Adaptive fuzzy backstepping output feedback control for a class of uncertain stochastic nonlinear system in pure-feedback form

Author's Accepted Manuscript

Adaptive Fuzzy Backstepping Output Feedback Control for a Class of Uncertain Stochastic Nonlinear System in Pure- Feedback Form Yang Gao, Shaocheng Tong, Yongming Li

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S0925-2312(13)00667-X http://dx.doi.org/10.1016/j.neucom.2013.06.036 NEUCOM13501

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Received date: 1 November 2012 Revised date: 9 March 2013 Accepted date: 3 June 2013 Cite this article as: Yang Gao, Shaocheng Tong, Yongming Li, Adaptive Fuzzy Backstepping Output Feedback Control for a Class of Uncertain Stochastic Nonlinear System in Pure- Feedback Form, Neurocomputing, http://dx.doi.org/ 10.1016/j.neucom.2013.06.036 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Adaptive Fuzzy Backstepping Output Feedback Control for a Class of Uncertain Stochastic Nonlinear System in Pure- Feedback Form Yang Gao, Shaocheng Tong, Yongming Li Department of Basic Mathematics, Liaoning University of Technology Jinzhou, Liaoning 121000, China Abstract: This paper is concerned with the problem of adaptive fuzzy output feedback for a class of uncertain stochastic pure-feedback nonlinear systems with immeasurable states. With the help of fuzzy logic systems to approximate the unknown nonlinear functions, and a fuzzy state observer is designed to estimate the unmeasured states. By incorporating the filtered signals into the backstepping recursive design, a fuzzy adaptive output feedback control scheme is developed. It is proved that all the signals of the closed-loop system are bounded in probability, and also that the observer errors and the output of the system converge to a small neighborhood of the origin by appropriate choice of the design parameters. Simulation studies are included to illustrate the effectiveness of the proposed approach. Keywords: Stochastic nonlinear pure-feedback systems, backstepping technique, state observer, adaptive fuzzy control. 1᧪ Introduction In the past decade, the topic of adaptive control design for stochastic nonlinear

This work was supported by the National Natural Science Foundation of China (Nos.61074014, 61203008), and Program for Liaoning Innovative Research Team in University (No. LT2012013). Corresponding author. E-mail: [email protected] 1

systems has been an intensive area of research [1-3]. Some nonlinear control design methods such as Lyapunov function approach and backstepping technique [4] were extended to the case of stochastic nonlinear systems, and a great number of the results have been reported in the literature, see for example [5]-[18] and references therein. [5] first proposed an adaptive backstepping control design approach for stochastic systems by a risk-sensitive cost criterion. [6] solved the output feedback stabilization problem of stochastic nonlinear systems by using the quadratic Lyapunov function, while [7] and [8] developed backstepping control design approaches for nonlinear stochastic systems with Markovian switching, and [9]-[15] studied the control problems of uncertain stochastic nonlinear systems with time delays or actuator failures, respectively. To handle the unmodeled dynamics involved in the considered stochastic systems, [16]-[18] developed several robust adaptive output-feedback controllers based on changing supply function and small-gain theorem, the obtained adaptive controllers not only guarantee the stability of the closed-loop systems, but also have strong robustness to the unmodeled dynamics. It should be pointed out that the above mentioned results are only suitable for those nonlinear systems, in which the nonlinearities are known exactly or can be linearly parameterized. Therefore, they can not be applied to those stochastic systems with structured uncertainties. In the recent years, many approximation-based adaptive backstepping control approaches have been developed for deterministic nonlinear systems via fuzzy-logic-systems (FLSs) [19] or neural-networks (NNs) [20], for example [21]-[35]. The works in [21]-[25] studied adaptive fuzzy or NN backstepping

2

controllers

for

single-input

and

single-output

(SISO)

uncertain

nonlinear

strict-feedback systems, the works in [26]-[27] investigated adaptive fuzzy and NN controllers for multiple-input and multiple-output (MIMO) uncertain nonlinear systems, and those in [28]-[30] considered adaptive fuzzy or NN backstepping controllers for uncertain SISO nonlinear systems in pure-feedback form. To solve the problem of the states unmeasured, [31]-[35] proposed adaptive fuzzy or NN output feedback control approaches for uncertain nonlinear systems by designing state observer. In general, adaptive fuzzy or NN backstepping controllers can handle uncertain nonlinear systems without the requirement of the matching condition, and do not require that the nonlinear uncertainties be linearly parameterized, the FLSs or NNs are employed to model them instead. Inspired by the above results on the deterministic nonlinear systems, many adaptive fuzzy or NN backstepping control schemes have been recently developed for uncertain stochastic nonlinear systems [36]-[44]. [36]-[38] proposed adaptive fuzzy or NN control approaches for a class of SISO stochastic nonlinear systems with or without time delays. [39]-[41] developed adaptive output feedback controllers for SISO or MIMO stochastic nonlinear with the immeasurable states, while [42]-[44] extended the above results to uncertain stochastic large-scale nonlinear systems and proposed several adaptive fuzzy or NN decentralized output feedback control approaches. Compared to their deterministic counterpart, the main challenges are: (i) in the It oˆ stochastic differentiation, there exists a second-order derivative term, which brings difficulties in controller design and analysis; (ii) the structure and the

