observer-based backstepping dynamic surface control for stochastic nonlinear strict-feedback systems

ORIGINAL ARTICLE

Observer-based backstepping dynamic surface controlfor stochastic nonlinear strict-feedback systems

Jiayun Liu

Received: 31 August 2012 / Accepted: 18 December 2012

� Springer-Verlag London 2013

Abstract An observer-based dynamic surface control

approach is proposed for a class of stochastic nonlinear

strict-feedback systems in order to solve the problem of

‘explosion of complexity’ in the backstepping design; that

is, the dynamic surface control approach is extended to the

stochastic setting. The circle criterion is applied to designing

a nonlinear observer, and so no linear growth condition is

imposed on nonlinear functions depending on system states.

It is proved that the closed-loop system is semi-globally

uniformly ultimately bounded in fourth moment, and the

ultimate boundedness can be tuned arbitrarily small. Two

examples are given to demonstrate the effectiveness of the

control scheme proposed in this paper.

Keywords Circle criterion � Dynamic surface control �Output-feedback � Nonlinear observer � Stochastic

strict-feedback systems

1 Introduction

Backstepping technique has been proved to be a powerful

tool in the nonlinear control area. After it obtains a series of

successes in the control of deterministic nonlinear systems

[1–18], it is natural to extend this technique to the case of

stochastic nonlinear systems. A backstepping design was

first developed by Pan and Basar [19] for strict-feedback

systems motivated by a risk-sensitive cost criterion. Since

then, a series of extensions have been made under different

assumptions or for different systems [20–25]. By using the

quadratic Lyapunov function instead of the classical qua-

dratic one, Authors in [26–30] solved the (adaptive) sta-

bilization problem of strict-feedback (or output-feedback)

stochastic nonlinear systems, and then this design idea was

extended to several different cases such as tracking control

[31], decentralized control [32, 33], and control of high-

order systems [34–36] or time-delay systems [37–40].

Recently, several output-feedback control schemes were

proposed for stochastic non-minimum-phase nonlinear

systems [41] and for stochastic nonlinear systems with

linearly bounded unmeasurable states [42, 43]. However,

these results still inherit the open problem of ‘explosion of

complexity’ in the backstepping design, which is even more

serious than that in deterministic systems owing to the

appearance of Hessian term in the infinitesimal generator.

This drawback makes it difficult to realize the designed

backstepping schemes, especially in the case when the order

of systems is more than two.

In fact, for deterministic systems, the above problem has

been solved well by the dynamic surface control (DSC)

approach, which was originally proposed in [44], and then

extended to the output-feedback control [45], adaptive

neural network control [46, 47] and decentralized control

[48], and control of semi-strict-feedback systems [49] or

time-delay systems [50–52]. Moreover, some applications

of DSC have been studied, for example, see [53–55]. The

underlying idea of DSC is to avoid differentiating the

virtual control variables by introducing a first-order filter in

each step of backstepping design procedure, which greatly

simplifies the traditional backstepping control algorithm.

Unfortunately, to the authors’ knowledge, until now no

works have been reported to extend the DSC to the sto-

chastic setting.

The purpose of this paper is to solve the aforementioned

problem, that is, to apply the DSC approach to stabilizing a

J. Liu (&)

The School of Science, Shandong Jianzhu University,

Jinan 250101, China

e-mail: [email protected]

123

Neural Comput & Applic

DOI 10.1007/s00521-012-1325-3

class of stochastic strict-feedback systems using the output-

feedback method. The main contributions of this paper are

listed as follows:

1. From the viewpoint of control method, the DSC

approach is utilized to solve the stabilization problem

of a class of strict-feedback stochastic nonlinear

systems. Instead of seeking the global stability in

probability, we prove that the closed-loop error signals

are semi-globally uniformly ultimately bounded

(SGUUB) in fourth moment and converge to a

sufficiently small residual set around the origin in

fourth moment.

2. Compared with the existing works on stochastic

control systems, where the output-feedback control

schemes are designed only for systems with the

standard output-feedback form [28, 29, 32, 33, 37,

39, 41] in which nonlinear functions depend only on

the system output or inverse dynamic, this paper will

investigate the output-feedback control for a more

general class of strict-feedback systems in which

nonlinear functions depend not only on the system

output, but also on the system states. In general, this

class of systems can be controlled only by the state-

feedback control schemes [26, 30].

3. As for the observer design, the circle criterion [56] is

introduced to solve the problem of nonlinear observer

design, so the linear growth condition imposed on

nonlinear functions is not required, which is different

from [42], where a high-gain linear observer is

designed for stochastic strict-feedback systems under

the assumption of linear growth condition.

The rest of this paper is organized as follows. In Sect. 2,

we present notations, definitions, and lemmas. Section 3

gives the problem formulation. Observer-based DSC design

procedure and stability analysis are given in Sect. 4. In Sect.

