Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay

Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay$ Yumei Sun a,b, Bing Chen a,n, Chong Lin a, Honghong Wang a a b

Institute of Complexity Science, Qingdao University, Qingdao, Shandong, China College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong, China

art ic l e i nf o

a b s t r a c t

Article history: Received 3 February 2016 Received in revised form 9 June 2016 Accepted 24 June 2016 Communicated by Shaocheng Tong

This paper addresses adaptive neural control for a class of non-strict-feedback stochastic nonlinear systems with time delays. An important structural property of radial basis function (RBF) neural networks (NNs) is introduced to overcome the design difficulty from the non-strict-feedback structure. The Lyapunov–Krasovskii functional is used for control design and stability analysis. Further, a backsteppingbased adaptive neural control strategy is proposed. The suggested adaptive neural controller guarantees that all the closed-loop signals are semi-globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a small neighborhood of the origin. Simulation results demonstrate the effectiveness of the proposed approach. & 2016 Elsevier B.V. All rights reserved.

Keywords: Neural adaptive control Stochastic nonlinear systems Non-strict-feedback Backstepping Time-delay Lyapunov–Krasovskii functional

1. Introduction During the past decades, many scholars have devoted much effort to approximation-based adaptive fuzzy or neural control for nonlinear systems. By their inherent ability in function approximation, neural networks (NNs) or fuzzy logical systems (FLSs) are used to model the unknown nonlinear functions in order to achieve control design for nonlinear systems. The work in [1] proposed an adaptive fuzzy control method for a class of nonlinear systems with unknown system functions. Some similar adaptive fuzzy control strategies were further proposed in [2–10] for nonlinear uncertain systems. On the other hand, a series of adaptive control design approaches were also developed for nonlinear systems by using RBF NNs as function approximators, for instance see [11–18] and the reference therein. Furthermore, adaptive neural or fuzzy control was discussed for multi-input and multioutput (MIMO) nonlinear strict-feedback systems in [18–20], respectively. In these researches, adaptive neural or fuzzy controllers are constructed recursively in the framework of the backstepping. Notice that time delay phenomena often occurs in many practical systems, such as physical, biological, and economical systems. The ☆ This work is supported in part by the National Natural Science Foundations of China under Project 61473160 and 61503223 and the Project of Shandong Province Higher Educational Science and Technology Program J15LI09. n Corresponding author. E-mail address: [email protected] (B. Chen).

existence of time delays is usually a main reason of instability and deteriorative performance of the controlled systems. Therefore, many researchers further extended the approximation-based adaptive control from delay-free systems to delayed systems. Some significant results have been reported for deterministic systems, for example, [21–28]. Ge and Tee [27] presented approximation-based adaptive neural control for a class of MIMO nonlinear time-delay systems, which is in block-triangular form. Liu et al. [4] further considered the coupling effect of time delays and dead-zone in the control design procedure. At the same time, some results were obtained for stochastic time-delay systems, for example see [28–33]. Liu and Xie [33] presented adaptive neural control schemes for a class of stochastic high-order nonlinear systems with time-varying delays. However, all the aforementioned control strategies only focus on the nonlinear systems in strict-feedback form [34–36] or in pure-feedback form [37]. To relax the restriction on system structure, Chen et al. [38–40] developed a variable separation technique by using the monotonically increasing functions as the bounding functions. An adaptive fuzzy control scheme was presented for non-strict-feedback nonlinear systems, in which each subsystem function, i.e. fi (·) contains all the state variables. Then, the control laws were proposed under the following assumptions: (i) the function fi (·) must be bounded by a strictly increasing smooth function and (ii) the function gi (·) does not contain the state variable xk for k ≥ i + 1. In practice, it is difficult to check if these assumptions are satisfactory, particularly, when the system functions are unknown. In

http://dx.doi.org/10.1016/j.neucom.2016.06.060 0925-2312/& 2016 Elsevier B.V. All rights reserved.

