p�Å5XÚZ6³°∗
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Áµ �©ïÄ
äkØ(½(��Åp�en�XÚ°¯K. ©¥æ
^È©{(ÜÈ©ígÑ1wì, ¦�4XÚ�)ªª
u�:NC?¿���, ¿é ÜZ6¢y°Z6³.
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cµ p�en�XÚ, Å, Z6³
1. ÚÚÚóóó
Cc5, Å5XÚ�ïÄ´nØ.~¹��+, ZyÑéõó
, XKrstic´Ú Deng[1,2]Ñ
Åî"XÚ9ÑÑ"XÚ� Backstepping�
O{, ¿^_`�{)û
aÅ5XÚ�Z6³¯K, cÙ´éÅ
î"XÚÑ Backstepping�O{. �ù
(J�éuî"/ª½ÑÑ
"/ª�XÚ, ép�en�XÚ, ù
{¿Ø·^, I, ÏéO�å»5)
ûùaXÚ��O. �5¿�´ Lin[4, 5, 7]32000cmJÑ�^O\È©��
O{5ïÄp�en�5XÚ, ¿JÑ
�Û½, Z6³9g·A�{,
Wang[10]éT{?1
í2, í2�õª9p�XÚ�¹. éÅp�en
�XÚ, © [11]JÑG� ½{. ù
Ñ�©�óJø
Ä:, �©Ò´3dÄ
:þïÄäkØ(½(��Åen�XÚ�°ì�O933 Ü6Ä�
O°Z6³ì.
2. ¯¯¯KKKJJJÑÑÑ
ÄXeÅp�en�5XÚ
dx1 = (x
p1
2 + f1(x1) + ∆f1(X))dt+ g
T
1 (x1)dB
...
dxi = (x
pi
i+1 + fi(Xi) + ∆fi(X))dt+ g
T
i (Xi)dB
...
dxn = (upn + fn(X) + ∆fn(X))dt+ gTn (X)dB
(1)
Ù¥ X = (x1, · · · , xn)T ∈ Rn´XÚ�G�, Xi = (x1, · · · , xi)T ´G��c i©þ|
¤�þ, u ∈ R´Ñ\, fi : Ri → R´1w¼ê, gi = (gi1, · · · , gir)T : Ri → Rr ´1w
þ¼ê, B´ rIOÕáBL§, ∆fi(X) : Rn → R´1w¼ê, LXÚ�Ø(
½5.
∗I[g,ÆÄ7(60674029)Úp�Æ�ƬÆ:;ïÄ7(20050358044)]Ï8
1
�©�¤�8IÒ´éþãXÚ�O1wƦ�Ù½, ¿
�3
Ü6Ä�O1wÆ÷v½5�Ä:þ¢yG�é Ü6Ä�³, =ée>�
XÚ�O°Z6³ì�O.
dx1 = (x
p1
2 + f1(x1) + ∆f1(X) + ξ1(x1)w)dt+ g
T
1 (x1)dB
...
dxi = (x
pi
i+1 + fi(Xi) + ∆fi(X) + ξi(X1)w)dt+ g
T
i (Xi)dB
...
dxn = (upn + fn(X) + ∆fn(X) + ξn(X)w)dt+ gTn (X)dB
(2)
e¡Ú\�©I^��b�ÚÄ�£.
b�1. pi´�Ûê, i = 1, · · · , n.
b�2. éu 1 ≤ i ≤ n, 3K1w¼ê ϕi(Xi), ¦�
|∆fi(X)| ≤ φi(|Xi|), ∀(X, t) ∈ Rn × R+ (3)
b�3. éu 1 ≤ i ≤ n, 3K1w¼ê µij(Xi), ¦�
|gij(Xi)| ≤ (|x1|
pi+1
2 + · · ·+ |xi|
pi+1
2 )µij(Xi) (4)
ïÄXÚ (1), ·kée¡�Å©§Ú\'�ÚnÚPÒ
dx = f(x, v)dt+ g(x, v)dw (5)
Ù¥ x ∈ Rn ´G�, v = v(x, t) : Rn × R+ → Rm ´Ñ\, w ´½Â3��VÇm
(Ω,F , {Ft}t≥0, P )� rIOÕáBL§,¼ê f : Rn×Rm → RnÚ g : Rn×Rm →
Rn×r éÙCþ÷vÛÜo¼^. ¿b½é?¿�Щ^ x0, ?¿�5k.ÿÑ
\ v, XÚd½Â3 [0,∞)� x(t), §´Ft·A, t−ëY�.
