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Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay$ Hongyun Yuea,n, Shuiqing Yub a
School of Science, Xi'an University of Architecture and Technology, Xi'an 710055, China b School of Statistics, Xian University of Finance and Economics, Xi'an 710100, China
Received 7 February 2015; received in revised form 21 October 2015; accepted 6 December 2015
Abstract This paper addresses the problem of adaptive tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay using fuzzy logic systems (FLS). In this paper, through a state transformation the system can be easily transformed into a system without the distributed input delay, the quadratic functions instead of the quartic functions often utilized in the existing results are used as Lyapunov functions to analyze the stability of systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms. FLS in Mamdani type are used to approximate the unknown nonlinear functions. Then, based on the backstepping technique, the adaptive fuzzy controller is designed. The proposed adaptive fuzzy controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error can converge to a small residual set around the origin in the mean square sense. Finally, the example is used to demonstrate the effectiveness of our results proposed in this paper. & 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
☆ This work is supported by National Natural Science Foundation of China Nos. 61403298, 11401457), the Youth Foundation of Xi'an University of Architecture and Technology (No. QN1436), the Talent Foundation of Xi'an University of Architecture and Technology (No. RC1425). n Corresponding author. E-mail addresses:
[email protected],
[email protected] (H. Yue),
[email protected] (S. Yu).
http://dx.doi.org/10.1016/j.jfranklin.2015.12.004 0016-0032/& 2015 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
H. Yue, S. Yu / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
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1. Introduction In past decade, the investigation for the systems with input delay has received some results [1– 7]. In References [1–3], the control schemes were proposed for the linear systems with input delay. In [4], authors proposed the state-feedback controller design scheme for a class of high order feedforward nonlinear systems with delayed inputs, in which the uncertainties were assumed to be dominated by higher-order nonlinearities multiplying by a constant growth rate and the designed controller could achieve global asymptotical stability. In [5], by using the appropriate Lyapunov–Krasovskii functional, the solution to the problem of globally asymptotically stabilizing a nonlinear system in feedback form with a known delay in the input was proposed. Moreover, in [6,7], authors developed the effective adaptive controller by using the NN or by using the FLS for the nonlinear system with input delay, respectively. However, to the authors' knowledge, there are no results in which the effective adaptive tracking controllers are proposed by using the FLS for the stochastic nonlinear systems with distributed input delay. In addition, the investigation on stability analysis and control design of stochastic systems has received increasing attention in the past few decades because stochastic disturbances often exist in practical systems, such as [8,10–13,19,21,24–29,32,35–38] and the references therein. In particular, many interesting results [10,11,26,27] have been proposed by using the well-known backstepping technique for a class of stochastic nonlinear strict-feedback (or output-feedback) systems. Ref. [12] first derived a backstepping design approach for stochastic nonlinear strictfeedback systems motivated by a risk-sensitive cost criterion. Since then, a series of extensions has been made under different assumptions or for different systems [13,21]. By using the quartic Lyapunov functions instead of the classical quadratic functions [10] solved the (adaptive) stabilization problem of stochastic strict-feedback (or output-feedback) systems, and then, this design idea was extended to several different cases [11,14,17,19,24,25,27,28,36–38]. It should be pointed out that up to now there are only several literatures using the classical quadratic Lyapunov functions to deal with the stochastic tracking control problem, but they are all with respect to a risk-sensitive cost criterion [9,21] or under the assumption that the disturbance covariance is bounded [31]. So far, only in [16] for the stochastic nonlinear systems, the authors design the effective tracking controller by combining the classical quadratic Lyapunov functions without using a risk-sensitive cost criterion, but in [16] the authors did not consider the stochastic nonlinear systems with distributed input delays. Another challenging problem is that most researches were concerned with the systems with linear parameterizations. In practice, since nonlinear parameters are commonly existing components in many deterministic practical control systems [20,23,33,34]. From a