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

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2

addition, the authors in [38,39] only studied the deterministic systems, but not the stochastic systems. The stochastic distribution was put into account in [40], while the diffusion terms ψ (·) s in those papers were supposed to be the function of the previous variable x¯ i . When the diffusion terms ψ (·) in the ith subsystems is a nonlinear function of the whole state variables, how to design the virtual control signal αi which is independent of the state variables xj , j = i + 1, … , n is the main difficulty to be overcome. In addition, to the best of our knowledge, so far, there has been no results about non-strict-feedback with time-delay to be reported. Motivated by the aforementioned discussion, in this research we will consider adaptive neural control for a class of unknown nonlinear delayed stochastic systems in non-strict-feedback structure. The main differences between our work and the ones in [38–40] are that unlike [38–40]: (1) the system function fi (·) s and the diffusion terms ψ (·) s are not required to be bounded by a strictly increasing smooth function and (2) the functions gi (·) s contain all the state variables. The system (6) is thus more general than that in [38–40]. Therefore, the proposed control strategy is easier to be implemented in practice than the existing results. In the process of constructing adaptive neural controller, a characteristic of RBF NNs shown in Lemma 4 is utilized to deal with the system functions which contain the whole state variable. A Lyapunov–Krasovskii functional is used to compensate for the effect of nonlinear time-delay terms. An adaptive neural tracking control scheme has been developed. The proposed adaptive neural controller guarantees that all the signals in the closed-loop are bounded and the tracking error converges to a small neighborhood of the origin.

For the purpose of introducing some useful concepts and lemmas, consider the following stochastic system:

∀ x ∈ Rn

(1)

where x is the state variable, w is r-dimensional independent standard Brownian motion and f and h are vector-value or matrixvalue function with appropriate dimensions. Definition 1. [41] For any given V (t , x ) ∈ C1,2, associated with the stochastic differential equation (1) define the operator L as follows:

LV =

1 ⎧ ∂2 V ⎫ ∂V ∂V f + Tr ⎨ hT 2 h⎬ + 2 ⎩ ∂x ⎭ ∂t ∂x

(2)

Definition 2. [42] The trajectory {x (t ) , t ≤ 0} of stochastic system (1) is called SGUUB in p-th moment, if for some compact set Ω ∈ Rn and any initial state x0 = x (t0 ), there exist a constant ε > 0, and time constant T = T (ε, x0 ) such that E (|x (t )|p ) < ε , for all t > t0 + T . Especially, when p = 2, it is usually called SGUUB in mean square.

C1,2 function C2 > 0, class

α¯1 (|x|) ≤ V (t , x) ≤ α¯ 2 (|x|) LV (t , x) ≤ − C1V (t , x) + C2

∀ t > t0

In the developed control design procedure, RBF NNs will be used to approximate the unknown nonlinear functions. A useful property of RBF NNs proposed in[44] is first listed as follows. It has been proved that with sufficiently large node number l, the RBF NNs W ⁎ T S (Z ) can approximate any continuous function f(Z) over a compact set Ωz ⊂ Rq , for given arbitrary accuracy ε > 0, such that

f (Z ) = W⁎ T S (Z ) + δ (Z ),

∀ z ∈ Ω z ⊂ R q,

(4)

where |δ (Z )| is the error approximation satisfying |δ (Z )| ≤ ε , ⁎ W = [w1, w2, … , wl ] ∈ Rl denotes the ideal constant weight vector, and defined as

W⁎ = arg min { sup |f (Z ) − W T S (Z )|} W ∈ R¯

l

Z ∈ ΩZ

and S (Z ) = [s1 (Z ) , … , sl (Z )] stands for the basis function vector, with l > 0 being the number of the neural networks nodes, and si(Z) are chosen as Gaussian function, namely,

⎡ (Z − μ ) T (Z − μ ) ⎤ i i ⎥ si (Z ) = exp ⎢ − , ⎢⎣ ⎥⎦ ηi2

i = 1, …, l

where μi = [μi1, … , μiq is the center of the receptive field and the width of Gaussian function.

]T

(5)

ηi is

Lemma 2. [45] For any real-valued continuous function f (x, y ), where (x, y ) ∈ Rm × Rn , there are smooth scalar functions a (x ) ≥ 0, b (y ) ≥ 0 such that

|f (x, y)| ≤ a (x) + b (y).

Lemma 4. Let x¯ q = [x1, x2, … , xq ]T and S (x¯ q ) = [s1 (x¯ q ) , … , sl (x¯ q )]T be the basis function vector. Then, for any positive integer k ≤ q , the following inequality holds:

∥ S (x¯ q )∥2 ≤ ∥ S (x¯ k )∥2 . The proof is omitted. Remark 1. This lemma provides a simple but useful characteristic of RBF NN. By which adaptive neural backstepping design method can be extended to the non-strict-feedback system (6) easily. In this work, we consider the following non-strict-feedback stochastic nonlinear systems with time delays:

dxi = [gi (x (t )) xi + 1 + fi (x (t )) + qi (x (t − τi ))] dt + φiT (x (t )) dw , i = 1, …, n − 1,

where Tr(A) is the trace of A.

