3URFHHGLQJVRIWKHQG&KLQHVH&RQWURO&RQIHUHQFH

-XO\;L
DQ&KLQD

ቅ⨹⻦᫔㌱㔕ᴿ䲆ᰬ䰪᧝࡬䇴䇗਀どᇐᙝ࠼᷆
嗏Ᲊߑθӄਂૂθ䠇ѳၒθᶒ֩⫆
བྷ䘎⎧һབྷᆖؑ᚟、ᆖᢰᵟᆖ䲒, བྷ䘎 116026
E-mail: shuanghe@dlmu.edu.cn

᪎ 㾷: ᴹ䲀ᰦ䰤っᇊᱟᤷ㌫㔏⣦ᘱ䖘䘩൘ᴹ䲀ᰦ䰤޵᭦ᮋࡠᵾ䳵Პ䈪ཛっᇊ⣦ᘱDŽᵜ᮷ѫ㾱⹄ウҶሿ⨳⻠ᫎ㜹ߢ㌫
㔏Ⲵᴹ䲀ᰦ䰤᧗ࡦ䰞仈DŽ俆‫ݸ‬ӻ㓽Ҷ䶎㓯ᙗ䘎㔝㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊᙗǃ␧ᵲࣘᘱ㌫㔏ԕ৺ᴹ䲀ᰦ䰤っᇊⲴᇊѹDŽ❦ਾ
䇮䇑Ҷሿ⨳⻠ᫎৼ〟࠶㌫㔏Ⲵᴹ䲀ᰦ䰤᧗ࡦಘˈᒦ࠶᷀ަっᇊᙗDŽᴰਾሩ㌫㔏䘋㹼ԯⵏ৺㔃᷌࠶᷀ˈ傼䇱Ҷᯩ⌅Ⲵᴹ᭸
ᙗDŽ

ީ䭤䈃: ሿ⨳⻠ᫎ㌫㔏ˈ␧ᵲࣘᘱ㌫㔏ˈ㜹ߢࣘᘱ㌫㔏ˈᴹ䲀ᰦ䰤っᇊᙗˈৼ〟࠶㌫㔏

Design and Analysis of Finite-time Controller of the Ball Impact System

LONG Xiaojun, YU Shuanghe, JIN Lina, DU Jialu
School of Information Science and Technology, Dalian Maritime University, Dalian 116026, P. R. China
E-mail: shuanghe@dlmu.edu.cn

Abstract: Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state
in finite time. In this paper, we study the finite-time control of the ball impact system. First, finite-time stability of nonlinear
continuous systems, hybrid dynamical systems and the definition of finite-time stability are mentioned. Then, we design the
finite-time controller and analyze stability of the impulsive double integral system of the ball impact system. Finally, we give
and analyze the simulation results of the system which verify the effectiveness of the finite-time controller.
Key Words: ball impact system, hybrid dynamical systems, impulsive dynamical systems, finite-time stability, double integral
systems