5, two simulation examples are provided to illustrate the

effectiveness of the proposed controller. In Sect. 6, we

conclude the work of this paper.

2 Notations, definitions, and lemmas

2.1 Notations

Throughout this paper, the following notations are adopted:

• R? denotes the set of all nonnegative real numbers; Rn

denotes the real n-dimensional space; Rn 9 r denotes

the real n 9 r matrix space;

• Tr(X) denotes the trace for square matrix X; kmin(X) and

kmax(X) denote the minimal and maximal eigenvalues

of symmetric real matrix, respectively;

• |X| denotes the Euclidean norm of a vector X, and the

corresponding induced norm for a matrix X is also

denoted by jXj; kXkF denotes the Frobenius norm of X

defined by kXkF ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

TrðXTXÞp

.

• Ci denotes the set of all functions with continuous ith

partial derivatives;

• K denotes the set of all functions: Rþ ! Rþ, which are

continuous, strictly increasing and vanish at zero; K1denotes the set of all functions which are of class K and

unbounded.

2.2 Definitions and lemmas

Consider the following stochastic nonlinear system

dvðtÞ ¼ f ðvðtÞÞdt þ gðvðtÞÞdwðtÞ ð1Þ

where v 2 Rn is the system state, w is an r-dimensional

independent standard Wiener process, and f : Rn ! Rn and

g : Rn ! Rn�r are locally Lipschitz and satisfy f(0) = 0,

g(0) = 0.

Definition 1 [29]. The trajectory v(t) of system (1) is said

to be SGUUB in pth moment, if for some compact set

X 2 Rn and any initial state v0 ¼ vðt0Þ 2 X, there exists an

e [ 0 and a time constant T ¼ Tðe; v0Þ such that

EkvðtÞkp\e for all t [ t0 ? T; especially, when p = 2, it

is usually called SGUUB in mean square.

Definition 2 [29]. For any given VðxÞ 2 C2, associated

with the stochastic system (1), the infinitesimal generator Lis defined as follows:

LVðvÞ ¼ oV

ovf ðvÞ ¼ 1

2Tr gTðvÞ o

2V

ov2gðvÞ

� �

:

Lemma 1 [29]. Suppose there exists a C2 function

V : Rn ! Rþ, two constants ‘[ 0 and �h [ 0, class K1functions �a1 and �a2 such that

�a1ðjvjÞ � VðvÞ� �a2ðjvjÞLVðvÞ � �‘VðvÞ þ �h

�

ð2Þ

for all v 2 Rn and t [ t0. Then, there is a unique strong

solution of system (1) for each v0 ¼ vðt0Þ 2 Rn and it

satisfies

E½VðvðtÞÞ� �Vðv0Þe�‘t þ�h

‘; 8t [ t0: ð3Þ

Remark 1 The proof of Lemma 1 can be easily obtained

using the same derivations as that in [30, Th. 4.1]. The

inequality (3) implies that under the conditions of Lemma

1, EV(v(t)) is globally uniformly ultimately bounded

(GUUB); especially, if the inequality (3) holds only for

v0 2 X, where X 2 Rn is a compact set, then EV(v(t)) is

SGUUB.


123

The following lemma will be used in this paper.

Lemma 2 (Young’s inequality) [26]. For 8ðx; yÞ 2 R2,

the following inequality holds

xy� �p

pjxjp þ 1

q�qjyjq ð4Þ

where �[ 0; p [ 1; q [ 1, and (p - 1)(q - 1) = 1.

3 Problem formulation

Consider the following stochastic nonlinear strict-feedback

system

dxi ¼P

iþ1

j¼1

ai;jxj þ uið�xiÞ þ fiðyÞ" #

dt þ giðyÞdx; i ¼ 1; . . .; n� 1;

..

.

dxn ¼P

n

j¼1

an;jxj þ unð�xnÞ þ fnðyÞ þ qu

" #

dt þ gnðyÞdx;

y ¼ x1

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

ð5Þ

where x ¼ ½x1; x2; . . .; xn�T 2 Rn; y 2 R and u 2 R are the

system state vector, output, and control input, respectively;

�xi ¼ ½x1; . . .; xi�T; ai;j and q are known constants; uið�xiÞ is

known smooth nonlinear functions with uið0Þ ¼ 0; fi : R!R and gT

i : R! Rr are unknown locally Lipschitz smooth

functions with fi(0) = 0 and gi(0) = 0; x is defined as in

the system (1).

Remark 2 The system model (5) is an extension of the

system addressed in [28]. However, the appearance of

uið�xiÞ makes the observer design more difficult than that in

[28].

Since fi(0) = 0 and gi(0) = 0, according to the well-

known mean value theorem, the following equalities hold

fiðyÞ ¼ yðtÞ dfiðsÞdsjs¼#fi

yðtÞ

giðyÞ ¼ yðtÞ dgiðsÞdsjs¼#gi

yðtÞ

where 0\#fi ; #gi\1. More generally, we make the

following assumption.

