On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems
We propose a general principle of comparison for stability-preserving mappings and establish sufficient conditions of stability for the Takagi – Sugeno continuous fuzzy systems.
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| Date: | 2009 |
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| Main Authors: | , , , , , |
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| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
2009
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3047 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509068897026048 |
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| author | Denisenko, V. S. Martynyuk, A. A. Slyn'ko, V. I. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. |
| author_facet | Denisenko, V. S. Martynyuk, A. A. Slyn'ko, V. I. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. |
| author_sort | Denisenko, V. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:07Z |
| description | We propose a general principle of comparison for stability-preserving mappings and establish sufficient conditions of stability for the Takagi – Sugeno continuous fuzzy systems. |
| first_indexed | 2026-03-24T02:35:14Z |
| format | Article |
| fulltext |
UDK 531.36
V. S. Denysenko, A. A. Mart¥ngk, V. Y. Sl¥n\ko
(Yn-t mexanyky NAN Ukrayn¥, Kyev)
OB OTOBRAÛENYQX, SOXRANQGWYX USTOJÇYVOST|
PO LQPUNOVU NEÇETKYX SYSTEM TAKAHY – SUHENO
The general comparison principle for stability preserving mappings is presented. Sufficient conditions of
the stability of the fuzzy continuous Takagi – Sugeno systems are established.
Navedeno zahal\nyj pryncyp porivnqnnq dlq vidobraΩen\, wo zberihagt\ stijkist\, ta vstanov-
leno dostatni umovy stijkosti neçitkyx neperervnyx system Takahi – Suheno.
1. Vvedenye. Odnym yz podxodov pry yssledovanyy neçetkyx system dyffe-
rencyal\n¥x uravnenyj qvlqetsq podxod, osnovann¥j na modely Takahy – Suhe-
no [1]. ∏ta model\ pozvolqet approksymyrovat\ neprer¥vnug vewestvennug
funkcyg g, opredelennug na zamknutom y ohranyçennom podmnoΩestve prost-
ranstva Rn
[2]. Prymenytel\no k dynamyke nelynejn¥x system upomqnut¥j
podxod pozvolqet approksymyrovat\ ysxodnug nelynejnug systemu s pomow\g
lokal\no lynejn¥x modelej v termynax predykatn¥x pravyl „esly-to”.
Obzor rezul\tatov po πtomu napravlenyg moΩno najty v stat\qx [3, 4], hde
ymeetsq obßyrnaq byblyohrafyq.
Aktual\noj zadaçej teoryy razr¥vn¥x dynamyçeskyx system qvlqetsq raz-
rabotka obwyx podxodov k yssledovanyg ustojçyvosty (sm. [5, 6]). Koncepcyq
matryçnoznaçn¥x otobraΩenyj, soxranqgwyx ustojçyvost\ dlq razr¥vn¥x
system, pozvolqet sformulyrovat\ unyfycyrovann¥j podxod y yssledovat\ us-
tojçyvost\ system Takahy – Suheno.
V dannoj rabote pryvodytsq obwyj pryncyp sravnenyq dlq otobraΩenyj,
soxranqgwyx ustojçyvost\, y rassmatryvagtsq neçetkye neprer¥vn¥e system¥
Takahy – Suheno. V kaçestve prymera yssleduetsq neçetkaq neprer¥vnaq syste-
ma Takahy – Suheno vtoroho porqdka.
2. Matryçnoznaçn¥e otobraΩenyq, soxranqgwye ustojçyvost\. Dyna-
myçeskug systemu budem opredelqt\ kak semejstvo dvyΩenyj, opys¥vaem¥x
sootvetstvugwej systemoj πvolgcyonn¥x uravnenyj. ∏volgcyq processa vo
vremeny T razlyçaetsq v zavysymosty ot sm¥sla T = R+ = 0, ∞[ ) (neprer¥v-
n¥j sluçaj) yly T = N = {0, 1, 2, … } (dyskretn¥j sluçaj). V metryçeskom
prostranstve ( , )X ρ lgb¥e dvyΩenyq system¥ opredelqgtsq naçal\n¥my us-
lovyqmy ( , )t a0 ∈ T × A, A X⊂ — otkr¥toe mnoΩestvo X. Dalee pryvedem ne-
kotor¥e neobxodym¥e ponqtyq y opredelenyq.
Opredelenye 1 [7]. Pust\ ( , )X ρ — metryçeskoe prostranstvo s podmno-
Ωestvom A X⊂ . OtobraΩenye p(⋅ ; a, t0) : Tt a0 , → X naz¥vaetsq dvyΩenyem
(pry uslovyy eho suwestvovanyq), esly ono opredelqetsq naçal\n¥my uslovyq-
my ( , )t a0 pry t ∈ t0 1, τ[ ) ∩ T = Tt a0 , y p t( 0 ; a, t0) = a , hde τ1 — koneçnoe
yly symvol beskoneçnosty.