1 㔠䇰 ᵜ᮷俆‫ݸ‬ӻ㓽Ҷ␧ᵲࣘᘱ㌫㔏ԕ৺ᴹ䲀ᰦ䰤っᇊ
Ⲵᇊѹˈᒦ㔉ࠪҶ䶎㓯ᙗ䘎㔝㌫㔏৺㜹ߢࣘᘱ㌫㔏Ⲵ
䘁Ӌᒤˈ䶎㓯ᙗ㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊ䰞仈㻛ᒯ⌋
ᴹ䲀ᰦ䰤っᇊᙗᇊ⨶DŽ❦ਾ䪸ሩሿ⨳⻠ᫎ㜹ߢ㌫㔏䇮
⹄ウˈᴹ䲀ᰦ䰤っᇊᱟᵾ䳵Პ䈪ཛっᇊᒦф㌫㔏䖘䘩
䇑Ҷᴹ䲀ᰦ䰤᧗ࡦಘˈᒦ࠶᷀ަっᇊᙗDŽᴰਾሩሿ⨳
൘ᴹ䲀ᰦ䰤޵᭦ᮋࡠᒣ㺑⣦ᘱDŽᴹ䲀ᰦ䰤っᇊнӵ䇙
⻠ᫎ㌫㔏䘋㹼ԯⵏˈᒦо᮷⥞[13]Ⲵԯⵏ㔃᷌䘋㹼∄
㌫㔏ᴹᘛ䙏Ⲵ᭦ᮋᙗˈ㘼фሩ㌫㔏Ⲵн⺞ᇊᙗ઼ᒢᢠ
䖳DŽ
ᴹ䖳ྭⲴᣁࡦ֌⭘[1]DŽሩҾ䘎㔝ᰦ䰤ࣘᘱ㌫㔏ˈᴹ䲀
ᰦ䰤っᇊ᜿ણ⵰䶎LipschitzianࣘᘱˈӾ㘼ሬ㠤Ҷ৽ੁ 2 ะᵢ⨼䇰
ᰦ䰤䀓Ⲵнୟаᙗ[2]DŽ ❦㘼൘ᴹ䲀ᰦ䰤᭦ᮋⲴᛵߥ 2.1 䶔㓵ᙝ䘔㔣㌱㔕Ⲻᴿ䲆ᰬ䰪どᇐ
л‫؍‬䇱Ҷࡽੁᰦ䰤䀓ⲴୟаᙗDŽ᮷⥞[3, 4]㔉ࠪҶ൘
㘳㲁䶎㓯ᙗ㌫㔏˖
䶎Lipschitzian䘎㔝Ⲵᛵߥлࡽੁᰦ䰤䀓Ⲵୟаᙗᆈ൘
Ⲵ‫࠶ݵ‬ᶑԦDŽ䪸ሩ䘎㔝㠚⋫㌫㔏ˈ᮷⥞[5]㔉ࠪҶ㌫㔏 ẋ(t) = f (x(t), t), x(t) ∈ U, t∈I
ᴹ䲀ᰦ䰤っᇊⲴᗵ㾱‫࠶ݵ‬ᶑԦDŽ䙊䗷֯⭘к⺞⭼઼л
⺞⭼Ⲵᵾ䳵Პ䈪ཛ࠭ᮠሩ䇱᰾Ҷ䶎㓯ᙗᰦਈ㌫㔏Ⲵᴹ U ѪRn ѝⲴањ䛫ฏˈf : U × I −→ Rn ᱟањ䘎㔝࠭
䲀ᰦ䰤っᇊ[6]DŽ䪸ሩৼ〟࠶㌫㔏,᮷⥞[7]ӻ㓽Ҷа㊫ส ᮠˈᒦф┑䏣ᖃᡰᴹt ∈ I ᰦˈfq (0, t) = 0DŽ‫ٷ‬ᇊ㌫㔏Ⲵ
ҾĀᴹ䲀ᰦ䰤࠶⿫৏⨶āⲴ䗃ࠪ৽侸᧗ࡦಘDŽҼ䱦ਟ㿲 䀓ާᴹୟаᙗDŽ
⍻㓯ᙗ㌫㔏൘䘎㔝ᰦнਈᖒᔿлˈ࡙⭘喀⅑ᙗ䇮䇑Ҷ ᇐ⨼1[5]φᖃфӵᖃᆈ൘䘎㔝࠭ᮠV : U −→ R+ ᒦ
‫ޘ‬ተᴹ䲀ᰦ䰤㿲⍻ಘ[8]DŽ᮷⥞[9]䘀⭘৽↕⌅⹄ウҶ䶎 ф┑䏣ԕлᶑԦᰦˈᑖᴹf (x(t), t) = f (x(t))Ⲵ㌫㔏ᱟ
㓯ᙗ㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊDŽ࡙⭘㓸ㄟ━⁑᧗ࡦᯩ⌅⹄ ᴹ䲀ᰦ䰤っᇊˈᒦф䇮ᇊᰦ䰤࠭ᮠT0 (t0 , x0 )൘0༴ᱟ䘎
ウн⺞ᇊࣘᘱ㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊ[10]DŽ᮷⥞[11]࡙⭘ 㔝Ⲵ˖
Ҷ喀⅑ᙗ⹄ウҶа㊫н⺞ᇊᔰ‫ޣ‬㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊ (1)V ᱟ䘎㔝Ⲵ˗
઼励ἂっᇊDŽ䘀⭘ḷ䟿઼⸒䟿ᵾ䳵Პ䈪ཛ࠭ᮠ䀓ߣҶ (2)ᆈ൘ᇎᮠc > 0ˈα ∈ (0, 1)ˈ઼ањᔰ䛫ฏD ⊆
㜹ߢࣘᘱ㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊᙗ䰞仈[12]DŽᵜ᮷ѫ㾱 U ֯ᗇ˖V̇ (x(t)) + c(V (x(t)))α ≤ 0ˈx(t) ∈ D \ {0}ˈࡉ
1
⹄ウҶሿ⨳⻠ᫎ㜹ߢ㌫㔏[13]Ⲵᴹ䲀ᰦ䰤᧗ࡦ䰞仈DŽ ᴹT0 (t0 , x0 ) ≤ c(1−α) V (x(t0 ))1−α ˈx(t0 )Ѫx(t)Ⲵࡍ࿻
٬DŽ
↔亩ᐕ֌ᗇࡠ䗭ᆱĀⲮॳзӪ᡽ᐕ〻āษޫ㓿䍩䍴ࣙ(2012921079)˗ ᇐ⨼2[5]:‫ٷ‬ᇊᆈ൘䘎㔝࠭ᮠV : U −→ R+ ઼ањ
ഭᇦ㠚❦、ᆖส䠁䍴ࣙ亩ⴞ(51079013)˗䗭ᆱⴱᮉ㛢঵儈ㅹᆖṑ、⹄䍴
↓ᇊ࠭ᮠr : R −→ R+ ┑䏣ԕлᶑԦ˖
ࣙ亩ⴞ(LT2010013)˗Ӕ䙊䜘ᓄ⭘ส⹰⹄ウ亩ⴞ(2012-329-225-070)˗ѝ
ཞ儈ṑสᵜ、⹄ъ࣑䍩䍴ࣙ亩ⴞ઼བྷ䘎⎧һབྷᆖՈ⿰、ᢰࡋᯠഒ䱏ษ (1)V ᱟ↓ᇊⲴ˗
㛢䇑ࡂ䍴ࣙ亩ⴞ䍴ࣙ(3132013334). (2)V̇ (x(t), t) ≤ −rV (x(t), t)ˈ∀(x, t) ∈ U × I ˗

