Robust adaptive fuzzy control for a class of stochastic nonlinear systems with dynamical uncertainties

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Journal of the Franklin Institute 349 (2012) 3121–3141 www.elsevier.com/locate/jfranklin

Robust adaptive fuzzy control for a class of stochastic nonlinear systems with dynamical uncertainties$ Tong Wang, Shaocheng Tongn, Yongming Li Department of Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121000, China Received 11 May 2012; received in revised form 1 August 2012; accepted 28 September 2012 Available online 12 October 2012

Abstract In this paper, a robust adaptive fuzzy output feedback control approach is developed for a class of uncertain stochastic nonlinear systems with unknown nonlinear functions, dynamical uncertainties and without the measurements of the states. The fuzzy logic systems are used to approximate the unknown nonlinear functions, and a fuzzy state observer is designed for estimating the unmeasured states. To solve the problem of the dynamical uncertainties, the dynamical signal combined with changing supply function is incorporated into the backstepping recursive design technique, and a new robust adaptive fuzzy output feedback control scheme is constructed. It is proved that all the solutions of the 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 design parameters appropriately. Two simulation examples are provided to demonstrate the effectiveness of the proposed control approach. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction In the past decades, many approximator-based adaptive backstepping control approaches have been developed to deal with uncertain nonlinear strict-feedback systems with unstructured uncertainties via fuzzy-logic-systems (FLSs) and neural-networks (NNs), see for example [1–13]. Works in [1–5] are for single-input and single-output $

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). n Corresponding author. E-mail address: [email protected] (S. Tong). 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.09.012

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(SISO) nonlinear systems, and works in [6–8] are for multiple-input and multiple-output (MIMO) nonlinear systems, while works in [9–13] are 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 neural networks or fuzzy systems are used to approximate unknown nonlinear functions, and based on the backstepping design technique, typically adaptive fuzzy or neural network controllers are constructed recursively. Two of the main features of these adaptive approaches are (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, the approximator-based adaptive backstepping control has become one of the most popular design approaches to deal with the control problem on the uncertain nonlinear systems without satisfying the matching condition. It is well known that stochastic disturbances often exist in various engineering, such as biological engineering, economical systems, paper machine and motor load. Their existence is a source of instability of the control systems. Thus, the investigations on the stochastic system control problem have received considerable attention in recent years, and many important results have been achieved, see for example [14–27] and references therein. [14] first proposed an adaptive backstepping control design approach for strict-feedback stochastic systems by a risk-sensitive cost criterion. [15] solved the output feedback stabilization problem of strict-feedback stochastic nonlinear systems by using the quadratic Lyapunov function, while [16,17] developed backstepping control design approaches for nonlinear stochastic systems with Markovian switching, [18–24] studied the control problems of uncertain stochastic nonlinear systems with time delays or actuator faults, respectively. Moreover, to solve the unmeasured states and the unmodeled dynamics, several different adaptive output-feedback controllers are developed for strictfeedback stochastic nonlinear systems by designing a linear state observer [25–27]. It should be pointed out that 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. Therefore, they cannot be applied to those stochastic systems with structured uncertainties. In order to handle the structured uncertainties included in the stochastic nonlinear strictfeedback systems, several adaptive backstepping control schemes have been developed by using fuzzy logic systems and neural networks. For example, [28,29] developed adaptive NN or fuzzy output feedback control approaches for a class of SISO stochastic nonlinear systems, while [30] extended the above results to a class of stochastic large-scale nonlinear systems. However, these adaptive fuzzy or NN control approaches did not consider the problem of the unmodeled dynamics; the designed controllers lacked the robustness to the unmodeled dynamics or dynamical disturbances. As stated in [31,32], the unmodeled dynamics or dynamical disturbances often exist in many practical nonlinear systems, and 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 is very important in control theory and applications. It should be mentioned that in recent years, the authors in [33,34] have developed two important output feedback control approaches for the fuzzy stochastic systems on the basis of the results in [35–40]. However, the proposed control schemes are designed based on the linearization technique or linear robust control theory, not based on the backstepping design technique.

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Therefore, they cannot be applied to control those nonlinear systems with the dynamical uncertainties and without satisfying the matching condition addressed in this paper. The purpose of this paper is to investigate the adaptive fuzzy output feedback control for a class of stochastic nonlinear systems with three types of uncertainties, i.e., unknown nonlinear functions, dynamical uncertainties and unmeasured states. In the control design, FLSs are employed to approximate the unknown nonlinear functions, and a fuzzy state observer is designed to estimate the unmeasured states. To solve the problem of the dynamical uncertainties, the dynamical signal and the changing supply function technique are incorporated into the backstepping recursive design technique, and a new robust adaptive fuzzy backstepping output feedback control scheme is constructed. The main advantages of the proposed adaptive fuzzy control approach are summarized as follows: (i) by designing a new fuzzy state observer, not a linear state observer used in [25–28], the proposed state observer can obtain the better estimations of the unmeasured states; (ii) by incorporating the dynamical signal and the changing supply function technique into the backstepping recursive design technique, the proposed adaptive fuzzy control approach can applied to a larger class of stochastic nonlinear systems and has the robustness to the dynamical uncertainties; and (iii) It is mathematically proved the proposed approach cannot only ensure that the resulting closed-loop system are bounded in probability, but also has the robustness to the dynamical uncertainties, which cannot be handled by the previous results in [28–30]. 2. System descriptions and preliminary results 2.1. System descriptions and basic assumptions In this paper, we consider the following uncertain stochastic nonlinear system: dz ¼ q1 ðz,yÞdt þ q2 ðz,yÞdw dxi ¼ ½xiþ1 þ fi ðx i Þ þ Di ðx,zÞdt þ gi ðxÞdw i ¼ 1,    ,n1, dxn ¼ ½u þ fn ðx n Þ þ Dn ðx,zÞdt þ gn ðxÞdw y ¼ x1 T