Lemma 1. [43] Suppose that there exists a V (t , x ) : R+ × Rn → R+ , two constants C1 > 0 and k∞-functions α¯1 and α¯ 2 such that

C2 C1

Lemma 3. [46] For 1 ≤ i ≤ n, define the set Ω zi as zi ∈ Ω zi , the inequality Ω zi ≜ {zi ∥ zi | ≥ 0.8814νi }. Then, for (1 − 4 tanh4 (zi/νi )) ≤ 0 holds, where νi is any positive constant.

2. Preliminary knowledge and system formulation

dx = f (t , x) dt + h (t , x) dw ,

E [V (t , x)] ≤ V (x 0 ) e−C1t +

(3)

for all x ∈ Rn and t > t0 . Then, there is an unique strong solution of system (1) for each x0 ∈ Rn and it satisfies

dxn = [gu (x (t )) u + fn (x (t )) + qn (x (t − τn ))] dt + φnT (x (t )) dw , y = x1 (t ),

(6)

where x = [x1, … , xn ]T ∈ Rn and y ∈ R are system state variable and output, respectively. The mappings gi : R n → R , fi : Rn → R , n n r qi : R → R and φi : R → R are assumed to be unknown smooth functions with fi (0) = 0 and qi (0) = 0, φi (0) = 0 and τi are unknown constant time delays. The control objective is to structure an adaptive NNs controller for system (6), such that (i) all the closed-loop signals remain uniformly ultimately bounded, and (ii) output y follows a desired reference signal yd. To this purpose, some assumptions are introduced as follows. Assumption 1. The sign of gi does not change, and there exist constants bm and bM such that

0 < bm ≤ |gi (x)| ≤ bM < ∞,

i = 1, …, n,

(7)

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

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without loss of generality, it is assumed that 0 < bm ≤ gi ≤ bM < + ∞.

LVzi = zi3 (gi xi + 1 + fi + qi (x (τi )) − Lαi − 1) ⎛ 3 + zi2 ⎜⎜ φi − 2 ⎝

Assumption 2. The reference signal yd(t) and its time derivatives up to the nth order are continuous and bounded. Remark 2. Obviously, fi , gi , qi , and φi are the functions of whole state variable. So, we call the system (6) to be in non-strict-feedback form. The existing approximate-based backstepping control schemes cannot be utilized to the system (6) because they are developed under the assumption of strict-feedback structure. In [40], an adaptive fuzzy control strategy was presented for stochastic non-strict-feedback nonlinear system under the assumption that the diffusion terms ψ (·) s are bounded by increasing functions. As shown later, this restriction is here removed by the structural character of RBF NN, and the proposed design procedure is more simple than that in [38–40]. To simplify the notations, the state variable x and the time variable t will be omitted from the corresponding functions if no confusion arises, and x (t − τi ) is denoted as x (τi ). Remark 3. From the above Lemma 3, for the time-delay term qi (x (τi )) there exist nonnegative smooth scalar functions ρij such that

∑ ρij (xj (τi )).

qi (x (τi )) ≤

⎞T ⎛ ∂αi − 1 ⎟ ⎜ φ φ − ∂xj j ⎟⎠ ⎜ i ⎝

i−1

∑ j=1

i = 1, …, n,

(9)

¯ where x¯ i = [x1, … , xi ]T , θ^i = [θ^1, … , θ^i ], y¯d(i) = [yd , yd′ , … , yd(i) ]T , yd(k ) is the kth derivative of yd, α0 = yd and θ^i is the estimation of the unknown constant θi, which is defined later. Step i: (1 ≤ i ≤ n − 1). By the above transformation and It o^