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 ε
(3)ᆈ൘ε > 0ˈ 0 (dz /r(z)) < +∞DŽ 3 ቅ⨹⻦᫔㌱㔕਀ᴿ䲆ᰬ䰪᧝࡬ಞ䇴䇗
ࡉ㌫㔏Ѫᴹ䲀ᰦ䰤っᇊDŽε
䇮ᇊᰦ䰤Ѫ˖ T0 (t0 , x0 ) =
∗ ∗
(t − t0 ) + 0 (dz /r(z))ˈt Ѫ㌫㔏ࡊ䗮ࡠ⑀䘁っᇊᰦ
Ⲵᰦ䰤ˈ⭡V (x(t∗ ), t∗ ) = ε⺞ᇊDŽ
к䘠ᇊ⨶Ѫ䶎㓯ᙗ䘎㔝㌫㔏Ⲵっᇊᙗˈ䘲⭘Ҿ␧
ᵲ㌫㔏൘䘎㔝䱦⇥っᇊᙗⲴࡔᇊˈնᱟ␧ᵲ㌫㔏Ⲵ⣦
ᘱՊ൘ⷜ䰤ਁ⭏ਈॆˈণ൘ḀӋᰦ࡫ᴹн䘎㔝⣦ᘱˈ↔
ᰦ䘎㔝㌫㔏Ⲵᴹ䲀ᰦ䰤っᇊᶑԦቡн䏣ԕᶕࡔᇊ␧ᵲ
㌫㔏ⲴっᇊᙗDŽഐ↔ˈ䴰㾱ሩк䘠䘎㔝㌫㔏Ⲵっᇊᙗ
䘋㹼ᢙኅDŽ
2.2 㜿ߨࣞᘷ㌱㔕Ⲻᴿ䲆ᰬ䰪どᇐ മ 1: ሿ⨳⻠ᫎ
␧ᵲࣘᘱ㌫㔏˖
ẋ(t) = fq(t) (x(t), t), x(t) ∈ Cq(t) , t ∈ I ྲമ1ᡰ⽪Ⲵሿ⨳⻠ᫎ㌫㔏ˈਟԕⴻ‫ڊ‬ањ䍘⛩Ⲵ