ð1Þ

i

where x i ¼ ½x1 ,x2 ,. . .,xi  2 R , 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(x,z) are the dynamic disturbances. fi x i Þ, i ¼ 1,2,. . .,n are unknown smooth nonlinear functions. q1(z,y), q2(z,y), Di(x,z) and gi(x) are uncertain functions; wAR is an independent standard Wiener process defined on a complete probability space. In this paper, it is assumed that the functions  fi x i Þ, gi(x), qi(z,y) and Di(x,z) satisfying the locally Lipschitz, and only the output y is available for measurement. Assumption 1. ([25–27]): For each 1rirn, there exist unknown positive constants pni such that jDi ðx,zÞjrpni ci1 ðyÞ þ pni ci2 ðjzjÞ jgi ðxÞjrpni ci3 ðyÞ where ci1(y), ci2(9z9) and ci3(y) are known nonnegative smooth functions with ci1(0) ¼ ci2(0) ¼ ci3(0) ¼ 0.

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By Assumption 1, there exist smooth functions ci1 , ci2 and ci3 such that ci1 ðyÞ ¼ yci1 ðyÞ, ci2 ðjzjÞ ¼ jzjci2 ðjzjÞ, ci3 ðyÞ ¼ yci3 ðyÞ

ð2Þ

Assumption 2. ([25–27]): For each z-subsystem in (1), there exist function Vz(z) and known kN functions a ðjzjÞ, aðjzjÞ, g(9y9) and   constant c0 40 and d 0 40 such that a ðjzjÞrVz ðzÞraðjzjÞ, ‘Vz rc0 Vz ðzÞ þ g y þ d 0 . Assumption 3. ([29]): There exist a set of known constants mi, i¼ 1,2,y,n for 8X1, X2ARi, the following inequality holds   fi ðX1 Þfi ðX2 Þrmi :X1 X2 :, 1rirn where :X1 X2 : expresses the 2-norm of vector X1X2. Control objective: The control task is to design an adaptive fuzzy output feedback controller using the output y and state estimations x^ i so that the system is bounded in probability and the outputs of the system can be regulated to a small neighborhood of the origin in probability.

2.2. Stability in probability Consider the following time-varying stochastic system: dx ¼ ðf ðt,xÞ þ gðt,xÞuÞdt þ hðt,xÞdw

ð3Þ

where w is an r-dimensional standard Brownian motion, xARn is the state, uAR is the control input, f,g:Rþ  Rn-Rn and h:Rþ  Rn-Rn  r. Definition 1. ([25–27]). For any given V(t,x) associated with the stochastic differential equation (3), define the differential operator ‘ as follows: ‘V ðxÞ ¼

  @V @V @V 1 @2 V þ f ðt,xÞ þ gðt,xÞu þ Tr hT ðt,xÞ 2 hðt,xÞ @t @x @x 2 @x

For control-free stochastic nonlinear system of the form: dx ¼ f ðt,xÞdt þ hðt,xÞdw

ð4Þ

the following stability notions introduced will be used throughout the paper. Definition 2. ([25–27]). The solution process{x(t),tZ0} of stochastic system (4) is said to be bounded in probability, if limc-1 sup0rto1 PfjxðtÞj4cg ¼ 0. Definition 3. ([25–27]). Consider the system (4) with, h(t,0)0. The equilibrium f(t,0)0 is globally stable in probability if for any e40, there exists a class k function g(  ) such that P{9x(t)9og(9x09)}Z1e, 8tZ0, x0ARn\{0}.

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Lemma 1. ([26]). Consider the stochastic system (4) and assume that f(t,0), h(t,0) are bounded uniformly in t. If there exist functions V(t,x), m1(  ), m2(  )AkN, constants c140, c2Z0, and a nonnegative function W(t,x), such that m1 ðjxjÞrV ðt,xÞrm2 ðjxjÞ, ‘V rc1 W ðt,xÞ þ c2 then (i) there exists an almost surely unique solution on [0,N)for the system (4). (ii) the solution process is bounded in probability, when W(t,x)ZcV(t,x) for some constants c40.

Lemma 2. ([26]): If Vz is an exp-ISpS Lyapunov function for a control system z_ ¼ f ðz,uÞ, then for any constants c0 2 ð0,c0 Þ, any initial conditions z0 ¼ z(t0) and r040, there exists a finite T0 ¼ T0(c0,r0,z0)Z0, a non-negative function D0(t0,t) defined for all tZt0 and a dynamical signal described by r_ ¼ c0 r þ r0 ðjujÞ þ d0 , rðt0 Þ ¼ r0 such that D0(t0,t) ¼ 0 for all tZt0þT0 and Vz ðzÞrrðtÞ þ Dðt0 þ T Þ for all tZt0 where the solutions are defined. Remark 2. Without loss of generality, we assume throughout the paper that g in Assumption 2 is a smooth function of the form   gðsÞ ¼ s2 g0 s2 with g0 a nonnegative function. Otherwise, for the non-decreasing function g, we have    1 2 1 gðsÞrg s þ rg s2 þ gð1Þ 2 2 From the smoothness of g there exists a g such that gðsÞ ¼ sgðsÞ, thus we have     gðsÞrg s2 þ gð1Þ : ¼ s2 g0 s2 þ gð1Þ   Replacing g(9x19) by x21 g0 x21 þ gð1Þ and redefining d 0 , the second inequality in Assumption 2 can be represented as   ‘Vz rc0 Vz ðzÞ þ x21 g0 x21 þ d 0 : From Assumption 2 and Lemma 1, for any constants c0 2 ð0,c0 Þ, any initial conditions z0 ¼ z(t0) and r040, there exists a finite T0 ¼ T0(c0,r0,z0)Z0, a non-negative function D0(t0,t) defined for all tZt0 and the dynamical signal described by   r_ ¼ c0 r þ x21 g0 x21 þ d0