â ItoˆÅ©, 2�ëY¼ê V (x, t)÷ÅXÚ (5)�CzÇ (q¡
infinitesimal generator)
LV := ∂V
∂t
+
∂V
∂x
f(x) +
1
2
Tr
{
gT
∂2V
∂x2
g
}
(6)
ùp TrL«Ý
�,. (½5XÚ� Lyapunov¼ê©LªØÓ�´, ª (6)p
O\
��©, ù~~ÅXÚ� Lyapunov�O5½(J. Ñ\—G
�½5 (ISS)´5XÚ°½5©Û�O�óä, §Äkd SontagJ
Ñ [8], © [3]òí2�ÅXÚ, Ú\
D(—G�½5 (NSS), © [9]JÑ
�Ñ\—G�½ (SISS).
½Â 1 ([9]) eéu?¿ ² > 0, 3 KLa¼ê β(·, ·), Ka¼ê γ(·), Ú~ê
d÷v
P
{
‖x(t)‖ < β(‖x0‖, t) + γ(sup0≤s≤t‖vs‖)
}
≥ 1− ², ∀t ≥ 0, ∀x0 ∈ Rn\{0} (7)
Ù¥ ‖vs‖ = infA⊂Ω,P (A)=0sup{‖v(x(w, s))‖ : w ∈ Ω\A}. XÚ�¡ÅÑ\G�½
(SISS)�.
2
e¡ÑÅÑ\G�½��â.
Ún 2 ([6]) ÄXÚ (5), b�3�½!ì����ëY¼ê V (x, t), ±9
K∞a¼ê χÚ Ka¼ê ρ, ¦�
LV ≤ −ρ(‖x‖) + χ(‖v‖), ∀t ≥ 0 (8)
KXÚ (5)´ÅÑ\G�½� (SISS).
Ñ�©^��(Ø.
Ún 3 ([6]) éXÚ dx = f(x)dt + h(x)dw, b½ f(x), h(x)´÷vÛÜo¼^
�, ¿
b½3�½»Ã., C2 ¼ê V : Rn → R, ~ê c ≥ 0Ú�½»
Ã.¼êW (x), ÷v
LV ≤ −W (x) + c (9)
KXÚ�G�´VÇk., ¿
é?¿ ² > 0, 3 β(·, ·) ∈ KL, γ ∈ K, ÷v
P
{
‖x(t)‖ ≤ β(‖x0‖, t) + γ(c)
}
≥ 1− ², ∀t ≥ 0, ∀x0 ∈ Rn\{0} (10)
5µeW (x) = bV (x), Kk E{V (x)} ≤ e−btV (0) + c/b, [3].
�©^�d YoungØ�ª���üÚn.