x(t+ ) = Fq(t) (x(t)) (1) ሿ⨳ਇ᧗ࡦ࣋֌⭘൘≤ᒣᯩੁк䘀ࣘˈ൘䘀ࣘⲴ䗷〻
, x(t) ∈ Dq(t)
q(t+ ) = Γ(q(t)) ѝՊоսҾ৏⛩༴Ⲵ඲ⴤ㺘䶒ਁ⭏⻠ᫎˈ൘┑䏣аᇊ
ᶑԦлՊਁ⭏৽ᕩDŽx1 ˈx2 ࠶࡛ԓ㺘ሿ⨳Ⲵս㖞઼䙏
ަѝˈI = [t0 , +∞)ˈt0 ≥ 0ˈx(t)ᱟ㌫㔏Ⲵ䘎㔝⣦ ᓖˈ㌫㔏ਟԕ㻛ㆰঅൠ㺘⽪Ѫ˖
ᘱˈq(t) ∈ Q = 1, 2, . . . , N ᱟ㌫㔏Ⲵ⿫ᮓ⣦ᘱˈᴹq ∈ 
QˈDq ⊆ Rn ˈCq ⊆ Rn ˈfq : Cq × I −→ Rn ˈᱟ䘎㔝 ẋ1 = x2
(5)
Ⲵ࠭ᮠ⸒䟿ˈ൘t ∈ I ᰦˈ┑䏣fq (0, t) = 0ˈFq : Dq −→ ẋ2 = uc − fc (x)
Rn ˈΓ : Q −→ Qˈx(t+ ) = limh→0+ x(t + h)ˈq(t+ ) = uc ԓ㺘䗃‫ˈ࣋ࡦ᧗Ⲵޕ‬fc (x)Ѫሿ⨳о඲ⴤ䶒᧕䀖ᰦⲴ
limh→0+ q(t + h)DŽ 䀖⛩঻࣋ˈfc (x)㺘⽪Ѫ˖
ᇐѿφ␧ᵲࣘᘱ㌫㔏ᱟᴹ䲀ᰦ䰤っᇊ˖ 
(1)ᵾ䳵Პ䈪ཛっᇊ˖ ∀t0 ∈ I,∀ε > 0ˈ ε > kc x1 + bc x2 , x1 > 0
fc (x) = (6)
0ˈq(t) ∈ Qˈᆈ൘δ(ε, t0 , q0 ) > 0ˈᖃt ≥ t0 ᰦˈ֯x0 ∈ 0, x1 ≤ 0
Bδ (0)ˈx(t) ∈ Bε (0)˗ kc > 0ˈbc > 0ˈ࠶࡛ԓ㺘ሿ⨳о඲ⴤ䶒᧕䀖ᰦⲴᕩᙗ
(2)ᴹ䲀ᰦ䰤᭦ᮋ˖ ∀t0 ∈ I ˈ q0 ∈ Qˈᖃx0 ∈ ઼䱫ቬ㌫ᮠDŽ
N (t0 , q0 )ᰦˈᆈ൘ᔰ䛫ฏN (t0 , q0 ) ⊆ C ˈᴹ˖ ྲ᷌ሿ⨳䘀ࣘࡠ඲ⴤ㺘䶒Ⲵ䙏ᓖབྷҾᡆ㘵ㅹ
(a)˖ᖃt ∈ I ᰦˈx(t)ᆈ൘˗ Ҿx2 > 0ቡՊਁ⭏⻠ᫎDŽᖃਁ⭏⻠ᫎᰦˈሿ⨳Պⷜ䰤
(b)˖ᆈ൘0 ≤ T (t0 , q0 , x0 ) < +∞ˈᖃᡰᴹt ≥ t0 + ৽ᕩˈ৽ᕩᶑԦѪ˖x1 ≥ 0фx2 ≥ x2 DŽ⻠ᫎਁ⭏ਾⲴ
T (t0 , q0 , x0 )ᰦˈx(t) = 0DŽ аⷜ䰤ˈ㌫㔏Ⲵ⣦ᘱਈ䟿Ⲵ٬Պਁ⭏ਈॆˈx+ +
1 ˈx2 Ѫ
T (t0 , q0 , x0 ) = inf{T (t0 , q0 , x0 ) : x(t) = ⻠ᫎਾ㌫㔏Ⲵ٬ˈᴹ˖
0, ᖃt ≥ t0 + T (t0 , q0 , x0 )}㻛ਛ‫␧ڊ‬ᵲ㌫㔏Ⲵ䇮ᇊᰦ  +
x1 = x1
䰤࠭ᮠDŽਖཆˈྲ᷌㌫㔏ᱟᴹ䲀ᰦ䰤っᇊᒦфCq =
x+
2 = −βx2
Rn ˈN (t0 , q0 ) = Rn ˈࡉ㌫㔏ᱟ‫ޘ‬ተᴹ䲀ᰦ䰤っᇊDŽྲ
᷌㌫㔏ᱟᴹ䲀ᰦ䰤っᇊˈࡉᱟ⑀䘁っᇊˈҏቡ䇱᰾Ҷˈ β ∈ [0, 1]ˈβ Ѫᚒ༽㌫ᮠDŽ
ᴹ䲀ᰦ䰤っᇊ∄⑀䘁っᇊᴤѕṬDŽ ሶк䘠㌫㔏߉ᡀ㜹ߢࣘᘱ㌫㔏Ⲵᖒᔿ˖
ᔿ(1)ᡰ⽪㌫㔏ˈᖃq(t)н䲿ᰦ䰤Ⲵਈॆ㘼ਈॆᰦˈ    
x˙1 x2
к䘠␧ᵲࣘᘱ㌫㔏ਟԕ㺘⽪Ѫ˖ ẋ = = , x∈C
x˙ u − fc (x)
 2 +  c  (7)
ẋ(t) = f (x(t), t), x(t) ∈ C, x1 x1
(2) x+ = = , x∈D
x(t+ ) = F (x(t)), x(t) ∈ D x+
2 −βx2

