adaptive neural network output feedback control of stochastic nonlinear systems with dynamical...

ORIGINAL ARTICLE

Adaptive neural network output feedback control of stochasticnonlinear systems with dynamical uncertainties

Tong Wang • Shaocheng Tong • Yongming Li

Received: 21 April 2012 / Accepted: 18 July 2012

� Springer-Verlag London Limited 2012

Abstract In this paper, a robust adaptive neural network

(NN) backstepping output feedback control approach is

proposed for a class of uncertain stochastic nonlinear sys-

tems with unknown nonlinear functions, unmodeled

dynamics, dynamical uncertainties and without requiring

the measurements of the states. The NNs are used to

approximate the unknown nonlinear functions, and a filter

observer is designed for estimating the unmeasured states.

To solve the problem of the dynamical uncertainties, the

changing supply function is incorporated into the back-

stepping recursive design technique, and a new robust

adaptive NN output feedback control approach is con-

structed. It is mathematically proved that the proposed

control approach can guarantee that all the signals of the

resulting closed-loop system are semi-globally uniformly

ultimately bounded in probability, and the observer errors

and the output of the system converge to a small neigh-

borhood of the origin by choosing design parameters

appropriately. The simulation example and comparison

results further justify the effectiveness of the proposed

approach.

Keywords Stochastic nonlinear systems � Dynamical

uncertainties � Neural network control � Changing supply

function � Stability analysis

1 Introduction

In the past decades, many approximator-based adaptive

backstepping control approaches have been developed to

deal with uncertain nonlinear strict-feedback systems with

unstructured uncertainties via neural networks (NNs) and

fuzzy logic systems (FLSs), see for example [1–13]. Works

in [1–5] are for single-input and single-output (SISO)

nonlinear systems, works in [6–8] are for multiple-input

and multiple-output (MIMO) nonlinear systems, and works

in [9–13] are for SISO or MIMO nonlinear systems with

immeasurable states, respectively. In general, these adap-

tive neural network or fuzzy backstepping control

approaches provide a systematic methodology of solving

control problems of unknown nonlinear systems, where

neural networks or fuzzy systems are used to approximate

unknown nonlinear functions, and based on the conven-

tional backstepping design technique, typically adaptive

fuzzy or neural network controllers are constructed recur-

sively. Two of the main features of these adaptive

approaches are the following: (1) They can be used to deal

with those nonlinear systems without satisfying the

matching conditions, and (2) they do not require the

unknown nonlinear functions being linearly parameterized.

Therefore, the approximator-based adaptive fuzzy back-

stepping control becomes one of the most popular design

approaches to a large class of uncertain nonlinear systems.

It is well known that stochastic disturbances often exist

in many practical systems. Their existence is a source of

instability of the control systems; thus, the investigations

on stochastic systems have received considerable attention

in recent years, and many important results have been

achieved, see for example [14–21] and references therein.

Pan and Basar [14] first proposed an adaptive backstepping

control design approach for strict-feedback stochastic

T. Wang (&) � S. Tong � Y. Li

Department of Mathematics, Liaoning University

of Technology, Jinzhou 121000, Liaoning, China

e-mail: [email protected]

S. Tong

e-mail: [email protected]

123

Neural Comput & Applic

DOI 10.1007/s00521-012-1099-7

systems by a risk-sensitive cost criterion. Deng and Krstic

[15] solved the output feedback stabilization problem of

strict-feedback stochastic nonlinear systems by using the

quadratic Lyapunov function, while Shi et al. [16] and Xia

et al. [17] developed backstepping control design approa-

ches for nonlinear stochastic systems with Markovian

switching. Moreover, to solve the unmeasured state prob-

lem, several different output feedback controllers are

developed for strict-feedback stochastic nonlinear systems

by designing a linear state observer [18–21]. It should be

pointed out that the aforementioned results are only suit-

able for those nonlinear systems with nonlinear dynamics

models being known exactly or with the unknown para-

meters appearing linearly with respect to known nonlinear

functions. Therefore, they cannot be applied to those sto-

chastic systems with structured uncertainties.

In order to handle the structured uncertainties included in

the stochastic nonlinear strict-feedback systems, several

adaptive backstepping control schemes have been developed

by using neural networks and fuzzy logic systems to

approximate the structured uncertainties. Chen and Jiao [22]

developed adaptive NN output feedback control approaches

for a class of SISO stochastic nonlinear systems, Chen et al.

[23] and Tong et al. [24] proposed adaptive NN and fuzzy

output feedback control approach for a more general class of

SISO stochastic nonlinear systems by introducing the

dynamic surface control technique, and Chen and Li [25] and

Chen et al. [26] extended the above results to a class of

stochastic large-scale nonlinear systems. However, these

adaptive NN or fuzzy control approaches did not consider the

problem of the unmodeled dynamics and dynamical distur-

bances (i.e., dynamical uncertainties), that is, the designed

controllers lacked the robustness to the ummodeled

dynamics or dynamical disturbances. As stated in [27] and

[28], the unmodeled dynamics or dynamical disturbances

often exist in many practical nonlinear systems, and they are

also the major source resulting in the instability of the control

systems. Therefore, to study the stochastic nonlinear systems

with consideration of dynamical uncertainties is very

important in control theory and applications.

The purpose of this paper is to investigate the adaptive

NN control for a class of stochastic nonlinear systems with

three types of uncertainties, that is, unknown nonlinear

functions, dynamical uncertainties and unmeasured states.

In the control design, NNs are employed to approximate

the unknown nonlinear functions, and a NN filter observer

is designed to estimate the unmeasured states. To solve the

problem of the dynamical uncertainties, the changing

supply function technique is incorporated into the back-

stepping recursive design technique, and a robust adaptive

NN backstepping output feedback control scheme is con-

structed. The main advantages of the proposed adaptive

NN backstepping output feedback control approach are

summarized as follows: (1) By incorporating the changing

supply function technique into the backstepping recursive

design technique, the proposed adaptive output feedback

control approach can be applied to a larger class of sto-

chastic nonlinear systems and has the robustness to the

dynamical uncertainties compared with the existing results

in [23], and (2) it is mathematically proved that the

resulting closed-loop system is SGUUB in probability and

the output converges to a small neighborhood of the origin.

2 Some notations and preliminary results

2.1 Stability in probability

Consider the following time-varying stochastic system:

dx ¼ ðf ðt; xÞ þ gðt; xÞuÞdt þ hðt; xÞdw ð1Þ

where w is an r-dimensional standard Brownian motion,

x [ Rn is the state, u [ R is the control input, f ; g : Rþ�Rn ! Rn, and h : Rþ � Rn ! Rn�r.