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such that jzjra 1 ðrðtÞ þ Dðt0 þ T ÞÞ: 2.3. Fuzzy logic Systems A 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 F1l

and

x2 is F2l

and . . . and

xn is Fnl ,

Then y is G l ,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 i mGl ðyÞare the membership functions of fuzzy sets Fil and Gl, N is the number of inference rules. Through singleton fuzzifier, center average defuzzification and product inference, the FLS can be expressed as PN Qn l ¼ 1 yl i ¼ 1 mFil ðxi Þ y ð xÞ ¼ P N Q n ð5Þ l ¼ 1 ½ i ¼ 1 mF l ðxi Þ i

where yl ¼ maxy2R mGl ðyÞ. Define the fuzzy basis functions as Pni ¼ 1 mF l ðxi Þ i j l ¼ PN n ½P i ¼ 1 mF l ðxi Þ l¼1 i

T

Denoting y ¼ ½y1 ,y2 ,. . .,yN  ¼ ½y1 ,y2 ,. . .,yN  and jðxÞ ¼ ½j1 ðxÞ,j2 ðxÞ,. . .,jN ðxÞT , then fuzzy logic system (5) can be rewritten as yðxÞ ¼ yT jðxÞ

ð6Þ

Lemma 3. ([41]): For any continuous functions f(x) defined over a compact set O and any given positive constants e, there exists a fuzzy logic system (6) and a ideal parameter vector y* such that supjf ðxÞynT jðxÞjre x2O

3. State observer design Note that in the system (1), the functions fi ðx i Þ,i¼ 1,2,y,n are unknown and the states x2,y,xn1 are unmeasured directly, thus a state observer should be established by using fuzzy logic systems to estimate the functions fi ðx i Þ, i¼ 1,2,y,n. By (4), it can be assumed that the unknown functions fi ðx^ i Þ, i¼ 1,2,y,n can be approximated by the following fuzzy logic systems f^ i ðx^ i jyi Þ ¼ yTi jðx^ i Þ, 1rirn,

ð7Þ

where x^ i ¼ ½x^ 1 , x^ 2 ,. . ., x^ i T are the estimates of x i ¼ ½x1 ,x2 ,. . .,xi T ,i ¼ 1,2,y,n, and x^ ¼ x^ n .

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The optimal parameter vectors yni is defined as yni ¼ argminyi 2Oi ½supx^ 2Ui jf^ i ðx^ i jyi Þfi ðx^ i Þj, 1rirn: i

where Oi and Ui are compact regions for yi and x^ i , respectively. The corresponding fuzzy minimum approximation error ei and approximation error di are defined by   ei ¼ fi ðx^ i Þf^ i ðx^ i yni Þ, di ¼ fi ðx^ i Þf^ i ðx^ i jyi Þ

Assumption 4. [11]There are unknown positive constants eni and dni such that jei jreni and jdi jrdni . Denote oi ¼ eidi, by Assumption 4, one has joi jreni þ dni ¼ oni , where oni is also an unknown constant. eni and oni can be estimated by the parameters adaptation laws to be designed in the next section. Write (1) in the state space form dz ¼ q1 ðz,yÞdt þ q2 ðz,yÞdw n X dx ¼ ðAx þ Ky þ Bi fi ðx i Þ þ D þ BuÞdt þ GðxÞdw i¼1

y ¼ Cx where

2

ð8Þ 3

k1

2

k1

3

2 3 0 6 7 B ¼ 4 ^ 5, 1

6 7 6 7 In1 A¼4 ^ 5, K ¼ 4 ^ 5, kn kn 0 . . . 0 ^ Dn ðx,zÞ Bi ¼ ½0    1    0T , C ¼ ½1    0    0,

D ¼ ½D1 ðx,zÞ

GðxÞ ¼ ½g1 ðxÞ    gn ðxÞT

Choose vector K such that matrix A is a strict Hurwitz, therefore, for any a given matrix Q ¼ QT40, there exists a positive definite matrix P¼ PT such that AT P þ PA ¼ Q

ð9Þ

Design a fuzzy state observer as x_^ ¼ Ax^ þ Ky þ

n X

 ^ i Þ þ Bu Bi f^ i xjy

i¼1

y^ ¼ C x^

ð10Þ

Define the observer error vector as e¼

xx^ pn

where pn ¼ maxf1,pni ,pin2 j1rirng.

ð11Þ

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From (7), (9) and (10), the observer error is expressed as

! n 1 X D 1 ^ Bi ½fi ðx i Þfi ðx^ i Þ þ fi ðx^ i Þf i ðx^ i jyi Þ þ n dt þ n G ðxÞdw de ¼ Ae þ n p i¼1 p p ¼ ðAe þ

1 1 ðF þ d þ DÞÞdt þ n G ðxÞdw n p p

ð12Þ

where F ¼ ½F1 ,. . .,Fn T ¼ ½ðf1 ðx1 Þf1 ðx^ 1 ÞÞ,. . .,ðfn ðx n Þfn ðx^ n ÞÞT and d ¼ ½d1 ,. . .,dn T . With (1), (9) and (11), one can obtain the following interconnected system dz ¼ q1 ðz,yÞdt þ q2 ðz,yÞdw  1 1 de ¼ Ae þ n ðF þ d þ DÞ dt þ n G ðxÞdw p p dy ¼ ½x2 þ f1 ðx1 Þ þ D1 ðx,zÞdt þ g1 ðxÞdw d x^ i ¼ ðx^ iþ1 þ f^ i ðx^ i jyi Þ þ ki ðx1 x^ 1 ÞÞdt, 1rirn1 d x^ n ¼ ðu þ f^ ðx^ jyn Þ þ kn ðx1 x^ 1 ÞÞdt n

n

ð13Þ

Consider the following Lyapunov function V0 ¼ eT Pe From (8) and (11), one has



 1 1 T ‘V0 rlmin ðQÞ:e: þ 2e P n ðF þ d þ DÞ þ Tr n2 GðxÞ PG ðxÞ p p 2

T



ð14Þ

By Assumption 3, Young’s inequality and the fact that p*Z1, one has the following inequalities 2 T 2 2 e Pdr2:e:U:P:U:d:r:e: þ :Pdn : pn

ð15Þ

n X 2 T 2 2 2 2 2 2 e PF r2:e:U:P:U:F :r:e: :P: þ :F : ¼ :e: :P: þ m2i :e: pn i¼1

ð16Þ

n X 2 T D 2 2 2 2 2 e PDr:e: :P: þ : : r:e: :P: þ 2 ½c2i1 ð yÞ þ c2i2 ðjzjÞ pn pn i¼1

ð17Þ



 n n   :P: X :P: X 1 T gi 2 ðxÞr n2 pn c ðy Þ2 Tr n2 GðxÞ PG ðxÞ r n2 p p i¼1 p i ¼ 1 i i3 n  n X  :P: X 2 r pci3 ðy Þ2 r:P: pci3 ðyÞy2 p i¼1 i¼1 Substituting (15)–(18) into (14), one obtains n n X X 2 2 ‘V0 rp0 :e: þ FðjzjÞ þ 2 c2i1 ðyÞ þ :P: pci3 ðyÞy2 þ d 1 i¼1

i¼1 n P

n P 2 m2i 1,FðjzjÞ ¼ 2 c2i2 ðjzjÞ where p ¼ maxfpn ,pn2 g,p0 ¼ lmin ðQÞ2:P:  n 2 i ¼ 1 i ¼ 1 d 1 ¼ :Pd : .

ð18Þ

ð19Þ and

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4. Adaptive fuzzy controller design and stability analysis 4.1. Adaptive fuzzy backstepping control design The adaptive fuzzy backstepping control design consists of n-steps, each step is based on the change of coordinates: z1 ¼ y, zi ¼ x^ i ai1 , i ¼ 2,. . .,n

ð20Þ

where ai1(U)ði ¼ 2,. . .,n Þ is an intermediate control. Step 1. From the second equation in (5), and according to It o^ ’s differentiation rule, one has   dy ¼ x2 þ y1nT j1 þ e1 þ D1 dt þ g1 dw   ¼ x^ 2 þ pn e2 þ F1 þ y1nT j1 þ e1 þ D1 dt þ g1 dw   ¼ z2 þ a1 þ pn e2 þ F1 þ y1nT j1 þ e1 þ D1 dt þ g1 dw ð21Þ Choose the Lyapunov function candidate as 1 1 ~T ~ 1 2 1 2 V1 ¼ V0 þ y2 þ y y1 þ p~ e~ þ 2 2l1 1 2r1 1 2r1

ð22Þ

where l140, r140 and r1 40 are design parameters. p ¼ maxfpn ,pn2 , ðpn Þ4=3 g, y~ 1 ¼ yn1 y1 , e~ 1 ¼ en1 ^e1 and p~ ¼ p^p are the parameters errors. y1, e^1 and p^ are the estimates of yn1 , en1 and p, respectively. From (19) and (20), the infinitesimal generator of V1 satisfies     ‘V1 r‘V0 þ y z2 þ a1 þ y1nT j1 þ y pn e2 þ F1 þ e1 þ D1 1 1 _ ~T _ ^1 r1 ~ p_^ þ g21 l1 1 y 1 y 1 r1 e~ 1 e 1 p 2 Using Assumptions 1 and 3 and Young’s inequality, one has 1 1 2 1 2 2 þ m1 :e: þ y2 þ c11 ð yÞy2 þ c212 ðjzjÞ y p e2 þ F1 þ D1 rpy þ 2 2 2 

n



2

ð23Þ



ð24Þ

1 2 1 2 1 2 g1 r pc13 ð yÞ ¼ pc13 ð yÞy2 ð25Þ 2 2 2 1 1 1 Substituting yz2 r y2 þ z42 þ , (24) and (25) into (23) gives 2 4 4   2 ‘V1 rp1 :e: þ FðjzjÞ þ c212 ðjzjÞ þ y a1 þ y1nT j1 þ en1 þ pC11 ð yÞ þ C12 ð yÞ 1 1 _ ~T _ ^1 r1 ~ p^_ þ d1 þ z42 l1 ð26Þ 1 y 1 y 1 r1 e~ 1 e 1 p 4 n P 1 1 1 2 2 where p1 ¼ p0   m21 , d1 ¼ d 1 þ , C12 ðyÞ ¼ y þ c11 ðyÞy þ 2 ci1 ðyÞy and C11 ð yÞ ¼ 2 2 4 i¼1 n P 1 2 2 :P: ci3 ðyÞy þ c13 ð yÞy þ y. 2 i¼1

