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ë©ö 1,2§¿Æ 1,2
1. ¥IÆ�êÆXÚÆïÄ�§XÚ:¢�¿§�® 100190
2. ¥IÆ�I[êÆ��ÆïÄ¥%§�® 100190
Á : XÚ�ÑÑ´L�kmSÑ\!ÑÑ�5¼ê f(·)¿U\ÅD(§ù��XÚ�5ARX
(NARX)XÚ§§£ã�2�aÄ�y§�/í25
ARXXÚ©NARXXÚ��Ú5XÚaq§6L�õ�mS�Ñ\ÚÑѧ§3?UØÓ©�©Ñé¼ê
f(·)�!§�Fݱ9XÚ���O§¿y²
§�r5©©¥ÑO¢~§�[(JÚnة۬ܩ
'c: 5 ARXXÚ§4íÛÜ��¦{§�g�O§r5©
Identification of Nonlinear Stochastic Systems: Strongly
ConsistentEstimates for Function Values, Gradients and System
Orders
Wenxiao Zhao1,2, Han-Fu Chen1,2
1. Key Laboratory of Systems and Control, Academy of Mathematics
and Systems Science,Chinese Academy of Sciences, Beijing 100190, P.
R. China
2. National Center for Mathematics and Interdisciplinary
Sciences,Chinese Academy of Sciences, Beijing 100190, China.
Abstract: The system, for which the current output is a
nonlinear function f(·) of its past inputs and outputs of a fixed
period oftime superimposed by a random noise, is called the
nonlinear ARX (NARX) system. It greatly extends the linear ARX
systemsand describes a large class of dynamic phenomena. Similar to
the linear case, the most remote past input and output the
currentoutput will depend on, define the system order. The system
order of NARX systems may change from place to place. The
paperproposes the estimates for the values of f(·), its gradients,
and orders and proves the strong consistency of the estimates.
Anumerical example is demonstrated, which is consistent with the
theoretical analysis.
Key Words: Nonlinear ARX system, recursive local least squares
estimator, order estimation, strong consistency.
1 Úó
ÄüÑ\üÑÑ (SISO)5 ARXXÚ§
yk+1 = f(yk,· · · ,yk+1−M , uk,· · · ,uk+1−M )+εk+1, (1)Ù¥ uk Ú
yk ©OXÚ�Ñ\Ñѧεk °ÄD
(§f(·)´�5¼ê§M ´ f(·)ê�®þ.©
XÚ (1)L�2�aÄ�y§±ºX5XÚ!HammersteinXÚ�ÙA~§ÏCc5ÙE£ïÄÉ�éõ'5©l5¼ê
f(·)�ï�ªw§yk©z©ëêz{Úëê
z{üa©ëêz{Ï~b� f(·) = f(·, θ)§Ù¥θëê§'Xļê�|ÜXê½k«/�Ú�(�ëê§lé f(·,
θ)�E£=zé θ��O§~X [12][16]¶ëêz{Ï~Øb�5¼êLõ�k�&E§��O f(·)3?¿½:�§~X
[3][4][15][17]©�©3ëêï�/e§¼ê f(·)3?¿½:�¼ê!Fݱ9XÚ��E£©
é5XÚ5`§�gÚëê�E£ïĤJ
©´LÚ�õ§'X AIC!BIC �Äu&EOK��
Email: [email protected], [email protected]ó��I[g,ÆÄ7]ϧ
81O
Òµ61273193§61134013§61104052§91130008.
