Robust adaptive fuzzy output feedback control for stochastic nonlinear systems with unknown control direction

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Neurocomputing 106 (2013) 31–41

Contents lists available at SciVerse ScienceDirect

Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Robust adaptive fuzzy output feedback control for stochastic nonlinear systems with unknown control direction$ Tong Wang, Shaocheng Tong n, Yongming Li Department of Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121000, China

a r t i c l e i n f o

abstract

Article history: Received 19 June 2012 Received in revised form 14 September 2012 Accepted 18 October 2012 Communicated by H. Zhang Available online 16 November 2012

This paper discusses the problem of adaptive fuzzy output feedback control for a class of uncertain stochastic nonlinear strict-feedback systems. The concerned systems have certain characteristics, such as unknown nonlinear functions, dynamical uncertainties, unknown control direction and unmeasured state variables. In this paper, the fuzzy logic systems are used to approximate the unknown nonlinear functions, and a ﬁlter state observer is developed for estimating the unmeasured states. To solve the problems of the dynamical uncertainties and the unknown control direction, the changing supply function and Nussbaum function techniques are incorporated into the backstepping recursive design technique, and a new robust adaptive fuzzy output feedback control approach is constructed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closedloop system are bounded in probability, and also that the observer errors and the output of the system can be regulated to a small neighborhood of the origin by choosing design parameters appropriately. A simulation example is provided to show the effectiveness of the proposed approach. & 2012 Elsevier B.V. All rights reserved.

Keywords: Stochastic nonlinear systems Fuzzy adaptive backstepping control Changing supply function Nussbaum function K-ﬁlters

1. Introduction In the past decades, many approximation-based adaptive backstepping control approaches have been developed to control uncertain nonlinear strict-feedback systems with unstructured uncertainties via fuzzy-logic-systems (FLSs) or neural-networks (NNs), see for example [1–19]. Works in [1–9] are for single-input and single-output (SISO) nonlinear systems, works in [10–12] are for multiple-input and multiple-output (MIMO) nonlinear systems, and works in [11–19] for SISO or MIMO nonlinear systems with immeasurable states, respectively. In general, these adaptive fuzzy or neural network backstepping control approaches provide a systematic methodology of solving control problems of unknown nonlinear systems, where fuzzy systems or neural networks are used to model the uncertain nonlinear systems, and then an adaptive fuzzy or neural network controller is developed based on the backstepping design principle. Two of the important features of these adaptive approaches include (i) they can be used to deal with those nonlinear systems without satisfying the matching conditions, and (ii) they do not require the unknown nonlinear functions being linearly parameterized. Therefore, nowadays, the approximation-

$ This work was supported by the National Natural Science Foundation of China (Nos.61074014, 61203008), and Program for Liaoning Innovative Research Team in University (LT2012013). n Corresponding author. E-mail address: [email protected] (S. Tong).

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.10.013

based adaptive fuzzy backstepping control has become one of the most popular design approaches in nonlinear control ﬁeld. It is well known that stochastic systems have been used in a variety of ﬁelds, such as chemical process, biology, ecology, and also can be found in control and information systems. The investigation on stochastic systems have received considerable attention in the past years, and a great number of the results have been reported in the literature, see for example [20–30]. Adaptive backstepping controllers are proposed in [20,21] for stochastic nonlinear systems in strict-feedback form. Adaptive backstepping controllers are developed in [22–27] for stochastic nonlinear systems with time delays or stochastic jump systems. Moreover, several adaptive output-feedback controllers are investigated in [28–30] for strictfeedback stochastic nonlinear systems by using linear state observer. However, the above mentioned results are only suitable for those nonlinear systems with nonlinear dynamics models being known exactly or with the unknown parameters appearing linearly with respect to known nonlinear functions, they can not be applied to those stochastic systems with structured uncertainties. In order to deal with the structured uncertainties included in the stochastic nonlinear systems in strict-feedback form, by combining the fuzzy logic systems and neural networks with the backstepping design technique, several adaptive NN or fuzzy backstepping control schemes have been developed. For example, [31–33] proposed adaptive fuzzy output feedback control approach for a class of SISO stochastic nonlinear systems, while [34 and 35] extended the above results to a class of stochastic large-scale

32

T. Wang et al. / Neurocomputing 106 (2013) 31–41

nonlinear systems. However, the aforementioned adaptive fuzzy or NN controllers have two main limitations as follows: One is that they do not consider the problem of unmodeled dynamics. The other is that they assume that the control directions are known. As stated in [15,18], the unmodeled dynamics and the unknown signs of control directions often exist in many practical nonlinear system, they are also the major source of resulting in the instability of the control systems. Therefore, to study the stochastic nonlinear systems with consideration of dynamical uncertainties and the unknown control directions is very important in control theory and applications. To handle the unknown control directions, Authors in [36,37] proposed adaptive fuzzy and NN backstepping control methods for stochastic nonlinear systems by using the Nussbaum function technique, and the stabilization properties of the control systems are achieved. However, the adaptive control schemes in [36,37] are restricted to a class of stochastic nonlinear systems without the unmodeled dynamics. Moreover, they need the assumption that all the states are available for the controllers design. Motivated by the above observation, in this paper, an adaptive fuzzy output feedback control scheme is investigated for a class of stochastic nonlinear systems without satisfying the matching condition. The considered stochastic nonlinear systems include unknown nonlinear functions, dynamical uncertainties and unknown control direction, and unmeasured states. In the control design, fuzzy logic systems are ﬁrst employed to model the uncertain nonlinear systems, and then a fuzzy ﬁlters is developed estimate the unmeasured states. To solve the problems of the unknown control direction and unmodeled dynamics, by introducing the Nussbaum functions and the changing supply function technique into the backstepping recursive design, a new robust adaptive fuzzy backstepping output feedback control scheme is constructed. It is demonstrated that all the variables of the resulting closed-loop system are bounded in probability, and that the observer errors and the output of the system converge to a small neighborhood of the origin by choosing design parameters appropriately.

Assumption 1. ([30,31]): For each 1r i rn, there exist unknown positive constantspni such that Di ðx, zÞ rpn c ðyÞ þpn c z i1

i

i

i2

g ðxÞ r pn c ðyÞ i i i3 whereci1 ðyÞ, ci2 z and ci3 ðyÞ are known nonnegative smooth functions with ci1 ð0Þ ¼ ci2 ð0Þ ¼ ci3 ð0Þ ¼ 0. Assumption 2. ([30,31]): For each z -subsystem in (1), there exist function V z ðzÞ and known k1 functions a 9z9 , a z , að9z9Þ, g 9y9 , cz and c0 such that a z r V z ðzÞ r a z and ‘V z r g y a z @V z =@z r c z and :q ðz,yÞ: r c z 2 z 0 Control objective: The control task is to design an adaptive output feedback controller using the output y and state estimations x^ i so that all the variables of the closed-loop system are bounded in probability and the outputs of the system can be regulated to a small neighborhood of the origin in probability. In order to cope with the unknown control direction, the Nussbaum gain technique is employed in this paper. Deﬁnition 1. ([36,37]). A function NðBÞ is called a Nussbaum-type function if it has the following properties: Z s 1 lim sup NðBÞdB ¼ 1 s-1 s 0 lim inf

s-1

1 s

Z

s

NðBÞdB ¼ 1 0

From the above deﬁnition, we know that the Nussbaum functions have inﬁnite gains and inﬁnite switching frequencies. There are many functions satisfying the above conditions, for example,exp x2 cos p=2 xÞ, x2 cosðxÞ and x2 sinðxÞ. Let us give a useful lemma on stochastic differential equation as the following dx ¼ f ðt,xÞdt þhðt,xÞdw

ð2Þ

where the deﬁnition of x and w are the same as in (1). Denote C 2,1 : Rn R þ ; R þ as the family of all nonnegative functions V ðx,t Þ A Rn R þ , which are continuously twice differentiable in x and one differentiable in t, and written as C 2,1 for simpliﬁcation.