Ún 4 ([10]) éu¢ê a ≥ 0, b > 0,m ≥ 1, ±eØ�ª¤á
a ≤ b+
( a
m
)m(m− 1
b
)m−1
(11)
Ún 5 ([4]) - x, y, zi, i = 1, · · · , l´¢Cþ, b½ g1 : R → R, g2 : Rl+1 → R ´
1wN�, Kké?¿��êm,nÚ¢ê N > 0, 3üK1w¼ê h1 : R2 → R,
h2 : Rl+1 → R÷ve�Ø�ª
|xm[(y + xg1(x))n − (xg1(x))n]| ≤ |x|
m+n
N
+ |y|m+nh1(x, y) (12)
|yn(zm1 + · · ·+ zml + ym)g2(z1, · · · , zl, y)|
≤ |z1|
m+n + · · ·+ |zl|m+n
N
+ |y|m+nh2(z1, · · · , zl, y) (13)
(x+ y)n ≤ 2n−1(|x|n + |y|n) (14)
3. äääkkkØØØ(((½½½555(((������°°°ììì���OOO
k?1XeIC
z1 = x1
zi = xi − αi−1(z1, · · · , zi), i = 2, . . . , n (15)
3
Ù¥ αi(z1, · · · , zi)´äk/ª −ziβ(z1, · · · , zi), i = 1, · · · , n− 1��Oì. u´
�XÚ±=z
dzi =
(
xpii+1 + fi −
i−1∑
k=1
∂αi−1
∂xk
(xpik+1 + fk)−
1
2
i−1∑
k,l=1
∂2αi−1
∂xk∂xl
gTk gl
+(∆fi −
i−1∑
k=1
∂αi−1
∂xk
∆fk)
)
dt+ (gTi −
i−1∑
k=1
∂αi−1
∂xk
gTk )dB (16)
Ù¥ xn+1 = u.
Ú\XePÒ
Fi = fi −
i−1∑
k=1
∂αi−1
∂xk
(xpik+1 + fk)−
1
2
i−1∑
k,l=1
∂2αi−1
∂xk∂xl
gTk gl
Gi = gi −
i−1∑
k=1
∂αi−1
∂xk
gk
∆i = ∆fi −
i−1∑
k=1
∂αi−1
∂xk
∆fk
� Lyapunov¼ê
V =
n∑
i=1
zp−pi+2i
p− pi + 2 (17)
Ù¥ p = max1≤i≤n{pi}, K
LV =
n∑
i=1
zp−pi+1i (x
pi
i+1 + Fi +∆i) +
1
2
n∑
i=1
(p− pi + 1)zp−pii GTi Gi
=
n∑
i=1
(zp−pi+1i (x
pi
i+1 − αpii ) + zp−pi+1i (αpii + Fi +∆i) +
1
2
(p− pi + 1)zp−pii GTi Gi)
âb� 1,2,3 9Ún 5, �3K1w¼ê ρi(z1, · · · , zi+1), ψij(z1, · · · , zi),
η1i (z1, · · · , zi+1), η2i (z1, · · · , zi+1) Ú?¿���~ê σ > 0¦�e¡ªf¤á
|zp−pi+1i (Fi +∆i)| ≤ |zp−pi+1i |
(
|Fi|+ |φi|+
i−1∑
k=1
|∂αi−1
∂xk
φk|
)
≤ σ + zp+1i ρi(z1, · · · , zi+1)
|Gij | ≤ (|x1|
pi+1
2 + · · ·+ |xi|
pi+1
2 )µ′ij(Xi)
≤ (|z1|
pi+1
2 + · · ·+ |zi|
pi+1
2 )ψij(z1, · · · , zi)
|zp−pi+1i (xpii+1 − αpii )| = |zp−pi+1i ((zi+1 + αi+1)pi − αpii )|
≤ zp+1i + zp+1i+1 η1i (z1, · · · , zi+1)
|p− pi + 1
2
zp−pii G
T
i Gi| ≤ zp+11 + · · ·+ zp+1i−1 + zp+1i η2i (z1, · · · , zi+1)
4
l
LV ≤
n∑
i=1
(zp+1i + z
p+1
i+1 η
1
i + z
p−pi+1
i α
pi
i + σ + z
p+1
i + · · ·+ zp+1i−1 + zp+1i η2i ) (18)
�J[
αi = −zi(n− i+ 2 + η1i−1 + ρi + η2i )
1
pi , i = 1, · · · , n (19)
Ù¥ η10 = 0, u = αn. K
LV ≤ −
n∑
i=1
zp+1i + nσ (20)
½n 6 ÄäkØ(½5(��Åp�en�XÚ (1), XJXÚ÷vb�
1,2,3, K1wÆ (19) ±¦�4XÚVÇ�Û½, =4XÚG�±V
Ǫu�:NC�?¿���, ¿
XÚG�3þ¿ÂeUêǪu
v
��ê.