gE£{ [1] ÚO2��¦!Å%C�ëê�4í{ [5][7][8]©¢Sþ§é5XÚ±Ó��XÚ�gÚëê�r�O
[7]©�é5XÚ5`§'ïÄffåÚ©e¡·{ü0�XÚ (1)ëêE£9��O�Ü©ó©
©z [3][4][15][17]ÄXÚ (1)�ëêE£©�{uk, yk}Nk=1êâ8§ϕ∗ ∈ R2M ½:§©z
[15]�E¡�`z�Xê£Direct weight optimization,DWO¤�{5�O f(·)3 ϕ∗:�©P
f(·)3 ϕ∗:��O f̂N (ϕ∗)§f̂N (ϕ∗)þÿÑÑ�5|ܧ
f̂N (ϕ∗) , w0 +N∑
k=1
wkyk, (2)
Ù¥ wN = [w0 w1 · · · wN ]T `ze¡þØ�¼ê����þ§
JN (w, ϕ∗, f) , E(f(ϕ∗)− f̂N (ϕ∗)
)2. (3)
du f(·)§¤± (3)Ã{�¦^©be f(·)áu,¼êa F§DWOE£{?Ú=ze¡�4�4`z¯K§
wN , argminw
maxg∈F
JN (w, ϕ∗, g). (4)
-
©z [3]ÄÄu DWO{�ëêE£§ [15]ØÓ:3uOK¼ê�ÀJ§[15]æ^þØ�¼ê§ [3]
æ^¦�OØ�u,½�VÇ4��{5�O�Xê§=
wN , argminw
P{|f(ϕ∗)− f̂N (ϕ∗)| > δ
}. (5)
©z [3][15]Ñ´Äu�½��Nþ�`z{§"yéN →∞{ì?5�?Ø©©z
[4]æ^k.Ñ\§¿3D(k.ÚXÚêÑ\Ñѽ�b�
^e§�EÄuؼê�4í{§¿y² f(ϕ∗)��OS�´VÇÂñ�©©z
[17]�EÄu*Ü��Å%C{§¿y²�OS�r/
Âñ� f(·)3½:�¼ê©b� f(·) ��ëY§FݼêP Of(·)§
|^�VÐm§é?¿¿©�� ² > 0 ±9 ϕ ÷v‖ϕ− ϕ∗‖ < ²kXe'X¤áµ
|f(ϕ)− f(ϕ∗)| = O (‖ϕ− ϕ∗‖) ,|f(ϕ)− f(ϕ∗)− Of(ϕ∗)T (ϕ− ϕ∗)| =
O
(‖ϕ− ϕ∗‖2
).
§éëêE£5`§Fݼê��O±�Ñ
5¼ê3½:�ÛÜ5�.§lJpï�
�°Ý§ù¯K3©z [3][4][15][17]¥¿?Ø©d§þã©zb� f(·)��®§Cc5§é
f(·)��E£ïĤJØäZy§X5z{ [2]!o¼êþ{ (Lipschitz number approach,
[10])!CĽ{ (False nearest neighbors approach, [11])!b�u�{[13][14]
�©8c§ù{"yî�nØ©Û¶õ'5E£ f(·)��§vkÓé¼ê f(·)lëêzï�½ëêï��Ý?1E£¶,§¤Ä
5¼ê f(·)��3�½Â (½£8m)´�©é5XÚ5`§Nõ¹e f(·)��Uó:�CzUC§·e¡ü~fµ
~ 1 ©¡� ARXXÚµ
yk+1 = f1(yk,· · · ,yk+1−M , uk,· · · ,uk+1−M )+εk+1, (6)
Ù¥
f1(yk, · · · , yk+1−M , uk, · · · , uk+1−M )
=
a(1)1 yk+· · ·+a(1)p1 yk+1−p1 +b(1)1 uk+· · ·+b(1)q1 uk+1−q1
,
if [yk, · · · , yk+1−M , uk, · · · , uk+1−M ]T ∈ X1,...
a(s)1 yk+· · ·+a(s)ps yk+1−ps +b(s)1 uk+· · ·+b(s)qs uk+1−qs
,
if [yk, · · · , yk+1−M , uk, · · · , uk+1−M ]T ∈ Xs,
Xi, i = 1, · · · , spØ�÷v⋃s
i=1 Xi = R2M©XÚ3Xi�� (pi, qi)§§3�mþvkÚ��©
~ 2 kóÀA5XÚ
yk+1 = f2(uk, uk−1, uk−2) + εk+1, (7)
Ù¥
f2(uk, uk−1, uk−2) =
ukuk−1uk−2, if uk > 1,
ukuk−1, if − 1 ≤ uk ≤ 1,uk, if uk < −1.