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(26) can be rewritten as   2 ‘V1 rp1 :e: þ FðjzjÞ þ c212 ðjzjÞ þ y a1 þ yT1 j1 þ p^ C11 ð yÞ þ C12 ð yÞ þ e^1     1 T 1 _ _ ^1 þ r1 ~ p1 p_^ þ d1 þ z42 þ y~ 1 yj1 l1 1 y 1 þ e~ 1 yr1 e 1 p 4 where p1 ¼ r1 C11 ðyÞy þ s1 p^ and s140 is a design parameter. Choose stabilizing control function a1 and the adaptation functions y1, e^1 as

ð27Þ

a1 ¼ c1 yrðrÞyg0 ðy2 ÞyT1 j1 ^pC11 ðyÞC12 ðyÞ^e1

ð28Þ

y_ 1 ¼ l1 yj1 t1 y1

ð29Þ

e_^1 ¼ r1 ys1 e^1

ð30Þ

where c140, t140 and s140 are design constants, and r(U) is a smooth nonnegative function to be designed later. Substituting (28)–(30) into (27) yields   1 2 ‘V1 rp1 :e: þ FðjzjÞ þ c212 ðjzjÞc1 y2 rðrÞy2 g0 y2 þ z42 4   s1 t1 ~ T s1 1 þ y 1 y1 þ e~ 1 e^ 1 þ r2 p~ p1 p^_  p~ p^ þ d1 ð31Þ l1 r1 r1 Step i (2rirn1): A similar procedure in step 1 is employed recursively for step i, one has i1 X   @ai1 dzi ¼ x^ iþ1 þ ki ðx1 x^ 1 Þ þ yTi ji dt d x^ j @x^ j j¼1

 

i1 X @ai1 j¼1

@yj

dyj 

i1 X @ai1 @ai1 @ai1 @ai1 ^ j dy d p^ d^e1  do ^ @^e1 @ o @y @^p j j¼2

 1 @2 ai1 @ai1  n T ¼ x^ iþ1 þ Hi þ y~ i ji þ oi  p e2 þ F1 þ d1 þ D1  g1 ðxÞ2 2 @y2 @y @ai1 _ @ai1 g1 ðxÞdw  p^ dt @^p @y where Hi ¼ ki 

i1 X @ai1 j¼1



@x^ j

! kj ðx1 x^ 1 Þ þ yTi ji 

ð32Þ

i1 i1 X X @ai1  @ai1 _ yj x^ jþ1 þ yTj jj  @x^ j @yj j¼1 j¼1

i1 X  @ai1 @ai1  @ai1 _ ^j x^ 2 þ yT1 j1  e_^1  o ^j @y @^e1 @o j¼2

Consider the following Lyapunov function candidate 1 1 ~T ~ 1 2 ~ Vi ¼ Vi1 þ z4i þ yi yi þ o 4 2li 2ri i

ð33Þ

^ i are the where li40 and ri40are design parameters. y~ i ¼ yni yi and o~ i ¼ oni o ^ i are estimates of yni and oni , respectively. parameters errors, yi and o

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From (47) and (48), one has

 @ai1  n T 3 p e2 þ F1 þ d1 þ D1 ‘Vi ¼ ‘Vi1 þ zi x^ iþ1 þ Hi þ y~ i ji þ oi  @y   1 @2 ai1 3 2 @ai1 2 1 T 1 _ 2 @ai1 _ ~i ~ io ^ z  g ðxÞ  g1 ðxÞ2  y~ i y_ i þ o p þ 1 2 @y2 2 i li ri @^p @y

3131

ð34Þ

By Assumption 1 and Young’s inequality, one obtains the following inequalities   @ai1  n @ai1 2 6 p e2 þ F1 þ d1 þ D1 rp z3i zi @y @y  2   1 1 2 @ai1 2 6 1 2 1 @ai1 1 1 2 þ þ m1 :e: þ zi þ y þ p c11 ð yÞ z6i þ c212 ðjzjÞ þ dn2 2 2 2 2 2 2 @y @y ð35Þ !   2 2 3 2 @ai1 2 2 1 3 @2 ai1 2 1 2 2 @ai1 3 @ ai1 zi 3zi g 1  zi g1 r zi pn c13 ð yÞyUy 2 2 2 2 2 @y @y @y @y  2 2 !  4 p @ai1 @ ai1 1 4 r z3i 9zi þ z3i c13 ð yÞy2 þ y2 ð36Þ 2 2 2 @y @y 3 1 Substituting z3i ziþ1 r z4i þ z4iþ1 and (35)–(36) into (34), one obtains 4 4 i1 i1 X   X tj ~ T iþ1 2 2 c12 ðjzjÞc1 y2  ‘Vi rpi :e: þ FðjzjÞ þ y j yj cj z4j rðrÞy2 g0 y2 þ 2 l j¼2 j¼1 j ! i1 i1 X X   sj @a s1 j1 ~ jo ^ j þ r1 ~þ z3j pi1 p_^ þ ði1Þy2 þ e~ 1 e^1 þ o 2 p r1 r @^ p j¼2 j j¼2