y² âc¡��E, � Lyapunov¼ê (17)÷Xd§ (1), (19)|¤�4X
ÚCzǵ
LV ≤ −
n∑
i=1
zp+1i + nσ (21)
âÚn 3, 4XÚVÇ�Û½, ¿
éu?¿ ² > 0, 3 β(·, ·) ∈ KL,
γ ∈ K, ÷v
P
{
‖z(t)‖ ≤ β(‖x0‖, t) + γ(nσ)
}
≥ 1− ², ∀t ≥ 0, ∀x0 ∈ Rn\{0} (22)
ù�·��, éu?¿ ² > 0, λ > 0, ·±� nσ < λ, ù� z(t)VǪu±�
:¥%, ± γ(λ)»�¥S, ù¥±��?¿�.
,¡, d YoungØ�ª
zp−pi+2i
p− pi + 2 ≤
zp+1i
(p+ 1)δ
p+1
p−pi+2
+
(pi − 1)δ
p+1
pi−1
p+ 1
(23)
ù�§d (21)·��
LV ≤ −(p+ 1)δ
p+1
p−pi+2V +
n∑
i=1
(pi − 1)δ(
p+1
pi−1+
p+1
p−pi+2 ) + nσ (24)
P
b = (p+ 1)δ
p+1
p−pi+2
c =
n∑
i=1
(pi − 1)δ(
p+1
pi−1+
p+1
p−pi+2 ) + nσ
5
KdÚn��5
E{V (t)} ≤ e−btV (0) + c
b
(25)
l
, ؽ ‖z(t)‖ = V (t) (´�y÷vêún),
E{‖z(t)‖} ≤ e−btV (0) + c
b
(26)
ÏL·��À�ëê δ, σ, K·±�yXÚ�G�êþ�?¿�. Óâ
·�IC´©Ó��, l
�XÚ�G�3·�E�1wÆe´ª
u�:NC�?¿¥S, XÚG��êþ�?¿�.
4. °°°ZZZ666³³³ììì������OOO
Ó�?1IC (15), �XÚ±=z
dzi =
(
xpii+1 + fi −
i−1∑
k=1
∂αi−1
∂xk
(xpik+1 + fk)−
1
2
i−1∑
k,l=1
∂2αi−1
∂xk∂xl
gTk gl
+(∆fi −
i−1∑
k=1
∂αi−1
∂xk
∆fk) + (ξi −
i−1∑
k=1
∂αi−1
∂xk
ξk)w
)
dt+ (gTi −
i−1∑
k=1
∂αi−1
∂xk
gTk )dB
Ù¥ xn+1 = u.
P
ψi = ξi −
i−1∑
k=1
∂αi−1
∂xk
ξk
� Lyapunov¼ê
V =
n∑
i=1
zp−pi+2i
p− pi + 2 (27)
Ù¥ p = max1≤i≤n{2pi + 1}. (Üþ¡�Ú½·k
LV =
n∑
i=1
zp−pi+1i (x
pi
i+1 + Fi +∆i + ψiw) +
1
2
n∑
i=1
(p− pi + 1)zp−pii GTi Gi
≤
n∑
i=1
(zp+1i + z
p+1
i+1 η
1
i + z
p−pi+1
i α
pi
i + σ + z
p+1
i + · · ·+ zp+1i−1
+zp+1i η
2
i +
1
2
z
2(p−pi+1)
i |ψi|2 +
1
2
‖w‖2)
�ÆXe
αi = −zi(n− i+ 2 + η1i−1 + ρi + η2i +
1
2
zp−2pi+1i |ψi|2)
1
pi , i = 1, · · · , n (28)
Ù¥ η10 = 0, u = αn, K
LV ≤ −
n∑
i=1
zp+1i +
n
2
‖w‖2 + nσ (29)
6
½n 7 ÄäkØ(½5(�Ú ÜZ6�Åp�en�XÚ (2),b½XÚ÷
vb� 1, 2, 3�^, K1wÆ (28), ±¦�4XÚ3 w = 0´VÇ�Û
½, ¿
±¢y4XÚ�G�é ÜZ6�³.