Assumption 1 [27]. There exist known continuous

functions �fiðyÞ and �giðyÞ such that the following inequali-

ties hold

jfiðyÞj � jyðtÞj�fiðyÞ ð6ÞjgiðyÞj � jyðtÞj�giðyÞ: ð7Þ

The objective of this paper is to design an observer-

based DSC approach for system (5), such that the closed-

loop error signals are SGUUB in fourth moment and

converge to a small residual set around the origin. To this

end, we rewrite the system (5) into the following matrix

form

dx ¼ ðAxþ UðxÞ þ FðyÞ þ BuÞdt þ GðyÞdx

y ¼ Cx

)

ð8Þ

where

A ¼

a1;1 a1;2 0 � � � 0

a2;1 a2;2 a2;3 � � � 0

..

. ... ..

.� � � ..

.

an�1;1 an�1;2 an�1;3 � � � an�1;n

an;1 an;2 an;3 � � � an;n

2

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

5

; UðxÞ ¼

/1ð�x1Þ/2ð�x2Þ

..

.

/nð�xnÞ

2

6

6

6

6

4

3

7

7

7

7

5

;

FðyÞ ¼

f1ðyÞf2ðyÞ

..

.

fnðyÞ

2

6

6

6

6

4

3

7

7

7

7

5

;B ¼

0

..

.

0

q

2

6

6

6

6

4

3

7

7

7

7

5

ðn�1Þ

; GðyÞ ¼

g1ðyÞg2ðyÞ

..

.

gnðyÞ

2

6

6

6

6

4

3

7

7

7

7

5

; C ¼

1

0

..

.

0

2

6

6

6

6

4

3

7

7

7

7

5

T

ðn�1Þ

:

The following further assumptions are used to design the

nonlinear observer using the circle criterion.

Assumption 2 [27]. There exists a matrix H and a known

vector-valued function J(x), such that UðxÞ ¼ HJðxÞ, where

J(x) satisfies

oJðxÞoxþ oJðxÞ

ox

� �T

� 0; 8x 2 Rn: ð9Þ

Assumption 3 [27]. Matrices A and H satisfy the

following linear matrix inequality (LMI):

ðAþ LCÞTPþ PðAþ LCÞ þ Q PH þ ðI þ KCÞTHTPþ ðI þ KCÞ 0

� �

� 0

ð10Þ

where P ¼ PT [ 0;Q ¼ QT [ 0;K ¼ ½k1; . . .; kn�T and

L ¼ ½l1; . . .; ln�T.

Remark 3 It is easily verified that the LMI (10) in

Assumption 3 is equivalent to the following inequality and

equality:

ðAþ LCÞTPþ PðAþ LCÞ� � Q; ð11Þ

PH ¼ �ðI þ KCÞT: ð12Þ

4 Observer-based DSC design and stability analysis

4.1 Nonlinear observer design

Using the circle criterion [56], we design the following

nonlinear observer for system (8)

dx̂ ¼ ðAx̂þ LðCx̂� yÞ þ Uðx̂þ KðCx̂� yÞÞ þ BuÞdt

ð13Þ

where K and L satisfy the LMI (10) in Assumption 3.


123

Define the observer error ~x ¼ x� x̂. From (8) and (13),

it follows that

d~x ¼ ððAþ LCÞ~xþ UðxÞ � UðvÞ þ FðyÞÞdt þ GðyÞdx

ð14Þ

where v ¼ x̂þ KðCx̂� yÞ.

Theorem 1 Consider the following Lyapunov candidate

for the observer error system (14)

V0 ¼1

2ð~xTP~xÞ2 ð15Þ

and then LV0 is bounded by

LV0� �kminðPÞkminðQÞ þ3

2�

4=31 jPj

8=3þ 3nffiffiffi

np

�22jPj

4

� �

j~xj4

þ y4 1

2�41

j �FðyÞj4þ 3nffiffiffi

np

�22

j �GðyÞj4� �

ð16Þ

where �1 [ 0 and �2 [ 0 are the design constants, �FðyÞ ¼½�f1ðyÞ; � � � ; �fnðyÞ�T and �GðyÞ ¼ ½�g1ðyÞ; . . .; �gnðyÞ�T.

Proof See ‘‘Appendix 1’’. h

4.2 DSC design

We give the following overall system consisting of the first

equation of the system (5) and the observer (13)

dy¼ a1;1x1þ a1;2x̂2þ a1;2~x2þu1ðyÞ þ f1ðyÞ

dtþ g1ðyÞdx

ð17Þ

dx̂i ¼X

i

j¼1

ai;jx̂j þ ai;iþ1x̂iþ1 � li~x1 þ uið�̂xi � �ki~x1Þ" #

dt;

i ¼ 2; . . .; n� 1 ð18Þ

dx̂n ¼X

n

j¼1

an;jx̂j þ qu� ln~x1 þ unðx̂� K~x1Þ" #

dt ð19Þ

where �̂xi ¼ ½x̂1; . . .; x̂i�T and �ki ¼ ½k1; . . .; ki�T. Obviously,

the above system can be designed using the backstepping

technique. However, to overcome the problem of ‘explo-

sion of complexity’, we introduce the DSC approach to the

following backstepping procedure.