practical point of view, when designing controller for a system, nonlinear parameterizations problem should be taken into account. On the other hand, from a theoretical view point, adaptive control of nonlinearly parameterized stochastic systems is also interesting, because it represents a new challenge to the theory of nonlinear adaptive control. However, to the best knowledge of the authors, there are very few results [41,42] in which the stochastic nonlinearly parameterized systems are investigated. In [41], the problem of adaptive stabilization by state-feedback for a class of stochastic nonholonomic systems in chained form with nonlinear parameterizations was solved, and in [42], the problem of adaptive state feedback stabilization for a class of stochastic nonlinearly parameterized nonholonomic systems in chained form with unknown control coefficients was investigated. Nevertheless, for the stochastic nonlinearly parameterized systems
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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with distributed input delays, how to design the controllers by combining the state transformation, the classical quadratic Lyapunov functions and the FLS is a challenging problem. Based on the above observation, in this paper, the problem of output tracking is revisited for stochastic strict-feedback nonlinearly parameterized systems with distributed input delay using fuzzy control. The system will be transformed into a system without the distributed input delay by using the state transformation with the integrator. Then, the separation principle is used to deal with the nonlinearly parameterized functions, quadratic functions are used as Lyapunov functions to analyze the stability of systems and the hyperbolic tangent functions are introduced to deal with the Hessian terms. Furthermore, the FLS are employed to approximate the unknown nonlinear functions. Finally, the adaptive backstepping approach is utilized to construct the fuzzy controller. The two main advantages of the scheme are that: (1) In this paper, the classical quadratic functions instead of the fourth moment approach often utilized in the most existing results [11,14,17,19,24,25,27,36–38] are used as Lyapunov functions to analyze the stability of the stochastic nonlinear systems, and the hyperbolic tangent functions are introduced to deal with the higher order Hessian terms. (2) Unlike the result [16], this is the first attempt to deal with the control problem of stochastic nonlinear systems with the distributed input delay and the unknown nonlinearly parameterized functions by using the classical quadratic functions, and the existence of the distributed input delay and the unknown nonlinear parameters such that the design of the adaptive fuzzy tracking controllers becomes much more difficult. (3) In [20], it was assumed that the nonlinearly parameterized functions satisfied the separation principle, and it should be pointed out that the functions separated out must be known, in this paper, we will relax the assumption, that is to say, the functions separated out may be unknown, and they can be dealt with by using the FLS. It can be proven that all the signals in the closed loop system are bounded in probability and the tracking error can converge to a small residual set around the origin in the mean square sense. Simulation results are provided to show the effectiveness of the proposed approach. 2. Problem formulation and preliminaries 2.1. Problem formulation Consider the following nonlinear system: 8 dx ¼ x þ f i x i ; ρi dt þ hTi ðx i Þ dω; > < i Riþ1 t dxn ¼ t τ uðsÞ ds þ f n x n ; ρn dt þ hTn ðx n Þ dω; > : y ¼ x1 ;
ð1Þ
where x ¼ ½x1 ; …; xn T A Rn is the state vector of the system with x i ¼ ½x1 ; …; xi T A Ri , τ and y A R are the distributed input delay and the output signal, respectively. ω is an r-dimensional standard Brownian motion defined on a complete probability space ðΩ; F ; PÞ with Ω being a Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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sample space, F being a s-field and P being a probability measure. ρi , i ¼ 1; 2; …; n, are unknown parameters, f i ðÞ and hi ðÞ are unknown smooth nonlinear functions. The control objective is to design an adaptive fuzzy state-feedback tracking controller for the system (1), such that (i) all the signals in the closed loop system are bounded in probability and (ii) the output y(t) follows the desired trajectories yid as expected in the mean square sense. To the end, h define a vector function as y Tdi ¼ yd ; …; yðdiÞ ; i ¼ 1; …; n, where yðdiÞ is the ith time derivative of yd . Consider an n-dimensional stochastic system dxðt Þ ¼ f ðxÞ dt þ gðxÞ dω;
8 t Z 0;