3

fundamental theory for stochastic interconnected systems are quite different and in fact much more complex than those in the deterministic case. Although a great progress has been made on adaptive fuzzy or NN backstepping control design for uncertain stochastic systems, the existing results are focused on the stochastic nonlinear systems in strict-feedback form, which cannot be applied to those uncertain stochastic nonlinear systems in pure-feedback form. As shown in [28]-[30], nonlinear pure-feedback systems have more representative form than nonlinear strict-feedback systems, since they have no affine appearance of the state variables to be used as virtual controls and actual control. This makes the control design of the nonlinear pure-feedback system quite difficult and challenging. Moreover, in practice, there are many systems falling into this category, such as mechanical systems, aircraft flight control systems, biochemical process and Duffing oscillator. Therefore, to study the adaptive output feedback control problem for stochastic nonlinear pure-feedback systems is important very important in both theory and real world applications. Motivated by the above observation, this paper investigates adaptive fuzzy output feedback control problem for a class of stochastic nonlinear systems in pure-feedback form. The considered stochastic nonlinear systems include unknown nonlinear functions and unmeasured states. In the control design, fuzzy logic systems are first employed to model the uncertain nonlinear systems, and then a fuzzy state observer is designed for estimate the unmeasured states. Based on the backstepping technique and by introducing the filtered signals into the backstepping control technique, an observer-based adaptive fuzzy output feedback control approach is developed. It is 4

proved that all the signals of the closed-loop system are bounded in probability, and also that the observer errors and the system output converge to a small neighborhood of the origin. 2. Problem formulation and Preliminaries 2.1 System Descriptions Consider the following stochastic uncertain nonlinear pure-feedback system:

­ dxi ° ®dxn ° y ¯ where x i

Fi ( x i , xi 1 )dt  gi ( y )T dZ , i 1,", n  1 Fn ( x n , u )dt  g n ( y )T dZ

.

(1)

x1

x1, x2 ,", xi T  Ri ,

i 1,2,", n is the state vector of system, u  R

and y  R are the input and output of system, respectively. Fi ( x i , xi1 ), i 1,2,", n are unknown smooth functions. Z is an independent standard Wiener process defined on a complete probability space.

Remark 1: Non-affine structure in the considered pure-feedback nonlinear system (1) covers many dynamic systems such as rolling mill systems, biological systems, aircraft flight, and mechanical systems. It can be seen that the pure-feedback system (1) has no affine appearance of state variables xi to be used as virtual controls D i , and of the actual control u itself. The cascade and non-affine properties make it quite difficult to find the explicit virtual controls and the actual control using the backstepping design. Let

Fi ( x i , xi 1 ) Fn ( x n , u )

xi 1  fi ( x i , xi 1 ), i 1,", n  1 u  fn ( xn , u)

5

.

System (1) is equivalent to

dxi

( xi 1  f i ( x i , xi 1 ))dt  g i ( y )T dZ , i 1,", n  1

dxn

(u  f n ( x n , u ))dt  g n ( y )T dZ

y

.

x1 (2)

Throughout this paper, the following standard assumptions are made for system (2). Assumption 1([40-42]): There exist known constants mi ! 0 , i

1,2,..., n such that

f i ( X 1 )  fi ( X 2 ) d mi X 1  X 2 . Assumption

gi  y

2

([16-17]):

There

exist

known

functions \ i  y

such

that

y\ i  y .

Our objective is to utilize fuzzy logic system to model the uncertain stochastic system (2) and design an adaptive fuzzy output feedback controller such that the closed-loop system is bounded in probability and the output of the system is as small as possible. 2.2 Fuzzy Logic Systems

A fuzzy logic system (FLS) consists of four parts: knowledge base, fuzzifier, fuzzy inference engine, and defuzzifier. The knowledge base is composed of a collection of fuzzy if-then rules of the following form: R l : If x1 is F1l and x 2 is F2l and ! and x n is Fnl ,

then y is G l , l where x

1,2, ! , N ,

( x1 , ! , x n ) T and y are the fuzzy logic system input and output,

respectively. Fuzzy sets Fi l and G l , associating with the fuzzy functions

P F ( xi ) and P G ( y ) , respectively. N is the rule number of if-then. l

l

i

With the singleton fuzzifier, center average defuzzification and product inference [45], the FLS can be expressed as

6

n N ¦l 1 yl –i 1 P F ( xi ) l

y ( x)

i

n N ¦l 1[–i 1 P F ( xi )]

,

(3)

l

i

max P G l ( y ) .

where y l

yR

Define the fuzzy basis functions as

Ml

n –i 1 P F ( x i ) l

i

N

l 1

Denoting T T

.