ަѝC, D ⊆ Rn ˈਟԕⴻࠪᖃq(t)‫؍‬ᤱнਈᰦˈ␧ᵲࣘ fc (x)ྲк䘠ᡰ⽪ˈ C = {x1 , x2 |x1 < 0ᡆ㘵x2 <

ᘱ㌫㔏ণѪ㜹ߢࣘᘱ㌫㔏ˈഐ↔ˈ㜹ߢࣘᘱ㌫㔏ᱟ␧ x2 }ˈD = {x1 , x2 |x1 ≥ 0фx2 ≥ x2 }DŽ
ᵲ㌫㔏Ⲵањ⢩ֻDŽ ᇐ⨼4φᔿ(7)ᡰ䘠Ⲵ㜹ߢࣘᘱ㌫㔏ˈԔx = x1 −
ᇐ⨼3[12]φ㘳㲁䶎㓯ᙗ㜹ߢࣘᘱ㌫㔏ˈ‫ٷ‬ᇊᆈ൘ x∗ ˈy = x2 ˈx∗ Ѫሿ⨳Ⲵっᇊս㖞ˈ䇮䇑ᴹ䲀ᰦ䰤᧗ࡦ
䘎㔝ਟᗞ࠭ᮠ˖V : Q −→ R+ ┑䏣V (0) = 0ˈV (x) > ಘѪ˖
0ˈx = 0ˈᒦф  α
−sgn(y)|y|α − sgn(x)|x| 2−α x1 ≤ 0
uc = α

V  (x)fc (x) ≤ −c(V (x))α , x∈C (3) −sgn(y)|y|α − sgn(x)|x| 2−α + fc (x) x1 > 0
(8)
V (x+ ) ≤ V (x), x∈D (4) ަѝα ∈ (0, 1]ˈsgn(.)Ѫㅖਧ࠭ᮠˈަᇊѹѪ
⎧
c > 0фα ∈ (0, 1)DŽ䴦䀓x(t) ≡ 0ᱟᴹ䲀ᰦ䰤っᇊDŽਖ ⎨ 1 z>0
ཆˈྲ᷌Q = Rn фV (·)ᱟᖴੁᰐ⭼ˈࡉ㌫㔏ᱟ‫ޘ‬ተᴹ sgn(z) = 0 z=0
⎩
䲀ᰦ䰤っᇊDŽ䇱᰾ྲ᮷⥞[12]ᡰ䘠DŽ −1 z<0