Pust\ Λ =
( , ) ,a t A T a tT
0 0∈ × (∪ × a{ } × t0{ } → X) y mnoΩestvo S � p(⋅{ ; a,
t0) ∈ Λ p t( 0 ; a , t0) = a} predstavlqet semejstvo dvyΩenyj, tohda korteΩ
mnoΩestv y prostranstv (T, X, A, S) budem naz¥vat\ dynamyçeskoj systemoj.
Esly T = R+ y vse dvyΩenyq p S∈ neprer¥vn¥ po t, to dynamyçeskaq
systema ( R+ , X, A, S) neprer¥vna. V sluçae, kohda πlement¥ mnoΩestva S ne
qvlqgtsq neprer¥vn¥my, dynamyçeskaq systema qvlqetsq razr¥vnoj.
Pust\ ( , )X1 1ρ y ( , )X2 2ρ — nekotor¥e metryçeskye prostranstva s metry-
kamy ρ1, ρ2 y (T, X1
, A1
, S1
) — razr¥vnaq dynamyçeskaq systema. Predpolo-
Ωym, çto kakym-lybo sposobom postroeno matryçnoznaçnoe otobraΩenye [8]
© V. S. DENYSENKO, A. A. MARTÁNGK, V. Y. SLÁN|KO, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5 641
642 V. S. DENYSENKO, A. A. MARTÁNGK, V. Y. SLÁN|KO
U t p( , ) : T × X1 → X2
, (1)
hde U — (m × m)-matryçnoznaçnaq funkcyq s πlementamy u t pij ( , ), i, j = 1, 2, …
… , m, y dlq lgboho dvyΩenyq p(⋅ ; a, t0) ∈ S1 funkcyq q(⋅ ; b , t0) = U ⋅( ;
p(⋅ , a, t0)) s naçal\n¥m znaçenyem b = U t a( , )0 qvlqetsq dvyΩenyem, dlq ko-
toroho Ta t, 0
= Tb t, 0
, b A∈ 2 � X2 y E Eq p⊂ , hde Eq y Ep — mnoΩestva toçek
razr¥va dlq dvyΩenyj q y p sootvetstvenno.
Pust\ S2 oboznaçaet mnoΩestvo dvyΩenyj q, kotoroe opredelqetsq varya-
cyej naçal\n¥x znaçenyj ( , )t a0 ∈ T × A1. V πtom sluçae (T, X
2
, A
2
, S2
) qv-
lqetsq razr¥vnoj dynamyçeskoj systemoj, poroΩdaemoj dvyΩenyqmy q. Pry
πtom mnoΩestvo S2 opredelqetsq tak:
S2 = q b t q t b t U t p t a t p a t S b U t a( ; , ) ( ; , ) , ( ; , ) , ( ; , ) , ( ; )⋅ = ( ) ⋅ ∈ ={ 0 0 0 0 1 0 ,
T T a A t Tb t a t, , , ,
0 0 1 0= ∈ ∈ } . (2)
Krome toho, oboznaçym çerez M A1 1⊂ y M A2 2⊂ nekotor¥e mnoΩestva, ynva-
ryantn¥e otnosytel\no mnoΩestv S1 y S 2 sootvetstvenno. Pry πtom mno-
Ωestvo M2 opredelym formuloj
M2 = q X q U t p p M t T∈ = ∈ ∈{ }∗ ∗
2 1( , ) dlq nekotor¥x y . (3)
Takym obrazom, funkcyq (1) ynducyruet otobraΩenye mnoΩestva S1 v mno-
Ωestvo S2, kotoroe oboznaçym çerez � , t.He. S2 = �( )S1 . Sleduq [9], pryve-
dem nekotor¥e vspomohatel\n¥e opredelenyq.
Opredelenye 2. MnoΩestvo M A⊂ naz¥vaetsq ynvaryantn¥m otnosy-
tel\no mnoΩestva S yly, çto to Ωe samoe, para (S, M) ynvaryantna, esly
a M∈ vleçet p t( ; a, t0) ∈ M dlq vsex t Ta t∈ , 0
, t T0 ∈ , y vsex p(⋅ ; a,
t0) ∈ S.
Opredelenye 3. Para (S, M) naz¥vaetsq ustojçyvoj, esly ona ynvaryat-
na y dlq lgboho ε > 0 y t T0 ∈ suwestvuet δ = δ ε( , )t0 > 0 takoe, çto
ρ p t( ( ; a , t M0), ) < ε dlq vsex t Ta t∈ , 0
y vsex p(⋅ ; a , t0) ∈ S, kak tol\ko
ρ( , )a M < δ. Esly δ = δ ε( ), to para ( S, M) naz¥vaetsq ravnomerno ustoj-
çyvoj.
Opredelenye 4. Para ( S, M ) naz¥vaetsq asymptotyçesky ustojçyvoj,
esly ona ustojçyva y dlq lgboho t T0 ∈ suwestvuet η = η( )t0 > 0 takoe,
çto
lim ( ; , ),
t
p t a t M
→∞
( )ρ 0 = 0
dlq vsex p(⋅ ; a, t0) ∈ S, kak tol\ko ρ( , )a M < η.