(4)

¦ (– P F ( x i ) ) n i 1

l

i

[ y1 , y 2 ,!, y N ] [T1 ,T 2 ,!,T N ] and M ( x) [M1 ( x ), ! , M N ( x)]T , then

fuzzy logic system (4) can be rewritten as y ( x)

fˆ ( x T ) T T M ( x) .

(5)

Lemma 1([45]): Let f ( x ) be a continuous function defined on a compact set : .

Then for any constant H >0, there exists a fuzzy logic system (5) such as fˆ ( x T )  H ( x) , H (x) d H .

f ( x) 3. Fuzzy state observer design

In this section, the state observer will be designed to estimate the unmeasured states in system (1), and then fuzzy adaptive output feedback control scheme is investigated based on the designed state observer. To begin with, rewrite (2) as ­ dxi °° ®dx n ° °¯ y

where 'f i

( xi 1  f i ( xˆ i , xˆi 1, f )  'f i ) dt  g i ( y )T dZ , i 1," , n  1 (u  f n ( xˆ n , u f )  'f n ) dt  g n ( y )T dZ

,

(6)

x1

f i ( x i , xi 1 )  f i ( xˆ i , xˆi 1, f ) , i 1,2,! , n  1 , 'f n

f n ( x n , u )  f n ( xˆ n , u f ) ;

xˆ i is the estimate of xi , which will be obtained by the state observer designed later.

7

xˆi , f and u f are the filtered signals for xˆ i and u , respectively. The filtered signals

defined by [30] and [46, 47] are as follows: xˆi , f

H L ( s ) xˆi , u f

H L ( s )u ,

(7)

where H L (s) is a Butterworth low-pass filter (LPF) with the cutoff frequency

ZC

1 rad/s for different values of n . The detailed descriptions about the

Butterworth low-pass filter can be referred to as Refs [30] and [46, 47]. Remark 2: The purpose of introducing the filtered signals xˆi , f and u f is to avoid the

so called algebraic loop problem existing in [28] and [29], and make it possible to design the state observer design and controller design for nonlinear systems in pure-feedback form. Assumption 3. There exist known constants W i , xˆ i 1  xˆ i 1, f d W i . ( W 1

1,2, ! , n , such that

i

0 ).

Remark 3: Based on the statements in [30] and [46, 47], most actuators have

low-pass property and the replacements xˆi , f | xˆi and u f | u are reasonable in the controller design. Therefore, it is reasonable in this paper to assume that xˆ i 1  xˆ i 1, f d W i , with W i being a known constant.

Rewrite (2) in the state space form x n

n 1

A x n  Ky  ¦ Bi ( f i ( xˆ i , xˆi 1, f )  'fi ) i 1

dZ  Bn ( f n ( xˆ n , u f )  'f n  u )  G ( y ) dt

where A

G( y)

ª  k1 « # « ¬« k n

º » , I » 0 ! 0¼»

>g1 ( y)" g n ( y)@

K

ª k1 º «#» , B n « » ¬«k n ¼»

y>\ 1 ( y )"\ n ( y )@

ª0 º «# » , « » ¬«1¼»

,

(8)

Bi

>0

" 1 " 0@ , T

y\ ( y ) .

Choose the vector K to make matrix A be a strict Hurwitz matrix. Thus, given a

8

QT ! 0 , there exists a matrix P

matrix Q

P T ! 0 satisfying

AT P  PA Q .

(9)

By Lemma 1, we can assume that the nonlinear functions in (6) can be approximated by the following fuzzy logic systems fˆi ( xˆ i , xˆi 1, f T i ) T iT M i ( xˆ i , xˆi 1, f ) , 1 d i d n . u f . The optimal parameter vector T i* is defined as

where xˆn 1, f

Ti*

arg minT i : i [sup( xˆ i , xˆ i1, f )U i1 uU i 2 fˆi ( xˆ i , xˆi 1, f Ti )  f i ( xˆ i , xˆi 1, f ) ] , 1 d i d n ,

where : i and U i1 u U i 2 are compact regions for T i and ( xˆ i , xˆi 1, f ) , respectively. Define f i ( xˆ i , xˆi 1, f )  fˆi ( xˆ i , xˆi 1, f T i ) , G i

Hi

f i ( xˆ i , xˆi 1, f )  fˆi ( xˆ i , xˆi 1, f Ti ) .