where f : Rn -Rn and g : Rn -Rnr are locally Lipschitz, x and ω are the same ones defined in Eq. (1). Define a differential operator L as follows: ∂V 1 ∂2 V f ðxÞ þ Tr gT 2 g ; LV ðxÞ ¼ ð2Þ ∂x 2 ∂x where VðxÞ A C2 . In addition, the following assumption and lemmas are needed in this paper. Assumption 1 (Wang and Chen [15]). The desired trajectory vectors y di are known and continuous, and y di A Ωdi A Riþ1 with Ωdi being known compact sets, i ¼ 1; 2; …; n. Lemma 1 (Wu et al. [20], Lin and Qian [23]). For any real-valued continuous function f ðx; yÞ, where x A Rn and y A Rn , there are smooth scalar functions aðxÞZ 0, bðyÞ Z 0, cðxÞ Z1 and dðyÞZ 1, such that f ðx; yÞ r aðxÞ þ bðyÞ; f ðx; yÞ r cðxÞdðyÞ: ð3Þ Lemma 2 (Wang and Chen [15]). For 1 r jr n and ϑj 40, consider the set Θϑj given by Θϑj ≔ zj zj r 0:2554ϑj . Then, for zj 2 = Θϑj , the inequalities 1 16 tanh2 zj =ϑj o0 are satisfied. 2.2. Fuzzy logic systems In this paper, the following rules are used to develop the adaptive fuzzy controller Rl : if x1 is and x2 is F l2 and … and xn is Fln, then y is Gl , l ¼ 1; 2; …; Q, where x ¼ ½x1 ; …; xn T and y are the FLS input and output, respectively. Fuzzy sets Fli and Gl , associated with the membership functions μFli ðxi Þ and μGl ðyÞ, respectively. Q is the rule number. Through singleton function, center average defuzzification, the FLS can be expressed as follows: PQ n l ¼ 1 Φl ∏i ¼ 1 μF li ðxi Þ yðxÞ ¼ PQ ; ð4Þ n l ¼ 1 ∏i ¼ 1 μF li ðxi Þ F l1
l function where x ¼ ½x1 ; x2 ; …; xn T A Rn , μFli ðxi Þ is the membership
T of F i , and Φl ¼ T arg supy A R μGl ðyÞ. Define ϕ ¼ Φ1 ; …; ΦQ and ξðxÞ ¼ ξ1 ðxÞ…; ξQ ðxÞ with the fuzzy basis
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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function ξl given by ∏ni ¼ 1 μFli ðxi Þ ξl ðxÞ ¼ PQ : n l ¼ 1 ∏i ¼ 1 μF li ðxi Þ
ð5Þ
Then, the fuzzy logic system (4) can be rewritten as yðxÞ ¼ ϕT ξðxÞ:
ð6Þ
Our first choice for the membership function is the Gaussian function μFli ðxi Þ ¼ exp 12 ððxi ali Þ=sli Þ2 Þ, where sli and ali are fixed parameters. It has been proven that when the membership functions are chosen as Gaussian functions, the above fuzzy logic system is capable of uniformly approximating any continuous nonlinear function over a compact set with any degree of accuracy. This property is shown by the following lemma. Lemma 3 (Wang [18]). Let f ðxÞ be a continuous function defined on compact set Ω. Then, for any constant ι40, there exists an FLS (6) such that sup f ðxÞ ϕT ξðxÞ r ι: ð7Þ xAΩ
3. Controller design In this section, we will use the recursive backstepping technique to develop the adaptive fuzzy tracking control laws as follows: αi ¼ ki zi
θ^ i ξTi ðZ i Þξi ðZ i Þzi ^ β i zi ; 2η2
" # 1 θ^ n ξTn ðZ n Þξn ðZ n Þzn ^ k n zn þ uðt Þ ¼ þ β n zn ; τ 2η2 _ θ^ i ¼ ξTi ðZ i Þξi ðZ i Þz2i si θ^ i ;
_ β^ i ¼ 2γz2i κi β^ i ;
ð8Þ
ð9Þ ð10Þ
where αi 1 is called the intermediate 1r i r n, η40, γ40, κi 40 and si 40 2 control function, 2 are designed parameters. θi ¼ ϕi , where ϕi and βi are unknown parameters and will be specified later. θ^ i and β^ i are the estimations of θi and βi , respectively, and the estimation errors are defined as θ~ i ¼ θi θ^ i and β~ i ¼ βi β^ i . The control gain ki satisfies ki 4 1λ ; i ¼ 1; 2; …; n 2; n with λ being a positive design parameter, and especially, k n 1 satisfies kn 1 4 1λ þ 1 ϖ 2 þ 2 with ϖ being a positive design parameter, ξi ðZ i Þ is a fuzzy basis function vector with Z i being the input vector. Note that when i¼ n, αn is the true control input u(t), α0 is equal to yd . The n-step adaptive fuzzy backstepping tracking control design is based on the following change of coordinates: 8 z ¼ y yd ; > < 1 zi ¼ xi αi 1 ; i ¼ 2; …; n 1; ð11Þ Rt Rt > : z ¼ x α þ u ð s Þ ds dι n n n1 tτ ι Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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Remark 1. It should be emphasized that for the systems with distributed input delay, it is very difficult to design the controllers, in this paper, the system (1) will be transformed into a system without the distributed input delay by using the state transformation with an integrator (11). Now, we propose the following backstepping-based design procedure. Step 1: From Eqs. (1) and (11) yields that dz1 ¼ dy dyd ¼ x2 þ f 1 ðx1 ; ρ1 Þ y_ d dt þ hT1 ðx1 Þ dω; choose the Lyapunov function candidate as 1 V 1 ¼ z21 ; 2 then, we can get 1 LV 1 ¼ z1 x2 þ f 1 ðx1 ; ρ1 Þ y_ d þ hT1 ðx1 Þh1 ðx1 Þ: 2 Using the triangular inequality and Lemma 1 gives that ε 1 z1 f 1 x1 ; ρ1 r jz1 jf 1 ðx1 ; ρ1 Þ r ν1 jz1 jn1 ðx1 Þ r ν21 z21 þ n21 ðx1 Þ; 2 2ε
ð12Þ
ð13Þ
ð14Þ
where ε is a positive design parameter, ν1 Z 1 and n1 ðx1 ÞZ 1 are unknown constant and unknown continuous function, respectively. Remark 2. In [20], it was assumed that f i x i ; ρi r ðjx1 j þ ⋯ þ jxi jÞbi x1 ; …; xi ; ρi and bi x1 ; …; xi ; ρi satisfied the separation principle, moreover, the functions separated out must be known. However, in this paper, the assumption is relaxed, the functions n1 ðx1 Þ separated out may be unknown. Then, they can be dealt with by using the FLS, and the details can be seen in the later. From Eqs. (13) and (14) gives that
1 LV 1 r z1 x2 þ H ðx1 Þ y_ d þ β1 z21 ¼ z1 x2 þ φ1 ðZ 1 Þ þ β1 z21 ; z1 where β1 ¼ 2ε ν21 , H ðx1 Þ ¼ 12 hT1 ðx1 Þh1 ðx1 Þ þ 2ε1 n21 ðx1 Þ and φ1 ðZ 1 Þ ¼
H 1 ðx 1 Þ z1
ð15Þ y_ d with Z 1 ¼ ½x1 ;