Definition 1 ([19, 20]) For any given V(t, x) associated

with the stochastic differential equation (1), define the

differential operator ‘ as follows:

‘VðxÞ ¼ oV

otþ oV

oxf ðt; xÞ þ oV

oxgðt; xÞu

þ 1

2Tr hTðt; xÞ o

2V

ox2hðt; xÞ

� �

For control-free stochastic nonlinear system of the form:

dx ¼ f ðt; xÞdt þ hðt; xÞdw ð2Þ

The following stability notions introduced will be used

throughout the paper.

Definition 2 ([19, 20]) The solution process {x(t), t C 0}

of stochastic system (2) is said to be bounded in proba-

bility, if limc!1 sup0� t\1 PfjxðtÞj[ cg ¼ 0.

Definition 3 ([20]) Consider the system (2) with

f(t, 0) : 0, h(t, 0) : 0. The equilibrium x(t) : 0 is

globally stable in probability if for any e[ 0, there exists a

class j function c(�) such that PfjxðtÞj\cðjx0jÞg� 1� e;8t� 0; x0 2 Rnnf0g.

Lemma 1 ([20]) Consider the stochastic system (2) and

assume that f(t, 0), h(t, 0) are bounded uniformly in t. If there

exist function V(t, x), l1(�), l2(�) [ j?, constants c1 [ 0,

c2 C 0, and a nonnegative function Z(t, x), such that

l1ðjxjÞ �Vðt; xÞ� l2ðjxjÞ; ‘V � � c1Zðt; xÞ þ c2;

then

1. There exists an almost surely unique solution on [0,

?) for the system (2).


123

2. The solution process is bounded in probability, when

Z(t, x) C cV(t, x) for any a constant c [ 0.

2.2 RBF neural networks

In this paper, the following RBF NN [1, 4, 7] is used to

approximate the continuous function h(X): Rq ? R,

hnnðXÞ ¼ WTuðXÞ ð3Þ

where the input vector X 2 X � Rq, weight vector W ¼½W1; . . .;Wl�T 2 Rl, the NN node number l [ 1, and

uðXÞ ¼ ½u1ðXÞ; . . .;ulðXÞ�T , with uiðXÞ being Gaussian

functions, which have the form

uiðXÞ ¼ exp�ðX � liÞTðX � liÞ

g2

" #; i ¼ 1; 2; . . .; l:

where li ¼ ½li1; . . .; liq�T is the center of the receptive field

and g is the width of the Gaussian function.

According to the literatures [1, 4] and [7], the NN (3)

can approximate any continuous function h(X) over a

compact set D � Rq to arbitrary any accuracy as

hðXÞ ¼ WTuðXÞ þ eðXÞ; 8X 2 D; ð4Þ

where W* is an ideal constant weight and e(X) is the

bounded approximation error and W* is defined as

W ¼ arg minW2X

supX2D

hðXÞ �WTuðXÞ��

2.3 System descriptions and basic assumptions

Consider the following uncertain stochastic nonlinear

system:

df ¼ q1ðf; yÞdt þ q2ðf; yÞdw

dxi ¼ ½xiþ1 þ fiðxiÞ þ Diðx; fÞ�dt þ giðxÞdw

i ¼ 1; . . .; n� 1;

..

.

dxn ¼ ½uþ fnðxnÞ þ Dnðx; fÞ�dt þ gnðxÞdw

y ¼ x1

ð5Þ

where xi ¼ ½x1; x2; . . .; xi�T 2 Ri; i ¼ 1; 2; . . .; n ðx ¼ xnÞare the states, and u and y are the control and output of the

system, respectively. f is the unmodeled dynamics, and

Diðx; fÞ are the dynamic disturbances. fiðxiÞ; i ¼ 1; 2; . . .; nare unknown smooth nonlinear functions. q1ðf; yÞ; q2ðf; yÞ;Diðx; fÞ and gi(x) are uncertain functions; w [ R is an

independent standard Wiener process defined on a com-

plete probability space. In this paper, it is assumed that the

functions fiðxiÞ; giðxÞ; qiðf; yÞ and Diðx; fÞ satisfy the

locally Lipschitz, and only the output y is available for

measurement.

Assumption 1 ([18, 20]) For each 1 B i B n, there exist

unknown positive constants pi such that

Diðx; fÞj j � pi wi1ðyÞ þ pi wi2ð fj jÞgiðxÞj j � pi wi3ðyÞ

where wi1ðyÞ; wi2ð fj jÞ and wi3ðyÞ are known nonnega-

tive smooth functions with wi1ð0Þ ¼ wi2ð0Þ ¼ wi3ð0Þ ¼ 0.

Assumption 2 ([19, 20]) For each f-subsystem in (5),

there exist function Vf(f) and known k? functions að fj jÞ;�að fj jÞ; að fj jÞ; cð yj jÞ; wf and w0 such that

að fj jÞ �VfðfÞ� �að fj jÞ; ‘Vf� cð yj jÞ � að fj jÞ;oVf=ofj j �wfð fj jÞ; q2ðf; yÞk k�w0ð fj jÞ:

Control objective: The control task is to design an

adaptive output feedback controller using the output of the

system y and state estimations xi, so that the system is

bounded in probability, and y can be regulated to a small

neighborhood of the origin in probability.

3 NN filter observer design

Note that fiðxiÞ in (5) are unknown functions; therefore, we

can assume that the nonlinear function fiðxiÞ can be

approximated by the following RBFs:

fi xi Wijð Þ ¼ WTi uiðxiÞ

where xi ¼ ½x1; x2; . . .; xi�T is the actual estimate of state xi, and

Wi ¼ arg minWi2Xi

supðxi;xiÞ2Ui

fi xi Wijð Þ � fiðxiÞ��

( ); 1� i� n:

where Xi and Ui are the bounded compact regions for Wi

and ðxi; xiÞ, respectively. In addition, the NN approxi-

mation error ei is defined as

fiðxiÞ ¼ fi xi Wi��

þ ei ð6Þ

Let e ¼ ½e1; e2; . . .; en�T be the NN approximation error

vector. By substituting (6) into (5), the system (5) can be

expressed as follows:

df ¼ q1ðf; yÞdt þ q2ðf; yÞdw

dxi ¼ ½xiþ1 þWTi uiðxiÞ þ ei þ Diðx; fÞ�dt þ giðxÞdw;

i ¼ 1; . . .; n� 1

dxn ¼ ½uþWTn unðxnÞ þ en þ Dnðx; fÞ�dt þ gnðxÞdw

y ¼ x1 ð7Þ

Rewritten (7) as

df ¼ q1ðf; yÞdt þ q2ðf; yÞdw dx ¼ ðAxþ Kyþ UT W

þ eþ Dþ BuÞdt þ GðxÞdwy ¼ Cx ð8Þ


123

where

A ¼�k1

..

.In�1

�kn 0 . . . 0

2664

3775; K ¼

k1

..

.

kn

2664

3775;

UT ¼

uT1

. ..

uTn

2664

3775

n�l

; D ¼

D1ðx; fÞ

..

.

Dnðx; fÞ

26664

37775;

W ¼ ½W1 ; . . .;Wn �Tl�1; l ¼ l1 þ . . .þ ln; C ¼ ½1; . . .; 0�;

GðxÞ ¼ ½g1ðxÞ; . . .; gnðxÞ�T ; B ¼ ½0; . . .; 1�T :

Choose vector K such that matrix A is a strict Hurwitz

matrix; therefore, for any given positive definite matrix

Q = QT [ 0, there exists a positive definite matrix

P = PT [ 0 such that

AT Pþ PA ¼ �Q ð9Þ

Note that the states x2; x3; . . .; xn in system (5) or (7) are

unmeasured; thus, the states of the system (5) should be

estimated by using the following filters.

Define a virtual state estimate as

v ¼ nþ NW þ k ð10Þ

The NN filters are designed as

_n ¼ Anþ Ky ð11Þ_N ¼ ANþ UT ð12Þ_k ¼ Akþ Bu ð13Þ

Define virtual observation error vector e as

e ¼ ½e1; e2; . . .; en�T ¼x� v

pð14Þ

where p ¼ max pi ; p2i ; 1j1� i� n

� �is an unknown

constant.

Remark 1 Note that the parameter vector W* is an

unknown and the virtual state estimate v cannot be used in

the control design. Instead, the actual state estimate x will

be used in the control design, which is defined as

x ¼ nþ NW þ k ð15Þ

where W is the estimate of W*.

From (8), (10)–(13) and (14), the observer error is

expressed as

de ¼ Aeþ eþ Dp

dt þ GðxÞ

pdw ð16Þ

To evaluate the designed NN filters (10)–(13), consider the

following Lyapunov function candidate as

V0 ¼1

2ðeT PeÞ2

From (9) and (16), one has

‘V0� � kminðQÞ � kminðPÞ ek k4þeT Pe � 2

p� eT Pðeþ DÞ

þ 1

p2Tr½GTð2PeeT Pþ eTPePÞG�

ð17Þ

From Assumption 1, and as the similar derivations to [18],

there exist smooth functions �wi1;�wi2 and �wi3 such that

wi1ðyÞ ¼ y�wi1ðyÞ; wi2ð fj jÞ ¼ fj j�wi2ð fj jÞ;wi3ðyÞ ¼ y�wi3ðyÞ

ð18Þ

By Assumption 1, Young’s inequality and the fact that

p* C 1, one has the following inequalities:

2

peT Pe � eT PD� 3

2Pk k8=3 ek k4þ4n

Xn

i¼1

w4i1ðyÞ þ w4

i2ð fj jÞ� �

ð19Þ1

p2Tr½GTð2PeeT Pþ eT PePÞG�

¼ 1

p2Tr½GPeðGPeÞT � þ 1

p2ðeTPeÞTrðGT PGÞ

� ð2 Pk k2þk2maxðPÞÞ ek k4þ 1

2

G

p

4 !

�ð2 Pk k2þk2maxðPÞÞ ek k4þ n

2ð2 Pk k2þk2

maxðPÞÞXn

i¼1

w4i3ðyÞ

ð20Þ2

peT Pe � eT Pe� 3

2Pk k8=3 ek k4þ 1

2dk k4 ð21Þ

where d[ 0 is an unknown boundary for ei, that is,

|ei| B d.

Substituting (19)–(21) into (17), one can obtain

‘V0� � p0 ek k4þ4nXn

i¼1

w4i1ðyÞ þ Uð fj jÞ

þ n

2ð2 Pk k2þk2

maxðPÞÞXn

i¼1

w4i3ðyÞ þ d0 ð22Þ

where p0 ¼ kminðQÞ � kminðPÞ � 3 Pk k8=3�2 Pk k2�k2max

ðPÞ; Uð fj jÞ ¼ 4nPn

i¼1 w4i2ð fj jÞ and d0 ¼ 1

2dk k4

.

Remark 2 It should be mentioned that if the system (5)

does not contain the unmodeled dynamics f, dynamic

disturbance Diðx; fÞ and dw = 0, then (22) is reduced to

‘V0� � p00V0 þ �D0, where p00 and �D0 are positive con-

stants, from which it follows that the NN filter observer

(10)–(13) is stable.


123

4 Adaptive NN controller design and stability analysis

In this section, a robust adaptive output feedback control

scheme will be developed based on the above-designed NN

filters, and the stability analysis of the closed-loop system

will be given.

4.1 Adaptive NN backstepping control design

From (13), one has

_ki ¼ kiþ1 � kik1; i ¼ 1; 2; . . .; n� 1 ð23Þ_kn ¼ u� knk1 ð24Þ

The adaptive NN backstepping control design consists of

n-steps; each step is based on the change of coordinates:

z1 ¼ y; zi ¼ ki � ai�1; i ¼ 2; . . .; n ð25Þ

where ai�1ð�Þ ði ¼ 2; . . .; nÞ is an intermediate control.

Step 1 From the second equation in (5), and according to

It o’s differentiation rule, one has

dy ¼ ðx2 þWT1 u1 þ e1 þ D1Þdt þ g1dw ð26Þ

Since x2 is unavailable, it is replaced by available filter

signals.

From (10), one has

x ¼ nþ NW þ kþ x� v ¼ nþ NW þ kþ pe ð27Þ

Therefore, using (27), x2 is expressed as

x2 ¼ n2 þ N2W þ k2 þ pe2 ð28Þ

Substituting (28) into (26) yields

dy ¼ ðn2 þ xW þ k2 þ pe2 þ e1 þ D1Þdt þ g1dw ð29Þ

where x ¼ ½uT1 ; 0; . . .; 0� þ N2:

Choose the Lyapunov function candidate as

V1 ¼ V0 þ1

4y4 þ 1

2~WTC�1 ~W þ 1

2r1

~d2 þ 1

2r2

~p2 ð30Þ

where C ¼ CT [ 0, r1 [ 0 and r2 [ 0 are design parame-

ters, and p ¼ max p; p2; ðpÞ4=3n o

. ~W ¼ W �W ; ~d ¼

d� d and ~p ¼ p� p are the parameters errors. W, d and p

are the estimates of W*, d and p, respectively.

From (29) and (30), the infinitesimal generator of V1

satisfies

‘V1� ‘V0 þ y3ðn2 þ xW þ k2 þ pe2 þ e1 þ D1Þ

þ 3

2y2g2

1 � ~WTC�1 _W � r�11

~d _d� r�12 ~p _p ð31Þ

Substituting (25) into (31) results in

‘V1� ‘V0 þ y3ða1 þ n2 þ xW þ z2Þ þ y3ðpe2 þ e1 þ D1Þ

þ 3

2y2g2

1 � ~WTC�1 _W � r�11

~d _d� r�12 ~p _p ð32Þ

Using Assumption 1 and Young’s inequality, one has

y3ðpe2 þ D1Þ�3

2py4 þ 1

4ek k4þ2�w4

11ðyÞy4 þ 2w412ð fj jÞ

ð33Þ3

2y2g2

1�3

2py2w2

13ðyÞ ¼3

2p�w2

13ðyÞy4 ð34Þ

Substituting y3z2� 34

y4 þ 14

z42 and (33)–(34) into (32) gives

‘V1� � p1 ek k4þ1

4z4

2þUð fj jÞ þ 2w412ð fj jÞ

þ y3 a1þ3

4yþ n2þxW þ dþW11ðyÞ þW12ðyÞp

� ~WTC�1 _W � r�11

~d _d� r�12 ~p _pþ d0 ð35Þ

where

p1 ¼ p0 �1

4; W12ðyÞ ¼

3

2yþ 3

2�w2

13ðyÞy;

W11ðyÞ ¼ 2�w411ðyÞyþ 4n

Xn

i¼1

�w4i1ðyÞy

þ n

2ð2 Pk k2þk2

maxðPÞÞXn

i¼1

�w4i3ðyÞy:

Let s1 ¼ xT y3; r1 ¼ y3 and p1 ¼ y3W12ðyÞ, then (35) can

be further rewritten as

‘V1� � p1 ek k4þ 1

4z4

2 þ Uð fj jÞ þ 2w412ð fj jÞ

þ y3 a1 þ3

4yþ n2 þ xW þ dþW11ðyÞ þW12ðyÞp

þ ~WTðs1 � C�1 _WÞ þ ~dðr1 � r�11

_dÞþ ~pðp1 � r�1

2_pÞ þ d0 ð36Þ

Choose stabilizing control function a1

a1 ¼ �3

4y�P1ðy2Þy� n2 � xW � d�W11ðyÞ

�W12ðyÞp ð37Þ

where P1(y2) is a smooth nonnegative function to be

designed later.

Substituting (37) into (36) yields

‘V1� � p1 ek k4þ 1

4z4

2 �P1ðy2Þy4 þ Uð fj jÞ þ 2w412ð fj jÞ

þ ~WTðs1 � C�1 _WÞ

þ ~dðr1 � r�11

_dÞ þ ~pðp1 � r�12

_pÞ þ d0 ð38Þ

Step 2 From (23) and (25), one has


123

dz2 ¼hk3 � k2k1 �

oa1

oy

� n2 þ xW þ k2 þ pe2 þ e1 þ D1Þ þ H2ð

� oa1

oWð _W � Cs1 þ ClWÞ � oa1

odð _d� r1r1 þ r1ldÞ

� oa1

opð _p� r2p1 þ r2lpÞ

� 1

2

o2a1

oy2g1ðxÞ2

idt � oa1

oyg1ðxÞdw ð39Þ

where l[ 0 is a design constant and

H2 ¼ �oa1

onðAnþ KyÞ � oa1

oNðANþ UTÞ � oa1

oWCðs1 � lWÞ

� oa1

odr1ðr1 � ldÞ � oa1

opr2ðp1 � lpÞ

Consider the following Lyapunov function candidate as

V2 ¼ V1 þ1

4z4

2 ð40Þ


‘V2 ¼ ‘V1 þ z32 k3 � k2k1 �

oa1

oyðn2 þ xW þ k2 þ pe2 þ D1Þ

�

þ H2 �oa1

oWð _W � Cs1 þ ClWÞ � oa1

odð _d� r1r1 þ r1ldÞ

� oa1


� 1

2

o2a1

oy2g1ðxÞ2 �

oa1

oye1

�þ 3

2z2

2

oa1

oy

2

g1ðxÞ2 ð41Þ

Using the similar derivations to step 1, one obtains the

following inequalities

�z32

oa1

oyðpe2 þ D1Þ�

3

2p

oa1

oy

4=3

z42

þ 1

4ek k4þ 3

4p

oa1

oy�w11ðyÞ

4=3

z42

þ 1

4y4 þ 1

4w4

12ð fj jÞ

ð42Þ

3

2z2

2

oa1

oy

2

g21

� 1

2z3

2

o2a1

oy2g2

1�9

4z2

oa1

oy

4

þ 1

4z3

2

o2a1

oy2

2" #

p�w413ðyÞz3

2

þ 1

2y4

ð43Þ

Substituting (42)–(43) into (41), one obtains

‘V2� � p2 ek k4þz32 z3 þ a2 � k2k1 �

oa1

oyðn2 þ xW þ k2Þ

�

� oa1

oWð _W � Cs1 þ ClWÞ

� oa1

odð _d� r1r1 þ r1ldÞ � oa1


þ 1

4z2 þ H2 þ pW22ðyÞ þ

oa1

oyd

�þ 3

4y4 þ 1

4w4

12ð fj jÞ

�P1ðy2Þy4 þ Uð fj jÞ þ 2w412ð fj jÞ þ ~WTðs2 � C�1 _WÞ

þ ~dðr2 � r�11

_dÞ þ ~pðp2 � r�12

_pÞ þ d0 ð44Þ

where

W22ðyÞ ¼3

2

oa1

oy

4=3

z2 þ3

4

oa1

oy�w11ðyÞ

4=3

z2

þ 9

4z2

oa1

oy

4

þ 1

4z3

2

o2a1

oy2

2 !

�w413ðyÞ;

p2 ¼ p1 �1

4; s2 ¼ s1 � z3

2

oa1

oyxT ;

r2 ¼ r1 þ z32

oa1

oy; p2 ¼ p1 þ z3

2W22ðyÞ:

Choose stabilizing control function a2

a2 ¼ �z2 � c2z2 � H2 þoa1

oyðn2 þ xW þ k2Þ � pW22ðyÞ

� oa1

oydþ k2k1 � ðD22 þ K22 þ A22Þz3

2 ð45Þ

where c2 [ 0 is a design constant, D22 ¼ oa1

oW Coa1

oy xT ; K22 ¼ � oa1

odr1

oa1

oy and A22 ¼ � oa1

op r2W22ðyÞ:Define

� oa1

oWð _W � Cs1 þ ClWÞ ¼

Xn

j¼2

D2jz3j ;

� oa1

odð _d� r1r1 þ r1ldÞ ¼

Xn

j¼2

K2jz3j ;

� oa1

opð _p� r2p1 þ r2lpÞ ¼

Xn

j¼2

A2jz3j :

where D2j ¼ oa1

oW C oaj�1

oy xT ; K2j ¼ � oa1

odr1

oaj�1

oy and A2j ¼� oa1

op r2Wj2ðyÞ:By using the fact z3

2z3� 34

z42 þ 1

4z4

3, and substituting (45)

into (44) yields

‘V2��p2 ek k4�c2z42þ

1

4z4

3þ3

4y4þ9

4w4

12 fj jð Þ

þXn

j¼3

ðD2jþK2jþA2jÞz32z3

j �P1ðy2Þy4þU fj jð Þ

þ ~WTðs2�C�1 _WÞþ~dðr2�r�11

_dÞþ ~pðp2� r�12

_pÞþd0

ð46Þ


123

Step iði¼3; . . .;n�1Þ A similar procedure in step 2 is

employed recursively for step i, one has

dzi ¼hkiþ1 � kik1 �

oai�1

oy

� n2 þ xW þ k2 þ pe2 þ e1 þ D1ð Þ þ Hi

� oai�1

oWð _W � Csi�1 þ ClWÞ

� oai�1

odð _d� r1ri�1 þ r1ldÞ

� oai�1

opð _p� r2pi�1 þ r2lpÞ

� 1

2

o2ai�1

oy2g1ðxÞ2

idt � oai�1

oyg1ðxÞdw ð47Þ

where

Hi ¼ �oai�1

onðAnþ KyÞ � oai�1

oNðANþ UTÞ

� oai�1

ok_k� oai�1

oWCðsi�1 � lWÞ

� oai�1

odr1ðri�1 � ldÞ � oai�1

opr2ðpi�1 � lpÞ

Consider the following Lyapunov function candidate as

Vi ¼ Vi�1 þ1

4z4

i ð48Þ


‘Vi ¼‘Vi�1 þ z3i

"ziþ1 þ ai � kik1 �

oai�1

oyn2 þ xW þ k2 þ pe2 þ D1

þ Hi �oai�1

oWð _W � Csi�1 þ ClWÞ

� oai�1

odð _d� r1ri�1 þ r1ldÞ

� oai�1

opð _p� r2pi�1 þ r2lpÞ � 1

2

o2ai�1

oy2g1ðxÞ2

� oai�1

oye1

#þ 3

2z2

i

oai�1

oy

2

g1ðxÞ2 ð49Þ

By Assumption 1 and Young’s inequality, one obtains the

following inequalities

�z3i

oai�1

oyðpe2 þ D1Þ�

3

2p

oai�1

oy

4=3

z4i

þ 1

4ek k4þ 3

4p

oai�1

oy�w11ðyÞ

4=3

z4i

þ 1

4y4 þ 1

4w4

12ð fj jÞ ð50Þ

3

2z2

i

oai�1

oy

2

g21 �

1

2z3

i

o2ai�1

oy2g2

1

� 9

4zi

oai�1

oy

4

þ 1

4z3

i

o2ai�1

oy2

2" #

p�w413ðyÞz3

i þ1

2y4

ð51Þ

Substituting (50) and (51) into (49), one obtains

‘Vi� �pi ek k4þz3i ziþ1þai� kik1�

oai�1

oy

�

ðn2þxW þk2ÞþHi�oai�1

oWð _W�Csi�1þClWÞ

�oai�1

odð _d� r1ri�1þ r1ldÞ

�oai�1

opð _p� r2pi�1þ r2lpÞþ 1

4ziþ pWi2ðyÞþ

oai�1

oyd

�

þ3ði�1Þ4

y4þ iþ7

4w4

12ð fj jÞ�P1ðy2Þy4

þ ~WTðsi�C�1 _WÞþUð fj jÞ�Xi�1

j¼2

cjz4j þ~dðri� r�1

1_dÞ

þ ~pðpi� r�12

_pÞþXn

j¼i

Xi�1

k¼2

ðDkjþKkjþAkjÞz3kz3

j þd0

ð52Þ

where

pi ¼ pi�1 �1

4; si ¼ si�1 � z3

i

oai�1

oyxT ; ri ¼ ri�1

þ z3i

oai�1

oy; pi ¼ pi�1 þ z3

i Wi2ðyÞ;

Wi2ðyÞ ¼3

2

oai�1

oy

4=3

zi þ3

4

oai�1

oy�w11ðyÞ

4=3

zi

þ 9

4zi

oai�1

oy

4

þ 1

4z3

i

o2ai�1

oy2

2 !

�w413ðyÞ:

Choose stabilizing control function ai

ai ¼�zi� cizi�Hiþoai�1

oyðn2þxW þ k2Þþ kik1

�Xi

k¼2

ðDkiþKkiþAkiÞz3k � pWi2ðyÞ�

oai�1

oyd ð53Þ

where ci [ 0 is a design constant.

Define

� oai�1

oWð _W � Csi�1 þ ClWÞ ¼

Xn

j¼i

Dijz3j ;

� oai�1

odð _d� r1ri�1 þ r1ldÞ ¼

Xn

j¼i

Kijz3j ;

� oai�1

opð _p� r2pi�1 þ r2lpÞ ¼

Xn

j¼i

Aijz3j :


123

where Dij ¼ oai�1

oW C oaj�1

oy xT ; Kij ¼ � oai�1

odr1

oaj�1

oy and Aij ¼� oai�1

op r2Wj2ðyÞ:By using the fact z3

i ziþ1� 34

z4i þ 1

4z4

iþ1, and substituting

(53) into (52) yields

‘Vi� � pi ek k4�Xi

j¼2

cjz4j þ

1

4z4

iþ1 þ3ði� 1Þ

4y4

þ iþ 7

4w4

12ð fj jÞ �P1ðy2Þy4 þ Uð fj jÞ

þXn

j¼iþ1

Xi

k¼2

ðDkj þ Kkj þ AkjÞz3kz3

j þ ~WTðsi � C�1 _WÞ

þ ~dðri � r�11

_dÞ þ ~pðpi � r�12

_pÞ þ d0 ð54Þ

Step n In the final step, the actual control input u will

appear. Consider the overall Lyapunov function candidate as

Vn ¼ Vn�1 þ1

4z4

n ð55Þ

Using the similar derivations in step i, one has

‘Vn� � pn ek k4þz3n

"u� knk1 �

oan�1

oyðn2 þ xW þ k2Þ þ Hn

� oan�1

oWð _W � Csn�1 þ ClWÞ � oan�1

odð _d� r1rn�1 þ r1ldÞ

� oan�1

opð _p� r2pn�1 þ r2lpÞ þ 1

4zn þ pWn2ðyÞ

þ oan�1

oyd

�þ 3ðn� 1Þ

4y4 þ nþ 7

4w4

12ð fj jÞ �P1ðy2Þy4

þ Uð fj jÞ þ ~WTðsn � C�1 _WÞ �Xn�1

j¼2

cjz4j þ ~dðrn � r�1

1_dÞ

þ ~pðpn � r�12

_pÞ þXn�1

k¼2

ðDkn þ Kkn þ AknÞz3kz3

n þ d0 ð56Þ

where

Hn ¼�oan�1

onðAnþ KyÞ � oan�1

oNðANþ UTÞ � oan�1

ok_k

� oan�1

oWCðsn�1 � lWÞ � oan�1

odr1ðrn�1 � ldÞ

� oan�1

opr2ðpn�1 � lpÞ

Wn2ðyÞ ¼3

2

oan�1

oy

4=3

zn þ3

4

oan�1

oy�w11ðyÞ

4=3

zn

þ 9

4zn

oan�1

oy

4

þ 1

4z3

n

o2an�1

oy2

2 !

�w413ðyÞ

sn ¼ sn�1 � z3n

oan�1

oyxT ; pn ¼ pn�1 þ z3

nWn2ðyÞ;

rn ¼ rn�1 þ z3n

oan�1

oy; pn ¼ pn�1 �

1

4:

Choose the actual control u, and the parameters adaptation

laws of W, d and p as

u ¼ � 1

4zn � cnzn � Hn þ

oan�1

oyðn2 þ xW þ k2Þ

þ knk1 �Xn

k¼2

ðDkn þ Kkn þ AknÞz3k

� pWn2ðyÞ �oan�1

oyd ð57Þ

_W ¼ Cðsn � lWÞ ð58Þ_d ¼ r1ðrn � ldÞ ð59Þ_p ¼ r2ðpn � lpÞ ð60Þ

where cn [ 0 is a design constant.

By substituting (57)–(60) into (56), one has

‘Vn� � pn ek k4þl ~WT W þ l~dd

þ l~ppþ 3ðn� 1Þ4

y4 �Xn

j¼2

cjz4j

þ nþ 7

4w4

12ð fj jÞ �P1ðy2Þy4 þ Uð fj jÞ þ d0 ð61Þ

By completing the squares

l ~WT W � � 1

2l ~W 2þ 1

2l Wk k2 ð62Þ

l~dd� � 1

2l~d2 þ 1

2ld2 ð63Þ

l~pp� � 1

2l~p2 þ 1

2lp2 ð64Þ

Substituting (62)–(64) into (61) results in

‘Vn� � pn ek k4� 1

2lð ~W 2þ~d2 þ ~p2Þ

þ 3ðn� 1Þ4

y4 �Xn

j¼2

cjz4j

�P1ðy2Þy4 þ �Uð fj jÞ þ d ð65Þ

where

�Uð fj jÞ ¼ Uð fj jÞ þ nþ 7

4w4

12ð fj jÞ

¼ 4nXn

i¼1

w4i2ð fj jÞ þ

nþ 7

4w4

12ð fj jÞ;

d ¼ d0 þ1

2l Wk k2þd2 þ p2� �

:

Remark 3 It should be mentioned that if the system (5)

does not contain the unmodeled dynamics f and the

dynamical disturbances Diðx; fÞ, then �Uð fj jÞ ¼ 0 in (65).

For this situation, we can choose P1ðy2Þ ¼ 0 in the

stabilizing control function a1, and (65) becomes


123

‘Vn� � pn ek k4� 1

4y4 �

Xn

j¼2

cjz4j

� 1

2l ~W 2þ~d2 þ ~p2� �

þ d

ð66Þ

From (66) and according to [22–27], it is easily con-

cluded that the proposed adaptive control scheme by step

1–step n can guarantee that the closed-loop system is

stable in probability. However, in this paper, the system

(5) contains the unmodeled dynamics f and the dynam-

ical disturbances Diðx; fÞ; therefore, it is necessary to

design P1(y2) to ensure the stability of the control

system.

4.2 Changing supply function design and stability

analysis

In the following, we will design the function P1(�) intro-

duced in step 1 by using the changing supply function

technique proposed by [19] and [20].

From (65), first choose a smooth nonnegative function

P1(�) such that

P1ðy2Þy4 � 3n

4y4�P10ðy2Þy4 ð67Þ

with P10(�) being a smooth nonnegative function to be

designed.

Choose parameter pn [ 0, and let V0 ¼ �pn ek k4

�Pn

j¼1 cjz4j , where c1 ¼ 1

4. Then, it follows from (65) and

(67) that

‘Vn�V0 � 1

2lð ~W 2þ ~d2 þ ~p2Þ �P10ðy2Þy4 þ �Uð fj jÞ

þ d

ð68Þ

To construct the nonnegative function P10(�), the following

Assumption 3 is introduced.

Assumption 3 ([20]) For the functions wf and w0, a(|f|),

wi2(|f|) given by Assumption 2, the following condition

holds

lim sups!0þ

w4i2ðsÞ þ w2

fðsÞw20ðsÞ

aðsÞ \1 ð69Þ

According to [20], from Assumptions 2 and 3, one can

construct continuous increasing functions n(�) and -ð�Þsatisfying nðsÞaðsÞ� 4�UðsÞ and -ðsÞaðsÞ� 2w2

fðsÞw20ðsÞ.

Lemma 2 ([20]) Under Assumption 3, and if

Z1

0

nða�1ðsÞÞ� �0

exp �Zs

0

½-ða�1ðsÞÞ��1ds

8<:

9=;ds\1 ð70Þ

Then there exist a nondecreasing positive function q(�)such that 8x 2 Rmi

1

4qðVfðfÞÞað fj jÞ � �Uð fj jÞ � 1

2q0ðVfðfÞÞw2

fð fj jÞw20ð fj jÞ

ð71Þ

Theorem 1 Consider the system (5). Under the

Assumptions 1–3 and the conditions of Lemma 2, if

lim sups!0þ

cðsÞs4

\1 ð72Þ

holds. Then, under the controller (57) and the control laws

(58)–(60), the closed-loop system has an almost surely

unique solution on [0, ?) and the solution process is

bounded in probability. Moreover, the output y can be

regulated into a small neighborhood of the origin in

probability.

Proof Suppose that q(�) is the supply function defined in

Lemma 2. Let

UðfÞ ¼ZVfðfÞ

0

qðtÞdt

By It o formula, Assumption 2, we have

‘UðfÞ ¼ qðVfÞ‘Vf þ1

2q0ðVfÞ

oVf

of

� �T

q2

2

� qðVfÞ cð yj jÞ � að fj jÞ½ � þ 1

2q0ðVfÞw2

fð fj jÞw2i0ð fj jÞ

� qðgð yj jÞÞcð yj jÞ � 1

2qðVfÞað fj jÞ

þ 1

2q0ðVfÞw2

fð fj jÞw20ð fj jÞ ð73Þ

where g ¼ �aða�1ð2cð�ÞÞÞ 2 k1.