(10)

Assumption 4: There exist unknown constants H i* and G i* , i 1,2, ! , n such that

H i d H i* and G i d G i* . H i  Gi , i

Denote wi

2,! , n , by Assumption 4, it is clear that there is an

unknown constant wi* ! 0 such that wi d wi*

H i*  G i* .

Based on the system (2), design a fuzzy state observer as ­ xˆi ° ° ® xˆn ° ˆ °¯ y

xˆi 1  fˆi ( xˆ i , xˆi 1, f T i )  ki  y  xˆ1 , i 1,", n  1

u  fˆn xˆ n , u f T n  kn  y  xˆ1

,

(11)

xˆ1

or ­ ° xˆ n ® ° yˆ ¯ where C Let e

>1

n 1

A xˆ n  Ky  ¦ Bi fˆi ( xˆ i , xˆi 1, f T i )  Bn ( fˆn ( xˆ n , u f T n )  u ) i 1

,

C xˆ n

! 0 ! 0@ .

x n  xˆ n be an observer error vector, then from (8) and (12), we have 9

(12)

n

de

( Ae  ¦ Bi [[ f i ( xˆ i , xˆi 1, f )  fˆi ( xˆ i , xˆi 1, f T i )]  'f i ])dt  G ( y )dZ i 1

,

(13)

( Ae  G  'F )dt  G ( y )dZ

where G

[G 1 ,! , G n ]T and 'F

[ 'f 1 , ! , 'f n ]T .

Theorem 1. Consider the following Lyapunov candidate V0 for system (13) V0

1 T (e Pe) 2 . 2

(14)

Then AV0 is bounded by 4

AV0 d  p0 e  ; 0  3n n y 4 \ ( y ) 8

where p0

4

O  3 P 3  4m 4 n 2  3n n P , ; 0

4

,

(15)

n 1 1 *4 G  4n¦W ic4 . 2 i 2

Proof. Using (13), and according to [3], we have

eT Pe{eT ( AT P  PA)e  2eT P'F  2eT PG }

AV0

2Tr{G ( y )T (2 PeeT P  eT PeP)G ( y )} 4

d O e  2eT PeeT P'F  2eT PeeT PG

,

(16)

 2Tr{G ( y )T (2 PeeT P  eT PeP)G ( y )} where O

Omin (Q)Omin ( P) .

Note that by Assumption 1, we have

'f i

f1 ( xi , xi1 )  f i ( xˆi , xˆi1, f ) ( f i ( xi , xi1 )  f i ( xˆi , xˆi1 ))  ( f i ( xˆi , xˆi1 )  f i ( xˆi , xˆi1, f )) , d mi e  mi xˆi1  xˆi1, f d mi e  W ic1

where W ic1

miW i 1

For the convenience of the later derivations, we give the following Lemma. Lemma 2 (Young’s Inequality): For any vectors x , y  R n , there exist the

inequality x T y d (a p / p) x

p

q

 (1 / qa q ) y ,

10

(17)

where q ! 1, p ! 1, ( p  1)( q  1) 1 . By Lemma 2 and Assumptions 3-4, we can obtain following inequalities as 2eT PeeT P'F d

3 P 2

2eT PeeT PG d

8 3

n 1

e  4m 4 n 2 e  4n ¦W ic4 , 4

4

(18)

i 2

3 P 2

8 3

4

e 

1 *4 G , 2

(19)

2Tr {G ( y )T (2 PeeT P  eT PeP)G ( y )} d 2n n G T ( y )(2 PeeT P  eT PeP)G ( y ) , 4

4

d 3n n y 4 \ ( y )  3n n P e where G *

[G1* ," , G n* ] , m

(20)

4

max^m1 , m2 ," , mn `.

By substituting (18)-(20) into (16) result in (15). 4. Adaptive Output Feedback Control Design

In this section, an adaptive fuzzy output feedback controller will be designed by the backstepping technique and the fuzzy state observer. The controller design in this paper is composed of n steps. Each of them is based on the change of coordinates

F1 Fi

y , xˆi  D i1 ( xˆ1 ," , xˆ i1 ,T1 ,T 2 ," ,T i1 , y )

(21)

where D i 1 , i 1," , n is an intermediate control function, and u (t ) is designed in the last step. For notation simplicity, D i 1 ( xˆ1 ,", xˆ i 1 ,T1 ,T 2 ,",Ti 1 , y ) and Mi ( xˆ i , xˆi 1, f ) will be denoted as D i 1 and Mi in the sequel. Step 1. By the first equation in (8), and by using F 2

11

xˆ2  D1 , we have

dF1

( x2  f1 ( x1 , x2 ))dt  g1 ( y )T dZ , ~ ( F 2  D1  e2  T1T M1  'f1  T1T M1  H1 )dt  g1 ( y )T dZ

~ where T1 is the estimate of T1* and T1

(22)

T1*  T1 .