yd ; y_ d T A ΩZ 1 R3 and ΩZ 1 being some known compact set in R3. Notice that in Eq. (15) the term H 1zð1x1 Þ is discontinuous at z1 ¼ 0. Therefore, it cannot be approximated by the FLS. Similar to [15], we introduce the hyperbolic tangent function tanh ϑz11 to deal with the term. Define H 1 ðx1 Þ 16 2 z1 φ 1 ð Z 1 Þ ¼ φ1 ð Z 1 Þ þ tanh ð16Þ H 1 ðx1 Þ; z1 z1 ϑ1 2 z1 where ϑ1 is a positive design parameter. Note that limz1 -0 16 tanh z1 ϑ1 H ðx1 Þ exists, thus, the nonlinear function φ ðZ 1 Þ can be approximated by an FLS ϕT1 ξ1 ðZ 1 Þ such that φ 1 ðZ 1 Þ ¼ ϕT1 ξ1 ðZ 1 Þ þ δ1 ðZ 1 Þ; furthermore, using the triangular inequality yields that θ1 z2 η2 λε2 z1 ϕT1 ξ1 ðZ 1 Þ þ δ1 ðZ 1 Þ r 2 ξT1 ξ1 z21 þ 1 þ þ 1 ; 2η 2λ 2 2
ð17Þ
ð18Þ
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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from Eqs. (15) to (18), we can get
θ1 T z1 η2 λε2 LV 1 r z1 z2 þ α1 þ 2 ξ1 ξ1 z1 þ þ β1 z21 þ þ 1 2η 2λ 2 2
z 1 þ 1 16 tanh2 H1; ϑ1
7
ð19Þ
where λ and η are the positive design parameters, and ε1 is the upper bound of δðZ 1 Þ. Remark 3. During the controller design process for the stochastic nonlinearly parameterized systems using the backstepping technique, the unknown parameter β1 ¼ 2ε ν21 is separated out from the nonlinearly parameterized function f 1 ðx1 ; ρ1 Þ by using 2 the separation principle, and θ1 ¼ ϕ1 is the unknown parameter of the fuzzy logic system which is used to approximate the unknown nonlinear continuous function φ 1 ðZ 1 Þ, the details can be seen in Eqs. (14) and (18), respectively. Therefore, in step 1, there are two parameters θ1 and β1 needing to be estimated in Eq. (19), and their parameter adaptive laws are defined in Eq. (10). Similarly, in step i two parameters θi and βi will be estimated online with i ¼ 2,…,n. Then, choose the following Lyapunov candidate as V 1 ¼ V1 þ
2 2 θ~ 1 β~ 1 ; þ 4η2 4γ
ð20Þ
then, we have
θ1 T z1 η2 LV 1 r z1 z2 þ α1 þ 2 ξ1 ξ1 z1 þ þ β1 z21 þ 2η 2λ 2
_ _ λε21 ~ θ^ 1 β^ 1 2 z1 ~ θ 1 2 β 1 þ 1 16 tanh þ H1; 2 2η 2γ ϑ1
ð21Þ
choosing the virtual controller α1 and the parameter adaptive laws defined in Eqs. (8) and (10) gives that
2 2 1 2 λε21 η2 s1 θ~ 1 s1 θ21 κ 1 β~ 1 η2 κ1 β21 LV 1 r z1 z2 k1 þ þ þ þ z1 þ 2λ 2 2 4η2 4η2 4γ 2 4γ
z 1 þ 1 16 tanh2 ð22Þ H1; ϑ1 moreover, in Eq. (22), the residual term z1 z2 will be dealt with in the next step. Remark 4. It should be pointed out that by combining the classical quadratic Lyapunov functions and the hyperbolic tangent functions, authors provide a new scientific and feasible control method for the stochastic nonlinear systems. The main technical obstacle in the design for stochastic systems is that the Itô stochastic involves not only the gradients but also n differentiation o the higher order Hessian terms
1 2
Tr gT ∂∂xV2 g . In order to handle the Hessian terms 2
conveniently, in the most existing results, the authors used the quartic functions 14 z41 to analyze the stability of the systems. In addition, for the stochastic tracking problem, in [9,21], the controllers are designed by using the quadratic Lyapunov function under a risk-sensitive cost Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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criterion. Unlike them in this paper, the classical quadratic functions 12 z21 are chosen as Lyapunov functions to investigate the stochastic tracking problem, and for the higher order Hessian terms, the hyperbolic tangent functions tanh
z1 ϑ1
are introduced to dealt with them so that the obstacle
for stochastic systems control by quadratic Lyapunov function is overcome. Step i: Considering zi ¼ xi αi 1 , where αi 1 defined in Eq. (8), then " # i1 X ∂αi 1 dzi ¼ dxi dαi 1 ¼ xiþ1 þ f i x i ; ρi W i f x j ; ρj dt ∂xj j j¼1 ! i1 X ∂α i 1 hT x j dω; þ hTi ðx i Þ ∂xj j j¼1
ð23Þ
where Wi ¼
i1 2 1X ∂ αi 1 T T h x j hl ð x l Þ 2 l;j ¼ 1 ∂xj ∂xl j
þ
i1 i1 i1 X X X ∂αi 1 ^_ ∂αi 1 ∂αi 1 _ ∂αi 1 ^_ xjþ1 þ T y dði 1Þ þ θj þ β j: ^ ^ ∂x ∂y j d ði 1 Þ j ¼ 1 ∂θ j j¼1 j ¼ 1 ∂β j
Choose the Lyapunov function candidate as 1 V i ¼ z2i ; 2
ð24Þ
then, we have
! i1 X ∂αi 1 LV i ¼ zi xiþ1 þ f i x i ; ρi W i f x j ; ρj þ zi 1 zi 1 zi ∂xj j j¼1 ! ! i1 i1 X X 1 T ∂αi 1 T ∂αi 1 h ðx i Þ þ h xj hj x j ; hi ðx i Þ 2 i ∂xj j ∂xj j¼1 j¼1
ð25Þ
using the triangular inequality and Lemma 1 gives that ε 1 zi f i x i ; ρi r jzi jf i ðx i ; ρi Þ r νi jzi jni ðx i Þ r ν2i z2i þ n2i ðx i Þ; 2 2ε i1 i1 i1 X X X ∂αi 1 ∂αi 1 zi f j x j ; ρj r z i f j x j ; ρj r ∂x ∂x j j j¼1 j¼1 j¼1 X i 1 i 1 i1 ∂αi 1 X ∂αi 1 ε X ∂αi 1 zi f j x j ; ρj r zi ν j nj x j r zi f j x j ; ρj r ν2j z2i ∂x ∂x 2 ∂xj j j j¼1 j¼1 j¼1 þ
i1 1 X ∂αi 1 2 nj x j ; 2ε j ¼ 1 ∂xj ð26Þ
where νj Z 1 and nj ðx j ÞZ 1, j¼ 1,…,i, are unknown constants and unknown continuous Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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functions, respectively. From Eqs. (25) and (26) gives that
Hi LV i r zi xiþ1 W i þ zi 1 þ þ βi z2i zi 1 zi ; zi P where βi ¼ 2ε ij ¼ 1 ν2j and
i1 1 X ∂αi 1 2 1 2 Hi ¼ nj x j þ ni ðx i Þ 2ε j ¼ 1 ∂xj 2ε ! ! i1 i1 X X 1 T ∂αi 1 T ∂αi 1 h ðx i Þ þ h xj hj x j : hi ðx i Þ 2 i ∂xj j ∂xj j¼1 j¼1
9
ð27Þ
ð28Þ
Define φi ð Z i Þ ¼ z i 1 þ