Opredelenye 5. Para (S, M) naz¥vaetsq ravnomerno asymptotyçesky us-
tojçyvoj, esly ona ravnomerno ustojçyva, suwestvuet δ > 0 y dlq lgboho
ε > 0 suwestvuet τ = τ ε( ) > 0 takoe, çto ρ p t( ( ; a, t M0), ) < ε dlq vsex t ∈
∈ t Ta t∈{ , 0
: t t− 0 ≥ τ} y vsex p(⋅ ; a, t0) ∈ S, kak tol\ko ρ( , )a M < δ.
Opredelenye 6. Para ( S, M ) naz¥vaetsq πksponencyal\no ustojçyvoj,
esly suwestvuet α > 0 y dlq vsex ε > 0, t T0 ∈ suwestvuet δ = δ ε( ) > 0
takoe, çto ρ p t( ( ; a , t M0), ) < ε e t t− −α( )0
dlq vsex t Ta t∈ , 0
y vsex p(⋅ ; a ,
t0) ∈ S, kak tol\ko ρ( , )a M < δ.
Opredelenye 7 [9]. OtobraΩenye (1) soxranqet ustojçyvost\ ot razr¥v-
noj dynamyçeskoj system¥ (T, X
1
, A
1
, S1
) k razr¥vnoj dynamyçeskoj systeme
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
OB OTOBRAÛENYQX, SOXRANQGWYX USTOJÇYVOST| PO LQPUNOVU … 643
(T, X2
, A2
, S2
) s ynvaryantn¥my mnoΩestvamy M A1 1⊂ y M A2 2⊂ soot-
vetstvenno, esly U t p( , ) udovletvorqet uslovyqm:
1)HHynvaryantnost\ par¥ (S2
, M2
) vleçet za soboj ynvaryantnost\ par¥
(S1, M1);
2)HHustojçyvost\ opredelennoho typa par¥ (S2
, M2
) vleçet za soboj us-
tojçyvost\ toho Ωe typa par¥ (S1, M1
)
;
3)HHymeet mesto vklgçenye E Eq p⊂ dlq lgb¥x razr¥vn¥x dvyΩenyj q y
p, dopustym¥x dlq zadann¥x razr¥vn¥x dynamyçeskyx system.
Sformulyruem teper\ osnovnug teoremu obobwennoho pryncypa sravnenyq
dlq otobraΩenyj, soxranqgwyx ustojçyvost\.
Teorema 1 [8]. PredpoloΩym, çto razr¥vnoj dynamyçeskoj systeme ( R+ ,
X1
, A1
, S1
) postavlena v sootvetstvye razr¥vnaq dynamyçeskaq systema srav-
nenyq ( R+ , X2
, A2
, S2
) s pomow\g matryçnoznaçnoho otobraΩenyq U t p( , ) :
R+ × X1 → X2
. Pust\ suwestvugt zamknut¥e mnoΩestva M Ai i⊂ , i = 1, 2,
kotor¥e vmeste s funkcyej U t p( , ) udovletvorqgt sledugwym uslovyqm:
1)HHdlq mnoΩestv �( )S1 y S2 ymeet mesto vklgçenye �( )S1 � S2
;
2)HHsuwestvugt funkcyy sravnenyq ψ , ψ , prynadleΩawye klassu Xana K,
takye, çto
ψ ρ1 1( , )p M( ) ≤ ρ2 2U t p M( , ),( ) ≤ ψ ρ1 1( , )p M( ) (4)
pry vsex t T∈ y p X∈ 1, hde ρ1 , ρ2 — nekotor¥e metryky, opredelenn¥e
na prostranstvax X1 y X2 sootvetstvenno.
Tohda:
1)HHyz ynvaryantnosty par¥ (S2
, M2
) sleduet ynvaryantnost\ par¥ (S1
,
M1
)
;
2)HHyz ustojçyvosty, ravnomernoj ustojçyvosty, asymptotyçeskoj ustoj-
çyvosty, ravnomernoj asymptotyçeskoj ustojçyvosty par¥ ( S2
, M2
) s l e -
dugt te Ωe typ¥ ustojçyvosty par¥ (S1
, M1
)
;
3)HHesly v ocenke (4) ψ ρ1 1( , )p M( ) = a p1( (p, M b
1)) , hde a > 0, b > 0, to yz
πksponencyal\noj ustojçyvosty par¥ (S2
, M2
) sleduet πksponencyal\naq us-
tojçyvost\ par¥ (S1
, M1
)
.
Dokazatel\stvo. Dokazatel\stvo utverΩdenyq 1. Pust\ para (S2
, M 2
)
ynvaryantna. V πtom sluçae dlq lgboho a M∈ 1 y lgboho dvyΩenyq p( ⋅ ; a,
t0) ∈ S1 dvyΩenyq q(⋅ ; b, t0) = U t( , p(⋅ ; a, t0)) ∈ S2, hde b = U t a( , )0 . ∏to sle-
duet yz uslovyqH1 teorem¥ y opredelenyq otobraΩenyq �( )S1 . Dalee, razr¥v-
nost\ dvyΩenyq p(⋅ ; a, t0) ∈ S1 v toçkax Ep sleduet yz razr¥vnosty dvyΩe-
nyj q S∈ 2 v toçkax Eq , y pry πtom E Eq p⊂ . Krome toho, yz ynvaryantnosty
par¥ (S2
, M2
) sleduet, çto q(⋅ ; b, t0) = U t p( , ) ∈ M2 pry vsex t Tb t∈ , 0
= Ta t, 0
.