Choose the Lyapunov function candidate as V1 V0 

1 ~2 1 4 1 ~T ~ F1  T1 T1  H1 , 2J 1 4 2J 1

where J 1 ! 0 and J 1 ! 0 are design parameters, H~1

(23)

H1*  Hˆ1 , Hˆ1 is the estimate of

H1* . By (22) and (23), we have 1 1 ~ ~ 1 AV1 d AV0  A( F14 )  T1TT1  H~1H~1 4 J1 J1 3 2 F1 g1 ( y )T g1 ( y ) . 2 ~ 1 1  H~1H~1  F13 ( F 2  D1  e2  'f1  T1T M1 )  T1T (M1F13  T1 ) 4

4

d  p0 e  ;  3n n y 4 \ ( y )  F13 H1* 

J1

(24)

J1

By Young’s inequality and Assumption 3, the following inequalities can be obtained: 3 4 1 4 F1  e , 4 4 3 1 3 1 3 4 F13'f1 d F1 (m1 e  W 2c ) d F14  m14 e  F14  W 2c4 , 4 4 4 4 3 2 T 3 4 F 1 g1 ( y ) g1 ( y ) F1\ 1 ( y )T \ 1 ( y ) , 2 2 3 1 F13 F 2 d F14  F 24 . 4 4

F13 e2 d

(25) (26) (27) (28)

By substituting (25)-(28) into (24), (24) becomes 1 1 4 4 4  m1 ) e  F13 (3F1  D1  3n n F1 \ ( y )  Hˆ1 tanh( F13 k ) 4 4 3 1 3 . (29)  F1\ 1 ( y )T \ 1 ( y )  T1T M1 )  F1 H1*  F13H1* tanh( F13 k )  F 24 2 4 ~ T 1 1  W 2c4  T1T ( 1  M1F13 )  H~1 ( Hˆ1  F13 tanh( F13 k ))  ; 0 4 J1 J1

AV1 d ( p0 

Design the intermediate control function D1 and the adaptive functions T1 and Hˆ1

12

as 4

D1

c1 F1  3F1  3n n F1 \ ( y )  Hˆ1 tanh( F13 k )

, 3 n 1  F1\ 1 ( y )T \ 1 ( y )  T1T M1  F1 (\ 1 ( y )) 4 2 2

T1 Hˆ1

(30)

J 1M1F13  V 1T1 ,

(31)

J 1F13 tanh( F13 k )  V 1Hˆ1 ,

(32)

where V 1 ! 0 and V 1 ! 0 . By substituting (30)-(32) into (29) and by using the following inequality 3

F1  F13 tanh( F13 k ) d 0.2785k

k c , ( k ! 0 ).

(29) becomes 4

AV1 d  p1 e  c1 F14 

where p1

V ~ V 1 4 n 1 4 F 2  ;1  1 T1TT1  1 H~1Hˆ1  y (\ 1 ( y )) 4 , J1 J1 2 4

1 m4 p0   1 , ;1 4 4

(33)

1 ; 0  H1*k c  W 2c4 . 4

Step i . ( 2 d i d n  1) . By (11) and (21), we have dF i

i 1 wD i 1 wD d xˆ j  ¦ i 1 dT j ˆj j 1 wT j 1 wx

i 1

( xˆi 1  ki e1  TiT Mi )dt  ¦ j

wD i 1 wD wD dHˆ j  ¦ i 1 d wˆ j  i 1 dy ˆj wy 1 wHˆ j j 2 ww

i 1

i 1

¦ j

wD ( xˆi 1  H i  T i Mi  wi  i 1 (e2  G1  'f1 ) wy

,

(34)

~T



wD 1 w 2D i 1 g1 ( y )T g1 ( y ))dt  i 1 g1 ( y )T dZ 2 wy 2 wy

where

wD i 1  i 1 wD i 1  xˆ  Tj ˆj j ¦ 1 wx j 1 wT j

i 1

Hi

ki e1  TiT Mi  ¦ j

i 1 i 1 wD wD wD ( i 1 )( xˆ2  T1T M1 )  ¦ i 1 Hˆ j  ¦ i 1 wˆ j ˆj wy j 1 wHˆ j j 2 ww

.