du uk ¤? ØÓ§XÚ��ØÓ©
éXÚ (1)§8vkw�ÓÄ5¼ê�!¼êÚFÝE£�©Ù§AO/§é5¼
ê3ØÓó:kØÓ��ïÄvk©�©�éþ
ã¯K§31�!Äuؼê�EÛÜ��¦
{ÚÛÜ&EOK§̂ ±�OXÚ�5¼ê3
½:��!¼ê±9Fݶ31n!§(ÜXÚ�
ê¼5§y²�!¼ê±9FÝ�O�r5¶
31o!ÏLý~f�y{9nØ�k�
5¶�§31Ê!o(Ú?Ø©
êÆÎÒµP (Ω, F ,P)Ä�VÇm§Bm Rm þ� Borel-σê©þ x(m) = [x1 · · ·xm]T
�îAp�êP ‖x(m)‖§ÎÒÿÝ ν(·)��C�êP ‖ν(·)‖var©éKS� {aN}N≥1 Ú {bN}N≥1§ÎÒ
aN ∼ bN ¿3��~ê c1 Ú c2§¦� c1bN ≤aN ≤ c2bN é?¿ N ≥ 1Ѥá"2
ÛÜ&EOKÚÛÜ��¦{
5¿�XÚ (1)¥¼ê f(·)�½Â R2M§½Â£8þ ϕk(M, M)Ú?¿½: x∗(2M)Xeµ
ϕk(M, M) , [yk · · · yk+1−M uk · · ·uk+1−M ]T , (8)x∗(2M) ,
[x∗1, · · · , x∗2M ]T . (9)
é?¿ 1 ≤ p ≤ M Ú 1 ≤ q ≤ M§½Âϕk(p, q) , [yk · · · yk+1−p uk · ·
·uk+1−q]T , (10)x∗(p, q) = [x∗1, · · · , x∗p, x∗M+1, · · · ,
x∗M+q]T . (11)�¼ê f(·)3: x∗(2M)�� (p0, q0), 1 ≤ p0 ≤
M, 1 ≤ q0 ≤ M§ù´f(x∗(2M)) =f
(x∗1, · · · , x∗p0 ,xT (M − p0),x∗M+1,· · · ,x∗M+q0 ,xT (M −
q0)
), (12)
Ù¥ x(M − p0)Ú x(M − q0)´ RM−p0 Ú RM−q0 ¥�?¿þ©
Xeb�µ
A1) f(·)3 x∗(2M)��ëY¿÷v∂f
∂x∗p06= 0, ∂f
∂x∗M+q06= 0.
P f(·) 3: x∗(2M) ?�FݼêOf(x∗(2M)) ,
[∂f∂x∗1
· · · ∂f∂x∗M∂f
∂x∗M+1· · · ∂f∂x∗2M
]T©
db�^�
Of(x∗(2M)) =
∂f
∂x∗1· · · ∂f
∂x∗p00 · · · 0︸ ︷︷ ︸M−p0
∂f
∂x∗M+1· · · ∂f
∂x∗M+q00 · · · 0︸ ︷︷ ︸M−q0
T
. (13)
-
é5XÚ§A1)Ú (13)w,¤á©,¡§é¿©�C x∗(2M) �: x(2M)§|
^�VÐm�
f(x(2M))
≈f(x∗(2M))+Of(x∗(2M))T (x(2M)−x∗(2M)). (14)
d (13)Ú (14)§XJ�� f(·)3: x∗(2M)¼êÚFÝ��O¿�ä ∂f∂x∗i
, i = 1, · · · , 2M ´Ä"§Ò±�OѼê3½:��§Ï�E3
½:�ÛÜ5�.´E£¯K�'©
3�E{c§·Ú\^µ
A2) À� bk = 1kδ§Ù¥ δ ∈(0, 12(2M+1)
)¶À�
w(·) é¡�VÇݼê§3 0 < ρ < 1¦�� ‖x‖ → ∞ k w(x) = O
(ρ‖x‖)§∫R2M w(x)xx
T dx > 0©�êâ8 {ϕk(M, M), yk+1}Nk=1©é?¿½
(p, q), 1 ≤ p ≤ M, 1 ≤ q ≤ M§3½: x∗(2M)�EXe\���¦{µ
θN+1(p, q)
=[θ0,N+1(p, q) θT1,N+1(p, q)
]T
, argminθ0(p,q)∈Rθ1(p,q)∈Rp+q
N∑
k=1
wk(x∗(2M))(yk+1 − θ0(p, q)
− θ1(p, q)T (ϕk(p, q)− x∗(p, q)))2
, (15)
Ù¥ wk(x∗(2M))½ÂXeµ
wk(x∗(2M))=1
b2Mkw
(1bk
(ϕk(M, M)−x∗(2M)))
. (16)
½Âþµ
Xk(p, q) ,[
1ϕk(p, q)− x∗(p, q)
]. (17)
´d (15)ª¤½Â�\���¦{§�Ý
∑Nk=1 wk(x
∗(2M))Xk(p, q)Xk(p, q)T ÛÉkOúªµ
θN+1(p, q) =
(N∑
k=1
wk(x∗(2M))Xk(p, q)Xk(p, q)T)−1
·(
N∑
k=1
wk(x∗(2M))Xk(p, q)yk+1
). (18)
?ÚÚ\Xe�OK¼êµ
LN+1(p, q) , σN+1(p, q) + aN · (p + q), (19)
Ù¥
σN+1(p, q) ,N∑
k=1
wk(x∗(2M))(yk+1−θ0,N+1(p, q)
−θ1,N+1(p, q)T (ϕk(p, q)−x∗(p, q)))2
, (20)
{aN}N≥1 ü N ª u à ¡ � � S�§ {θ0,N+1(p, q)}N≥1 Ú {θ1,N+1(p,
q)}N≥1 ´d{ (15)����OS�©���O½ÂXeµ
(pN+1, qN+1) , argmin1≤p≤M1≤q≤M
LN+1(p, q). (21)
5P 1 ±þ{¥� w(·)~¡Ø¼ê§~�ؼêkpdݼê§d
wk(x∗(2M)) =1
(2π)M1
b2Mkexp
{−1
2
M∑
i=1
(yk+1−i − x∗i
bk
)2
−12
M∑
j=1
(uk+1−j − x∗M+j
bk
)2 .