2. System descriptions and preliminary results 2.1. System descriptions and basic assumptions Consider the following uncertain stochastic nonlinear system in strict feedback form: dz ¼ q1 ðz,yÞdt þ q2 ðz,yÞdw dxi ¼ ½xi þ 1 þ f i x i þ Di ðx, zÞdt þ g i ðxÞdw i ¼ 1,. . .,n1,

Lemma 1. ([36]). Consider the stochastic system (2), assume that there exists functions V ðx,t Þ A C 2,1 , smooth function B1 : R þ -R and Nussbaum type even function NðUÞ; let w be a nonnegative random variable, MðtÞ be a real valued continuous local martingale with Mð0Þ ¼ 0 such that Z t V ðx,t Þ r w þ ect b0 NðB1 ÞB_ 1 þ B_ 1 Þect dt þ Mðt 0

dxn ¼ ½b0 ZðyÞu þf n x n þ Dn ðx, zÞdt þ g n ðxÞdw y ¼ x1

ð1Þ

where x i ¼ ½x1 ,x2 ,. . .,xi T A Ri , i ¼ 1,2,. . .,n (x ¼ x n ) are the states, u and y are the control and output of the system, z is unmodeled dynamics and Di ðx, zÞ are the dynamic disturbances. f i x i , i ¼ 1,2,. . .,n are unknown smooth nonlinear functions. q1 ðz,yÞ, q2 ðz,yÞ, Di ðx, zÞ and g i ðxÞ are uncertain functions; ZðyÞ a 0is a known smooth nonlinear function; b0 a 0 is unknown constant and the sign of b0 is unknown; w A R is an independent standard Wiener process deﬁned on a complete probability space. In this paper, it is assumed that the functionsf i x i ,g i ðxÞ,qi ðz,yÞ and Di ðx, zÞ satisfying the locally Lipschitz condition, and only the outputyis available for measurement.

Then the functions V ðx,t Þ, B1 ðtÞ and be bounded in probability.

Rt

0

ðb0 NðB1 ÞB_ 1 þ B_ 1 Þdt must

2.2. Fuzzy logic systems A fuzzy logic system (FLS) consists of four parts: the knowledge base, the fuzziﬁer, the fuzzy inference engine, and the defuzziﬁer. The knowledge base is composed of a collection of fuzzy. If-then rules of the following form: Rl : If x1 is F l1 and x2 is F l2 and y and xn is F ln , Then y is Gl ,l ¼ 1,2,. . .,N where x ¼ ðx1 ,x2 . . .xn ÞT and y are FLS input and output, respectively,mF l ðxi Þ and mGl ðyÞ are the memberi

ship function of fuzzy sets F li and Gl , N is the number of inference rules. Through singleton fuzziﬁer, center average defuzziﬁcation

T. Wang et al. / Neurocomputing 106 (2013) 31–41

and product inference, the FLS can be expressed as PN Qn l ¼ 1 yl i ¼ 1 mF li ðxi Þ yðxÞ ¼ PN Qn l ¼ 1 ½ i ¼ 1 mF l ðxi Þ

The system (7) is equivalent to ð3Þ

i

where yl ¼ maxy A R mGl ðyÞ. Deﬁne the fuzzy basis functions as Qn i ¼ 1 mF l ðxi Þ fl ¼ PN Qn i l ¼ 1 ½ i ¼ 1 mF l ðxi Þ

y

yT ¼ ½y1 ,y2 ,. . .,yN ¼ ½y1 , y2 ,. . ., yN

and fðxÞ ¼ ½f1 ðxÞ, f2 ðxÞ, , fN ðxÞT ,then fuzzy logic system (3) can be rewritten as T

yðxÞ ¼ y fðxÞ

ð4Þ

Lemma 2. ([38]). For any continuous function f ðxÞ deﬁned over a compact set O and any given positive constant e, there exist a n FLS (4) and a ideal parameter vector y such that nT supf ðxÞy fðxÞr e xAO

3. Filter state observer design Note that the functions f i ðUÞ in (1) are unknown, and the states x2 ,y,xn are not available for the controller design. Therefore, the FLSs are employed to approximate f i ðUÞ and a ﬁlter state observer is established to estimate the unmeasured states. on Lemma 2, we assume that the nonlinear functions Based f i x i in (1) can be approximated by the following FLSs T T f i x i 9yi ¼ yi fi x i , f^ i x^ i yi ¼ yi fi x^ i 1 ri r n where x^ i ¼ ½x^ 1 , x^ 2 ,. . ., x^ i T are the actual estimations of x i ¼ ½x1 , x2 ,. . .,xi T ,i ¼ 1,2,. . .,n. In this paper, denote x^ ¼ x^ n . n The optimal parameter vectors yi is deﬁned as i yni ¼ argminyi A Oi supx , x^ A U U f^ i x^ i yi f i x i , 1 r i rn i

i1

i

dz ¼ q1 ðz,yÞdt þq2 ðz,yÞdw dx ¼ Ax þ Ky þ YT ðy,uÞW þ e þ DÞdt þGðxÞd wy ¼ Cx where W ¼ b0n

i

Denoting

where Oi , U i1 , U i2 are bounded compact regions for yi , x i and x^ i , respectively. In addition, the approximation error ei is deﬁned as n ð5Þ f i x i ¼ f^ i x^ i 9yi þ ei , 1r i rn By substituting . (5) into (1), the system (1) can be expressed as

Deﬁne a virtual state estimate as

w^ ¼ x þ OT W where O ¼ ½l, X. Similar to [18], the ﬁlters are designed as

x_ ¼ Ax þ Ky

ð11Þ

X_ ¼ AX þ FT

ð12Þ

l_ ¼ Al þ BZðyÞu

2

FT ¼ 4

fT1 &

fTn

ð7Þ

2 3 k1 6 ^ 7 7 K ¼ 4 5, 5, 0 kn 3

0

...

3

2

5

6 ,D ¼ 4

nl

D1 ðx, zÞ ^

Dn ðx, zÞ

ð13Þ T

T

where x ¼ ½x1 ,. . ., xn , l ¼ ½l1 ,. . ., ln , X ¼ Deﬁne observation error vector e as e ¼ ½e1 ,e2 ,. . .,en T ¼

½X1 ,. . ., Xn Tnl .

xw^ pn

ð14Þ

where pn ¼maxpni ,pni 2 ,1; 1 ri r n is an unknown constant. Remark 1. Note that the parameter vector W is unknown, so the virtual state estimate w^ cannot be used in the control design. In fact, the actual state estimate x^ will be used in the control design, which is deﬁned as x^ ¼ x þ Xy þ b^ l ð15Þ n where y and b^ 0 are the estimate of y and b0 , respectively. From (7) and (10)–(14), the observer error is expressed as

eþD GðxÞ ð16Þ de ¼ Ae þ n dt þ n dw p p

To evaluate the property of the ﬁlters (11)–(13), consider the Lyapunov function as 1 T 2 e Pe 2

ð17Þ

‘V 0 rlmin ðQ Þ lmin ðP Þ:e: þ eT PeU þ

In1

ð10Þ

T

4

ð6Þ

dz ¼ q1 ðz,yÞdt þ q2 ðz,yÞdw

k1 6 A¼4 ^ kn

ð9Þ

From (9), (14) and (16)–(17), one can obtain the ‘ inﬁnitesimal generator of V 0

Rewrite (6) as

where 2

0ðn1Þ1 ZðyÞu, FT ðyÞ 1

AT P þPA ¼ Q

i ¼ 1,. . .,n1,

n

ðl þ 1Þ1

Choose vector K such that matrix A is a strict Hurwitz matrix, therefore, for any a given matrix Q ¼ Q T 4 0, there exists a positive deﬁnite matrix P ¼ P T 40 such that

V0 ¼

dz ¼ q1 ðz,yÞdt þq2 ðz,yÞdw nT dxi ¼ ½xi þ 1 þ yi fi x^ i þ ei þ Di ðx, zÞdt þg i ðxÞdw

dx ¼ ðAx þKyþ FT y þ e þ D þ Bb0 ZðyÞuÞdt þ GðxÞdw y ¼ Cx

, YT ðy,uÞ ¼

ð8Þ

0

i2

nT dxn ¼ ½b0 ZðyÞu þ yn fn x^ n þ en þ Dn ðx, zÞdt þ g n ðxÞdw y ¼ x1

33

3 7 n n n 5, y ¼ ½y1 . . .yn Tl1 , l ¼ l1 þ . . . þ ln ,

C ¼ ½1,. . .,0, GðxÞ¼ ½g 1 ðxÞ,. . .,g n ðxÞT , B ¼ ½0,. . .,1T . e ¼ ½e1 , e2 . . .en T .

h i 1 Tr GT 2PeeT P þ eT PeP G pn2

2 eT P ðe þ DÞ pn ð18Þ

From Assumption 1, and as the similar derivations to [30,31], there exist smooth functions ci1 , ci2 and ci3 such that ci1 ðyÞ ¼ yci1 ðyÞ, ci2 z ¼ zci2 z , ci3 ðyÞ ¼ yci3 ðyÞ ð19Þ For the sake of the following derivations, we recall the following Young’s inequality. Lemma 3. ([33]). For any vectors x,y A Rn , the following inequality holds: ap 1 p q xT yr :x: þ q :y: qa p where a 4 0, p 41, q 41 and ðp1Þðq1Þ ¼ 1.

34

T. Wang et al. / Neurocomputing 106 (2013) 31–41

By Assumption 1, Lemma 3 and the fact that pn Z 1, one has the following inequalities 2 T 3 1 3 1 4 8=3 4 4 8=3 4 e Pe eT P e r :P: :e: þ :e: r :P: :e: þ d pn 2 2 2 2

ð20Þ

n X 2 T 3 8=3 4 4 4 e Pe eT P D r :P: :e: þ 4n ½ci1 ðyÞ þ ci2 z pn 2 i¼1

ð21Þ

h i 1 Tr GT 2 PeeT P þ eT PeP G n 2 p

1 G 4 2 4 2 r 2:P: þ lmax ðP Þ :e: þ : n : 2 p n X n 2 4 2 2 2 2:P: þ lmax ðP Þ c4i3 ðyÞ r 2:P: þ lmax ðP Þ :e: þ 2 i¼1

4

n X

ð22Þ

c4i1 ðyÞ þ F z

ð29Þ

Substituting Eq. (29) into (27) yields n dy ¼ x2 þ oy þ b0 l2 þpn e2 þ e1 þ D1 dt þ g 1 dw

ð30Þ

T ¼ ½f1 ,0,. . .,0 þ X2 .

where o Choose the Lyapunov function candidate as

ð23Þ

8=3 2 2 p0 ¼ lmin ðQ Þ lmin ðP Þ3:P: 2:P: lmax ðP Þ, F z n X 1 c4i2 z and d0 ¼ d4 ¼ 4n 2 i¼1

3 2 2 ~ T 1 _ 1 ~ _^ 1 _ y g 1 y G y r 1 d dr 2 p~ p^ 2

(1) does not contain the unmodeled dynamics z, dynamic disturbance Di ðx, zÞ and dw ¼ 0, then (23) is reduced to ‘V 0 r p00 V 0 þ d0 , from which, it follows that the virtual state estimations by (10) are bounded.

3 2 2 ~ T 1 _ 1 ~ _^ 1 _ y g 1 y G y r 1 d dr 2 p~ p^ 2

ð33Þ

3 4 1 4 4 y3 pn e2 þ D1 r py4 þ :e: þ 2c11 ðyÞy4 þ2c12 z 2 4

ð34Þ

2 3 2 2 3 2 2 3 y g 1 r py c13 ðyÞ ¼ pc13 ðyÞy4 2 2 2

ð35Þ

Substituting (34) and (35) into (33) gives 4 ‘V 1 rp :e: þ F z þ2c4 z þy3 ðb0 ða1 þz2 Þ þ x2 1

12

T n ~ _^ 1 ~ _^ þ oy þ C11 ðyÞ þ C12 ðyÞp þ dÞy~ G1 y_ r 1 1 d dr 2 p p þ d0

where p1 ¼ p0 14, C12 ðyÞ ¼ 32 yþ 4

C11 ðyÞ ¼ 2c11 ðyÞy þ 4n

n X

In this section, a robust adaptive output feedback control scheme will be developed based on the above designed fuzzy ﬁlters, and the stability analysis of the control system will be given. 4.1. Adaptive fuzzy backstepping control design From (13), one has

l_ i ¼ li þ 1 ki l1 , i ¼ 1,2,. . .,n1

ð24Þ

l_ n ¼ ZðyÞukn l1

ð25Þ

The adaptive fuzzy backstepping control design consists of n-steps, each step is based on the change of coordinates: ð26Þ

where ai1 ðUÞði ¼ 2,. . .,n Þ is an intermediate control. Step 1: From the second equation in (1), and according to It o^ ‘s differentiation rule, one has nT dy ¼ x2 þ y1 f1 þ e1 þ D1 dt þg 1 dw ð27Þ

þ C11 ðyÞ þ C12 ðyÞp^ T _^ 1 _^ ~ þ d^ Þ þ y~ t1 G1 y_ þ d~ s1 r 1 1 d þ p p1 r 2 p þd0

ð37Þ

where t1 ¼ oT y3 , s1 ¼ y3 , p1 ¼ y3 C12 ðyÞ. Choose stabilizing control function a1

1 a1 ¼ NðB1 Þ P1 y2 y þ x2 þ oy þ d^ þ C11 ðyÞ þ C12 ðyÞp^ þ y3 2

ð38Þ

p NðB1 Þ ¼ exp B21 cos B1 2

1 B_ 1 ¼ y3 P1 y2 y þ x2 þ oy þ d^ þ C11 ðyÞ þ C12 ðyÞp^ þ y3 2 where P1 ðy2 Þ is a smooth nonnegative function designed later. Substituting (38)–(40) into (37) yields 4 ‘V 1 rp1 :e: þ F z þ2c412 z P1 y2 y4 þ b0 y3 z2 T _ þ b NðB ÞB_ þ B_ þ y~ t G1 y_ þ d~ s r 1 d^

ð28Þ

n X 4 n 2 2 2:P: þ lmax ðP Þ ci3 ðyÞy: 2 i¼1

Then (36) can be rewritten as 4 ‘V 1 rp1 :e: þ F z þ2c412 z þy3 ðb0 ða1 þz2 Þ þ x2 þ oy

0

From (10), one has

4

ð36Þ

3 2 2 c13 ðyÞy,

ci1 ðyÞy þ

i¼1

4. Adaptive fuzzy controller design and stability analysis

n n x ¼ x þ Xy þ b0 l þ xw^ ¼ x þ Xy þb0 l þpn e

ð32Þ

Using Assumption 1 and Lemma 3, one has

Remark 2. Note that according to Lemma 2, the fuzzy logic systems f^ i x^ i 9yi can well approximate the unknown functions f i x i in system (1), thus d0 is a small constant. If the system

z1 ¼ y, zi ¼ li ai1 , i ¼ 2,. . .,n

ð31Þ

Substituting (26) into (32) results in ‘V 1 r ‘V 0 þy3 b0 ða1 þ z2 Þ þ x2 þ oyn þ y3 pn e2 þ e1 þ D1 þ

where

1 4 1 ~ T 1 ~ 1 ~2 1 2 y þ y G yþ p~ d þ 4 2 2r 1 2r 2

n where G ¼ GT 40, r 1 40, r 2 40are design parameters, y~ ¼ y y,~ ^ ^ d ¼ dd and p~ ¼ pp^ are the parameters errors; y, d and p^ are the n estimates of y , d and p, respectively. Herep ¼ maxpn ,pn2 , ðpn Þ4=3 . From Eqs. (30) and (31), the inﬁnitesimal generator of V 1 satisﬁes ‘V 1 r ‘V 0 þy3 x2 þ oyn þ b0 l2 þpn e2 þ e1 þ D1

þ

i¼1 n X n 2 2 2:P: þ lmax ðP Þ þ c4i3 ðyÞ þ d0 2 i¼1

n

x2 ¼ x2 þ X2 y þb0 l2 þ pn e2

V1 ¼ V0 þ

where d is an unknown positive constant and :e: r d. Substituting Eqs. (20)–(22) into (18), one obtains

‘V 0 r p0 :e: þ 4n

Therefore, using (28), x2 is expressed as

1

1

1

1

1 6 _^ þ p~ p1 r 1 2 p y þ d0 2

1

ð39Þ

ð40Þ to

be

1

ð41Þ

T. Wang et al. / Neurocomputing 106 (2013) 31–41

1 ~ _~ _^ þ p~ p1 r 1 2 p þ r 3 b 0 b 0 þ d1 @a1 4 r p2 :e: þ z32 z3 þ a2 k2 l1 x2 þ oy þ b^ 0 l2 @y @a _ @a1 _ 1 ^ þ H2 y Gt1 þ Gmy dr 1 s1 þ r1 md^ ^ @y @d 1 @a1 _ @a ^ 2 p1 þ r 2 mp^ þ z2 þ p^ C22 ðyÞ þ 1 d^ pr 4 @y @p^ 2 4 3 4 1 4 þ y þ c12 z P1 y y þ F z þb0 NðB1 ÞB_ 1 4 4 T _ 4 þ B_ 1 þ 2c z þ y~ t2 G1 y_ þ d~ s2 r 1 d^

2

By using 2ab ra2 þb , one has b0 y3 z2 r

1 6 1 4 1 4 y þ z2 þ b0 2 4 4

ð42Þ

Substituting (42) into (41) yields 4 ‘V 1 r p1 :e: þ F z þ 2c412 z P1 y2 y4 T 1 þ z42 þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ y~ t1 G1 y_ 4 1 _^ ^_ ~ þ d~ s1 r 1 1 d þ p p1 r 2 p þ d1

ð43Þ

4

where d1 ¼ d0 þ 14 b0 .

12

Step 2: From (23) and (25), one has @a1 dz2 ¼ l3 k2 l1 x þ oyn þ b0 l2 þpn e2 þ e1 þ D1 þH2 @y 2 @a _ @a1 _ y Gt1 þ Gmy 1 d^ r1 s1 þ r1 md^ ^ @y @d # 1 @2 a @a1 _ @a1 1 2 ^ ^ g ðxÞdw pr 2 p1 þr 2 mp g ðxÞ dt 2 @y2 1 @y 1 @p^

1 _^ 2 r 2 p

þ p~ p

C22 ðyÞ ¼

ð44Þ

H2 ¼ @@ax1 ðAx þKyÞ @@aX1 AX þ FT @@ay1 G t1 my @a^1 r 1 s1 md^ @d @@ap^1 r 2 p1 mp^ Consider the Lyapunov function candidate as ð45Þ

where r 3 4 0is a design parameter. Denote b~ 0 ¼ b0 b^ 0 is the parameters error. b^ is the estimate of b . 0

0

From (44) and (45), one has @a ‘V 2 ¼ ‘V 1 þ z32 l3 k2 l1 1 x2 þ oyn þ b0 l2 þpn e2 þ D1 @y @a _ @a1 _ þ H2 y Gt1 þ Gmy 1 d^ r1 s1 þr 1 md^ ^ @y @d # 1 @2 a @a1 _ 1 2 @a1 ^ 2 p1 þr 2 mp^ pr g ðxÞ e 1 2 @y2 1 @y @p^

2 3 @a1 ~ _~ þ z22 g 1 ðxÞ2 þ r 1 3 b0b0 2 @y Using the similar derivations in step 1, one can obtain

3 @a1 n @a1 4=3 4 1 4 p e2 þ D1 r p z2 þ :e: z32 2 4 @y @y

4=3 3 @a1 1 1 4 þ p c11 ðyÞ z42 þ y4 þ c12 z 4 4 4 @y

ð49Þ

4=3 3 @a1 4=3 3 @a1 9 @a1 4 z2 z2 þ c11 ðyÞ z2 þ 2 @y 4 @y 4 @y !2 ! 4 1 @2 a1 1 @a c13 ðyÞ, p2 ¼ p1 , t2 ¼ t1 z32 1 oT , þ z32 4 4 @y @y2

s2 ¼ s1 þ z32

1 4 1 ~2 z þ b 4 2 2r 3 0

1

_^ b~ 0 r 1 3 b0 u2 þ d1

where

where m 4 0 is a design constant and

V2 ¼ V1 þ

35

@a1 @a1 , p2 ¼ p1 þ z32 C22 ðyÞ, u2 ¼ z32 l2 @y @y

Choose stabilizing control function a2 @a a2 ¼ z2 c2 z2 H2 þ 1 x2 þ oy þ b^ 0 l2 þ k2 l1 @y @a1 ^ ðD22 þ L22 þ A22 Þz32 p^ C22 ðyÞ d ð50Þ @y where c2 40 is a design constant. D22 ¼ @a1 =@y G @a1 =@y oT , L22 ¼ ð@a1 =@d^ Þr1 ð@a1 =@yÞ and A22 ¼ @a1 =@p^ r2 C22 ðyÞ. Deﬁne

n X @a1 _ y Gt1 þ Gmy ¼ D2j z3j , @y j¼2

n X @a1 _^ dr1 s1 þ r1 md^ ¼ L2j z3j , ^ @d j¼2

ð46Þ

n X @a1 _ ^ 2 p1 þr 2 mp^ ¼ pr A2j z3j : @p^ j¼2

where

D2j ¼

ð47Þ

2 !2 3

3 2 @a1 2 2 1 3 @2 a1 2 49 @a1 4 1 3 @2 a1 5 4 1 pc13 ðyÞz32 þ y4 z2 z z g 1 z2 g r þ 2 2 2 4 4 2 @y2 2 @y @y @y2 1

@a @a1 @aj1 T @a @a G o , L2j ¼ ^1 r1 j1 , A2j ¼ ^1 r2 Cj2 ðyÞ @y @y @y @p @d

(50) into (49) and using the fact z32 z3 r By substituting 3=4 z42 þ 1=4 z43 , one has

1 3 9 4 ‘V 2 r p2 :e: c2 z42 þ z43 þ y4 þ c412 z þ

n X

4

4

D2j þ L2j þ A2j z32 z3j P1

4 2 4 y y þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ F z

j¼3

ð48Þ By substituting (47), (48) into (46), one has @a 4 ‘V 2 r p2 :e: þ z32 l3 k2 l1 1 x2 þ oyn þ b0 l2 @y @a _ @a1 _ þ H2 y Gt1 þ Gmy 1 d^ r1 s1 þr 1 md^ ^ @y @d 1 @a1 _ @a ^ 2 p1 þr 2 mp^ þ z2 þ pC22 ðyÞ þ 1 d pr 4 @y @p^ 2 4 3 4 1 4 þ y þ c12 z P1 y y þ F z þ b0 NðB1 ÞB_ 1 þ B_ 1 4 4 T _ 4 þ 2c z þ y~ t G1 y_ þ d~ s r 1 d^ 12

1

1

1

T _^ þ y~ t2 G1 y_ þ d~ s2 r 1 1 d 1 _^ _^ ~ þ p~ p2 r 1 2 p þ b 0 u2 r 3 b0 þ d1

ð51Þ

Step iði ¼ 3, ,n1Þ: A similar procedure in step 2 is employed recursively for step i, one has @a n dzi ¼ li þ 1 ki l1 i1 x2 þ oy þb0 l2 þ pn e2 þ e1 þ D1 @y @a _ @a þHi i1 y_ Gti1 þ Gmy i1 d^ r 1 si1 þ r 1 md^ ^ @y @d @a _ @ai1 _ i1 ^ ^ ^ pr 2 pi1 þ r 2 mp b0 r 3 ui1 þ r 3 mb^ 0 @p^ @b^ 0

36

T. Wang et al. / Neurocomputing 106 (2013) 31–41

# 1 @2 ai1 @a 2 g ðxÞ dt i1 g 1 ðxÞdw 2 @y2 1 @y

ð52Þ

where @a @ai1 @a @a ðAx þKyÞ i1 AX þ FT i1 l_ i1 G ti1 my @x @X @l @y @a @a @a i1 r 1 si1 md^ i1 r 2 pi1 mp^ i1 r 3 ui1 mb^ 0 @p^ @d^ @b^

Hi ¼

Choose stabilizing control function ai @a ai ¼ zi ci zi Hi þ i1 x2 þ oy þ b^ 0 l2 þ ki l1 @y i X @a ðDki þ Lki þ Aki þY ki Þz3k p^ Ci2 ðyÞ i1 d^ @y k¼2 where ci 40 is a design constant and U2i ¼ 0. Deﬁne

0

Consider the Lyapunov function candidate V i ¼ V i1 þ

1 4 z 4 i

ð53Þ

n X @ai1 _ y Gti1 þ Gmy ¼ Dij z3j , @y j¼i

n X @ai1 _^ dr1 si1 þr 1 md^ ¼ Lij z3j ^ @d j¼i

From (52) and (53), one has @a ‘V i ¼ ‘V i1 þ z3i zi þ 1 þ ai ki l1 i1 x2 þ oyn þ b0 l2 þ pn e2 þ D1 þ Hi @y

0

3 @ai1 1 @2 ai1 @a g ðxÞ2 i1 e1 þ z2i @y 2 @y2 1 2 @y

2

2

g 1 ðxÞ

0

ð54Þ

ð55Þ

ð56Þ

By substituting (55), (56) into (54), one can obtain @a 4 ‘V i r pi :e: þ z3i zi þ 1 þ ai ki l1 i1 x2 þ oy þ b^ 0 l2 @y @a _ @a þ Hi i1 y_ Gti1 þ Gmy i1 d^ r 1 si1 þ r 1 md^ @y @d^ @a _ @ai1 _ i1 ^ ^ 2 pi1 þ r 2 mp^ b0 r 3 ui1 þ r 3 mb^ 0 pr ^ ^ @p @b0 1 @ai1 ^ 3ði1Þ 4 i þ 7 4 y þ d þ c12 z þ zi þ p^ Ci2 ðyÞ þ 4 4 4 @y 2 4 T ^_ P1 y y þb0 NðB1 ÞB_ 1 þ B_ 1 þ y~ ti G1 y_ þ d~ si r 1 1 d

n X @aj1 T @ai1 @aj1 , o , Lij ¼ r ^ 1 @y @y j ¼ i @d

n X @ai1 @ai1 @aj1 r 2 Cj2 ðyÞ and Uij ¼ r3 l2 ^ ^ @y @ p j¼i j ¼ i @b0

(59) into (58) and using the fact z3i zi þ 1 r By substituting 3=4 z4i þ 1=4 z4i þ 1 , one has i X

4

‘V i r pi :e:

cj z4j þ

j¼2

1 4 3ði1Þ 4 i þ 7 4 z y þ þ c12 z 4 iþ1 4 4

n i X X Dkj þ Lkj þ Akj P1 y2 y4 þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ F z þ

cj z4j þd1

_^ ~ 1 _^ þ Y kj z3k z3j þ y~ ti G1 y_ þ d~ si r 1 1 d þ p pi r 2 p _^ þ b~ 0 ui r 1 3 b0 þ d1

V n ¼ V n1 þ

T

ð59Þ

1 4 z 4 n

ð60Þ

Using the similar derivations in step i, one has

4

‘V n r pn :e: þ z3n ZðyÞukn l1

@an1 x2 þ oy þ b^ 0 l2 þ Hn @y

@a _ @an1 _ y Gtn1 þ Gmy n1 d^ r1 sn1 þ r1 md^ @y @d^ @a @an1 _ ^ 2 pn1 þ r2 mp^ n1 b_^ 0 r 3 un1 þ r 3 mb^ 0 pr @p^ @b^ 0

j¼ik¼2

þ p^ Cn2 ðyÞ þ

ð57Þ

j¼2

where

4=3 3 @ai1 4=3 3 @ai1 9 @ai1 4 zi Ci2 ðyÞ ¼ zi þ c11 ðyÞ zi þ 2 4 4 @y @y @y !2 ! 2 4 1 @ ai1 1 c13 ðyÞ pi ¼ pi1 þ z3i 4 4 @y2 @ai1 T ti ¼ t o si @y @ai1 @a ¼ si1 þz3i pi ¼ pi1 þz3i Ci2 ðyÞ and ui ¼ ui1 z3i i1 l2 @y @y 3 i1 zi

Aij ¼

@y

G

Step n : In the ﬁnal step, the actual control input u will appear. Consider the overall Lyapunov function as

n X i1 X 1 _^ _^ ~ Dkj þ Lkj þAkj z3k z3j þ p~ pi r 1 2 p þ b 0 ui r 3 b0 þ

þ F z

n X @ai1

j ¼ iþ1 k ¼ 2

1 4 y 2

i1 X

Dij ¼

j¼i n X

2 !2 3

3 2 @ai1 2 2 1 3 @2 ai1 2 49 @ai1 4 1 3 @2 ai1 5 z z g 1 zi g r þ zi 2 i 2 4 i @y 4 @y @y2 1 @y2 4

j¼i

where

4=3

3 @ai1 n @ai1 4=3 4 1 3 @ai1 4 z3i zi þ :e: þ p c11 ðyÞ z4i p e2 þ D1 r p @y @y @y 2 4 4

pc13 ðyÞz3i þ

n X @ai1 _ ^ 2 pi1 þ r 2 mp^ ¼ pr Aij z3j , @p^

n X @a _ Uij z3j i1 b^ 0 r 3 ui1 þ r 3 my ¼ @b^

By Assumption 1 and Lemma 3, one has

1 1 4 þ y4 þ c12 z 4 4

j¼i

@a _ @ai1 _ y Gti1 þ Gmy i1 d^ r1 si1 þ r 1 md^ ^ @y @d @a _ @ai1 _ ^ 2 pi1 þ r2 mp^ i1 b^ 0 r 3 ui1 þ r 3 mb^ 0 pr ^ @p^ @b

ð58Þ

3ðn1Þ 4 n þ 7 4 1 @an1 ^ y þ d þ c12 z P1 y2 y4 zn þ @y 4 4 4

n1 X T _^ þ F z þ y~ tn G1 y_ þ b0 N ðB1 ÞB_ 1 þ B_ 1 cj z4j þ d~ sn r 1 1 d j¼2

_^ ~ þ p~ pn r 1 2 p þ b0

n1 _^ X un r 1 ðDkn þ Lkn þ Akn þ Y kn Þz3k z3n þ d1 3 b0 þ k¼2

ð61Þ where @an1 @an1 ðAx þ KyÞ AX þ FT @x @X @a @an1 _ @an1 G tn1 my n1 r 1 sn1 md^ l ^ @l @y @d

Hn ¼

T. Wang et al. / Neurocomputing 106 (2013) 31–41

@an1 @an1 r 2 pn1 mp^ r 3 un1 mb^ 0 , ^ @p^ @b 0

Cn2 ðyÞ ¼

þ

4=3 3 @an1 4=3 3 @an1 9 @an1 4 zn zn þ c11 ðyÞ zn þ 2 4 4 @y @y @y

1 3 @2 an1 z 4 n @y2

!2 !

1 4

4

c13 ðyÞ pn ¼ pn1 , tn ¼ tn1

@an1 T o , pn ¼ pn1 þ z3n Cn2 ðyÞ, sn ¼ sn1 @y @an1 @an1 and un ¼ un1 z3n þ z3n l2 @y @y

z3n

Choose actual control u, and parameters adaptation laws y, d^ , p^ and b^ 0 as follows: 1 1 @an1 zn cn zn Hn þ x2 þ oy þ b^ 0 l2 þ kn l1 u¼ 4 ZðyÞ @y ! n X @an1 ^ 3 ^ ðDkn þ Lkn þ Akn þY kn Þzk pCn2 ðyÞ d ð62Þ @y k¼2

y_ ¼ G tn my

_

ð63Þ

d^ ¼ r1 sn md^

ð64Þ

p_^ ¼ r 2 pn mp^

ð65Þ

_ b^ 0 ¼ r 3 un mb^ 0

ð66Þ

where cn 40 is a design constant. By substituting (62)–(66) into (61), one has T

4

‘V n r pn :e: þ my~ y þ md~ d^ þ mp~ p^ þ mb~ 0 b^ 0 þ þ

n X

3ðn1Þ 4 y cj z4j 4 j¼2

n þ7 4 c12 z P1 y2 y4 þ F z þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ d1 4

1 2

1 2

In the following, we will design the function P1 ðUÞ introduced in step 1 by using the changing supply function technique proposed by [28] and [30]. From (72), ﬁrst choose a smooth nonnegative function P1 ðUÞ such that

ð69Þ

1 2

1 2

mb~ 0 b^ 0 r mb~ 0 þ mb0 2

ð73Þ

with P10 ðUÞ being a smooth nonnegative function to be designed. 4 P Choose parameter pn 40, and let V 0 ¼ pn :e: nj¼ 1 cj z4j , where c1 ¼ 1=4. Then it follows from (72) and (73) that 2 2 1 2 ‘V n rV 0 m :y~ : þ d~ þ p~ 2 þ b~ 0 P10 y2 y4 þ F z 2 þ b0 N ðB1 ÞB_ 1 þ B_ 1 þd

ð74Þ

To construct the nonnegative function P10 ðUÞ, the following Assumption 3 is introduced. Assumption 3. ([30]). For the functions cz and c0 ,að9z9Þ,ci2 9z9 given by Assumption 2, the following condition holds

c4i2 ðsÞ þ c2z ðsÞc20 ðsÞ

aðsÞ

o1

ð75Þ

According to [30], from Assumptions 1–3, one can construct continuous increasing functions xðUÞ and $ðUÞ satisfying

ð70Þ

ð71Þ

Then there exist a non-decreasing positive function rðUÞ such that 8x A Rmi

n 3ðn1Þ X 2 2 1 2 y4 cj z4j 2 4 j¼2 P1 y2 y4 þ F z þb0 NðB1 ÞB_ 1 þ B_ 1 þ d ð72Þ 4

‘V n r pn :e: m :y~ : þ d~ þ p~ 2 þ b~ 0 þ

0

where

nþ7 4 , c12 z ¼ 4n F z ¼ F z þ

n X

c4i2 z

4 i¼1 2 n þ7 4 1 2 c12 z , d ¼ d1 þ m :yn : þ d2 þ p2 þ b0 þ 4 2

Remark 3. It should be mentioned that if the system (1) does not contain the unmodeled dynamics zand the dynamical distur bances Di ðx, zÞ, then F 9z9 ¼ 0 in (72). For this situation, we can

ð76Þ

0

1 1 r V z ðzÞ a z F z Z r0 V z ðzÞ c2z z c20 z 4 2

ð77Þ

Consider the system (1). Under Assumptions 1–3 and the conditions of Lemma 4, and limsup s-0 þ

3n 4 y Z P10 y2 y4 4

Lemma 4. ([30]). Under Assumption 3, and if Z s Z 1 ½x a 1 ðs Þ0 exp ½$ a 1 ðt Þ1 dt ds o 1

Substituting (68)–(71) into (67) results in

xðsÞaðsÞ Z 4FðsÞ and $ðsÞaðsÞ Z 2c2z ðsÞc20 ðsÞ.

mp~ p^ r mp~ 2 þ mp2 2

4.2. Changing supply function design and stability analysis

limsup

ð68Þ

1 2 1 md~ d^ r md~ þ md2 2 2

1 2

From the above inequality and Lemma 3, and using the similar proof to [2,7,8,16,37], it is easily concluded that the proposed adaptive control scheme by Step 1–Step n can guarantee that all the variables in closed-loop system are bounded in probability. However, in this paper, the system (1) contains the unmodeled dynamics z and the dynamical disturbances Di ðx, zÞ, therefore, it is necessary to design P1 ðy2 Þ to ensure the stability of the control system.

s-0 þ

my~ y r m:y~ :2 þ m:yn :2

1 2

þ b0 N ðB1 ÞB_ 1 þ B_ 1 þd

ð67Þ

By completing the squares T

choose P1 y2 ¼ 0 in the stabilizing control function a1 , and (72) becomes n X 2 2 1 4 1 2 ‘V n rpn :e: y4 cj z4j m :y~ : þ d~ þ p~ 2 þ b~ 0 4 2 j¼2

P1 y2 y4

37

gðsÞ s4

o1

ð78Þ

Suppose that rðUÞ is the supply function deﬁned in Lemma 4. Let Z V z ðzÞ UðzÞ ¼ rðtÞdt 0

By It o^ formula and Assumption 2, we have

@V z T 1 2 ‘U ðzÞ ¼ r V z ‘V z þ r0 V z : q2 : 2 @z 1 2 2 r r V z g y a z þ r0 V z cz z ci0 z 2

38

T. Wang et al. / Neurocomputing 106 (2013) 31–41

1 1 2 2 r r Z y g y r V z a z þ r0 V z cz z c0 z ð79Þ 2 2 1 2gðUÞÞÞA k1 . where Z ¼ a a To show the last step of (79), we consider the following two cases separately: Case 1. : If g y r 1=2 a z , then in this case, we have that 1 r V z g y a z r r V z a z 2 Case 2. If g y Z 1=2 a z , then in this case, we have V z ðzÞ r a 9z9 r Z 9y9 , and r V z g y a z r r Z y Þg y r V z a z For these two cases, it follows that 1 r V z g y a z r r Z y g y r V z a z : 2

From (83) and (84), we have ct d e W rect b0 NðB1 ÞB_ 1 þ B_ 1 þdÞdt þ ect ðM 1 Þdw

ð85Þ

Integrating (85) over ½0,T, we get Z T d W ðT Þ r ecT W ð0Þ þ þ ecT ect ðb0 NðB1 ÞB_ 1 þ B_ 1 Þdt c 0 Z T þ ecT ect M 1 dw

ð86Þ

0

RT According to [36], we know that ecT 0 ect ðM 1 Þdw is a real valued continuous local martingale. Therefore, by Lemma 1 one can draw the conclusion that the functionsWðTÞ, B1 ðtÞ and RT ecT 0 ect ðb0 NðB1 ÞB_ 1 þ B_ 1 Þdt must be bounded in probability. RT (ii). Deﬁne s ¼ sup 0 Eðb0 N ðB1 ÞB_ 1 þ B_ 1 Þdt, then Z T Z T ecT Eðb0 NðB1 ÞB_ 1 þ B_ 1 Þect dt r E b0 NðB1 ÞB_ 1 þ B_ 1 ÞecðtT Þ dt r s 0

Consider the Lyapunov function candidate for the entire system W ¼ V n þ UðzÞ

ð80Þ

Then, it follows from (72) and (79) that 2 2 1 2 ‘W r V 0 m :y~ : þ d~ þ p~ 2 þ b~ 0 P10 y2 y4 þ F z 2 1 þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ d þ r Z y g y r V z a z 2 2 2 1 0 2 2 2 0 1 ~ þ r V z cz z c0 z r V m :y : þ d~ þ p~ 2 þ b~ 0 2 2 1 P10 y2 y4 r V z a z þ r Z y g y þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ d 4 ð81Þ Then from (78), we can construct a smooth non-decreasing function P10 s2 A k1 such that gðtÞ P10 s2 Z r ZðsÞ sup 4 , hence P10 y2 y4 Z r Z y g y t t A ð0,s Thus, by (81) we have 1 2 2 1 2 ‘W r V 0 m :y~ : þ d~ þ p~ 2 þ b~ 0 r V z a z 2 4 ð82Þ þ b0 N ðB1 ÞB_ 1 þ B_ 1 þ d Deﬁne c0 ¼ sup ½ r V z a 9z9=4Uðz Þ, then (82) can be further tZ0 rewritten as

‘W r cW þb0 NðB1 ÞB_ 1 þ B_ 1 þ d

ð83Þ

0

ð87Þ where EðUÞ is the expectation operator. Taking expectation on (86) and using (87), we have EW ðT Þ r EW ð0Þ þ n P

d þs c

ð88Þ

d z4i r EW rEW then, 14 E ð0Þ þ c þ s. P Let Xi ¼¼1 EWð0Þ þ d=c þ s; then E ni¼ 1 z4i r 4X. Therefore, there exists a compact set Oz , such that for any given X Z EWð0Þ þðd=cÞ þ s, zi A Oz .