Hi W i; zi
ð29Þ
h iT with Z i ¼ x1 ; …; xi ; y di ; θ^ 1 ; …; θ^ i 1 ; β^ 1 ; …; β^ i 1 A ΩZ 1 R4i 1 and ΩZ i being some known compact set in R4i 1 . Similar to step 1, define H i 16 zi φ i ðZ i Þ ¼ φi ðZ i Þ þ tanh2 Hi; zi zi ϑi where ϑi is a positive design parameter. Thus, the nonlinear function φ ðZ i Þ can be approximated by an FLS ϕTi ξi ðZ i Þ such that φ i ðZ i Þ ¼ ϕTi ξi ðZ i Þ þ δi ðZ i Þ;
ð30Þ
then, using θi z2 η2 λε2 zi ϕTi ξi ðZ i Þ þ δi ðZ i Þ r 2 ξTi ξi z2i þ i þ þ i ; 2η 2λ 2 2 yields that
θi z2 η2 λε2 LV i r zi ziþ1 þ αi þ 2 ξTi ξi zi þ i þ þ i þ βi z2i zi 1 zi 2η 2λ 2 2
z i þ 1 þ 16 tanh2 Hi; ϑi where εi is the upper bound of δi ðZ i Þ with δi ðZ i Þ r εi . Choose the Lyapunov function as V i ¼ Vi þ
2 2 θ~ i β~ i ; þ 4η2 4γ
from Eqs. (8) and (10) yields that
2 2 1 2 λε2i η2 si θ~ i si θ2i κi β~ i κ i β2i LV i r zi ziþ1 zi zi 1 ki þ 2 þ 2 þ zi þ 2λ 2 2 4η 4η 4γ 4γ
z i þ 1 16 tanh2 Hi: ϑi
ð31Þ
ð32Þ
ð33Þ
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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Step n 1: Considering zn 1 ¼ xn 1 αn 2 , where αn 2 defined in Eq. (8), then " # 2 nX ∂αn 2 dzn 1 ¼ dxn 1 dαn 2 ¼ xn þ f n 1 x n 1 ; ρn 1 f x j ; ρj W n 1 dt ∂xj j j¼1 ! nX 2 ∂αn 2 T T þ hn 1 ðx n 1 Þ h x j dω; ð34Þ ∂xj j j¼1 where Wn1 ¼
2 2 1 nX ∂ αn 2 T T h x j hl ðx l Þ 2 l;j ¼ 1 ∂xj ∂xl j
þ
nX 2 nX 2 ∂αn 2 ^_ ∂αn 2 ∂αn 2 _ ∂αn 2 ^_ xjþ1 þ T y d ðn 2 Þ þ θj þ β j: ^ ^ ∂xj ∂y dðn 2Þ j ¼ 1 ∂θ j j¼1 j ¼ 1 ∂β j nX 2
Choose the Lyapunov function candidate as 1 V n 1 ¼ z2n 1 ; 2 then, we have
LV n 1 ¼ zn 1 xn þ f n 1 x n 1 ; ρn 1
ð35Þ
ð36Þ ! ∂αn 2 f x j ; ρj þ zn 2 zn 2 zn 1 Wn1 ∂xj j j¼1 nX 2
! ! nX 2 nX 2 1 T ∂αn 2 T ∂αn 2 h þ ðx n 1 Þ h xj hj x j hn 1 ð x n 1 Þ 2 n1 ∂xj j ∂xj j¼1 j¼1 Z t Z t ¼ zn 1 zn þ αn 1 uðsÞ ds dι þ f n 1 x n 1 ; ρn 1 W n 1 tτ ι ! nX 2 ∂αn 2 f x j ; ρj þ z n 2 z n 2 z n 1 ∂xj j j¼1 ! ! nX 2 nX 2 1 T ∂αn 2 T ∂αn 2 h þ ðx n 1 Þ h xj hj x j ; hn 1 ð x n 1 Þ 2 n1 ∂xj j ∂xj j¼1 j¼1 ð37Þ using the triangular inequality and Lemma 1 gives that zn 1 f n 1 x n 1 ; ρn 1 r jzn 1 jf n 1 ðx n 1 ; ρn 1 Þ r νn 1 jzn 1 jnn 1 ðx n 1 Þ ε 1 r ν2n 1 z2n 1 þ n2n 1 ðx n 1 Þ; 2 2ε nX 2 nX 2 ∂αn 2 ∂αn 2 zn 1 f x j ; ρj r zn 1 f x j ; ρj ∂xj j ∂xj j j¼1 j¼1 nX 2 2 2 nX nX zn 1 ∂αn 2 f j x j ; ρj r zn 1 ∂αn 2 f j x j ; ρj r zn r ∂xj ∂xj j¼1 j¼1 j¼1 Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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2 2 ∂αn 2 ε nX 1 nX ∂αn 2 2 2 2 ν j nj x j r 1 ν z þ nj x j ; 2 j ¼ 1 j n 1 2ε j ¼ 1 ∂xj ∂xj 1 1 ð38Þ zn 1 zn r z2n 1 þ z2n ; 2 2 where νj Z 1 and nj ðx j Þ Z 1, j ¼ 1; …; n 1, are unknown constants and unknown continuous functions, respectively. Moreover, if the u(t) is a continuous function on the closed interval ½t τ; t, by using the integral mean value theorem, we can obtain Z t Z t
2 Z t Z t ϖ 1 ϖ zn 1 uðsÞ ds dιr z2n 1 þ uðsÞ ds dι ¼ z2n 1 2 2ϖ t τ ι 2 tτ ι Z t
2 1 ϖ τ4 2 u ðχ Þ; χ A ½ι; t ; ιA ½t τ; t ; þ uðχ Þðt ιÞ dι ¼ z2n 1 þ ð39Þ 2ϖ t τ 2 8ϖ where ϖ40 is a design parameter. From Eqs. (37) to (39), one gets
1 ϖ Hn 1 1 LV n 1 r zn 1 zn 1 þ zn 1 þ αn 1 W n 1 þ zn 2 þ þ z2n 2 2 2 zn 1 τ4 2 u ðχ Þ; þβn 1 z2n 1 zn 2 zn 1 þ 8ϖ P where βn 1 ¼ 2ε nj ¼ 11 ν2j and
2 1 nX ∂αn 2 2 1 2 nj x j þ nn 1 ð x n 1 Þ Hn 1 ¼ 2ε j ¼ 1 2ε ∂xj ! ! nX 2 nX 2 1 T ∂αn 2 T ∂αn 2 h þ h hj : hn 1 2 n 1 j ¼ 1 ∂xj j ∂xj j¼1
ð40Þ
ð41Þ
Define φn 1 ð Z n 1 Þ ¼ z n 2 þ
Hn 1 W n 1; zn 1
ð42Þ
h iT with Z n 1 ¼ x1 ; …; xn 1 ; y dn 1 ; θ^ 1 ; …; θ^ n 2 ; β^ 1 ; …; β^ n 2 A ΩZ n 1 R4n 5 and ΩZ n 1 being some known compact set in R4n 5 . Similar to step 1, define
Hn 1 16 zn 1 φ n 1 ðZ n 1 Þ ¼ φn 1 ðZ n 1 Þ þ tanh2 Hn 1; zn 1 zn 1 ϑn 1 where ϑn 1 is a positive design parameter. Thus, the nonlinear function φ ðZ n 1 Þ can be approximated by an FLS ϕTn 1 ξn 1 ðZ n 1 Þ such that φ n 1 ðZ n 1 Þ ¼ ϕTn 1 ξn 1 ðZ n 1 Þ þ δn 1 ðZ n 1 Þ;
ð43Þ
then, using θn 1 T z2n 1 η2 λε2n 1 2 þ þ ; ξ ξ z þ zn 1 ϕTn 1 ξn 1 ðZ n 1 Þ þ δn 1 ðZ n 1 Þ r n 1 n 1 n 1 2η2 2λ 2 2 Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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yields that
1 ϖ θn 1 T z2n 1 η2 λε2n 1 ξ ξ z LV n 1 r zn 1 zn 1 þ zn 1 þ αn 1 þ þ þ þ n 1 2 2 2η2 n 1 n 1 2λ 2 2
1 τ4 2 zi u ðχ Þ þ 1 þ 16 tanh2 ð44Þ þ z2n þ βn 1 z2n 1 zn 2 zn 1 þ Hi; 2 8ϖ ϑi where εn 1 is the upper bound of δi ðZ n 1 Þ with δn 1 ðZ n 1 Þ r εn 1 . Choose the Lyapunov function as V n1 ¼ Vn1 þ
2 2 θ~ n 1 β~ n 1 ; þ 4η2 4γ
ð45Þ
from Eqs. (8) to (10) yields that
1 1 1 ϖ 2 τ4 2 u ðχ Þ zn 1 zn 2 LV n 1 r z2n k n 1 zn 1 þ 2 2λ 2 2 8ϖ 2 2 λε2 sn 1 θ~ n 1 sn 1 θ2n 1 κn 1 β~ n 1 κ n 1 β2n 1 η2 þ þ þ n1 þ 2 2 4η 4γ 4γ 2 2
4η z n1 ð46Þ þ 1 16 tanh2 Hn 1: ϑn 1 Rt Rt Step n: Considering zn ¼ xn αn 1 þ t τ ι uðsÞ ds dι, similar to Step i, we can have dzn ¼ dxn dαn 1 " #
Z t 1 nX ∂αn 1 þ τuðtÞ uðsÞ ds dt ¼ τuðt Þ þ f n x n ; ρn f x j ; ρj W n dt ∂xj j tτ j¼1 ! nX 1 ∂αn 1 T T h x j dω; þ hn ðx n Þ ð47Þ ∂xj j j¼1 ε Xn ν2 and where βn ¼ j¼1 j 2
1 1 nX ∂αn 1 2 1 2 nj x j þ nn ð x n Þ Hn ¼ 2ε j ¼ 1 2ε ∂xj ! ! 1 nX 1 1 T nX ∂αn 1 T ∂αn 1 h þ h hj : hn ð48Þ 2 n j ¼ 1 ∂xj j ∂xj j¼1 Finally, define 2 2 1 2 θ~ n β~ n V n ¼ zn þ 2 þ ; 2 4η 4γ