Step 1. Define the new variable z1 = y. From (17), we

design the first virtual control a1 as follows

a1 ¼1

a1;2�c1y� yNðyÞ � a1;1x1 � u1ðyÞ� �

ð20Þ

where c1 [ 0 is a design constant, and NðyÞ is given by

NðyÞ ¼ 1

2�41

j �FðyÞj4 þ 3nffiffiffi

np

�22

j �GðyÞj4 þ 3

2j�g1ðyÞj2: ð21Þ

Introduce a new state variable f2 and let a1 pass through a

first-order filter f2 with time constant i2 to obtain f2

i2_f2 ¼ �f2 þ a1; f2ð0Þ ¼ a1ð0Þ: ð22Þ

Step i (i ¼ 2; . . .; n� 1). Define the new variable

zi ¼ x̂i � fi. From (18), we have

_zi ¼ _̂xi � _fi

¼X

i

j¼1

ai;jx̂j þ ai;iþ1x̂iþ1 � li~x1 þ uið�̂xi � �ki~x1Þ

� 1

iið�fi þ ai�1Þ: ð23Þ

From (23), we design the ith virtual control as follows

ai ¼1

ai;iþ1

�cizi �X

i

j¼1

ai;jx̂j þ li~x1 � uið�̂xi � �ki~x1Þ

þ 1

iið�fi þ ai�1Þ

!

ð24Þ

where ci [ 0 is a design constant. Similarly, introduce a

new state variable fiþ1 and let ai pass through a first-order

filter fiþ1 with time constant iiþ1 to obtain fiþ1

iiþ1_fiþ1 ¼ �fiþ1 þ ai; fiþ1ð0Þ ¼ aið0Þ: ð25Þ

Step n. Define a new variable zn ¼ x̂n � fn. From (19), it

follows that

_zn ¼ _̂xn� _fn

¼X

n

j¼1

an;jx̂jþqu� ln~x1þunðx̂�K~x1Þ�1

inð�fnþ an�1Þ:

ð26Þ

From (26), the control law is designed as

u ¼ 1

q

� cnzn �X

n

j¼1

an;jx̂j þ ln~x1 � unðx̂� K~x1Þ

þ 1

inð�fn þ an�1Þ

!

ð27Þ

where cn [ 0 is a design constant.

4.3 Stability analysis

Define the filter error .iþ1 ¼ fiþ1 � ai, and then x̂iþ1 � ai

may be expressed as

x̂iþ1 � ai ¼ ðx̂iþ1 � fiþ1Þ þ ðfiþ1 � aiÞ¼ ziþ1 þ .iþ1

ð28Þ

where from (22) and (25), it follows that

d.2 ¼ � .2

i2

þ B2ð�z2; ~xÞ� �

dt þ C2ðz1Þdw ð29Þ

d.iþ1 ¼ � .iþ1

iiþ1

þ Biþ1ð�zi; ~xÞ� �

dt þ Ciþ1ð�zi; ~xÞdw ð30Þ


123

where �zi ¼ ½z1; . . .; zi�T, and

B2ð�z2; .2; ~xÞ ¼ �oa1

oya1;1x1 þ a1;2x̂2 þ a1;2~x2 þ u1ðyÞ þ f1ðyÞ� �

� 1

2

o2a1

oy2g1ðyÞTg1ðyÞ ð31Þ

C2ðz1Þ ¼ �oa1

oyg1ðyÞT ð32Þ

Biþ1ð�ziþ1; �.iþ1; ~xÞ ¼ �oai

oya1;1x1 þ a1;2x̂2 þ a1;2~x2

�

þ u1ðyÞ þ f1ðyÞÞ �oai

ox̂_̂x

�X

i

j¼2

oai

of̂j

_̂fj �1

2

o2ai

oy2g1ðyÞTg1ðyÞ;

i ¼ 2; . . .; n� 1 ð33Þ

Ciþ1ð�ziþ1; �.iþ1; ~xÞ ¼ �oai

oyg1ðyÞT; i ¼ 2; . . .; n� 1 ð34Þ

are continuous functions.

Adding and subtracting (a1,2a1)dt and (ai,i?1ai)dt in the

right side of (17) and (23), respectively, and then substi-

tuting (20), (24), and (27) into (17), (23), and (26),

respectively, together with (28), we have the final closed-

loop error system described completely by

d~x ¼ ððAþ LCÞ~xþ UðxÞ � UðvÞ þ FðyÞÞdt þ GðyÞdxdy ¼ �c1y� yNðyÞ þ a1;2~x2 þ f1ðyÞ þ a1;2z2 þ a1;2.2

þ g1ðyÞdw_zi ¼ �cizi þ ai;iþ1ziþ1 þ ai;iþ1.iþ1; i ¼ 2; . . .; n� 1

_zn ¼ �cnzn

d.iþ1 ¼ ð�.iþ1

iiþ1þ Biþ1Þdt þ Ciþ1dw; i ¼ 1; . . .; n� 1:

9

>

>

>

>

=

>

>

>

>

;

:

ð35Þ

The main results of this paper can be summarized by the

following theorem.