 1 T @ai1 _ T  y~ i y_ i þ z3i ai þ Hi þ y~ i ji þ pCi1 ð yÞ þ oni þ Ci2 ð yÞ p^ li @^p 1 4 1 _ s1 ~ i  p~ p^ þ di ~ io ð37Þ þ ziþ1 þ o 4 ri r1 where  1 1 1 @ai1 2 3 pi ¼ pi1   m21 , di ¼ di1 þ dn2 , Ci2 ð yÞ ¼ z i þ zi 2 2 2 @y    2  2 2 ! @ai1 2 3 1 @ai1 1 @ai1 4 4 3 3 @ ai1 9zi Ci1 ð yÞ ¼ zi þ c ð yÞ zi þ þ zi c13 ð yÞy2 : 2 @y 11 2 @y @y @y2

^ i as Choose stabilizing control function ai and the adaptation functions yi, o ^ i Ci2 þ ai ¼ ci zi Hi yTi ji ^pCi1 ð yÞo y_ i ¼ li ji z3i ti yi

i1 X @aj1 3 @ai1 pi r1 z Ci1 @^p @^p j j¼2

ð38Þ

ð39Þ

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_^ i ¼ ri z3 si o ^i o i

ð40Þ

where ci40, ti40 and si40 are design constants. Then substituting (38)–(40) into (37) yields 2

‘Vi rpi :e: þ FðjzjÞ þ þ

i X tj

l j¼1 j

T y~ j yj

i X   iþ1 2 c12 ðjzjÞ cj z4j rðrÞy2 g0 y2 2 j¼2

i X sj s1 ~ jo ^j þ þ e~ 1 e^1 þ o r1 r j¼2 j

~ r1 2 p

i X @aj1

@^p

j¼2

!  3

zj

pi p_^



1 s1 þði1c1 Þy2 þ z4iþ1  p~ p^ þ di 4 r1

ð41Þ

where pi ¼ pi1 þ r1 z3i Ci1 ðyÞ. Step n: In the final step, the actual control input u will appears. Consider the overall Lyapunov function as 1 1 ~T ~ 1 2 yn yn þ Vn ¼ Vn1 þ z4n þ o~ 4 2ln 2rn n

ð42Þ

Using the similar derivations in step i, one can choose actual control u, and parameters ^ and p^ as follows: adaptation laws y, o ^ n Cn2 þ u ¼ cn zn Hn yTn jn ^pCn1 ð yÞo

n1 X @aj1 3 @an1 pn r1 z Cn1 @^p @^p j j¼2

ð43Þ

y_ n ¼ ln jn z3n tn yn

ð44Þ

_^ n ¼ rn z3 sn o ^n o n

ð45Þ

p_^ ¼ pn ¼ pn1 þ r2 z3n Cn1 ðyÞ



2

ð46Þ

where cn40, tn40, sn40 are design constants and Cn2 ðyÞ ¼ @an1 =@y z3n þ 1=4zn . By substituting (43)–(46) into (42), one has n X   nþ1 2 2 c12 ðjzjÞ þ ðn1c1 Þy2 rðrÞy2 g0 y2 ‘Vi rpi :e:  cj z4j þ FðjzjÞ þ 2 j¼2 þ

n X sj s1 s1 T ~ jo ^ j  p~ p^ þ dn y~ j yj þ e~ 1 e^1 þ o l r r r1 1 j¼1 j j¼2 j n X tj

ð47Þ

By completing the squares n 2

tj :yj : tj :y~ j : tj ~ T tj T  y j yj ¼ y~ j ynj y~ j r þ lj lj 2lj 2lj

ð48Þ

s1 s1 e~ 21 s1 e1n2 þ e~ 1 e^1 r r1 2r1 2r1

ð49Þ

~ 2j sj ojn2 sj o sj ~ jo ^ j r o þ rj 2rj 2rj

ð50Þ

2

T. Wang et al. / Journal of the Franklin Institute 349 (2012) 3121–3141



s1 s1 s1 2 s1 2 p p~ p^ ¼  p~ ðp~ þ pÞr p~ þ r1 r1 2r1 2r1

3133

ð51Þ

Substituting (48)–(51) into (47) results in 2

‘Vn rpn :e: 

n X

  cj z4j þ FðjzjÞ þ ðn1c1 Þy2 rðrÞy2 g0 y2

j¼2 n n X 1 ~ 2 1 2 X 1 2 1 2 ~j þ m :y j : þ p~ e~ 1 þ o 2l 2r 2r 2r j 1 j 1 j¼1 j¼2

! þd

ð52Þ

where n t :yn :2 n s o n2 X

 s1 e1n2 X s1 2 j j j j þ þ þ p m ¼ min tj ,s1 ,sj ,s1 , d ¼ dn þ 2l 2r 2r 2r j 1 j 1 j¼1 j¼2

FðjzjÞ ¼ FðjzjÞ þ

n X nþ1 2 nþ1 2 c12 ðjzjÞ ¼ 2 c12 ðjzjÞ c2i2 ðjzjÞ þ 2 2 i¼1

4.2. Changing supply function design and stability analysis Next, we will design the function r(s) introduced in the first step of the backstepping, using the changing supply function design technique proposed by [42]. From the definition of FðjzjÞ, one gets FðjzjÞ ¼ 2

n X i¼1

X n þ 1 2  1 nþ1 2 c12 a ðrðtÞ c2i2 ða 1 ðrðtÞ þ Dðt0 ,T ÞÞÞ þ c12 ðjzjÞr2 2 2 n

c2i2 ðjzjÞ þ

i¼1

ð53Þ þDðt0 ,T ÞÞÞrPðrÞ þ PðDðt0 ,T ÞÞ  1   2  1 Pn 2 with PðsÞ ¼ 2 i ¼ 1 ci2 a ð2sÞÞ þ n þ 1=2 c12 a ð2sÞÞ. From (6) and Assumption 2, there exists a nonnegative function P0 such that P(s) ¼ sP0(s), which together with (53) gives FðjzjÞrrP0 ðrÞ þ PðDðt0 ,T ÞÞ

ð54Þ

Noting that P(D(t0,T)) ¼ 0 for all tZt0. By (52) and (54), one has   ‘Vn rcVn rðrÞy2 g0 y2 þ rP0 ðrÞ þ PðDðt0 ,T ÞÞ þ d

ð55Þ

where 1

c ¼ minf2ðc1 n þ 1Þ,4c2 ,. . .,4cn ,m,2pn :P: g: Choose Lyapunov function as Z r rðsÞds V ¼ Vn þ 0

From Remark 1 and (55), one gets       ‘V rcVn rðrÞy2 g0 y2 þ rP0 ðrÞ þ P Dðt0 ,T ÞÞ þ d þ rðrÞ c0 r þ y2 g0 y2 þ d0 Þ

ð56Þ

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It follows from the nondecreasing property of r(s) that  Z r c0 4d0 rðsÞdsrrðrÞr,  rrðrÞ þ rðrÞd0 rr d0 4 c0 0

ð57Þ

Choose r(r) satisfying rðrÞ ¼

4 P 0 ð rÞ c0

Combining (56)–(58) gives  c c c0 0 0 ‘V rcVn  rðrÞr þ r  rðrÞ þ P0 ðrÞ  rðrÞr þ PðDðt0 ,T ÞÞ þ d þ rðrÞd0 2 4 4 rcV þ PðDðt0 ,T ÞÞ þ X

ð58Þ

ð59Þ

where  n c o 4d0 0 c ¼ min c, d0 , X¼d þr 2 c0 Define W ¼ cVP(D(t0,T)). Then, it is easy to see that W is positive-definite and radically unbounded in its ^ p^ Þ and satisfies arguments ðe,z,y,^e1 , o, ‘V rW þ X

ð60Þ

where e ¼ ½e1 ,. . .,en T , z ¼ ½z1 ,. . .,zn T , p^ ¼ ½^p1 ,. . ., p^ n T , d^ ¼ ½d^ 1 ,. . ., d^ n T , ^ ¼ ½o ^ 2 ,. . ., o ^ n T y ¼ ½y1 ,. . .,yn T , o By Lemma 1, the closed-loop system has an almost surely unique solution on [0,N), and moreover, the solution of the closed-loop system is bounded in probability and for any given e40, there exist a k‘ function b and a k function bc such that 8tZ0, one has  ^ Þjob jðeð0Þ,zð0Þ, p^ ð0Þ,^e1 ð0Þ, oð0Þ,yð0ÞÞj,tÞ ^ Pfjðe,z, p^ ,^e1 , o,,y þ bc ðXÞgZ1e ^ where ðeð0Þ,zð0Þ, p^ ð0Þ,^e1 ð0Þ, oð0Þ,yð0ÞÞa0. From the definition of d, X can be made small if we choose the design parameters appropriately. The aforementioned analysis and the changing supply function design are summarized in the following theorem. Theorem 1. For stochastic nonlinear system with unmodeled dynamics (1), under Assumptions 1–4, the fuzzy state observer (10) and the controller (43), with the intermediate control (38) and parameter laws (29)–(30), (39)–(40) and (44)–(46) guarantee that the closed-loop system (1) is bounded in probability.

Simulation study In this section, the simulation example and comparisons with the adaptive control method in [29] are provided to illustrate the effectiveness of the proposed adaptive fuzzy control approach.

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Example 1. consider the following stochastic nonlinear system:  1 dz ¼ ðz þ x1 Þdt þ pffiffiffi dw 2zcosx1 dx1 ¼ ðx2 þ sinx1 þ 0:1z þ 0:5x1 Þdt þ ðx1 sinx2 Þdw dx2 ¼ ðu þ x1 þ x1 þ zsinx2 Þdt þ ðx1 cosx2 Þdw y ¼ x1

ð61Þ

with the notations of Assumption 1, we can take c11(y)¼ 9y9, c21(y)¼ 9y9, c12(z) ¼ 9z9, c22(z)¼ 9z9, c13(y) ¼ c23(y)¼ 9y9, pni ¼ 1. Define fuzzy membership as follows: mF l ðx^ 1 Þ ¼ exp½ðx^ 1 3 þ l Þ2 =16,l ¼ 1,. . .,5 1

mF l ðx^ 1 , x^ 2 Þ ¼ exp½ðx^ 1 3 þ l Þ2 =4  exp½ðx^ 2 3 þ l Þ2 =4, l ¼ 1,. . .,5 2

We obtain fuzzy basis functions as follows: j1j ðx^ 1 Þ ¼

exp½ðx^ 1 3 þ j Þ2 =16 , j ¼ 1,. . .,5: 5 P 2 exp½ðx^ 1 3 þ nÞ =16 n¼1

j2j ðx^ 1 , x^ 2 Þ ¼

exp½ðx^ 1 3 þ j Þ2 =4  exp½ðx^ 2 3 þ j Þ2 =4 , j ¼ 1,. . .,5: 5 P 2 2 exp½ðx^ 1 3 þ nÞ =4  exp½ðx^ 2 3 þ nÞ =4 n¼1

The fuzzy logic systems can be expressed in the form f^ 1 ðx^ 1 jy1 Þ ¼ yT1 j1 ðx^ 1 Þ ¼

5 X

yT1j j1j ðx^ 1 Þ, f^ 2 ðx^ 2 jy2 Þ ¼ yT2 j2 ðx^ 1 , x^ 2 Þ ¼

j¼1

5 X

yT2j j2j ðx^ 1 , x^ 2 Þ

j¼1

where yT1 ¼ ½y11 ,y12 ,y13 ,y14 ,y15 , yT2 ¼ ½y21 ,y22 ,y23 ,y24 ,y25 , j1 ðx^ 1 Þ ¼ ½j11 ðx^ 1 Þ,j12 ðx^ 1 Þ,j13 ðx^ 1 Þ,j14 ðx^ 1 Þ,j15 ðx^ 1 ÞT , j2 ðx^ 1 , x^ 2 Þ ¼ ½j21 ðx^ 1 , x^ 2 Þ,j22 ðx^ 1 , x^ 2 Þ,j23 ðx^ 1 , x^ 2 Þ,j24 ðx^ 1 , x^ 2 Þ,j25 ðx^ 1 , x^ 2 ÞT Choose actual control u, and parameters adaptation laws y, d^ and p^ as follows: a1 ¼ c1 yrðrÞyg0 ðy2 ÞyT1 j1 ^pC11 ðyÞC12 ðyÞ^e1 ^2 þ u ¼ c2 z2 H2 yT2 j2 ^pC21 ð yÞC22 o

@a1 _ p^ @^p

ð62Þ ð63Þ

with y_ 1 ¼ l1 yj1 t1 y1 , y_ 2 ¼ l2 j2 z32 t2 y2 , e_^1 ¼ r1 ys1 e^1 , p_^ ¼ r1 C11 ðyÞy þ r1 z32 C21 ðyÞ þ s1 p^ : For the z-system, we choose Lyapunov function as Vz ¼ (1/2)z2, then we have       ‘Vz r 1=4 z2 þ 1=2 x21 , which implies that Assumption 2 is satisfied for c0 ¼ 1=2 ,   P d 0 ¼ 0, g(9y9)¼ (1/2)y2, g0(9y9)¼ (1/2),a ðjzjÞ ¼ aðjzjÞ ¼ 1=2 z2 . FromFðjzjÞ ¼ 2 2i ¼ 1 c2i2

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3 ðjzjÞ þ c212 ðjzjÞ¼ (7/2)z2, it is easy to obtain P(s)¼ 14s and P0(s)¼ 14, then we can choose 2 c0 ¼ (1/4) and r(r)¼ 224. Selecting Q ¼ I, k1 ¼ 12, k2 ¼ 20, by solving (9) to obtain the positive definite matrix  0:0437 0:0250 P¼ . 0:0250 1:1750 In this simulation, we choose the design parameters c1 ¼ c2 ¼ 0.1, r1 ¼ r2 ¼ 0.1, l1 ¼ l2 ¼ 0.5, t1 ¼ t2 ¼ 1.2, s1 ¼ s2 ¼ 1, r1 ¼ s1 ¼ 0:8, the initial conditions are chosen as yT1 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9, yT2 ð0Þ ¼ ½0:1,0:3,0:5,0:7,0:9, x1(0) ¼ 0, x2(0) ¼ 0.2, x^ 1 ð0Þ ¼ 0, x^ 2 ð0Þ ¼ 0, z(0) ¼ 0.5, d^ ¼ 0, p^ ¼ 0. The simulation results are shown in Figs. 1–4, where Fig. 1 is the trajectories of x1 and x^ 1 ; Fig. 2 is the trajectories of x2 and x^ 2 ; Fig. 3 is the trajectories of :y: and z; Fig. 4 is the trajectory of u. Example 2. In order to illustrate the robustness of the proposed control approach to the unmodeled dynamics, we use the adaptive fuzzy control approach in [29] to control the nonlinear stochastic system (61) in Example 1. In the simulation, we use the same fuzzy logic systems, state observer and the initial conditions as in Example 1. The design parameters included in the adaptive control scheme in [29] are: k1 ¼ 10, k2 ¼ 12, g1 ¼ 1, g2 ¼ 2, s1 ¼ 0.1, s2 ¼ 0.12, s1 ¼ 0:01, s2 ¼ 0:02, c1 ¼ 4, c2 ¼ 5, v1 ¼ 0.2, v2 ¼ 0.3, Z0 ¼ 10, Z1 ¼ 0.5, Z2 ¼ 0.2, k ¼ 0.05, g1 ¼ 5, g2 ¼ 6. The simulation results are shown by Figs. 5–7. From Figs. 5–7, one can conclude that the control scheme of [29] cannot guarantee the stability of the control due to the dynamical uncertainties. From the simulation results in Figs. 1–7, it clearly shows that the proposed fuzzy output feedback control approach can achieve good control performances even if the controlled

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

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

Fig. 3. :y: (solid) and z(dotted).

nonlinear stochastic system contains the unknown functions, the unmodeled dynamics and unmeasured states. 6. Conclusions In this paper, a fuzzy state-observer-based adaptive fuzzy output feedback control approach has been proposed for a class of uncertain stochastic nonlinear systems with

3138

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Fig. 4. Controller u.

Fig. 5. x1 (solid) and x^ 1 (dotted).

unknown functions, dynamic uncertainties and without the direct measurements of state variables. In the design, fuzzy logic systems are utilized to approximate the unknown functions and a fuzzy state observer is developed. By using the fuzzy state observer and based on the principle of the adaptive backstepping technique and combining with the changing supply function and the dynamical signal, a new adaptive fuzzy output feedback control scheme is synthesized. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system

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Fig. 6. x2 (solid) and x^ 2 (dotted).

Fig. 7. Controller u.

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.

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