To show the last step of (73), we consider the following

two cases separately:

Case 1 If cð yj jÞ � 12að fj jÞ, then in this case, we have that

qðVfÞ cð yj jÞ � að fj jÞ½ � � � 1

2qðVfÞað fj jÞ

Case 2 If cð yj jÞ � 12að fj jÞ, then in this case, we have

VfðfÞ� �að fj jÞ� gð yj jÞ, and

qðVfÞ cð yj jÞ � að fj jÞ½ � � qðgð yj jÞÞcð yj jÞ � qðVfÞað fj jÞ

For these two cases, it follows that

qðVfÞ cð yj jÞ � að fj jÞ½ � � qðgð yj jÞÞcð yj jÞ � 1

2qðVfÞað fj jÞ:

Consider the Lyapunov function candidate for the entire

system


123

Z ¼ Vn þ UðfÞ

Then, it follows from (68) and (73) that

‘Z�V0 � 1

2l ~W 2þ~d2 þ ~p2� �

�P10ðy2Þy4 þ �Uð fj jÞ þ d

þ qðgð yj jÞÞcð yj jÞ � 1

2qðVfÞað fj jÞ þ

1

2q0ðVfÞw2

fð fj jÞw20ð fj jÞ

�V0 � 1

2l ~W 2þ~d2 þ ~p2� �

�P10ðy2Þy4 � 1

4qðVfÞað fj jÞ

þ qðgðjyjÞÞcðjyjÞ þ d ð74Þ

Then from (72), we can construct a smooth nondecreasing

function P10ðs2Þ 2 k1 such that P10ðs2Þ� qðgðsÞÞsupt2ð0;s�

cðtÞt4 , and hence P10ðy2Þy4� qðgðjyjÞÞcðjyjÞ:

Thus, by (73) we have

‘Z�V0 � 1

2l ~W 2þ~d2 þ ~p2� �

� 1

4qðVfÞað fj jÞ þ d

Define

Z1 ¼ V0 � 1

2l ~W 2þ~d2 þ ~p2� �

� 1

4qðVfÞað fj jÞ

Then, it is easy to see that Z1 is positive definite and radically

unbounded in its arguments ðe; z; d;W ; pÞ and satisfies

‘Z� � Z1 þ d ð75Þ

where e¼ ½e1; . . .;en�T ; z¼ ½z1; . . .; zn�T ; p¼ ½p1; . . .; pn�T ;d¼ ½d1; . . .; dn�T ; W ¼ ½W1; . . .;Wn�T :

By Theorem 1, the closed-loop system has an almost

surely unique solution on [0, ?), and moreover, the solu-

tion of the closed-loop system is bounded in probability,

and for any given e [ 0, there exist a j‘ function b and a jfunction / such that 8t� 0, one has

P ðe; z; p; d;WÞ�� \b ðeð0Þ; zð0Þ; pð0Þ; dð0Þ;Wð0ÞÞ

�� ; t� �n

þ /ðdÞg� 1� e

where ðeð0Þ; zð0Þ; pð0Þ; dð0Þ;Wð0ÞÞ 6¼ 0. From the defini-

tion of d, it can be made small if we choose the design

parameters appropriately.

5 Simulation study

In this section, the simulation example is provided to

illustrate the effectiveness of the proposed adaptive NN

control approach.

Example Consider the following stochastic nonlinear system:

df¼ð�3fþ x21Þdtþ 1ffiffiffi

2p fcosx2

dw

dx1¼ðx2þ sinx1þ0:1fþ0:5x1Þdtþðx1 sinx2Þdw

dx2¼ðuþ x1þ x22þ fsinx2Þdtþðx1 cosx2Þdw

y ¼ x1 ð76Þ

with the notations of Assumption 1, we can take w11ðyÞ¼yj j; w21ðyÞ¼ yj j; w12ðfÞ¼ fj j; w22ðfÞ¼ fj j; w13ðyÞ¼w23ðyÞ¼ yj j; pi ¼ 1:

For f-system, with the choice of Lyapunov function

Vf ¼ 14f4, we can verify ‘Vf� � 3

4f4 þ 1

4x8

1, which implies

that Assumption 2 is satisfied for cðsÞ ¼ 14

s8 and

aðsÞ ¼ 34

s4. From (67), and by applying Lemma 2, it is easy

to obtain that q(s) = 100, P10ðy2Þ ¼ 25y4; P1ðy2Þ ¼ 27y4:

Selecting Q = I, k1 = 3, k2 = 4, by solving (9) to obtain

the positive definite matrix P ¼ 0:2083 0:1250

0:1250 1:2083

:

According to (37) and (57)–(60), we have the following

controller and parameters adaptation laws:

a1 ¼ �3

4y�P1ðy2Þy� n2 � xW � d�W11ðyÞ

�W12ðyÞp ð77Þ

u ¼ � 1

4z2 � c2z2 � H2 þ

oa1

oyðn2 þ xW þ k2Þ þ k2k1

� ðD22 þ K22 þ A22Þz32 � pW22ðyÞ �

oa1

oyd

ð78Þ

with _W ¼ Cðsn� lWÞ; _d¼ r1ðrn� ldÞ; _p¼ r2ðpn� lpÞ:In this simulation, we choose the design parameters

c2 = 1, r1 = r2 = 1, l = 0.5, C = 10 9 I10910, where

I10910 is the identity matrix, and the initial conditions

are chosen as WT1 ð0Þ ¼ ½0:1; 0:3; 0:5; 0:7; 0:9�; WT

2 ð0Þ¼ ½�0:1;�0:3;�0:5; � 0:7;�0:9�; x1ð0Þ ¼ 0:1; x2ð0Þ ¼0:5; fð0Þ ¼ 0:5; d ¼ 0; p ¼ 0:

The simulation results are shown by Figs. 1, 2, 3 and 4,

respectively. From the Figs. 1, 2, 3 and 4, we can see that

the proposed adaptive control approach can guarantee that

all the variables in the closed-loop system are bounded and

the output y = x1 can converge to a small neighborhood of

zero.

In order to illustrate the robustness of the proposed

adaptive control approach against dynamic uncertainties,

we let P1(y2) = 0 in (77), the simulation results are shown

by Figs. 5, 6, 7 and 8, respectively.

From the Figs. 5, 6, 7 and 8, we can see that if

P1(y2) = 0 in the proposed control approach, then the

control scheme cannot guarantee that stability of the con-

trol system.

6 Conclusions

In this paper, an observer-based adaptive NN output

feedback control approach has been proposed for a class of

uncertain stochastic nonlinear systems with unknown


123

Fig. 2 x2 (solid) and x2 (dotted)

Fig. 3 Unmodeled dynamics f

Fig. 1 x1(solid) and x1(dotted)


123

Fig. 4 Controller u




123

functions, dynamic uncertainties and without the direct

measurements of state variables. In the design, NNs are

utilized to approximate the unknown functions, and a NN

filter observer is developed. By using the filter observer and

based on the principle of the adaptive backstepping tech-

nique and changing supply function, a new robust adaptive

NN output feedback control scheme is synthesized. It has

been proved that the proposed control approach can guar-

antee that all the signals of the resulting closed-loop system

are SGUUB in probability, and the observer errors and the

output of the system converge to a small neighborhood of

the origin by choosing appropriate design parameters. The

simulation studies have justified the effectiveness of the

proposed control approach. The future researches will

mainly concentrate on the tracking output feedback control

problem and the tolerant-control design problem for the

large-scale stochastic nonlinear systems based on the result

of this paper.

Acknowledgments This work was supported by the National Nat-

ural Science Foundation of China (No. 61074014), the Outstanding

Youth Funds of Liaoning Province (No. 2005219001) and Program

for Liaoning Innovative Research Team in University.

References

1. Polycarpou MM, Mears MJ (1998) Stable adaptive tracking

of uncertain systems using nonlinearly parametrized on-line

approximators. Int J Control 70(3):363–384

2. Chen WS, Jiao LC, Wu JS (2012) Globally stable adaptive robust

tracking control using RBF neural networks as feed forward com-

pensators. Neural Comput Appl. doi:10.1007/s00521-010-0455-8

3. Wang D, Huang J (2005) Neural network-based adaptive dynamic

surface control for a class of uncertain nonlinear systems in strict-

feedback form. IEEE Trans Neural Netw 16(1):195–202

4. Tong SC, Li YM (2007) Direct adaptive fuzzy backstepping

control for a class of nonlinear systems. Int J Innov Comput Inf

Control 3(4):887–896

5. Chen B, Liu XP, Liu K, Lin C (2010) Fuzzy-approximation-

based adaptive control of strict-feedback nonlinear systems with

time delays. IEEE Trans Fuzzy Syst 18(5):883–892

Fig. 7 Unmodeled dynamics f

Fig. 8 Controller u


123

http://dx.doi.org/10.1007/s00521-010-0455-8

6. Li TS, Li RH, Wang D (2011) Adaptive neural control of non-

linear MIMO systems with unknown time delays. Neurocom-

puting 78(1):83–88

7. Zhang HG, Quan YB (2001) Modeling, identification and control

of a class of nonlinear system. IEEE Trans Fuzzy Syst 9(2):

349–354

8. Li TS, Li RH, Li JF (2011) Decentralized adaptive neural control

of nonlinear interconnected large-scale systems with unknown

time delays and input saturation. Neurocomputing 74(14–15):

2277–2283

9. Hua CC, Guan XP, Shi P (2007) Robust output tracking control

for a class of time-delay nonlinear using neural network. IEEE

Trans Neural Netw 18(2):495–505

10. Hua CC, Wang QG, Guan XP (2009) Adaptive fuzzy output-

feedback controller design for nonlinear time-delay systems with

unknown control direction. IEEE Trans Syst Man Cybern B

39(2):363–374

11. Tong SC, Li YM (2009) Observer-based fuzzy adaptive control

for strict-feedback nonlinear systems. Fuzzy Sets Syst 160(12):

1749–1764

12. Tong SC, Li CY, Li YM (2009) Fuzzy adaptive observer back-

stepping control for MIMO nonlinear systems. Fuzzy Sets Syst

160(19):2755–2775

13. Chen WS, Li JM (2008) Decentralized output-feedback neural

control for systems with unknown interconnections. IEEE Trans

Syst Man Cybern B 38(1):258–266

14. Pan Z, Basar T (1999) Backstepping controller design for non-

linear stochastic systems under a risk-sensitive cost criterion.

SIAM J Control Optim 37(3):957–995

15. Deng H, Krstic M (1999) Output-feedback stochastic nonlinear

stabilization. IEEE Trans Autom Control 44(2):328–333

16. Shi P, Xia YQ, Liu GP, Rees D (2006) On designing of sliding

mode control for stochastic jump systems. IEEE Trans Autom

Control 51(1):97–103

17. Xia YQ, Fu M, Shi P (2009) Adaptive backstepping controller

design for stochastic jump systems. IEEE Trans Autom Control

54(12):2853–2859

18. Wu ZJ, Xie XJ, Zhang SY (2007) Adaptive backstepping con-

troller design using stochastic small-gain theorem. Automatica

43(4):608–620

19. Wu ZJ, Xie XJ, Zhang SY (2007) Stochastic adaptive backstep-

ping controller design by introducing dynamics signal and

changing supply function. Int J Control 79(2):1635–1643

20. Liu SJ, Zhang JF, Jiang ZP (2007) Decentralized adaptive output-

feedback stabilization for large-scale stochastic nonlinear sys-

tems. Automatica 43(2):238–251

21. Xie S, Xie L (2000) Decentralized stabilization of a class of

interconnected stochastic nonlinear systems. IEEE Trans Autom

Control 45(1):132–137

22. Chen WS, Jiao LC (2010) Adaptive NN backstepping output-

feedback control for stochastic nonlinear strict-feedback systems

with time-varying delays. IEEE Trans Syst Man Cybern B

40(3):939–950

23. Chen WS, Jiao LC, Du ZB (2010) Output-feedback adaptive

dynamic surface control of stochastic nonlinear systems using

neural network. IET Control Theory Appl 4(12):3012–3021

24. Tong SC, Li Y, Li YM, Liu YJ (2011) Observer-based adaptive

fuzzy backstepping control for a class of stochastic nonlinear

strict-feedback systems. IEEE Trans Syst Man Cybern B 41(6):

1693–1704

25. Li J, Chen WS, Li JM (2011) Adaptive NN output-feedback

decentralized stabilization for a class of large-scale stochastic

nonlinear strict-feedback systems. Int J Robust Nonlinear Control

21(3):452–472

26. Chen WS, Jiao LC, Wu JS (2012) Decentralized backstepping

output-feedback control for stochastic interconnected systems

with time-varying delays using neural networks. Neural Comput

Appl. doi:10.1007/s00521-011-0590-x

27. Jiang ZP, Praly L (1998) Design of robust adaptive controllers for

nonlinear systems with dynamic uncertainties. Automatica

34(7):825–840

28. Wu ZJ, Xie XJ, Shi P (2008) Robust adaptive output-feedback

control for nonlinear systems with output unmodeled dynamics.

Int J Robust Nonlinear Control 18(11):1162–1187


123

http://dx.doi.org/10.1007/s00521-011-0590-x

adaptive neural network output feedback control of stochastic nonlinear systems with dynamical...

Documents