Poskol\ku mnoΩestva M1
, M2 zamknut¥ y v¥polnqetsq neravenstvoH(4), dvy-
Ωenye p(⋅ ; a, t0) prynadleΩyt M1 pry vsex t Ta t∈ , 0
. Otsgda sleduet ynva-
ryantnost\ par¥ (S1, M1).
Dokazatel\stvo utverΩdenyq 2. PredpoloΩym, çto para (S2
, M2
) ustoj-
çyva. V πtom sluçae, sohlasno opredelenyg ustojçyvosty, dlq lgboho ε2 > 0 y
t0 ∈ +R suwestvuet δ2 = δ ε2 0 2( , )t > 0 takoe, çto ρ2 q t( ( ; b, t0) , M2) < ε2 pry
vsex q(⋅ ; b, t0) ∈ S2 y vsex t Tb t∈ , 0
, kak tol\ko ρ2 2( , )b M < δ ε2 0 2( , )t .
Poskol\ku para (S2
, M2
) ustojçyva, dlq lgboho ε1 > 0 y lgboho t0 ∈ +R
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
644 V. S. DENYSENKO, A. A. MARTÁNGK, V. Y. SLÁN|KO
v¥berem ε2 = ψ ε( )1 y δ1 = ψ δ−1
2( ) . Esly teper\ predpoloΩyt\, çto ρ1(a,
M1) < δ1, to sohlasno (4) poluçym
ρ2 2( , )b M ≤ ψ ρ1 1( , )a M( ) < ψ δ( )1 = ψ ψ δ−( )1
2( ) = δ2,
t.He. ρ2 2( , )b M < δ2. Otsgda sleduet, çto dlq vsex dvyΩenyj q(⋅ ; b , t0) ∈ S2
verna ocenka ρ2 2( , )q M < ε2 pry vsex t Tb t∈ , 0
. Vozvrawaqs\ vnov\ k ocenkam
(4), naxodym, çto pry vsex p(⋅ ; a, t0) ∈ S1 y pry vsex t Ta t∈ , 0
= Tb t, 0
, hde b =
= U t a( , )0 , v¥polnqgtsq neravenstva
ρ1 1( , )p M ≤ ψ ρ− ( )( )1
2 2U t p M( , ), < ψ ε−1
2( ) = ψ ψ ε− ( )1
1( ) = ε1,
t.He. ρ1 1( , )p M < ε1, kak tol\ko ρ1 1( , )a M < δ1. Otsgda sleduet, çto para (S1,
M1) ustojçyva.
Yzvestno, çto dvyΩenye system¥ asymptotyçesky ustojçyvo, esly ono us-
tojçyvo y prytqhyvagwee. PredpoloΩym, çto para (S2
, M2
) prytqhyvagwaq.
V πtom sluçae dlq lgboho t0 ∈ +R suwestvuet ∆2 = ∆2 0( )t takoe, çto pry vsex
q(⋅ ; b, t0) ∈ S2 ymeet mesto predel\noe sootnoßenye
lim ( ; , ),
t
q t b t M
→∞
( )ρ2 0 2 = 0,
kak tol\ko ρ2 2( , )b M < ∆2. Yn¥my slovamy, dlq lgboho ε2 > 0 suwestvuet τ =
= τ ε( 2 , t0
, q) > 0, q = q(⋅ ; b , t0) ∈ S2
, takoe, çto ρ2 2( , )q M < ε2 pry vsex
t Tb t∈ +, 0 τ , kak tol\ko ρ2 2( , )b M < ∆2
. Sohlasno uslovyg 1 teorem¥ 1, dlq
lgboho dvyΩenyq p(⋅ ; a, t0) ∈ S1 poloΩym b = U t a( , )0 . Tohda q(⋅ ; b, t0) =
= U t( , p) ∈ S2. Dalee, dlq lgboho ε1 > 0 v¥berem ε2 = ψ ε( )1 y poloΩym ∆1 =
= ψ−1
2( )∆ . Pry πtom dlq lgboho dvyΩenyq p S∈ 1 poluçym
ρ2 2( , )b M ≤ ψ ρ1 1( , )a M( ) < ψ( )∆1 = ψ ψ−( )1
2( )∆ = ∆2
,
t.He. ρ2 2( , )b M < ∆2
, kak tol\ko ρ1 1( , )a M < ∆1 y t Ta t∈ +, 0 τ = Tb t, 0 + τ . Sledo-
vatel\no, ρ2 q t( ( ; b, t0) , M2) < ε2 pry vsex t Tb t∈ +, 0 τ . Vozvrawaqs\ k ocenke
(4), naxodym
ρ1 1( , )p M ≤ ψ ρ− ( )1
2 2( , )q M < ψ ε−1
2( ) = ψ ψ ε− ( )1
1( ) = ε1,
t.He. ρ1 1( , )p M < ε1. Poπtomu para (S1
, M1
) prytqhyvagwaq. Takym obrazom,
esly para (S2
, M2
) asymptotyçesky ustojçyva, to para (S1
, M1
) takΩe asymp-
totyçesky ustojçyva.