5P 2 d (16)ª§e ϕk(M, M)C x∗(2M)§{ (15)éA��Ò§K�§Ï¡{ (15)|^½:
x∗(2M)/C�0êâ�ÛÜ��¦{©|^ݦ_úª§´{ (18)±�¤4í/ª©(16)ª� bk ~¡Ú©²;�ÛÜ5�O{'
[9]§(16)¥�ÚëêXêâþ�O\
UC§²;�ÛÜ5�O{Ú~§Ï
Ã{4íO©
5P 3 d (15)ª��� θ0,N+1(p, q)Ú θ1,N+1(p, q)©O
f(·)3½:�¼ê9ÙFÝ��O©(19)¤½Â�OK¼êÀ5XÚ'OK¼ê (X AIC�) 35/e�í2§üö�«O3u
(20) ª¥�ؼê§Ï¡ (19)ÛÜ&EOK©
3©Û{Âñ5c§·?ÚÚ\^©
d§Äk½Âþµ
Φ1(ϕk(M,M))
, [f(yk, · · · , yk+1−M , uk, · · · , uk+1−M ) yk · · · yk+2−M
]T ,Φ2(ϕk(M,M))
, [0 uk · · ·uk+2−M ]T ,Φ(ϕk(M, M)) , [Φ1(ϕk(M,M))T Φ2(ϕk(M,
M))T ]T ,ξk+1 , [εk+1 0 · · · 0 uk+1 0 · · · 0]T .
lXÚ (1)L«G�m/ªµ
ϕk+1(M, M) = Φ(ϕk(M, M)) + ξk+1. (22)
?ÚXeb�µ
A3) 3 RM þ�þê ‖ · ‖ν ±9~ê 0 < λ <1, c1 > 0, c2 > 0±9
γ > 0§¦�
‖Φ1(x)‖ν≤λ‖s‖ν+c1M∑
i=1
|ti|γ+c2, ∀ x ∈ R2M , (23)
Ù¥ s , [s1 · · · sM ]T ∈ RM , t , [t1 · · · tM ]T ∈RM , x , [sT
tT ]T ∈ R2M©
A4) Ñ\&Ò {uk}k≥0 ÕáÓ©Ù�ÅS�§÷v 0 < E|uk|γ < ∞§kVÇݼê
fu(·)§fu(·)3 RþëY�©
-
A5) D( {εk}k≥0 ÕáÓ©ÙÅS�§Eεk =0, 0 < Eε2k < ∞§kVÇÝ
fε(·)§fε(·)3 RþëY�¶{uk}k≥0Ú {εk}k≥0pÕá©
A6) E‖Y0‖ < ∞§Ù¥ Y0 , [y0, y−1, · · · , y1−M ]T©|^Ýê�'(ا
±y²5X
Ú!Hammerstein XÚ�Ñ÷v^ A3)©d (22) §{ϕk(M, M)}k≥0 �u (R2M , B2M
)�ê¼ó©Äu±þb�§ê¼ó {ϕk(M,M)}k≥0ke¡5©Ún 1 ([17])XJ A3)-A6)¤á§Kk
(i) {ϕk(M, M)}k≥0 AÛH{ê¼ó§ =3(R2M , B2M )þ�VÇÿÝ PIV(·)±9~ê c1
> 0Ú 0 < ρ1 < 1§¦� ‖Pk(·)−PIV(·)‖var ≤ c1ρk1§Ù¥ Pk(·)
ϕk(M,M)�©Ù�Ñ�VÇÿÝ©
(ii) PIV(·)kVÇݼê fIV(·)§fIV(·)3 R2M þ�©
(iii) {ϕk(M, M)}k≥0´ α-·Ü�§·ÜXêP{α(k)}k≥0§¿3~ê c2 > 0Ú 0 <
ρ2 < 1§¦� α(k) ≤ c2ρk2©
5P 4 H{5�yê¼ó�ìC²5§·Ü¿XÅS��ìCÕá5©'½Â!5ë [9]©
3 E£{�r5
|^ {ϕk(M,M)}k≥0�H{5Ú·Ü5¿(Ü·ÜÅL§�4½n§éÛÜ��¦
{�e¡ü(Ø©