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 ൘䈕᧗ࡦಘⲴ֌⭘лˈሿ⨳⻠ᫎ㜹ߢ㌫㔏Պᴹ䲀ᰦ䰤 2 − α + 2−α
2 1
= |x | + (β)2 y 2 ≤ V (x, y)
っᇊࡠ(x∗ , 0)DŽ 2 2
䇷᱄φ⭡ᔿ(5∼8)ˈ㌫㔏ਟԕ㻛᧿䘠Ѫྲлৼ〟࠶ ণV (x(t+ ), y(t+ )) ≤ V (x(t), y(t))ˈ┑䏣ᇊ⨶3ᶑԦ2DŽ
㌫㔏Ⲵᖒᔿ˖  㔬кᡰ䘠ˈ൘᧗ࡦಘuc Ⲵ֌⭘лˈሿ⨳⻠ᫎ㌫㔏
ẋ = y Ѫᴹ䲀ᰦ䰤っᇊDŽ
ẏ = uc − fc (x)
4 Եⵕ㔉᷒਀࠼᷆
xˈy Ѫ㌫㔏Ⲵ⣦ᘱ䟿ˈuc Ѫ㌫㔏Ⲵ᧗ࡦ䗃‫ޕ‬DŽ
Ѫ䇱᰾㌫㔏Ⲵっᇊᙗˈ䇮䇑㌫㔏Ⲵᵾ䳵Პ䈪ཛ࠭ ሩሿ⨳ࣘᘱ㌫㔏䘋㹼ԯⵏˈ ਆkc = 8ˈ bc =
ᮠѪ˖ 10ˈx2 = 0.1ˈx∗ = 0.1DŽᖃα = 1/3ˈβ = 1ᰦˈ㌫㔏Ⲵ⣦
2 − α 2−α2 1 ᘱ䖘䘩ᴢ㓯ˈᵾ䳵Პ䈪ཛ࠭ᮠᴢ㓯઼᧗ࡦಘuc Ⲵ䗃ࠪ
V (x, y) = |x| + y2
2 2 䖘䘩࠶࡛ྲമ2∼4ᡰ⽪DŽᖃα = 1/3ˈβ = 1/2ᰦˈ㌫㔏
ࡉᴹ˖ Ⲵ⣦ᘱ䖘䘩ᴢ㓯ˈᵾ䳵Პ䈪ཛ࠭ᮠᴢ㓯઼᧗ࡦಘuc Ⲵ
V̇
α
= sgn(x)|x| 2−α · ẋ + y · ẏ 䗃ࠪ䖘䘩࠶࡛ྲമ5∼7ᡰ⽪DŽᖃα = 1ˈβ = 1ᰦˈ㌫㔏
α
= sgn(x)|x| 2−α y + y · (uc − fc (x)) Ⲵ⣦ᘱ䖘䘩ᴢ㓯ྲമ8ᡰ⽪DŽᖃα = 1ˈβ = 1/2ᰦˈ㌫
α α
= sgn(x)|x| 2−α y + y(−sgn(y)|y|α − sgn(x)|x| 2−α ) 㔏Ⲵ⣦ᘱ䖘䘩ᴢ㓯ྲമ9ᡰ⽪DŽ
= y · (−sgn(y)|y|α ) 㤕ሶ᧗ࡦಘ߉ᡀྲлᖒᔿ˖
= −|y|α+1 ≤ 0  α
−k1 sgn(y)|y|α − k2 sgn(x)|x| 2−α x1 ≤ 0
uc = α
⭡кᔿਟᗇ˖ −k3 sgn(y)|y|α − k4 sgn(x)|x| 2−α + fc (x) x1 > 0
2
V (k 2−α x, ky) = 2−α
2 |k
2−α 2−α
x| + 12 (ky)2 ᒦਆk1 = 2ˈk2 = 10ˈk3 = 12ˈk4 = 10ˈα = 1ˈ᧗ࡦ
2
= k ( 2 |x| 2−α + 12 y 2 )
2 2−α (9) ಘⲴᖒᔿѪ˖
= k 2 V (x, y) 
−2y − 10x x1 ≤ 0
uc =
V̇ (k 2−α x, ky) = −|ky|α+1 −12y − 10x + f x1 > 0
= k α+1 (−|y|α+1 ) (10)
= k α+1 V̇ (x, y) ণѪ᮷⥞[13]ѝⲴᔿ(11)ˈԯⵏ㔃᷌ྲമ14∼16ᡰ⽪DŽ
Ӿമ2઼മ14ਟԕⴻࠪˈമ2ᡰ⽪㌫㔏Ⲵ⣦ᘱ䖘䘩
ަѝk > 0,ഐѪ䳶v = {(x, y) : V (x, y) = 1}ᱟ
Ⲵጠ٬ᴤሿˈ⻠ᫎ⅑ᮠ䖳ቁˈ䇱᰾ᴹ䲀ᰦ䰤っᇊ᧗ࡦ
㍗䳶ˈഐ↔Ӿᔿ(9)ਟԕ᧘ሬࠪˈV ᱟᖴੁᰐ⭼ⲴDŽ৸
ಘⲴ᧗ࡦ᭸᷌ᴤྭDŽമ3ˈ6ᡰ⽪Ⲵᴹ䲀ᰦ䰤᧗ࡦ㌫㔏
ഐѪV̇ ᱟ䘎㔝ⲴˈᡰԕV̇ ൘㍗䳶v кਟԕ䗮ࡠᴰབྷ٬DŽ
Ⲵᵾ䳵Პ䈪ཛ࠭ᮠⲴ䖘䘩ˈ൘ᴹ㜹ߢⲴⷜ䰤ˈ࠭ᮠ٬
ᇊѹc = − max(x,y)∈v V̇ (x, y)ˈc > 0ˈ−V̇ ᱟ↓ᇊⲴ
нབྷҾࡽаᰦ࡫Ⲵ࠭ᮠ٬ˈ傼䇱Ҷᴹ䲀ᰦ䰤᧗ࡦಘ┑
ф(0, 0) ∈ vDŽ
1 䏣ᴹ䲀ᰦ䰤っᇊᙗⲴࡔᯝᶑԦDŽ
Ԕk = [V (x, y)]− 2 ˈ⭡ᔿ(9)ᗇ:
മ2઼മ5∄䖳ˈ മ8઼മ11∄䖳ˈ ࠶࡛Ѫα =
V (k 2−α x, ky) = k 2 V (x, y) 1/3઼α = 1ˈβ ٬н਼Ⲵ∄䖳DŽβ = 1/2ᰦˈሿ⨳Ⲵ
1
= ([V (x, y)]− 2 )2 · V (x, y) = 1 ⻠ᫎ⅑ᮠ䖳ቁˈ㌫㔏ਟԕᴤᘛⲴࡠ䗮っᇊ⣦ᘱDŽമ4઼
മ7࠶࡛ᱟ⴨ᓄⲴ᧗ࡦಘuc Ⲵԯⵏᴢ㓯ˈ᧗ࡦ᭸᷌൷䖳
ᡰԕ(k 2−α x, ky) ∈ vDŽ ྭDŽ
⭡ᔿ(10)ᗇ˖ മ2઼മ8∄䖳ˈ മ5઼മ11∄䖳ˈ ࠶࡛Ѫβ =
V̇ (x, y) 2−α 1
1઼β = 1/2ˈα٬н਼Ⲵ∄䖳DŽα = 1ᰦˈሿ⨳Ⲵ⻠
α+1 = V̇ ([V (x, y)]− 2 x, [V (x, y)]− 2 ]y) ᫎ⅑ᮠ䖳ቁˈնᱟ㌫㔏Ⲵ䈳㢲ᰦ䰤䖳䮯ˈ૽ᓄ䙏ᓖធDŽ
V (x, y) 2