Dokazatel\stvo utverΩdenyj o ravnomernoj ustojçyvosty y ravnomernoj
asymptotyçeskoj ustojçyvosty provodytsq po toj Ωe sxeme, no s tem otlyçyem,
çto velyçyn¥ δ2, ∆2 v¥byragtsq nezavysym¥my ot t0 ∈ +R .
Dokazatel\stvo utverΩdenyq 3. PredpoloΩym, çto para (S2
, M2
) πkspo-
nencyal\no ustojçyva. V πtom sluçae suwestvuet α2 > 0 y dlq lgboho ε2 > 0
suwestvuet δ2 = δ ε2 2( ) > 0 takoe, çto dlq lgb¥x dvyΩenyj q(⋅ ; b , t0) ∈ S2
pry vsex t Tb t∈ , 0
ymeet mesto ocenka
ρ2 0 2q t b t M( ; , ),( ) < ε α
2
2 0e t t− −( )
,
kak tol\ko ρ2 2( , )b M < δ2
. Sohlasno uslovyg 1 teorem¥, dlg lgboho p S∈ 1
suwestvuet dvyΩenye q(⋅ ; b, t0) = U t p( , ) ∈ S2
, hde b = U t a( , )0 . Dalee, dlq
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
OB OTOBRAÛENYQX, SOXRANQGWYX USTOJÇYVOST| PO LQPUNOVU … 645
lgboho ε1 > 0 v¥berem ε2 = a bε1 . Pust\ α1 = α2 /b y δ1 = ψ δ−1
2( ) . Dlq dvy-
Ωenyj p t( ; a, t0) ∈ M1, dlq kotor¥x ρ1 1( , )a M < δ1, sohlasno ocenke (4) po-
luçym
ρ2 2( , )b M ≤ ψ ρ1 1( , )a M( ) < ψ δ( )1 = ψ ψ δ−( )1
2( ) = δ2,
t.He. ρ2 2( , )b M < δ2. Sledovatel\no,
ρ2 0 2q t b t M( ; , ),( ) < ε α
2
2 0e t t− −( )
pry vsex t Tb t∈ , 0
. Sohlasno predpoloΩenyg teorem¥, v ocenke (4) sleduet
poloΩyt\ ψ ρ1 1( , )p M( ) = a p p M b( , )1( ) , hde a > 0, b > 0. Poπtomu
ρ1 1( , )p M ≤
ε
α
2
1 2
0
a
e
b
b
t t
− −/ ( )
= ε α
1
1 0e t t− −( )
pry vsex t Ta t∈ , 0
. Takym obrazom, para (S1, M1) πksponencyal\no ustojçyva.
3. Ustojçyvost\ neçetkoj system¥ Takahy – Suheno. Pust\ v fazovom
prostranstve Rn
zadan¥ neçetkye mnoΩestva Mi s funkcyqmy prynadleΩ-
nosty µi : Rn → [0, 1], i = 1, 2, … , r. Otnosytel\no funkcyj prynadleΩnosty
dopolnytel\no predpoloΩym, çto µi x( ) ≥ 0, i = 1, 2, … , n, y
i
n
i x=∑ 1
µ ( ) > 0
pry vsex x n∈R .
Neçetkaq systema Takahy – Suheno, opys¥vaemaq predykatn¥my pravylamy
„esly-to”, formalyzuetsq sledugwym obrazom:
esly x Mi∈ , to Si , i = 1, 2, … , r,
hde x n∈R — fazov¥j vektor system¥, — nekotoraq dynamyçeskaq (vozmoΩno,
razr¥vnaq) systema. Zametym, çto kaΩdomu pravylu stavytsq v sootvetstvye
dynamyka lokal\no lynejnoj podsystem¥ v obwej modely system¥ takym obra-
zom, çto podsystem¥ qvlqgtsq nezavysym¥my.
V πtoj stat\e rassmatryvaetsq sluçaj, kohda Si , i = 1, 2, … , r, — lynejn¥e
neprer¥vn¥e dynamyçeskye system¥.
Pust\ systema Takahy – Suheno opys¥vaetsq naborom neçetkyx predykatn¥x
pravyl Ri , i = 1, 2, … , r:
esly x Mi∈ , to dx
dt
= A xi , (5)
x t( )0 = x0, i = 1, 2, … , r,
hde x t n( ) ∈R pry vsex t ∈R+ , Ai , i = 1, 2, … , r, — postoqnn¥e (n × n)-mat-
ryc¥.
B πtom sluçae polnaq dynamyka neçetkoj system¥ Takahy – Suheno opys¥-
vaetsq nelynejnoj systemoj
dx
dt
= 1
1 1i
r
i i
r
i i
x
x A x
= =∑ ∑
µ
µ
( )
( ) . (6)
Pust\ Ω ⊂ Rn
— otkr¥taq svqznaq okrestnost\ sostoqnyq ravnovesyq x = 0
system¥ (6). Ustojçyvost\ sostoqnyq ravnovesyq x = 0 nelynejnoj system¥ (6)
ponymaetsq v klassyçeskom sm¥sle.