Theorem 2 Under Assumptions 1–3, consider the closed-

loop adaptive system (35) consisting of plant (5), observer

(13), virtual control variables (20), (24), filters (22), (25),

and control law (27); then for any initial condition satis-

fying12ð~xTð0ÞP~xð0ÞÞ2 þ 1

4

Pni¼1 z4

i ð0Þ þ 14

Pn�1i¼1 .4

iþ1ð0Þ�M0,

where M0 is any positive constant, there exist ci; ii; and li,

such that the closed-loop signals ~x; zi; .iþ1 are SGUUB in

fourth moment. Moreover, the ultimate boundedness of

above closed-loop signals can be tuned arbitrarily small by

choosing design parameters.

Proof See ‘‘Appendix 2’’. h

5 Simulation examples

In this section, we give two simulation examples to illus-

trate the effectiveness of the proposed DSC method.

Example 1 Consider the following second-order system:

dx1 ¼ ½0:5x1 þ 1:5x2 � x31 þ f1ðyÞ�dt þ g1ðyÞdw

dx2 ¼ ½2uþ 2x1 � 2x2 þ x31 � x5

2 þ f2ðyÞ�dt þ g2ðyÞdw

y ¼ x1

9

=

;

ð36Þ

where f1ðyÞ ¼ y2; g1ðyÞ ¼ y1þy2 ; f2ðyÞ ¼ y sinðyÞ and

g2ðyÞ ¼ y lnð1þ y2Þ. When the initial states x1(0) = 0.5,

x2(0) = -0.5, and the control u = 0, the response curves

of states are shown in Fig. 1, from which we can see that

the states cannot converge.

Now, we begin to demonstrate the effectiveness of the

proposed control method. Compared with system (8), it can

be easily shown that

A ¼ 0:5 1:52 �2

� �

;UðxÞ ¼ �x31

x31 � x3

2

� �

¼ �1 0

1 �1

� �

x31

x52

� �

¼ HJðxÞ

where J(x) satisfies Assumption 2. It is easy to verify that

when

P ¼ 1:5 1

1 1

� �

; L ¼ �1

�2

� �

; and K ¼ �0:51

� �

;

the LMI (10) holds. Based on the control method proposed

in Sect. 4, the nonlinear observer is designed as follows

_̂x1 ¼ 0:5x̂1 þ 1:5x̂2 � ðx̂1 � yÞ � ½x̂1 � 0:5ðx̂1 � yÞ�3_̂x2 ¼ 2x̂1 � 2x̂2 � 2ðx̂1 � yÞ þ ½x̂1 � 0:5ðx̂1 � yÞ�3

�½x̂2 þ ðx̂1 � yÞ�5 þ 2u

9

>

=

>

;

:

ð37Þ

The virtual control function a1 and real control u are

designed as follows

a1¼2

3

"

�c1y�y1

2�41

j �FðyÞj4þ6ffiffiffi

2p

�22

j �GðyÞj4þ3

2j�g1ðyÞj2

� �

�0:5yþx31Þ#

u¼1

2�c2z2�2x̂1þ2x̂2þ2ðx̂1�yÞ�½x̂1�0:5ðx̂1�yÞ�3

þ½x̂2þðx̂1�yÞ�5þ 1

i2

ð�f2þa1Þ�

9

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

;

ð38Þ

where z2 ¼ x̂2� f2; �FðyÞ ¼ ½y; sinðyÞ�T; �GðyÞ ¼ ½1=ð1þ y2Þ;lnð1þ y2Þ�T; �g1ðyÞ ¼ 1=ð1þ y2Þ, and the first-order filter is

given by

i2f2 ¼ �f2 þ a1; f2ð0Þ ¼ a1ð0Þ: ð39Þ


123

In simulation, the design parameters are chosen as

c1 ¼ c2 ¼ �1 ¼ �2 ¼ 1 and i2 ¼ 10. The initial conditions

are set to be x1ð0Þ ¼ 0:5; x2ð0Þ ¼ �0:5; x̂1ð0Þ ¼ 0; x̂2ð0Þ¼ 0.

The simulation results are shown in Fig. 2, where it can

be seen that the states x1 and x2 can converge to a

small neighborhood around the origin, and the estimates

of x1 and x2, that is, x̂1 and x̂2, are also bounded.

Moreover, the filter signal f2 and control input u are all

bounded.