ᆈ൘˖ 0.6
X1
V̇ (x,y) X2
sup α+1
0.4

(x,y)=(0,0) V (x,y) 2 0.2

2−α 1
= sup V̇ ([V (x, y)]− 2 x, [V (x, y)]− 2 ]y) 0
(x,y)=(0,0)
−0.2
= sup V̇ (x, y) = −c
(x,y)∈v −0.4

−0.6
α+1
ᡰԕˈV̇ (x, y) ≤ −cV (x, y) 2 ˈc > 0ˈণV̇ (x, y) + −0.8
α+1
cV (x, y) 2 ≤ 0ˈ৸ഐѪα ∈ (0, 1)ˈഐ↔ α+1 2 ∈ (0, 1)ˈ −1
0 1 2 3 4 5 6 7 8 9 10
┑䏣ᇊ⨶3ѝⲴᶑԦ1DŽ t /s

৸ഐѪx(t+ ) = x(t)ˈy(t+ ) = −βy(t)ˈρ ∈ (0, 1]ˈ

ᡰԕˈ മ 2: α = 1/3ˈβ = 1ᰦˈ㌫㔏⣦ᘱⲴԯⵏᴢ㓯
2 − α + 2−α
2 1
V (x+ , y + ) = |x | + (y + )2
2 2