Pust\ E — prostranstvo symmetryçn¥x (n × n)-matryc, K E⊂ — konus po-
loΩytel\no poluopredelenn¥x symmetryçn¥x matryc, B : E → E , BX =
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
646 V. S. DENYSENKO, A. A. MARTÁNGK, V. Y. SLÁN|KO
= ( )tr X I , hde tr X — sled matryc¥ X, I — edynyçnaq matryca. Pust\ F : E →
→ E — lynejn¥j operator.
Opredelenye 8. Postoqnnaq γ ≥ 0 naz¥vaetsq konstantoj pozytyvnos-
ty operatora F : E → E otnosytel\no konusa K , esly operator F + γ B
qvlqetsq poloΩytel\n¥m otnosytel\no konusa K.
PokaΩem, çto dlq operatora F X = A X + X AT
suwestvuet konstanta po-
zytyvnosty γ. Oboznaçym çerez Γ pereseçenye konusa K y edynyçnoj sfer¥
Sn n( )/−1 2 � Rn n( )/+1 2
. MnoΩestvo Γ est\ kompakt y funkcyq g(X , Y) =
= − +tr tr
tr tr
( ) ( )AXY X A Y
X Y
T
neprer¥vna na Γ × Γ∗ , poπtomu dostyhaet na πtom
kompakte supremuma γ ≥ 0, t.He. − +tr tr
tr tr
( ) ( )AXY X A Y
X Y
T
≤ γ. Otsgda tr ( )AXY +
+ tr ( )X A YT + γ tr X tr Y ≥ 0 pry vsex X ∈K , Y ∈ ∗K . Takym obrazom, F + γ B ≥
K
≥
K
0.
Teper\ v¥qsnym, yz kakyx uslovyj moΩno najty konstantu pozytyvnosty γ
operatora F. Yz opredelenyq qsno, çto esly proyzvol\naq matryca X prynad-
leΩyt konusu K poloΩytel\no poluopredelenn¥x symmetryçn¥x matryc, to
y matryca
G = ( )F B+ γ X = A X + X AT + γ( )tr X I
dolΩna prynadleΩat\ konusu K. Sleduet otmetyt\, çto, blahodarq strukture
konusa K [10], dostatoçno delat\ proverku lyß\ dlq matryc vyda X = xxT
,
x
n∈R .
Pust\
A =
a a a
a a a
a a a
n
n
n n nn
11 12 1
21 22 2
1 2
…
…
… … … …
…
,
tohda dlq matryc X = xxT
, x = (x1, … , xn
T) , poluçym
G = ( )F B+ γ xxT = A xxT + xx AT T + γ tr( )xx IT( ) .
Pust\
G x( ) =
g g g
g g g
g g g
n
n
n n nn
11 12 1
12 22 2
1 2
…
…
… … … …
…
,
tohda dlq toho, çtob¥ matryca G b¥la poloΩytel\no poluopredelennoj, ne-
obxodymo y dostatoçno, çtob¥ vse hlavn¥e mynor¥ πtoj matryc¥ b¥ly neotry-
catel\n¥, t.He.
∆1( )x = g11 0≥ , ∆2( )x = g g11 12 – g12
2 ≥ 0, … , ∆n x( ) = det G ≥ 0.
Yz πtyx neravenstv y nuΩno yskat\ konstantu pozytyvnosty γ. Otmetym, çto
mynor¥ ∆1
, ∆2
,H… , ∆n qvlqgtsq formamy sootvetstvenno vtoroho, çetverto-
ho,H… , 2n-ho porqdka ot x.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
OB OTOBRAÛENYQX, SOXRANQGWYX USTOJÇYVOST| PO LQPUNOVU … 647
Pust\ operator¥ Fi : E → E, Fi X = A Xi + X Ai
T
, i = 1, 2, … , r, a γ i — soot-
vetstvugwye konstant¥ pozytyvnosty operatorov Fi otnosytel\no K. Obo-
znaçym
wi = sup
( )
( )x
i
i
r
i
x
x∈
=∑Ω
µ
µ
1
, i = 1, 2, … , r, µ = inf
( )
( )x
i
r
i i
i
r
i
x
x∈
=
=
∑
∑Ω
1
1
γ µ
µ
.
Dlq toho çtob¥ prymenyt\ teoremu 1 s cel\g analyza ustojçyvosty sostoq-
nyq x = 0 system¥ (6), formalyzuem nekotor¥e ponqtyq yz p. 2 dlq system¥ (6).
Prostranstvo ( , )X1 1ρ =∆ Rn( , ⋅ ) , hde ⋅ — evklydova norma vektora,
T = R+ y A1 = Rn
. Semejstvo dvyΩenyj S1 system¥ (6), opys¥vagwee dyna-
myçeskug systemu, poluçaetsq varyacyej naçal\noho znaçenyq x0 na A1 y t
na R+ . Sledovatel\no, ( R+ , Rn , A1
, S1
) qvlqetsq koneçnomernoj dynamyçes-
koj systemoj.