5. Simulation study In this section, a simulation example is given to illustrate the effectiveness of the proposed adaptive fuzzy output feedback control approach. Example: Consider the following stochastic nonlinear system:

1 dz ¼ 3z þx21 dt þ pﬃﬃﬃ zcosx2 dw 2 dx1 ¼ ðx2 þsinx1 þ 0:1z þ 0:5x1 Þdt þ ðx1 sinx2 Þdw dx2 ¼ b0 ð10:8sinðyÞÞu þ x1 þx22 þ zsinx2 Þdt þ ðx1 cosx2 Þdw y ¼ x1

ð89Þ

where b0 ¼ 1. By Assumption 1, we can take c11 ðyÞ ¼ 9y9, c21 ðyÞ ¼ 9y9, c12 ðzÞ ¼ 9z9, c22 ðzÞ ¼ 9z9, c13 ðyÞ ¼ c23 ðyÞ ¼ 9y9, pnij ¼ 1. Deﬁne fuzzy membership as follows:

where n o 2 c ¼ min pn :P: ,4c1 , ,4cn , m:G:, mr 1 , mr 2 , mr 3 ,c0 : Theorem 1. Consider the system (1) with intermediate control functions (38), (50), (58) and controller (62); let the parameters be updated by the adaptive laws (63)–(66), while Assumptions (1)–(3) are true. Then one has the following properties: ^ b^ 0 and z are bounded in (i) All involved signals zi , y, d^ , p, probability. (ii) Given any X Z EWð0Þ þðd=cÞ þ s, the variables zi converge to the compact deﬁned as ! n X 4 Oz ¼ zi ,9E zi r4X i¼1

mF l x^ i ¼ exp½ x^ i 3 þl 2 =16, i ¼ 1,2 l ¼ 1, ,5 i

The fuzzy logic systems can be expressed in the form 5 X T f^ 1 ðx^ 1 y1 Þ ¼ y1 f1 x^ 1 ¼ yT1j f1j x^ 1 , j¼1 5 X T yT2j f2j x^ 1 , x^ 2 f^ 2 ðx^ 2 y2 Þ ¼ y2 f2 x^ 1 , x^ 2 ¼ j¼1

where

yT1 ¼ ½y11 , y12 , y13 , y14 , y15 , yT2 ¼ ½y21 , y22 , y23 , y24 , y25 , f1 x^ 1 ¼ ½f11

ct

Proof: (i). Multiplying (80) by e and by It o^ formula leads to ð84Þ d ect W ¼ ect ðcW þ ‘W Þdt þ ect ðM 1 Þdw where n X @V @V @ai1 @V GðxÞ @V g 1 ðxÞ þ g 1 ðxÞ þ q ðz,yÞ M1 ¼ @z1 @z @e pn @z 2 @y i i¼2

x^ 1 , f12 x^ 1 , f13 x^ 1 , f14 x^ 1 , f15 x^ 1 T

f2 x^ 1 , x^ 2 ¼ ½f21 x^ 1 , x^ 2 , f22 x^ 1 , x^ 2 ,

f23 x^ 1 , x^ 2 , f24 x^ 1 , x^ 2 , f25 x^ 1 , x^ 2 T :

mF l1 x^ 1 mF l2 x^ 2

fij x^ 1 , x^ 2 ¼ P5

, i ¼ 1,2, j ¼ 1, ,5 ^ ^ l ¼ 1 mF l x1 mF l x2 1

2

T. Wang et al. / Neurocomputing 106 (2013) 31–41

Fig. 1. x1 and x^ 1 .

Fig. 3. z and :y:.

Fig. 2. x2 and x^ 2 .

Fig. 4. Controller u.

For the z-system, by choosing the Lyapunov function as 4 4 V z ¼ 1=4 z , then we have ‘V zr3=2 z þ 1=4 x81 , and from Assumption 2,we can get gðsÞ ¼ 1=4 s8 , aðsÞ ¼ 3=2 s4 , rðsÞ ¼ 50, P10 y2 ¼ 25=2 y4 , P1 y2 ¼ 14y4 . Select Q ¼ I,k1 ¼ 12, k2 ¼ 20, by solving (9) to obtain the positive deﬁnite matrix

0:04370:0250 P¼ 0:02501:1750 The intermediate control function a1 is

1 a1 ¼ NðB1 Þ 14y5 þ x2 þ oy þ d^ þ C11 ðyÞ þ C12 ðyÞp^ þ y3 2 with

B_ 1 ¼ y3

1 P1 y2 yþ x2 þ oy þ d^ þ C11 ðyÞ þ C12 ðyÞp^ þ y3 2

p NðB1 Þ ¼ exp B21 cos B1 : 2

39

ð90Þ

Control input and the parameter adaptive laws are 1 1 @a1 u¼ z2 c2 z2 H2 þ x2 þ oy þ b^ 0 l2 þ k2 l1 10:8 sinðyÞ 4 @y

@a1 ^ ðD22 þ L22 þA22 þY 22 Þz32 p^ C22 ðyÞ d ð91Þ @y

y_ ¼ G t2 my _

ð92Þ

d^ ¼ r1 s2 md^

ð93Þ

p_^ ¼ r 2 p2 mp^

ð94Þ

_ b^ 0 ¼ r 3 u2 mb^ 0

ð95Þ

In this simulation, the design parameters are chosen asc2 ¼ 12,

m ¼ 0:8, r1 ¼ r 2 ¼ 0:1, r 3 ¼ 10, G ¼ 8 I1010 , where I1010 is the identity matrix. The initial conditions are chosen as x1 ð0Þ ¼0:2, T x2 ð0Þ ¼0:4,zð0Þ ¼ 0:5, d^ ¼ 0, p^ ¼ 0, b^ 0 ¼ 0, y1 ð0Þ ¼ ½0:1,0:3,0:5, T 0:7,0:9, y2 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9. The simulation

40

T. Wang et al. / Neurocomputing 106 (2013) 31–41

results are shown by Figs. 1–4, respectively, where Fig.1 is the trajectory of x1 and x^ 1 , Fig.2 is the trajectory of x2 and x^ 2 , Fig.3 is the trajectories of z and :y:, and Fig.4 is the trajectory of controller u. Remark 4. It should be mentioned that the gain of control input u in (89) is not a constant. Thus, the adaptive fuzzy output feedback control approaches in [19] and [33] cannot be applied to control system (89). From the above the simulation results, one can conclude that the proposed adaptive fuzzy output feedback control approach can achieve the good control performance even if the controlled nonlinear stochastic system contains the unknown functions, the unmodeled dynamics, unmeasured states and unknown control direction.

6. Conclusions In this paper, an observer-based adaptive fuzzy output feedback control approach has been proposed for a class of uncertain stochastic nonlinear systems with unknown functions, dynamic uncertainties, the unknown control direction, and without the direct measurements of state variables. In the design, fuzzy logic systems are utilized to approximate the unknown functions and a ﬁlter state observer is developed to estimate the unmeasured states. By using the ﬁlter observer and based on the principle of the adaptive backstepping technique, and combining the Nussbaum function and changing supply function design, a new adaptive fuzzy output feedback control scheme has been synthesized. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are bounded 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.

Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.neucom.2012. 10.013.

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T. Wang et al. / Neurocomputing 106 (2013) 31–41

Tong Wang, received the B.S. degree from the Department of Mathematics, Shanxi University, Taiyuan, China, in 2010. He is now working towards the M.E. degree in control theory and control engineering 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.

41 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. Now, he is pursuing his Ph.D. degree in transportation information engineering and control in Dalian Maritime University. His current research interests include fuzzy and neural networks control, and nonlinear adaptive control.