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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

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'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.

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=

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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 ,

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~ 2 kóÀA5XÚ

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(x∗1, · · · , x∗p0 ,xT (M − p0),x∗M+1,· · · ,x∗M+q0 ,xT (M − q0)

), (12)

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Xeb�µ

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∂x∗p06= 0, ∂f

∂x∗M+q06= 0.

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[∂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)

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^�VÐm�

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(p, q), 1 ≤ p ≤ M, 1 ≤ q ≤ M§3½: x∗(2M)�EXe\���¦{µ

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=[θ0,N+1(p, q) θT1,N+1(p, q)

]T

, argminθ0(p,q)∈Rθ1(p,q)∈Rp+q

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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

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k=1

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·(

N∑

k=1

wk(x∗(2M))Xk(p, q)yk+1

). (18)

?ÚÚ\Xe�OK¼êµ

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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)

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LN+1(p, q). (21)

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(2π)M1

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{−1

2

M∑

i=1

(yk+1−i − x∗i

bk

)2

−12

M∑

j=1

(uk+1−j − x∗M+j

bk

)2 .

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UC§²;�ÛÜ5�O{Ú~§Ï

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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

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‖Φ1(x)‖ν≤λ‖s‖ν+c1M∑

i=1

|ti|γ+c2, ∀ x ∈ R2M , (23)

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(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

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{�e¡ü(Ø©

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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)

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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)

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(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)

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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

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=

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b(3)1 uk, if uk < −1,

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1 1.5 2 2.5 3 3.5 450

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80

90

100

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100

110

120

130

L N(q

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