Ún 2 XJ A1)-A6)¤á§¿ fIV(·)3: x∗(2M)��ëY§Ké?¿ ² > 0k
1N
N∑
k=1
wk(x∗(2M))(f(ϕk(M,M))− f(x∗(2M))
− Of(x∗(2M))T (ϕk(M, M)− x∗(2M)))
=1
2(1− 2δ)b2N
∫
R2Mw(x)xT
∂2f
∂x∗(2M)2xdx · fIV(x∗(2M))
+ o(b2N ) + o
(1
N12−²bMN
)a.s. (24)
1N
N∑
k=1
wk(x∗(2M))(ϕk(M, M)− x∗(2M))(f(ϕk(M,M))
− f(x∗(2M))− Of(x∗(2M))T (ϕk(M, M)− x∗(2M)))
=1
2(1− 3δ)b3N
∫
R2Mw(x)xxT
∂2f
∂x∗(2M)2xdx · fIV(x∗(2M))
+ o(b3N ) + o
(1
N12−²bM−1N
)a.s. (25)
N−1∑
k=1
wk(x∗(2M))εk+1 = O(N
12+Mδ+²
), a.s. (26)
N−1∑
k=1
wk(x∗(2M))(ϕk(M,M)− x∗(2M))εk+1
= O(N
12+(M−1)δ+²
), a.s. (27)
½Âµ
AN (1, 1) =1N
N∑
k=1
wk(x∗(2M)),
AN (1, 2) =1
N1−δ
N∑
k=1
wk(x∗(2M))(ϕk(M, M)− x∗(2M))T ,
AN (2, 1) =AN (1, 2)T ,
AN (2, 2) =1
N1−2δ
N∑
k=1
wk(x∗(2M))(ϕk(M, M)− x∗(2M))
· (ϕk(M, M)− x∗(2M))T .Ún 3 XJ A1)-A6)¤á§¿ fIV(·)3:
x∗(2M)��ëY§Kk
[AN (1, 1) AN (1, 2)AN (2, 1) AN (2, 2)
]−→
N→∞fIV(x∗(2M))
·[1 00 11−2δ
∫R2M w(x)xx
T dx
]> 0 a.s. (28)
½n 1 XJ A1)-A6)¤á§¿ fIV(·)3: x∗(2M)��ëY§KéÛÜ��¦{k
θN+1(M, M)−[f(x∗(2M)) Of(x∗(2M))T
]T −→N→∞
0 a.s.
(29)
y²µl (18)ª�
θN+1(M, M)−[f(x∗(2M)) Of(x∗(2M))T
]T
=
(Nk∑
i=1
wi(x∗(2M))Xi(M,M)Xi(M, M)T)−1
(Nk∑
i=1
wi(x∗(2M))Xi(M,M)ξi+1
)
=[N−δ 0
0 I
] [AN (1, 1) AN (1, 2)AN (2, 1) AN (2, 2)
]−1 [BN (1)BN (2)
], (30)
Ù¥
ξi+1 , εi+1 + f(ϕi(M,M))− f(x∗(2M))− Of(x∗(2M))T (ϕi(M, M)−
x∗(2M)),
BN (1) =1
N1−δ
N∑
k=1
wk(x∗(2M))(f(ϕk(M, M))−f(x∗(2M))
−Of(x∗(2M))T (ϕk(M, M)−x∗(2M))+εk+1),
BN (2) =1
N1−2δ
N∑
k=1
wk(x∗(2M))(ϕk(M, M)− x∗(2M))
·(f(ϕk(M, M))−f(x∗(2M))
− Of(x∗(2M))T (ϕk(M, M)−x∗(2M))+εk+1).
|^Ún 2�[BN (1)BN (2)
]−→
N→∞
[00
]a.s.