then, similar to Step i, from (9), (10) and (49) yields that
2 2 1 2 λε2n η2 sn θ~ n sn θ2n κ n β~ n κ n β2n LV n r kn þ þ þ zn þ 2λ 2 2 4η2 4η2 4γ 4γ
z n þ 1 16 tanh2 Hn: ϑn
ð49Þ
ð50Þ
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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4. Stability analysis Theorem 1. For stochastic nonlinear system (1), under Assumption 1, Lemma 1, the controllers (8)–(9) and the parameter adaptive laws (10) guarantee that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in probability. Moreover, the tracking error in probability may be made arbitrarily small by appropriately adjusting the design parameters. In order to prove this theorem, we need to use the above mentioned Lemma 2. Proof. Now, we assume that for j ¼ 1; …; n; ϑj ¼ ϑ, where ϑ40 is an arbitrary small constant. Then, in Lemma 2, the mentioned set Θϑj can be rewritten as Θϑ . τ Remark 5. From Eq. (44), it can be seen that boundedness of the residual term 8ϖ u2 ðχ Þ cannot be sure, thus, the boundedness of the whole closed-loop system cannot be proven directly. Then, in the following, we will firstly prove that the true controller signal u(t) is bounded. 4
~ If jzn j r 0:2554ϑ, according to Eq. (10), it is obvious that θ^ n and nβ^ n are bounded. Further, o θn and β~ n are bounded as θn and βn are all constants. Define θ^ n ¼ max θ^ n ð0Þ; ð0:2554ϑÞ2 =sn and n o β^ n ¼ max β^ n ð0Þ; 2γ ð0:2554ϑÞ2 =κn , respectively. Then, we have θ^ n r θ^ n , β^ n r β^ n . Combining Eq. (9) gives that " # ! 1 ^ n ξT ðZ n Þξn ðZ n Þzn ^n 1 θ θ n uðtÞ ¼ kn zn þ þ β^ n zn r þ β^ n 0:2554ϑ: k þ ð51Þ τ n 2η2 τ 2η2 If jzn j40:2554ϑ, using Lemma 2 gives that 1 16 tanh2 zϑn H n o0. Then, from Eq. (50) yields that
1 2 λε2n 1 1 η2 1 1 2 2 LV n r kn 2 sn θ~ n þ 2 sn θ2n þ κn β~ n þ κn β2n ; ð52Þ zn þ 2λ 4η 4η 4γ 4γ 2 2 1 40. Now, denote ψ ¼ min 2kn 1λ ; sn ; κn , where k n satisfies kn 4 1λ, so we can deduce kn 2λ one gets LV~ n r ψ V~ n þ C n ; ð53Þ λε2
where Cn ¼ 4γ1 κn β2n þ 4η12 sn θ2n þ 2n þ η2 . Note that Cn is bounded, therefore, based on the conclusion of [22] and Eq. (53), it follows that zn, θ~ n and β~ n are bounded in probability. Therefore, we can get zn r N with N being the upper bound of zn. Further, θ^ n and β^ n are bounded as θn and βn are constants. Then, according to Eq. (9) we can get that u(t) is bounded and uðtÞr M with M being the upper bound of u(t). Finally, let ( ! 1 θ^ 1;n1 ^ u 1 ¼ max kn þ 2 þ β n 0:2554ϑ; Mg; τ 2η z n ¼ maxf0:2554ϑ; N g; then, we have uðt Þ ru 1 ;
jzn j r z n :
2
ð54Þ ð55Þ
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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In the following, we will prove that all the other signals in the closed-loop system are bounded, and the proof process is divided into the following three cases. P Case 1: For 1 r jr n 1; zj 2 = Θϑ . In this case, zj 40:2554ϑ. Let V~ n 1 ¼ nj ¼ 11 V j , then, according to (22), (33) and (46) yields that LV~ n 1 r
nX 1 j¼1
þ
k j z2j þ
nX 1 λε2 j j¼1
2
þ
nX 1
nX 1 nX 1 1 1 1 2 2 2 κ s θ þ β s θ~ j j j j 2 2 j j 4η 4γ 4η j¼1 j¼1 j¼1
nX 1 nX 1 z η 1 ~2 1 2 τ4 2 j κ j β j þ zn þ u ðχ Þ þ 1 16 tanh2 Hj; 4γ 2 2 8ϖ ϑ j¼1 j¼1 j¼1 nX 1 2
ð56Þ
1 1 1 ϖ where for j ¼ 1; 2…; n 2, k j ¼ kjP 2λ and k n 1 ¼ 2knzj 1 2λ 2 2 . It follows from the n definitions of Hj and Lemma 2 that j ¼ 1 1 16 tanh ϑ H j o0, then, from (56) and (55), we have
LV~ n 1 r
nX 1 j¼1
k j z2j þ
nX 1 λε2 j j¼1
2
þ
nX 1
1 s θ2 2 j j 4η j¼1
nX 1
þ
nX 1 nX 1 1 1 1 ~2 1 2 τ4 2 ~2 κ j β2j κ s þ z þ u θ β j j j 4γ 4η2 4γ j 2 n 8ϖ 1 j¼1 j¼1 j¼1
þ
nX 1 nX 1 nX 1 η 1 1 ~2 ~2 κj β þ C; r k j z2j s θ j j 2 4η 4γ j 2 j¼1 j¼1 j¼1 j¼1 nX 1 2
ð57Þ
where for j ¼ 1; 2; …; n 2, k j satisfy kj 4 λ1j , and for j ¼ n 1, k n 1 satisfied k n 1 4 1λ þ 12 þ ϖ2 . Thus, we can deduce k j 40 and k n 1 40. Now, denote ψ n 1 ¼ min 2k 1 ; …; 2k n 1 ; sj ; κj , one gets LV~ n 1 r ψ n 1 V~ n 1 þ C; P P P P 2 λε2 τ4 2 with C ¼ nj ¼ 11 2j þ nj ¼ 11 4η12 sj θ2j þ nj ¼ 11 4γ1 κj β2j þ 12 z 2n þ 8ϖ u 1 þ nj ¼ 11 η2 .
ð58Þ
Note that C is bounded, therefore, based on the conclusion of [22,27] and Eq. (58), we can get that zj, θ~ j and β~ j , j ¼ 1; …; n 1, are bounded in probability. Further, θ^ j and β^ j are bounded as θj and βj are constants. Since z1 and yd are bounded, y is also bounded. Using Eq. (8) with i¼ 1, and noting that z1, θ^ 1 , β^ 1 and ξ1 ðZ 1 Þ are all bounded, we can conclude that α1 is bounded. Consequently, it follows from x2 ¼ z2 þ α1 that x2 is bounded. Following in the same way, αj 1 and xj, j ¼ 3; …; n, can be proven to be bounded. Thus, in case 1, all the signals in the closedloop system are bounded in probability. Furthermore, similar to [32] or based on Eq. (58) and Dynkin's formula [30], we have
E V~ n 1 r e ψ n 1 t E V~ n 1 ð0Þ þ ψ n11 C; 8 t Z 0; ð59Þ this Eq. (59) together with E kZ n 1 k2 r 2E V~ n 1 indicates that