Sostavlqgwye dynamyçeskoj system¥ sravnenyq opredelqgtsq tak: T =
= R+ , ( , )X2 2ρ =∆ Rn n×( , ⋅ )E , hde A E = tr ( )A AT
, A2 = Rn n×
. Sledova-
tel\no, ( R+ , Rn n× , A2
, S2
) — dynamyçeskaq systema sravnenyq.
V kaçestve matryçnoznaçnoho otobraΩenyq U t p( , ) budem prymenqt\ pros-
tejßee U x( ) = xxT
, otobraΩagwee Rn
na hranycu konusa K. V πtom sluçae
matryçnoe uravnenye sravnenyq
dV
dt
=
i
r
i i
i
r
i iw w V
= =
∑ ∑+ −
1 1
F Bγ µ , V t( )0 = U0
, (7)
heneryruet semejstvo dvyΩenyj S2
, opredelqemoe sootnoßenyem (2).
Poskol\ku dlq system¥ (7) sostoqnyem ravnovesyq qvlqetsq U = 0, param
(S2
, M2
) y (S1
, M1
) sootvetstvugt par¥ (S2
, 0) y (S1
, 0). Nakonec zametym,
çto dlq system¥ (6) y system¥ sravnenyq (7) E Eq p⊂ , tak kak Eq = ∅, Ep ≠
∅.
Teper\ m¥ moΩem dokazat\ sledugwee utverΩdenye.
Teorema 2. Pust\ neçetkaq systema Takahy – Suheno (6) takova, çto v¥-
polnqetsq neravenstvo
max Re ( )
, ,j n
j
i
r
i i i
i
r
i iw A I I A w E
= … = =
∑ ∑⊗ + ⊗ + −
1 1 1
2
λ γ µ < 0,
hde λ j ( )⋅ , j = 1, 2, … , n2
, — sobstvenn¥e znaçenyq sootvetstvugwej matry-
c¥, E = eij i j
n{ } =, 1
2
,
eij =
1 1 1 1
0 1 1 1
, ( , ) ( , ) mod ( ),
, ( , ) ( , ) mod ( )
i j n
i j n
= +
≠ + ,,
ƒ — kronekerovo proyzvedenye. Tohda sostoqnye ravnovesyq x = 0 system¥H(6)
asymptotyçesky ustojçyvo.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
648 V. S. DENYSENKO, A. A. MARTÁNGK, V. Y. SLÁN|KO
Dokazatel\stvo. Dlq otobraΩenyq U : Rn → E , U x( ) = xxT , ymegt mes-
to neravenstva
dU
dt
=
1
1 1i
r
i i
r
i i
x
x U
= =∑ ∑
µ
µ
( )
( )F =
i
r
i i i i
r
i i
i
r
i
x U x U
x
= =
=
∑ ∑
∑
+ −
1 1
1
µ γ γ µ
µ
( )( ) ( )
( )
F B B
≤
≤
i
r
i i
i
r
i iw w U
= =
∑ ∑+ −
1 1
F Bγ µ .
Neobxodymo otmetyt\, çto mnoΩestvo reßenyj dyfferencyal\noho nera-
venstva
dU
dt
≤ F U, U t( )0 = U0 , hde F — lynejn¥j operator, obrazuet dyna-
myçeskug systemu. Dlq πtoho dostatoçno pokazat\, çto suwestvuet verxnee re-
ßenye πtoho neravenstva. Dejstvytel\no, vsledstvye matryçnoho pryncypa
sravnenyq suwestvuet nekotoroe reßenye V t+( , t0 , U0) uravnenyq
dV
dt
= F V
takoe, çto U t( , t0 , U0) ≤ V t+( , t0 , U0) ∀ >t t0 , hde U t( , t0 , U0) — reßenye
dyfferencyal\noho neravenstva
dU
dt
≤ F U. Poπtomu v kaçestve verxneho re-
ßenyq moΩno vzqt\ V t+( , t0 , U0).
Netrudno proveryt\, çto pry otobraΩenyy U x( ) = xxT
teoremaH1 ymeet
mesto. Tak, oçevydno, çto �( )S1 � S2, a v¥byraq
ψ ρ1 1( , )p M( ) = ψ x( ) = x 2
, ψ ρ1 1( , )p M( ) = ψ x( ) = x 2
,
ubeΩdaemsq, çto ψ ρ1 1( , )p M( ) ≤ ρ2 U( (t, p), 0) ≤ ψ ρ1 1( , )p M( ), tak kak ρ2 U( (t,
p), 0) = xxT
E
= x 2
. Teper\, sohlasno teoremeH1, moΩno utverΩdat\, çto
otobraΩenye U x( ) = xxT
soxranqet ustojçyvost\.
Vsledstvye svojstv kronekerov¥x proyzvedenyj uslovye teorem¥H2 haranty-
ruet v¥polnenye neravenstva
max Re
, ,j n
j
i
r
i i
i
r
i iw w
= … = =
∑ ∑+ −
1 1 1
2
λ γ µF B < 0.