2(ÜÚn 3Ò½n(ؤ᩠¥
-
½n 2 XJ A1)-A6)¤á!fIV(·)3: x∗(2M)��ëY§¿ {aN}N≥1÷v
N1−4δ
aN−→
N→∞0,
aNN1−2δ
−→N→∞
0, (31)
Ù¥ δ > 0÷vb�^ A2)§KÛÜ&EOK�����O´r�§=
(pN , qN ) −→N→∞
(p0, q0) a.s. (32)
y²µ½n�y²ÌÉ©z [6]'u5XÚ�gE£'(Ø�éu©·�ÑÌg´©
PÝ
∑N
i=1 wi(x∗(2M))Xi(M, M)Xi(M, M)T
��A� λ(M,M)min (N)§|^Ún 3�
λ(M,M)min (N) ∼ N1−2δ. (33)
5¿ p, q, p0, q0 þ��ê§y (32) y² {(pN , qN )}N≥1 �?¿4:Ñ�u (p0, q0)
=©b� (p′, q′) ´ {(pN , qN )}N≥1 �4:§l3 {(pN , qN )}N≥1 �f�§P {(pNk
, qNk)}k≥1§¦� (pNk , qNk) −→
k→∞(p′, q′)©5¿� {(pN , qN )}N≥1 ±9
(p′, q′)þ��ê§l3K > 0¦�
(pNk , qNk) = (p′, q′), ∀ k ≥ K. (34)
y (p′, q′) = (p0, q0)§·y²±en«¹þØUu)µ(i) p′ < p0¶(ii) q′ <
q0¶(iii) p′ + q′ >p0 + q0©ÄkÄ (i)©be p′ < p0§â��O�½Â§
w,k
0 ≥ LNk+1(p′, q′)− LNk+1(p0, q0).
,¡§�
LNk+1(p′, q′)− LNk+1(p0, q0)
=σNk+1(p′, q′)− σNk+1(p0, q0) + aNk(p′ + q′ − p0 − q0)
≥λ(M,M)min (Nk)(
c +aNk
λ(M,M)min (Nk)
(p′ + q′ − p0 − q0))
,
Ù¥ c > 0© |^ (33) ¿5¿� (31)§ ÒkLNk+1(p
′, q′)−LNk+1(p0, q0) −→k→∞
∞©þãgñÒ�yp′ ≥ p0©aq/±y² q′ ≥ q0±9 p′+q′ ≤ p0 +q0§?� (32)© ¥
5P 5 aN � CN1−3δ§Ù¥ C > 0~ê©d½n1 Ú½n 2 §|^{ (15) Ú
(21)§±Ó��¼ê f(·)3½:�!¼ê±9FÝ�r�O©
4 ý~f
ÄkóÀAXÚ
yk+1 = f(uk, uk−1, uk−2) + εk+1, (35)
Ù¥
f(uk, uk−1, uk−2)
=
b(1)1 uk + b
(1)2 uk−1 + b
(1)3 uk−2, if uk > 1,
b(2)1 uk + b
(2)2 uk−1, if − 1 ≤ uk ≤ 1,
b(3)1 uk, if uk < −1,
Ù¥¤këê b(j)i þ� 1§{uk}k≥1Ú {εk}k≥1
pÕá� iidÅS�§©OÑl©Ù N (0, 22)ÚN (0, 0.12)©w,XÚ
(35)3ØÓ½ÂþkØÓ�©b���þ. 4§Ä: ϕ∗ = [0.5 0.5 0.5 0]T©Äuêâ8 {uk,
yk+1}N=2000k=1 Úëê δ = 0.01§¿À�aN = 0.01N1−3δ§ã 1± N = 2000 σN
(q)ÚLN (q), q = 1, 2, 3, 4 �Cz§ã 2 ± b
(2)1 Ú
b(2)2 ��OS�©lã 1Úã 2§¤k�OÑÂñ�ý©
1 1.5 2 2.5 3 3.5 450
60
70
80
90
100
σ N(q
)
1 1.5 2 2.5 3 3.5 490
100
110
120
130
L N(q
)
ã 1: σN (q)Ú LN (q)
0 500 1000 1500 2000−2
−1
0
1
2
3
4
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ë©z
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