ð60Þ E kZ n 1 k2 r 2E V~ n 1 r 2e ψ n 1 t E V~ n 1 ð0Þ þ 2ψ 1 C; 8t Z 0; where Z n 1 ¼ ½z1 ; z2 ; …; zn 1 T . Case 2: For 1 r jr n 1; zj r 0:2554ϑ. The boundedness of θ^ j and β^ j for 1r j r n 1 are obtained according to zj r 0:2554ϑ and Eq. (10). Moreover, for 1 r jr n 1, θ~ j and β~ j are bounded as θj and βj are all constants. According to Assumption 1, yd ; y_ d ; …; yðdnÞ are all Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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bounded, thus, it follows from z1 ¼ x1 yd that x1 is also bounded. In addition, using Eq. (8) and noting that zj ; θ^ j , β^ j and ξj ðZ j Þ are all bounded, one gets that αj , j ¼ 1; …; n 1, are also bounded. Furthermore, in the light of z2 ¼ x2 α1 , the boundedness of x2 is ensured. Similarly, we can conclude that the state variables xj, j ¼ 3; …; n 1, in the closed-loop system are bounded in probability. Furthermore, we have 2 E kZ n 1 k2 r ð0:2554Þ2 ϑ ; ð61Þ where ϑ is a n 1dimensional vector with ϑ ¼ ½ϑ; …; ϑT . Case 3: Some zm A Θϑ , while some zj 2 = Θϑ with m; j ¼ 1; 2; …; n 1. Define Σ M and Σ J as the index sets of subsystems consisting of zm A Θϑ and zj 2 = Θϑ , respectively. Then, for jA Σ J , choose the Lyapunov function candidate as X V~ Σ J ¼ V j; j A ΣJ
by the controller design process, we have the infinitesimal generator of V Σ J as follows: nX 1 s θ2 nX 1 κ β2 nX 1 s θ 1 κ β ~ 2 nX ~2 λε2j j j j j j j j j þ þ 2 4η2 j A Σ J 4γ 4η2 4γ j A ΣJ j A ΣJ j A ΣJ j A ΣJ j A ΣJ
nX 1 2 nX 1 nX 1 η 1 τ4 2 þ mðzn 1 Þ z2n þ u ðχ Þ þ þ zj zjþ1 zj zj 1 2 2 8ϖ j A ΣJ j A ΣJ j A ΣJ nX 1 z j þ 1 16 tanh2 Hj; ϑ j A ΣJ
LV~ Σ J r
nX 1
k j z2j þ
nX 1
ð62Þ
1 1 where k j ¼ k j 2λ , j ¼ 1; 2…; n 2, k n 1 ¼ k n 1 2λ 12 ϖ2 and ( 1; if n 1 A Σ J ; mðzn 1 Þ ¼ 0; if n 1 A Σ M : z P Then, we can have j A Σ J 1 16 tanh2 ϑj H j o0. Further, similar to the proof of Theorem 1 in [15], the last term of Eq. (62) can be expressed as ! nX 1 nX 1 nX 1 z2j 2 þ λzjþ1 zj zjþ1 zj zj 1 r 4λ j A ΣJ j A ΣJ jþ1 A Σ M ;j A Σ J ! nX 1 nX 1 z2 nX 1 z2j j 2 þ λzj 1 r þ 2λð0:2554ϑÞ2 : ð63Þ þ 4λ 2λ j A ΣJ j 1 A Σ ;j A Σ jþ1 A Σ ;j 1 A Σ M
J
M
M
Based on the above discussion, Eq. (62) can be rewritten as
nX 1 1 s θ 1 κ β ~ 2 nX ~2 1 2 nX j j j j ~ LV Σ J r þ CΣJ ; kj z 2λ j j A Σ J 4η2 4γ j A ΣJ j A ΣJ where
C Σ J ¼ 12 z 2n þ
2λð0:2554ϑÞ2 þ
η2 2.
τ4 2 8ϖ u 1
þ
Further, since
ð64Þ
Pn 1
Pn 1 sj θ2j Pn 1 κj β2j Pn 1 λε2j j A ΣJ 2 þ j A Σ J 4η2 þ j A Σ J 4γ þ jþ1 A Σ M ;j 1 A Σ M 1 1 1 kj 4 λ ; j ¼ 1; 2; …; n 2 and kn 1 4 λ þ 2 þ ϖ2 , we can get
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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1 k j 2λ ¼ k j 1λ 40. Now, letting ψ J ¼ min 2k j 2 λ1j ; sj ; κj gives that LV~ Σ J r ψ J V~ Σ J þ CΣ J :
ð65Þ
Similar to case 1, it can be shown that zj , β~ j and θ~ j are, in probability, bounded for jA Σ J . For m A Σ M , we know that zm are bounded. Similar to case 2, by combining with the conclusions of j A Σ J , we can obtain that the variables xm, θ^ m and β^ m are bounded in probability. Furthermore, for m A Σ M , we can get 2 2 E Z Σ M r ð0:2554Þ2 ϑΣ M ; ð66Þ 2 2 where Z Σ M ¼ Σ m A Σ M z2m and ϑΣ M ¼ Σ m A Σ M ϑ2 . Then, similar to [32], using Eq. (65) yields that 2 ð67Þ E Z Σ J r 2E½V~ Σ J r 2e ψ J t E V~ Σ J ð0Þ þ 2ψ J 1 C Σ J : Finally, from Cases 1–3, we can conclude that 2 2 E kZ n 1 k2 r max 2e ψ n 1 t E V~ n 1 ð0Þ þ 2ψ n11 C; ð0:2554Þ2 ϑ ; ð0:2554Þ2 ϑΣ M ð68Þ þ2e ψ J t EV~ Σ J ð0Þ þ 2ψ J 1 C Σ J ; which means that Z n 1 eventually in probability, converges to the following set: Ξ χ ≔ E kZ n 1 k2 r Π χ ; where
2 2 Π rχ ¼ max 2ψ n11 C; ð0:2554Þ2 ϑ ; ð0:2554Þ2 ϑΣ M þ 2ψ J 1 C Σ J :
ð69Þ
This concludes the proof.□ Remark 6. From (55) and (69) yields that the error signals z1 ; z2 ; …; zn are all bounded in probability. Moreover, according to Eq. (69), in theory, the tracking error in probability may be made arbitrarily small by appropriately adjusting the design parameters η1 ; η2 ; s1 ; s2 ; κ 1 ; κ 2 ; k1 ; k2 and γ. However, how to choose the optimal parameters to get the optimal tracking performance is still an open problem. In the simulation, the design parameters are set using a trial-and-error method. Remark 7. In [15], the effective adaptive fuzzy tracking controllers have been constructed by combining the FLS and the hyperbolic tangent functions for the deterministic nonlinear systems with unknown state time delays. Moreover, the main advantage of the proposed control scheme was that the number of online adjustable parameters depended on the order of the original system, which makes the design scheme easier to be implemented in practical applications. In this paper, the adaptive fuzzy control methods proposed in [15] have been extended to a class of stochastic nonlinearly parameterized systems with distributed input delay. However, the systems addressed in this paper are very different from the ones studied in [15]. The main differences among them lies in that the stochastic disturbance problem, the nonlinear parameterizations problem and the distributed input delay problem were not taken into account in [15].
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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2 1.5
the signals x and yd 1
1 0.5 0 −0.5 −1 −1.5 −2
0
5
10
15
20
25
30
35
40
45
50
time (s)
Fig. 1. The trajectories of x1 (solid line) and yd (dashed line).