(35) 13

Choose the following Lyapunov function candidate as: Vi 1 

Vi

1 ~2 1 4 1 ~T ~ Fi  Ti Ti  wi , 2J i 4 2J i

(36)

~ wi*  wˆ i and Ti

~ where J i ! 0 and J i ! 0 are design parameters. w i

Ti*  Ti are

the parameter errors, while wˆ i and T i are the estimate of wi* and T i* , respectively. By (34) and (36), we have AVi

1 1 ~ ~ 1 ~ ~ w AVi 1  A( F i4 )  T i TT i  w 4 Ji Ji i i ~ wD AVi 1  F i3 ( xˆi 1  H i  T i T Mi  wi  i 1 (e2  G1  'f1 ) wy

.

(37)

~ ~ wˆ Ti TTi w 1 w 2D i 1 3 2 wD i 1 2 T T  g1 ( y ) g1 ( y ))  F i (  i i ) g1 ( y ) g1 ( y )  2 wy Ji Ji 2 wy 2

By Assumptions 1, 3 and 4, and Young’s inequality, we can obtain that wD i 1 3 wD 1 4 e2 d ( i 1 ) 3 F i4  e , wy 4 wy 4 4

 F i3

 F i3

(38)

wD i 1 wD i 1 3 F i 'f1 'f1 d wy wy d

3 wD wD i 1 3 F i m1 e  W 2c d ( i 1 ) 3 F i4 , 4 wy wy



4



m14 4 3 wD i 1 3 4 1 4 e  ( ) F i  W 2c 4 4 4 wy 4



(39) 3 wD 1 wD i 1 G1 d ( i 1 ) 3 F i4  G1*4 , 4 wy 4 wy 4

 F i3

3 2 wD i 1 2 1 w 2D i 1 Fi ( g1 ( y )T g1 ( y ) ) g1 ( y )T g1 ( y )  F i3 wy wy 2 2 2 w 2D i 1 2 3 1 4 wD 1 9 ) ) F i  y (\ 1 ( y )) 4 d ( F i ( i 1 ) 4  F i3 ( 2 wy 2 wy 4 4 3 4

1 4

F i3 F i 1 d F i4  F i41 . By (38)-(42) and using mathematical induction, we have 14

(40)

,

(41)

(42)

i 1 ~ wD 4 AVi d  pi 1 e  ¦ c j F 4j  ; i 1  F i3 ( xˆi 1  H i  T i T Mi  wi  3( i 1 ) 3 F i wy j 1 4

wD w 2D i 1 2 n  i 1 4 9 1  F i  ( F i ( i 1 ) 4  F i3 ( y (\ 1 ( y )) 4 ) ))  2 wy wy 4 4 2 ~ ~ wˆ T TT w 1 m4 1 1 1 1 4  i i  i i  (  1 ) e  y 4 (\ 1 ( y )) 4  F i41  G1*4  W 2c4 Ji Ji 4 4 2 4 4 4 i 1 V V j ~T V ~T w T j T j  1 H~1Hˆ1  ¦ j w j j J J J 1 j 2 1 j j

i 1

¦ j

, (43)

i 1

d  pi e  ¦ c j F 4j  F i3 ( xˆi 1  H i  wˆ i tanh( F i3 / k )  3( 4

j 1

wD i 1 3 ) Fi wy 4

wD w 2D i 1 2 ni 4 9 1 1  F i  ( F i ( i 1 ) 4  F i3 ( y (\ 1 ( y )) 4  F i41 ) ))  2 wy wy 4 4 2 4 i 1  ~ T ~ ( wˆ i  F 3 tanh( F 3 / k ))  V j T~TT  ; i  T i T ( i  F i3Mi )  w ¦ j j i i i

Ji



where ; i

Ji

j 1

Jj

i 1 V V1 ~ ~T w H1Hˆ1  ¦ j w j j J1 j 2 J j

1 1 ; i 1  wi*k c  G1*4  W 2c4 , pi 4 4

pi 1 

1 m14  . 4 4

Design the intermediate control function D i and the adaptive functions T i and wˆ i as: wD i 1 3 ) Fi wy 4

Di

ci F i  H i  wˆ i tanh( F i3 / k )  3(

wD w 2D i 1 2 9 1  F i  ( F i ( i 1 ) 4  F i3 ( ) ) wy wy 2 4 4

Ti wˆ i

,

(44)

J i F i3Mi  V iT i ,

(45)

J i F i3 tanh( F i3 / k )  V i wˆ i ,

(46)

where ci ! 0 , V i ! 0 and V i ! 0 . Substituting (44)-(46) into (43) yields i

AVi d  pi e  ¦ c j F 4j  4

j 1

ni 4 1 y (\ 1 ( y )) 4  F i41 2 4

i V V ~ V ~T w  ; i  ¦ j T jTT j  1 H~1Hˆ1  ¦ j w j j J J J j 1 j 2 1 j j i

.