Example 2 Consider the following third-order system:

dx1 ¼ ½x2þ f1ðyÞ�dtþ g1ðyÞdw

dx2 ¼ �x2 þ x3 � 13x3

2þ f2ðyÞ

dtþ g2ðyÞdw

dx3 ¼ u� 12x3� 1

3x3

3� x22x3þ f3ðyÞ

dtþ g3ðyÞdw

y¼ x1

9

>

>

>

=

>

>

>

;

ð40Þ

where f1(y), g1(y), f2(y), g2(y) are kept as in Example 1,

f3ðyÞ ¼ y cosðyÞ, and g3ðyÞ ¼ yffiffiffi

y3p

. When the initial states

0 2 4 6 8 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (s)

x 1,

x 2

0 2 4 6 8 10

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time (s)

estim

ate

of

x 1,

x 2

0 2 4 6 8 10−2.5

−2

−1.5

−1

−0.5

0

time (s)

ζ 2

0 2 4 6 8 10−8

−6

−4

−2

0

2

4

6

time (s)

u

x1

x2

estiamte of x1

estimate of x2

Fig. 2 Response curves of

system (36)

0 2 4 6 8 100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

time (s)

x1

0 2 4 6 8 10−0.5

0

0.5

1

1.5

2

2.5

3

time (s)

x2


system (36) with u = 0


123

x1(0) = 0.5, x2(0) = 0, x3(0) = - 0.5, and the control u =

0, the response curves of states are shown in Fig. 3, from which

we can see that the states still cannot converge. Similarly,

compared with system (8), it can be easily shown that

A ¼0 1 0

0 �1 1

0 0 � 12

2

4

3

5;UðxÞ ¼0

� 13

x32

� 13

x33 � x2

2x3

2

4

3

5

¼� 1

20 0

0 �1 0

0 0 �1

2

4

3

5

013

x32

13

x33 þ x2

2x3

2

4

3

5 ¼ HJðxÞ

where J(x) satisfies Assumption 2. It is easy to verify that

when

P ¼1 0 0

0 1 0

0 0 1

2

4

3

5; L ¼� 9

2

� 154

� 158

2

4

3

5; and K ¼0

0

0

2

4

3

5;

the LMI (10) holds. Based on the control method proposed

in Sect. 4, the nonlinear observer is designed as follows

_̂x1 ¼ x̂2 � 92ðx̂1 � yÞ

_̂x2 ¼ x̂3 � x̂2 � 154ðx̂1 � yÞ � 1

3x̂3

2

_x3 ¼ u� 12

x̂3 � 158ðx̂1 � yÞ � 1

3x̂3

3 � x̂22x̂3

9

=

;

: ð41Þ

The virtual control functions a1,a2, and real control u are

designed as follows

a1 ¼ �c1y� y 12�4

1

j �FðyÞj4 þ 6ffiffi

2p

�22

j �GðyÞj4 þ 32j�g1ðyÞj2

�

a2 ¼ �c2z2 þ x̂2 þ 154ðx̂1 � yÞ þ 1

3x̂3

2 þ 1i2�f2 þ a1ð Þ

u ¼ �c3z3 þ 12

x̂3 þ 158ðx̂1 � yÞ þ 1

3x̂3

3 þ x̂22x̂3 þ 1

i3ð�f3 þ a2Þ

9

>

=

>

;

ð42Þ

where z2 ¼ x̂2� f2; z3 ¼ x̂3� f3; �FðyÞ ¼ ½y; sinðyÞ;cosðyÞ�T;�GðyÞ ¼ ½1=ð1þ y2Þ; lnð1þ y2Þ;

ffiffi

½p

3�y�T; �g1ðyÞ ¼ 1=ð1þ y2Þ,and the first-order filters are given as

i2f2 ¼ �f2 þ a1; f2ð0Þ ¼ a1ð0Þi3f3 ¼ �f3 þ a2; f3ð0Þ ¼ a2ð0Þ

�

: ð43Þ

In simulation, the design parameters are chosen as c1 ¼c2 ¼ c3 ¼ �1 ¼ �2 ¼ 1 and i2 ¼ 10. The initial conditions

0 2 4 6 8 10−15

−10

−5

0

5

10

15

time (s)

x 1, x

2, x

3

0 2 4 6 8 10−15

−10

−5

0

5

10

15

time (s)

estim

ate

of x

1, x

2, x

3

0 2 4 6 8 10−15

−10

−5

0

5

10

time (s)

ζ 1,

ζ 2

0 2 4 6 8 10−500

−400

−300

−200

−100

0

100

200

300

400

time (s)

u

ζ1

ζ2

estimate of x1

estimate of x2

estimate of x3

x1

x2

x3


system (40)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−50

0

50

100

150

200

250

300

350

400

450

time (s)

x 1, x2, x

3

x1

x2

x3

Fig. 3 Response curves of system (40) with u = 0


123

are set to be x1ð0Þ ¼ 0:5; x2ð0Þ ¼ 0; x3ð0Þ ¼ �0:5; x̂1ð0Þ ¼x̂2ð0Þ ¼ x̂3ð0Þ ¼ 0. The simulation results are shown in

Fig. 4, where it can be seen that the control performance is

still fairly satisfactory.