5. Simulation To show the feasibility of the developed scheme, the simulation example is given in the following. The developed adaptive fuzzy controllers are applied to the following stochastic nonlinear system with distributed input delay. 8 dx1 ¼ x2 þ ρx11 dt þ 0:7x1 dω; > > < R 2 t ð70Þ dx2 ¼ t τ uðsÞ ds þ ln 1 þ ρ2 x2 dt þ 2x21 dω; > > :y¼x ; 1 in the simulation, τ ¼ 0:05, ρ1 ¼ 0:9, ρ2 ¼ 0:5 and the initial states are chosen as x1 ð0Þ ¼ 0:03, θ^ 1 ð0Þ ¼ θ^ 2 ð0Þ ¼ 0, β^ 1 ð0Þ ¼ β^ 2 ð0Þ ¼ 0 and x2 ð0Þ ¼ 0. The simulation objective is to apply the developed adaptive fuzzy controller such that the boundedness of all the signals in the closedloop system is guaranteed and the system output y follows the reference signal yd to a small neighborhood of zero with yd ¼ sin t þ sin ð0:5tÞ. The virtual controller, the true controller and the parameter adaptive laws are chosen as α1 ¼ k1 z1
θ^ 1 ξT1 ðZ 1 Þξ1 ðZ 1 Þz1 ^ β 1 z1 ; 2η21
ð71Þ
! 1 θ^ 2 ξT2 ðZ 2 Þξ2 ðZ 2 Þz2 ^ k 2 z2 þ þ β 2 z2 ; uðt Þ ¼ τ 2η22
ð72Þ
_ θ^ i ¼ ξTi ðZ i Þξi ðZ i Þz2i si θ^ i ;
ð73Þ
_ β^ i ¼ 2γz2i κi β^ i ;
ð74Þ
select the design parameters as η1 ¼ η2 ¼ 0:1, s1 ¼ 0:08, s2 ¼ 0:5, κ1 ¼ κ2 ¼ 0:5, k1 ¼ 15, k2 ¼ 16 and γ ¼ 2:5. Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
H. Yue, S. Yu / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
18 0.25 0.2
the tracking error
0.15 0.1 0.05 0 −0.05 −0.1
0
5
10
15
20
25
30
35
40
45
50
time (s)
Fig. 2. The tracking error z1. 600 400
the controller u(t)
200 0 −200 −400 −600 −800 0
5
10
15
20
25
30
35
40
45
50
35
40
45
50
time (s)
Fig. 3. The trajectory of u(t). 1.5 1 0.5
the state x2
0 −0.5 −1 −1.5 −2 −2.5 −3
0
5
10
15
20
25
30
time (s)
Fig. 4. The state x2.
Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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3
the estimations of β1 and β2
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
45
50
time (s)
Fig. 5. The trajectories of β^ 1 (solid line) and β^ 2 (dashed line).
For the stochastic nonlinearly parameterized system (70), authors make many simulation results by choosing different random seeds ω, all of them show that the signals in the closed-loop system are bounded and the system output y follows the reference signal yd to a small neighborhood of zero. In this paper, one case of them is listed and the figures are shown in Figs. 1–6. From Figs. 1 and 2, it can be seen that good tracking performance is obtained. The boundedness of u and x2 are illustrated in Figs. 3 and 4, respectively. The adaptive parameters β^ 1 , β^ 2 , θ^ 1 and θ^ 2 are also bounded by Figs. 5 and 6. Remark 8. In the paper, during the controller design process, quadratic functions instead of quartic functions often applied in the existing results [25,27,32,35–38] are used as Lyapunov functions to analyze the stability of the stochastic nonlinear systems, and then, the hyperbolic tangent functions are introduced to deal with the higher order Hessian terms such that the long-standing obstacle for stochastic systems control by quadratic Lyapunov function is overcome. Remark 9. It should be emphasized that the unknown parameters β1 and β2 are separated out from the unknown functions f 1 ðx1 ; ρ1 Þ and f 2 ðx 2 ; ρ2 Þ, respectively. And the parameter adaptive laws are defined in Eq. (10), in this simulation, the boundedness of β^ 1 and β^ 2 is shown in Fig. 5. Moreover, in many references, such as [34,35,39] and so on, more parameters were needed to be adjusted online in controller design procedure for each neural network or each fuzzy logic system. However, in this paper, a technique is introduced to reduce the number of adjustable 2 parameters coming from the fuzzy logic systems, i.e., the unknown constant θi ¼ ϕi is used as an estimated parameter, which results in only one adaptive parameter θ^ i for each fuzzy logic system, thus, in this simulation, the only two parameters θ1 and θ2 which come from the fuzzy logic systems need to be estimated online for the two-order nonlinear systems. Nevertheless, it should be pointed out that in the proposed controllers, the number of adjustable parameter depends on the order of the system. If the order of system is added online, the number of adjustable parameter will be increased. In the existing results [7,36,40], for the nth order nonlinear systems, the effective controllers which contain less adjustable parameters have been Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
H. Yue, S. Yu / Journal of the Franklin Institute ] (]]]]) ]]]–]]]
20 0.7
the estimations of θ1 and θ
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Fig. 6. The trajectories of θ^ 1 (solid line) and θ^ 2 (dashed line).
designed. Thus, for the stochastic nonlinear systems, by using the classical quadratic Lyapunov functions how to construct the controllers containing less adoption parameters remains to be further investigated.
6. Conclusion In this paper, the problem of adaptive fuzzy tracking control has been addressed for a class of stochastic nonlinearly parameterized systems with distributed input delay. The adaptive fuzzy controller has been constructed by using the backstepping approach and the separation principle. The proposed controller ensures that all the signals of the resulting closed-loop system are bounded in probability, and the tracking error converges to a small neighborhood of the origin in the mean square sense. However, in the proposed controller, the number of adjustable parameter depends on the order of the system, thus, by combining the classical quadratic Lyapunov functions how to design the controller containing less adoption parameters to reduce the computation burden is very significant. In addition, it should be pointed out that in the nonlinear functions f i ðx i ; ρi Þ, the parameters ρi are time-invariant, if ρi ðtÞ are time-varying, i.e., for the stochastic nonlinear system with the functions f i ðx i ; ρi ðtÞÞ, how to design the controller is also worth studying. References [1] Y. Roh, J. Oh, Robust stabilization of uncertain input-delay systems by sliding mode control with delay compensation, Automatica 35 (1999) 1861–1865. [2] Z. Lin, H. Fang, On asymptotic stability of linear systems with delayed input, IEEE Trans. Autom. Control 52 (2007) 998–1013. [3] B. Zhou, H.J. Gao, Z.L. Lin, G.R. Duan, Stabilization of linear systems with distributed input delay and input saturation, Automatica 48 (2012) 712–724. [4] X.F. Zhang, E.K. Boukas, Y.G. Liu, L. Baron, Asymptotic stabilization of high-order feedforward systems with delays in the input, Int. J. Robust Nonlinear Control 20 (12) (2010) 1395–1406. Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004
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Please cite this article as: H. Yue, S. Yu, Adaptive fuzzy tracking control for a class of stochastic nonlinearly parameterized systems with distributed input delay, Journal of the Franklin Institute. (2016), http://dx.doi.org/ 10.1016/j.jfranklin.2015.12.004