(47)

Step n . In this step, the actual control input u will be obtained. By (11) and (21), we 15

have ~ wD (u  H n  T nT Mn  wn  n1 (e2  'f1  G1 ) wy

dF n

1 w 2D n1 wD  g1 ( y)T g1 ( y))dt  n1 g1 ( y)T dZ 2 2 wy wy

,

(48)

where

wD n1  n1 wD n1  xˆ  Tj ˆj j ¦ 1 wx j 1 wT j

n1

k n e1  T nT M n  ¦

Hn

j

n1 n 1 wD wD wD ( n1 )( xˆ2  T1T M ( xˆ1 , xˆ2, f ))  ¦ n1 Hˆ j  ¦ n1 wˆ j ˆ ˆj wy j 1 wH j j 2 ww

.

Choose the following Lyapunov function candidate: Vn 1 

Vn

1 4 1 ~T ~ 1 ~2 wn . Fn  Tn Tn  4 2J n 2J n

(49)

Design controller u , and the adaptive functions Tˆn and wˆ n as cn F n  H n  wˆ n tanh F n3 / k  3(

u

wD n 1 3 ) Fn wy 4

1 9 1 wD w 2D n 1 2 ) )  F n  ( F n ( n 1 ) 4  F n3 ( 4 4 4 wy wy 2

Tn wˆ n

,

(50)

J n F n3M n  V nT n ,

(51)

J n F n3 tanh( F n3 / k )  V n wˆ n .

(52)

Similar to the derivations in Step i , we have n

n

n V V j ~T V ~T w , T j T j  1 H~1Hˆ1  ¦ j w j j J J J j 2 1 j j 1

AVn d  pn e  ¦ c j F 4j  ; n  ¦ 4

j 1

where pn

pn 1 

j

(53)

1 m14  . 4 4

By the completion of squares, we have

V j ~T T T Jj j j

~

V j Tj V j ~T * ~ T j (T j  T j ) d  Jj 2J j

16

2



V j T *j 2J j

2

,

(54)

V1 ~ V H~ 2 V H *2 H1Hˆ1 d  1 1  1 1 , J1 2J 1 2J 1

(55)

~ 2 V w*2 Vj ~ V jw j ˆ w jZ j d   j j . Jj 2J j 2J j

(56)

Substituting (54)-(56) into (53) results in n

n

AVn d  pn e  ¦ c j F  ¦ 4

4 j

j 1

Let U

~

V j Tj

j 1

2J j

2

~ V H~ 2 n V w  1 1 ¦ j j P. 2J 1 j 2 2J j 2

(57)

min( 2 pn / O2min P ,4c1 , " ,4cn , V 1 , " , V n , V 1 , " , V n ) , and n

P ;n  ¦ j 1

V j T *j 2J j

2

V H *2 n V w  1 1 ¦ j j . 2J1 j 2 2J j *2

then (57) can be rewritten as A V d  UV  P .

(58) Based on (58), and using the same arguments as [36]-[43], we can follow that all the signals of the closed-loop system are bounded in probability. Moreover, by adjusting the design parameters, the observer errors and the output of the system can be made arbitrarily small. The above design procedures and analysis are summarized in the following theorem: Theorem 2: For the stochastic nonlinear system (1) under assumptions 1-4, the state

observer (11), the controller (50) with the intermediate control (44), (55) and parameters adaptive laws (44), (45) and (46), (51) and (52) guarantee that all the signals in the closed-loop system are bounded in probability, Moreover, the observer errors and the output of the system can be made arbitrarily small by choosing 17

appropriate design parameters. 5. Simulation Study

In this section, a simulation example is presented to illustrate the effectiveness of the proposed adaptive fuzzy control approach. Example : Consider a nonlinear system governed by the following form:

­ ° dx1 ° ° ®dx2 ° ° y ° ¯

x12 (sin x1 )5 x2   )dt  0.5 y 2 dZ 10 20 9 x x (sin x1 )5 u 3 (u  2 1   )dt  0.5 y 2 dZ . 10 20 10 x1 (

(59)

Choosing fuzzy membership functions for the variables as xˆ1 , xˆ2 , xˆ2 f and u f as follows:

P F ( xˆ1 ) exp[ l 1

P F ( xˆ2 f ) exp[ 

( xˆ1  5  2l ) 2 ] , P F l ( xˆ2 ) 2 2

( xˆ2 f  3  l ) 2

l 3

5

] , P F l (u f ) 4

exp[

exp[ 

( xˆ2  3  l ) 2 ], 5

(u f  5  3l ) 2 7

],l

1, " ,5 .