y² ìcÜ©, � Lyapunov¼ê (27)÷Xd§ (2), (28)|¤�4XÚ
CzÇ
LV ≤ −
n∑
i=1
zp+1i +
n
2
‖w‖2 + nσ (30)
KâÚn 3, 3 w = 04XÚVÇ�Û½, ¿
éu?¿ ² > 0, 3
β(·, ·) ∈ KL, γ1, γ2 ∈ K, ÷v
P
{
‖z(t)‖ ≤ β(‖x0‖, t) + γ1(nσ) + γ2(‖w‖)
}
≥ 1− ²,∀t ≥ 0, ∀x0 ∈ Rn\{0} (31)
Ó�ÚcÜ©�, ·±��
E{V (t)} ≤ e−btV (0) + c
b
+
n
2b
sup0≤s≤t‖w‖2 (32)
=
E{‖z(t)‖} ≤ e−btV (0) + c
b
+
n
2b
sup0≤s≤t‖w‖2 (33)
ù�·��, éu?¿ ² > 0, λ > 0, ·±� nσ < λ, ù� z(t)VǪu±
�:¥%, ± γ1(λ) + γ2(‖w‖)»�¥S, ù¥±��?¿�C± γ2(‖w‖)�
¥, l
3VÇ¿Â�)k.�G�. ,¡, ⪠(33), ê�þ
´k.�. Óâ·�IC´©Ó��, l
�XÚ3·�E�1w
Æe´÷v¦�, =¢yZ6³.
5. (((ØØØ
�©Ä
äkØ(½5(��Åp�en�XÚ�°¯K, ѽb
��cJe, ÏLæ^È©ìÈ©íg�E5�Ñ
1wÆ, ¿é
äk ÜZ6Ñ\�¹e, Ó��E5�Ñ
1wÆ, ¢y°Z6³
¯K.
ëëë©©©zzz
[1] Deng H. and Krstic´ M. Stochastic nonlinear stabilization — Part I: A backstepping
design. Syst. Contr. Lett., 1997, 32: 143–150.
[2] Deng H. and Krstic´ M. Output feedback stochastic nonlinear stabilization. IEEE
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1237–1253.
7
[4] Lin W. and Qian C J. Adding one power integrator: a tool for globalstabilization
of high–order lower–trangular systems. Syst. Contr. Lett., 2000, 39: 339–351.
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[6] Liu S J., Zhang J F. and Jiang Z P. A notion of stochastic input–to–state stabil-
ity and its application to stability of cascaded stochastic nonlinear systems. Acta
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tomat. Contr., 1989, 34: 435–443.
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tems. Proc. of 40th IEEE Conference on Decision and Control, Orlando, Florida,
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[11] , od¬, �². p�Å5XÚG�" ½. XÚó§>fEâ,
2006, 28: 1729–1731.
Robust and Disturbance attenuation Control Design for a Class of
Stochastic High-order Nonlinear Systems
Wang Xing-Hu and Ji Hai-Bo
(Dept. of Automation, Univ. of Sci. & Tech. of China, Hefei 230027)
Abstract: We study the robust control design of stochastic high-order lower-
triangular systems with structure uncertainties. By power integrator technique
and backstepping, we give a controller that can make the states go into the ar-
bitrary small neighborhoods asymptotically. Further in the existence of external
disturbance, we also design a smooth controller to implement the disturbance at-
tenuation.
Keywords: stochastic high-order nonlinear systems, robust control, disturbance
attenuation
8
Discuss the Relationship between the College Students’ Body Mass Grade and the Level of Physical Health
Text3:
