›jCsJs]Y<nmqilpxzs8lkh–iY6xzsJc]abcfe]Y<X"cž

0 )D ( 6 „Cv Mf‰

KLK

O

¡B‰I‚d‰

K

GB‰I‚z„W‰mž

@

„AwžGŸž šqis]e|xmh–xzW{ E©žBUxzs]e<¡40

%3)2 2! 3( !26 F Q!6 6#&'!("

1 !6'# ž4SY7be|cUhk ? yiyilkhkY<{ B@x’e|TiYX"xze]hp}c<¡4vd‰Cžd_dyis]hkinCYs

—

Y<s]lpxzn4¡-JY Ojms5B¡#‰

KLK

‚dž

@R A B ž—

SžDE%jW¡i_bab:e]TWY<c]hkcšjz™‡rG[ }jm:e|s]jCl–lkYs~cœx©yWxms|xmX!Ye]s|hp}%xzyiyWs]j:xm}~T ž 0 )D( 6 „ K MQvmˆmˆC„

O ¡

‰6ˆ

K

GB‰6ˆ

P

‚dž

@

‚A4ž%žB@xzs~c]TWxzlklQ¡E©žCs]Y6}65bhQ¡!ž • xmle|jm©xmW{ ? žjCs]a:e]j’%c 5bhQ¡ ($! 3L! # %#&'!(" ' L Q#

3 ! '( 6! '!'

1 Q62

žOlklkhkcFEjms|šjbjd{#¡H‰

KK

viž

@

ASž:SžPY<jms|nmhkjmq!xzW{ B ž  ž:_dXh–e]T‡ž#

—

cfe|xmgih–lkh–eua©jz™H™›Y<Y<{dgWxC}65 c]adcue|YX"cž #&6 !( 3 2

! !6 ‰<„ Mu‰

K ƒ K O

¡iv P ‰ — v

PP

ž

@P

A_#ž

—

[¬ž J%hp}qilkY<c|}q ¡ ! :# ! !6$' # $ L)+' 62 >L61

žŽ_byis|h–inCYs

— Y<s]lpxzn4¡

ZY<s]lkh–‡¡4vzˆCˆi‰mž

9S

@ƒA4žGwžšrJxzs]e]hkinze|jm ¡ ! 2! 6 3 Q ! '#&6!( ž xmX©gis|hk{inmY%ih:8mY<s|c]h–eua rOs|Y<c|c¡

vmˆmˆ

R ž

@K

A4žzwž’rJxzs]e]hkinme]jm xzW{  ž’ZšjCiiYe<¡

H ∞

xmW{©Z[uZ%\cfe|xmgih–lkhk˜<x’e|h–jC©jm™4{dYlpxIac]abcfe]Y<X"cjm™WiY<qde]s~xzl

euabyBYmž

#&6!( 3 % ! ! 6

‚mv MQvmˆmˆ

R O

¡ivzƒC„&GivzƒmƒWž

@

‰<ˆA ? ž  qWxm{ds~x’e6¡P\  x nmYWYs~xzlkh–˜6x’e]hkjm jm™e]TiY OjmqWlkxGqYs~x yWxzs~xzX!Ye|s]hk˜<xze]hkjm žšrJxzs]e)[¬œe]TiY

™›s~xm}e]hkjmWxmlUhk{dY6xzl%xmyiyis|jCxm}~T e]j _b[f_W\cfadcfe]Y<X!c<ž #&'!( 3 2 !!6 ‚zˆ MQvmˆmˆC„

O ¡

‰6„C‚

— ‰ R

ƒiž

@

‰m‰Až4ž _bh–l:8’xi¡ ? ž xzefe|x xz4{ _HžrŽžOZTWxzefe~xm}~TWxms]abaCxW¡9JY s|Y<c]qil–e|cjm e|TiY¤_babCe|TiY<c]hpc©jm™rO[

}jm:e]s|jmlklkYs~cž

/ 2

0

)D % R+P

MQvzˆCˆCv

O

¡iv

R

‰'GdvC‚mviž

@

‰6vA ž ž

xzlkY:e|h–iYxz4{ B žšTihp{ixzX gWxzs~xzX¡JrO[ }jm:e]s|jmlšjz™qWWcue~xzgilkY)e]hkXY{dY<lkxIa c]abcfe]Y<X"cž

12!( 6 -,L!6! '-, ("( )D26 ‰ v Mf‰

KK-P

O ¡ „&G P&R

ž

@

‰<„A • ž ž BTWxminxzW{©žB žq‡¡

H ∞

rO[ }jm:e]s|jmlkl–Y<sO{dY<c]hknm ™›jmsOs|qiWxIUxIayis]jd}Y<c|cfY6c8he|T e|h–X!Y

{iYlpxIamž /6 0 2

R ‰ MQvzˆCˆCv

O

¡d„i‰

P

Gb„:vmviž

@‰ R A !ž BTijCq ¡Wž  ž

j’abl–YC¡:xmW{ !žPlkj8mY<s<ž

)+' 3 ( 6%

ž#rOs|Y:e|hk}Y Exml–l>¡%yiyBYs

_ixm{i{dlkYwh:8mY<s<¡ J W¡#‰

KLK

ž

STS

´U2V'V'W'X

Unité de recherche INRIA Rocquencourt

Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399