6 Conclusions

In this paper, we extend the DSC approach to a class of

stochastic nonlinear systems with the standard strict-feed-

back form. The proposed approach further simplifies the

backstepping design procedure where differentiating the

virtual control functions is avoided by introducing a first-

order filter in each step of backstepping design. This work

further enlarges the type of systems for which the DSC

approach can be utilized.

There are still some problems that need to be investi-

gated in the future. For example, it is interesting to extend

the proposed control approach to the more general class of

systems such as pure-feedback stochastic systems and

higher-order stochastic systems. Another interesting prob-

lem is how to combine the DSC approach with some other

approximators, such as fuzzy logic systems, to develop

some more efficient control schemes.

Acknowledgments The author would like to thank the anonymous

reviewers for their comments that improve the quality of the paper.

This work was supported by the Research Award Fund for

Outstanding Young Teachers in Higher Education Institutions of

Shandong Province.

Appendix 1: The proof of Theorem 1

Proof: Define l = x - v and Kðx; lÞ ¼ JðxÞ � Jðx� lÞ.From this together with UðxÞ ¼ HJðxÞ by Assumption 2, it

follows that

UðxÞ � UðvÞ ¼ HðJðxÞ � Jðx� lÞÞ ¼ HKðx; lÞ ð44Þ

lT ¼ xT � ðx̂� KC~xÞT ¼ ~xTðI þ KCÞT ð45Þ

and then, by the Mean Value Theorem, we have Kðx; lÞ ¼R 1

0oJos js¼x�sllds which, together with Assumption 2,

implies that

lTKðx;lÞ ¼ 1

2lT

Z

1

0

oJ

osþ oJ

os

� �T" #

s¼x�sl

ds

0

@

1

Al�0 ð46Þ

and then, recalling (12) and (44), we have

~xTPðUðxÞ � UðvÞÞ ¼ ~xTPHKðx; lÞ ¼ �lTKðx; lÞ� 0:

ð47Þ

Along the trajectory of (14), one has

LV0 ¼ ð~xTP~xÞ~xT½ðAþ LCÞTPþ PðAþ LCÞ�~xþ 2ð~xTP~xÞ~xTPðUðxÞ � UðvÞÞ þ 2ð~xTP~xÞ~xTPF

þ 2TrfGTð2P~x~xTPþ ~xTP~xPÞGg: ð48Þ

where for convenience, F(y) and G(y) are denoted by F and

G, respectively. Substituting (11) and (47) into (48) yields

LV0� � ð~xTP~xÞ~xTQ~xþ 2ð~xTP~xÞ~xTPF

þ 2TrfGTð2P~x~xTPþ ~xTP~xPÞGg: ð49Þ

Using Young’s inequality (see 4), together with (6) and (7),

we have

2ð~xTP~xÞ~xTPF� 2jPj2j~xj3jFj � 3

2�

4=31 jPj

8=3j~xj4

þ 1

2�41

y4j �FðyÞj4 ð50Þ

2TrfGTð2P~x~xTPþ ~xTP~xPÞGg� 3nffiffiffi

np

�22

y4j �GðyÞj4

þ 3nffiffiffi

np

�22jPj

4j~xj4 ð51Þ

where �1 and �2 are the positive design constants. The detailed

derivation of the inequality (51) is similar to [28, Eq. (A.7)].

Substituting (50) and (51) back into (49) yields (16).

Appendix 2: The proof of Theorem 2

Consider the following Lyapunov function

V ¼ 1

2ð~xTP~xÞ2 þ 1

4

X

n

i¼1

z4i þ

1

4

X

n�1

i¼1

.4iþ1 ð52Þ

and then along the trajectory of (35), noting Theorem 1, we

have

LV� �kminðPÞkminðQÞþ3

2�

4=31 jPj

8=3þ3nffiffiffi

np

�22jPj

4

� �

j~xj4

þy4 1

2�41

j �FðyÞj4þ3nffiffiffi

np

�22

j �GðyÞj4� �

�y4NðyÞ

þ3

2y2gT

1 ðyÞg1ðyÞþa1;2y3~x2�X

n

i¼1

ciz4i

þX

n�1

i¼1

ai;iþ1z3i ziþ1þ

X

n�1

i¼1

ai;iþ1z3i .iþ1�

X

n�1

i¼1

1

iiþ1

.4iþ1

þX

n�1

i¼1

.3iþ1Biþ1þ

3

2

X

n�1

i¼1

.2iþ1TrfCT

iþ1Ciþ1g: ð53Þ

Since for any M0 [ 0, the sets Pi :¼f12ð~xTP~xÞ2þ

14

Pij¼1 z4

j þ14

Pi�1j¼1.