The Butterworth low-pass filter is chosen as H L ( s ) 1 /( s 2  1.414 s  1) . According to [45], the fuzzy logic systems are constructed as fˆ1 ( xˆ1 , xˆ2 f T1 ) T1T M ( xˆ1 , xˆ2 f ) and fˆ2 ( xˆ1 , xˆ2 , u f T 2 ) T 2T M ( xˆ1 , xˆ2 , u f ) . Setting k1

25 and k 2 ­° xˆ1 ® °¯ xˆ2

20 , then we obtain the state observer as xˆ2  25 y  xˆ1  fˆ1 ( xˆ1 , xˆ2, f T1 )

u  fˆ2 xˆ1 , xˆ2 , u f T 2  20 y  xˆ1

.

(60)

Intermediate control function D1 and controller u , the adaptive laws T1 , T 2 , Hˆ1 and

Zˆ 2 are as follows D1

4

c1 F1  3F1  3n n F1 \ ( y )  Hˆ1 tanh( F13 k )

, 3 n 1  F1\ 1 ( y )T \ 1 ( y )  T1T M1  F1 (\ 1 ( y )) 4 2 2

18

(61)

1 wD c2 F 2  H 2  wˆ 2 tanh F 23 / k  F 2  3( 1 ) 3 F 2 4 wy 4

u

1 9 1 wD w 2D  F n  ( F 2 ( 1 ) 4  F 23 ( 21 ) 2 ) 4 4 4 wy wy

T1 Hˆ1

,

J 1M1F13  V 1T1 ,

(63)

J 1 F13 tanh( F13 k )  V 1Hˆ1 ,

T2 wˆ 2

(62)

(64)

J 2 F 23M 2  V 2T 2 ,

(65)

J 2 F 23 tanh( F 23 / k )  V 2 wˆ 2 .

(66)

The design parameters in the above control scheme (61)-(66) are chosen as J 1 1 ,

J 2 1 , J 1 1.8 , J 2 1.5 , c1 1 , c2 initial condition are chosen as x1 (0)

2 , V1 V 2

2 and V 1 V 2

1.4 . The

0 , x2 (0) 1 , xˆ1 (0) 0 , xˆ2 (0) 0 , the other

parameters are chosen as 0. The simulation results are shown by the following Figs. 1-3, respectively. From the simulation results in Figs. 1-3, it can be seen that the proposed fuzzy output feedback control approach can achieve good control performances.

Fig. 1. State xˆ1 (dotted) and x1 (solid).

19

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

Fig. 3. Controller u . Remark 4: It should be mentioned that the existing adaptive output control

approaches in [38], [39] and [40] can also be applied to the considered system (59) if the term of u 3 / 10 does not exist, i.e., system (59) bacomes a strict-feedback nonlinear system. However, since the considered system (59) in this paper contains the term of u 3 / 10 , and it is a pure-feedback system. Therefore, the above mentioned adaptive output feedback control approaches [38], [39] and [40] can not be applied to control the system (59). So far no output feedback control approaches for stochastic nonlinear pure-feedback systems have been reported.

20

6. Conclusion

In this paper, an adaptive fuzzy backstepping output feedback control approach has been proposed for a class of uncertain SISO stochastic nonlinear systems in pure-feedback form. With the help of fuzzy logic systems to approximate the unknown nonlinear functions and by the filter transforms, a fuzzy state observer has been designed to estimate the unmeasured states. By applying the backstepping design approach, a fuzzy adaptive output feedback control scheme has been developed. It has been proved that all the signals of the closed-loop system are bounded in probability, and also that the observer errors and the system output converge to a small neighborhood of the origin by appropriate choice of the design parameters. Simulation studies have been included to illustrate the effectiveness of the proposed approach. Reference

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Yang Gao, received the B.S. degree from the School of Computer Science, Southwest Petroleum University, Chengdu, China, in 2008. He is now working towards the M.E. degree in applied mathematics from Liaoning University of Technology, Jinzhou, China. His current research interests include fuzzy control, adaptive control, and stochastic control.

Shaocheng Tong, received the B.A. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.A. degree in fuzzy mathematics from Dalian Marine University, PRC, and the Ph.D degree in fuzzy control from Northeastern University, PRC, in 1982, 1988, and 1997, respectively. Currently, he is a Professor in the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou, PRC. His research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control. Yongming Li, received the B.S. degree and the M.S. degree in applied mathematics from Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively. He is currently a lecturer in the Department of Basic Mathematics, Liaoning University of Technology. His current research interests include 26

fuzzy and neural networks control, and nonlinear adaptive control.

27