Poπtomu sostoqnye ravnovesyq V = 0 matryçnoho uravnenyq sravnenyq (7)
asymptotyçesky ustojçyvo.
Poskol\ku sostoqnye ravnovesyq V = 0 matryçnoho uravnenyq sravnenyq (7)
asymptotyçesky ustojçyvo y konus K normal\n¥j, dynamyçeskaq systema, op-
redelqemaq dyfferencyal\n¥m neravenstvom
dU
dt
≤ F U, asymptotyçesky us-
tojçyva. Vsledstvye teorem¥H1 sostoqnye ravnovesyq x = 0 system¥ (6) takΩe
asymptotyçesky ustojçyvo.
Teorema 2 dokazana.
4. Prymer. Na ploskosty R2
vvedem sledugwye mnoΩestva: G 1 — ot-
kr¥taq pravaq poluploskost\ y G 2 — zamknutaq levaq poluploskost\. Opre-
delym neçetkye mnoΩestva M1 y M2 funkcyqmy prynadleΩnosty µ1( )x y
µ2( )x :
µ1( )x =
1
0 9
1
2
pry
pry
x G
x G
∈
∈
,
, ,
µ2( )x =
0 9
1
1
2
, ,
.
pry
pry
x G
x G
∈
∈
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
OB OTOBRAÛENYQX, SOXRANQGWYX USTOJÇYVOST| PO LQPUNOVU … 649
Rassmotrym neprer¥vnug neçetkug systemu Takahy – Suheno (6) so struk-
turn¥my matrycamy
A1 =
−
2 0 1
0 1 1
,
,
, A2 =
1 0 1
0 1 2
−
− −
,
,
. (8)
Netrudno ubedyt\sq, çto γ1 = γ2 = 4,01, w1 = w2 = 0,526, µ = 4,01 y
max Re ( )
, ,j
j
i
i i i
i
i iw A I I A w E
= … = =
∑ ∑⊗ + ⊗ + −
1 4 1
2
1
2
λ γ µ = – 0,63496 < 0.
Sledovatel\no, sostoqnye ravnovesyq x = 0 neçetkoj system¥ Takahy – Suheno
(6) so strukturn¥my matrycamy (8) asymptotyçesky ustojçyvo.
Otmetym, çto dlq yssledovanyq neçetkyx system (6) yzvesten kryteryj
asymptotyçeskoj ustojçyvosty, v osnove kotoroho leΩyt naxoΩdenye obwej
poloΩytel\no opredelennoj matryc¥, udovletvorqgwej matryçn¥m uravneny-
qm Lqpunova, no dlq suwestvovanyq takoj obwej matryc¥ neobxodymo, çtob¥
vse podsystem¥ b¥ly asymptotyçesky ustojçyv¥. V rassmatryvaemom prymere
matryc¥ A1 y A2 qvlqgtsq neustojçyv¥my, y poπtomu prymenyt\ πtot kry-
teryj nevozmoΩno, no s pomow\g teorem¥ 2 moΩno yssledovat\ systemu (6) na
ustojçyvost\, çto y b¥lo sdelano v¥ße. Takym obrazom, stabylyzacyq v neçet-
kyx systemax Takahy – Suheno vozmoΩna daΩe tohda, kohda lokal\n¥e podsys-
tem¥ neustojçyv¥e.
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ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 5
|
| id | umjimathkievua-article-3047 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:35:14Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0e/135823831c801e3fe4a06753ac45f50e.pdf |
| spelling | umjimathkievua-article-30472020-03-18T19:44:07Z On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems Об отображениях, сохраняющих устойчивость по Ляпунову нечетких систем Такаги - Сугено Denisenko, V. S. Martynyuk, A. A. Slyn'ko, V. I. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. We propose a general principle of comparison for stability-preserving mappings and establish sufficient conditions of stability for the Takagi – Sugeno continuous fuzzy systems. Наведено загальний принцип порівняння для відображень, що зберігають стійкість, та встановлено достатні умови стійкості нечітких неперервних систем Такагі - Сугено. Institute of Mathematics, NAS of Ukraine 2009-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3047 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 5 (2009); 641-649 Український математичний журнал; Том 61 № 5 (2009); 641-649 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3047/2843 https://umj.imath.kiev.ua/index.php/umj/article/view/3047/2844 Copyright (c) 2009 Denisenko V. S.; Martynyuk A. A.; Slyn'ko V. I. |
| spellingShingle | Denisenko, V. S. Martynyuk, A. A. Slyn'ko, V. I. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. Денисенко, В. С. Мартынюк, А. А. Слынько, В. И. On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems |
| title | On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems |
| title_alt | Об отображениях, сохраняющих устойчивость по Ляпунову нечетких систем Такаги - Сугено |
| title_full | On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems |
| title_fullStr | On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems |
| title_full_unstemmed | On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems |
| title_short | On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems |
| title_sort | on the mappings preserving the lyapunov stability of takagi–sugeno fuzzy systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3047 |
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