4jþ1�M0g;i¼1;...;n are compact.

Therefore, Bi?1 and Tr{Ci?1T Ci?1} have their maximum

on Pi, denoted by Mi?1 and Ni?1, respectively.


123

Using Young’s inequality (see Lemma 2), we have

a1;2y3~x2�3

4ð�3a1;2Þ4=3y4 þ 1

4�43

j~xj4 ð54Þ

X

n�1

i¼1

ai;jz3i ziþ1�

3

4

X

n�1

i¼1

ðdiai;jÞ4=3z4i þ

1

4

X

n

i¼2

1

d4i�1

z4i ð55Þ

X

n�1

i¼1

ai;jz3i .iþ1�

3

4

X

n�1

i¼1

ð1iai;jÞ4=3z4i þ

1

4

X

n�1

i¼1

1

14i

.4iþ1

X

n�1

i¼1

.3iþ1Biþ1�

3

4

X

n�1

i¼1

ðmiMiþ1Þ4=3.4iþ1 þ

1

4

X

n�1

i¼1

1

m4i

ð56Þ

3

2

X

n�1

i¼1

.2iþ1TrfCT

iþ1Ciþ1g�3

4

X

n�1

i¼1

ðniNiþ1Þ4=3.4iþ1 þ

1

4

X

n�1

i¼1

1

n4i

:

ð57Þ

Substituting (54–57) and (21) into (53), we have

LV�� kminðPÞkminðQÞ�3

2�

4=31 jPj

8=3�3nffiffiffi

np

�22jPj

4� 1

4�43

� �

j~xj4

� c1�3

4ða1;2�3Þ4=3�3

4ða1;2d1Þ4=3�3

4ða1;211Þ4=3

� �

y4

�X

n�1

i¼2

ci�3

4ðai;iþ1diÞ4=3� 1

4d4i�1

�3

4ðai;iþ11iÞ4=3

!

z4i

� cn�1

4d4n�1

!

z4n

�X

n�1

i¼1

1

iiþ1

�3

414=3

i �3

4ðmiMiþ1Þ4=3�3

4ðniNiþ1Þ4=3

� �

.4iþ1

þ1

4

X

n�1

i¼1

1

m4i

þ1

4

X

n�1

i¼1

1

n4i

: ð58Þ

Choose the design parameter �1;�2;�3;ci;di;1i;mi;ni such

that

kminðPÞkminðQÞ �3

2�

4=31 jPj

8=3 � 3nffiffiffi

np

�22jPj

4 � 1

4�43

¼ k0 [ 0 ð59Þ

c1 �3

4ða1;2�3Þ4=3 � 3

4ða1;2d1Þ4=3 � 3

4ða1;211Þ4=3 ¼ c0

1 [ 0

ð60Þ

ci �3

4ðai;iþ1diÞ4=3 � 1

4d4i�1

� 3

4ðai;iþ11iÞ4=3 ¼ c0

i [ 0;

i ¼ 2; . . .; n� 1 ð61Þ

cn �1

4d4n�1

¼ c0n [ 0 ð62Þ

1

iiþ1

� 3

414=3

i � 3

4ðmiMiþ1Þ4=3 � 3

4ðniNiþ1Þ4=3 ¼ i0

iþ1 [ 0;

i ¼ 1; . . .; n� 1: ð63Þ

Substituting (59–63) into (58) yields

LV � � k0j~xj4 �X

n

i¼1

c0i z4

i �X

n�1

i¼1

i0iþ1.

4iþ1

þ 1

4

X

n�1

i¼1

1

m4i

þ 1

4

X

n�1

i¼1

1

n4i

� � ‘V þ �h

ð64Þ

where

‘ ¼min 2=k2maxðPÞ; 4c0

1; . . .; 4c0n; 4i0

2; . . .; 4i0n

� �

;

�h ¼ 1

4

X

n�1

i¼1

1

m4i

þ 1

4

X

n�1

i¼1

1

n4i

:

Based on (64), we easily obtain

dðEVÞdt¼ EðLVÞ� � ‘EV þ �h: ð65Þ

Let ‘[ �h=M, then d(EV)/dt B 0 on EV = M. Thus,

V B M is an invariant set, that is, if EV(0) B M, then

EV(t) B M for all t [ 0. Thus, (65) holds for all

V(0) \ M and all t [ 0.

Based on Lemma 1, inequality (64) further implies that

0�EVðtÞ�Vð0Þe�‘t þ �h

‘; 8t� 0: ð66Þ

The above inequality means that EV(t) is eventually

bounded by �h‘. Thus, recalling (52), all error signals of the

closed-loop system, that is, ~x; zi; .iþ1 are SGUUB in the

sense of fourth moment. Moreover, by adjusting the design

parameters, we can increase the value of ‘ and reducing the

value of �h. In other words, the boundedness of closed-loop

error signals above can be made arbitrarily small. This

completes the whole proof.

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