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 2
0.8

0.7 1.5

0.6
1
0.5
0.5
0.4

0.3 0

0.2 −0.5

0.1
−1
0
0 1 2 3 4 5 6 7 8 9 10
−1.5
t /s 0 1 2 3 4 5 6 7 8 9 10
t /s

മ 3: α = 1/3ˈβ = 1ᰦˈᵾ䳵Პ䈪ཛ࠭ᮠԯⵏᴢ㓯
മ 7: α = 1/3ˈβ = 1/2ᰦˈ᧗ࡦಘuc ԯⵏᴢ㓯

2
0.8
X1
1 0.6 X2

0.4
0
0.2
−1
0

−2 −0.2

−0.4
−3
−0.6

−4
0 1 2 3 4 5 6 7 8 9 10 −0.8
t /s
−1
0 5 10 15
t /s

മ 4: α = 1/3ˈβ = 1ᰦˈ᧗ࡦಘuc ԯⵏᴢ㓯

മ 8: α = 1ˈβ = 1ᰦˈ㌫㔏⣦ᘱⲴԯⵏᴢ㓯

0.6
X1
X2 0.8
0.4

0.7
0.2

0.6
0

0.5
−0.2

0.4
−0.4

0.3
−0.6

0.2
−0.8

0.1
−1
0 1 2 3 4 5 6 7 8 9 10
t /s 0
0 5 10 15
t /s

മ 5: α = 1/3ˈβ = 1/2ᰦˈ㌫㔏⣦ᘱⲴԯⵏᴢ㓯
മ 9: α = 1ˈβ = 1ᰦˈᵾ䳵Პ䈪ཛ࠭ᮠԯⵏᴢ㓯

0.8
2

0.7
1
0.6

0.5 0

0.4
−1

0.3
−2
0.2

0.1 −3

0
0 1 2 3 4 5 6 7 8 9 10 −4
0 5 10 15
t /s
t /s

മ 6: α = 1/3ˈβ = 1/2ᰦˈᵾ䳵Პ䈪ཛ࠭ᮠԯⵏᴢ㓯 മ 10: α = 1ˈβ = 1ᰦˈ᧗ࡦಘuc ԯⵏᴢ㓯

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 7
0.8
X1
0.6 X2 6

0.4 5

0.2
4
0
3
−0.2

−0.4 2

−0.6
1

−0.8
0
−1 0 1 2 3 4 5 6 7 8 9 10
0 5 10 15 t /s
t /s

മ 15: ᵾ䳵Პ䈪ཛ࠭ᮠԯⵏᴢ㓯
മ 11: α = 1ˈβ = 1/2ᰦˈ㌫㔏⣦ᘱⲴԯⵏᴢ㓯

12

0.8
10

0.7
8

0.6
6

0.5
4

0.4
2

0.3
0

0.2
−2

0.1
−4
0 1 2 3 4 5 6 7 8 9 10
0
0 5 10 15
t /s
t /s

മ 16: ᧗ࡦಘuc ԯⵏᴢ㓯

മ 12: α = 1ˈβ = 1/2ᰦˈᵾ䳵Პ䈪ཛ࠭ᮠԯⵏᴢ㓯

1.5
5 ᙱ㔉
1 ᵜ᮷ѫ㾱⹄ウҶሿ⨳⻠ᫎ㜹ߢ㌫㔏Ⲵᴹ䲀ᰦ䰤
0.5 っᇊ䰞仈ˈ 㘼㜹ߢࣘᘱ㌫㔏ਟԕⴻ‫ڊ‬ᱟ␧ᵲࣘᘱ㌫
0
㔏Ⲵ⢩ֻˈᖃ㌫㔏ᴹ㜹ߢⲴᰦ‫ˈى‬㌫㔏⣦ᘱՊਁ⭏
−0.5
ਈॆˈ ൘䘉⿽ᛵߥлˈ ㌫㔏Ⲵᴹ䲀ᰦ䰤᭦ᮋᱟ൘
䶎LipschitzianⲴᛵߥлᆼᡀⲴˈо䘎㔝Ⲵ䶎㓯ᙗ㌫㔏
−1
⴨∄ቡ䴰㾱ᴤཊⲴ䲀ࡦᶑԦDŽᵜ᮷㔉ࠪҶ㜹ߢࣘᘱ㌫
−1.5
㔏Ⲵᴹ䲀ᰦ䰤っᇊᙗᇊ⨶ˈ࡙⭘ᴹ䲀ᰦ䰤っᇊᙗ䇮䇑
−2
0 5 10 15 Ҷሿ⨳⻠ᫎ㜹ߢ㌫㔏Ⲵᴹ䲀ᰦ䰤᧗ࡦಘˈᒦሩަっᇊ
t /s
ᙗ䘋㹼࠶᷀ˈᴰਾ㔉ࠪҶ㌫㔏Ⲵԯⵏ䗷〻৺㔃᷌࠶᷀DŽ
ԯⵏ㔃᷌㺘᰾ˈᴹ䲀ᰦ䰤っᇊ᧗ࡦಘⲴ᧗ࡦ᭸᷌ᴤྭˈ
മ 13: α = 1ˈβ = 1/2ᰦˈ᧗ࡦಘuc ԯⵏᴢ㓯 ᒦ䘋а↕傼䇱Ҷ䈕ᯩ⌅Ⲵᴹ᭸ᙗDŽ

৸㘹ᮽ⥤
2.5
X1
2 X2 [1] G.Chen, Y.Yang, Finite time stability of a class of hybrid
1.5 dynamical systemsˈIET Control Theory and Applica-
1
tions, 6(1): 8-13, 2012.
0.5

0
[2] V.T.Haimo, Finite time controllers, SIAM Journal on
−0.5 Control and Optimization, 24(4): 760-770, 1986.
−1 [3] M.Kawski, Stabilization of nonlinear systems in the
−1.5
plane, Systems and Control Letters, 12(2): 169-175,
−2

−2.5
1989.
0 1 2 3 4 5 6 7 8 9 10
t /s [4] R.P.Agarwal, V.Lakshmikantham, Uniqueness and
Nonuniqueness Criteria for Ordinary Differential Equa-
മ 14: ㌫㔏⣦ᘱԯⵏᴢ㓯 tions, Singapore: World Scientific Press, 1993.
[5] S.P.Bhat, D.s.Bernatein, Finite-time stability of continu-

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 ous autonomous systems, SIAM Journal on Control and
Optimization, 38(3): 751-766, 2000.
[6] E.Moulay, W.Perruquetti, Finite time stability of nonlin-
ear systems, 42nd IEEE Conference On Decision and
Control, 4: 3641-3646, 2003.
[7] Y.G.Hong, J.Huang, On an output feedback finite-time
stabilization problem, IEEE Transactions on Automatic
Control, 46(2): 305-309, 2001.
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of second order nonlinear systems, Proc. Third World
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[9] X.Q.Huang, W.Lin, Global finite-time stabilization of a
class of uncertain nonlinear systems, Automatica, 41(5):
881-888, 2005.
[10] S.H.Yu, X.H.Yu, Continuous finite-time control for
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[11] Y.Orlov, Finite time stability and robust control syn-
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[12] G.N.Sergey, Wassim M.H., Finite-time stabilization
of nonlinear impulsive dynamical systems, Nonlinear
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[13] N.Roberto, G.S.Ricardo, Passivity-based Controllers
for a Class of Hybrid Systems with Applications to Me-
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