⎞T ∂αi − 1 ⎟ φj ⎟ dw, ∂xj ⎠

where

Lαi − 1 =

∑ ∂αi − 1 (gj xj + 1 + f j j=1

+ α0 = yd ,

+ qj (x (τj )) ) +

∂xj

i−1

i−1

j=1

(j ) j = 0 ∂yd

∑ ∂αi^− 1 θ^j̇ + ∑

∂α0 ∂xk

∂θj

= 0 and

∂α0 ∂θ^

∂αi − 1

1 2

i−1

∑ p, q = 1

i−1

3 2 zi φi − 2 i

∑ j =1

∑ j=1

2

∂αi − 1 φ ∂xj j

≤

i

∂α i − 1 q (x (τj )) ≤ ∂x j j

∂ 2αi − 1 T φ φ ∂xp ∂x q p q

= 0.

k

At the ith step, consider a Lyapunov function candidate as 1

and

= − 1.

i−1

3 4 zi φi − 4li2

∑ j=1

4

∂αi − 1 φ ∂xj j

+

3 2 li 4

(13)

n

∑ ∑ |zi3 ∂α i − 1 |ρjk (xk (τj )) j =1 k=1 i

∑

∂x j

4/3 3n 4 ⎛ ∂α i − 1 ⎞ ⎟⎟ + z i ⎜⎜ 4 ⎝ ∂x j ⎠

n

i

∑∑ k=1 j =1

⎛ 3 3 ⎜ LVzi ≤ zi3 ⎜ fi + gi αi − L′αi − 1 + gi zi + zi φi − ⎜ 4 4l i ⎝ i

+

∑ j=1

+

1 4 ρ (xk (τj )). 4 jk

(14)

4/3⎞ 3n ⎛ ∂αi − 1 ⎞ ⎟ ⎟ ⎟+ zi ⎜ 4 ⎝ ∂xj ⎠ ⎟ ⎠

i

n

∑∑ j=1 k=1

4

i−1

∑ ∂αi − 1 φj j=1

∂xj

1 4 ρ (xk (τj )) 4 jk

1 3 g zi4+ 1 + li2. 4 i 4

(15)

To compensate for the nonlinear time-delay terms in (15), construct a Lyapunov–Krasovskii function VQ i as follows: i

n

t

∑ ∑ e − r (τ j ) ∫

VQ i =

e rσ

4 ρjk (xk (σ ))

t − τj

j=1 k=1

4

dσ

Its time derivative is i

i ∑ j=1

with Wi = candidate as

Vi = Vzi + VQ i +

yd(j + 1) ,

Vzi = 4 zi4 .

(11) ∂αi − 1 ∂xi

(12)

n

∑∑ j=1 k=1

(10)

i−1

j=1

3 1 g zi4 + gi zi4+ 1 4 i 4

zi3 gi zi + 1 ≤

VQ̇ i = − rVQ i + Wi −

dzi = dxi + dαi − 1 = (gi xi + 1 + fi + qi (x (τi )) − Lαi − 1) dt

j=1

∑

By using Young's inequality and Remark 3, the following inequalities hold:

formula, the ith subsystem of the error systems is described by

∑

∂αi − 1 φ ∂xj j

The li in (13) is positive design parameter. Then substituting (12), (13) and (14) into (11) yields

In this section, an adaptive tracking controller is constructed for system (6) via backstepping and adaptive neural control approach. The recursive design procedure contains n steps, in which the virtual control signal αi (i = 1, … , n − 1) will be designed for the ith subsystem by a suitable Lyapunov function Vi. The real control law u will be constructed at the final step n. The backstepping design procedure is developed based on the following coordinate transformation:

i−1

2

i−1

∂α i ∑ j = 1 ∂ix− 1 qj (x (τj )) j

with L′αi − 1 = Lαi − 1 +

3. Main result and adaptive control design

⎛ + ⎜⎜ φi − ⎝

⎠

∂xj

j=1

j =1

zi = xi − αi − 1 (x¯ iT− 1, θ^i − 1, y¯d(i − 1) T ),

∂xj

∑ ∂αi − 1 qj (x (τj )) + zi3 gi zi + 1,

− zi3

≤

¯T

j=1

i

(8)

j=1

⎞

i−1

∑ ∂αi − 1 φj ⎟⎟

3 + fi − L′αi − 1) + zi2 φi − 2

zi3 (gi αi

=

−z i3

n

3

1 4 ρ (xk (τj )) 4 jk

n 1 4 ∑k = 1 4 erτj ρjk (xk (t )).

(16)

Choose a Lyapunov function

bm ˜ 2 θi 2ri

(17)

In light of (15) and (16) the differential operator L of Vi is LVi ⎛ ⎜ 3 3 ≤ z i3 ⎜ fi + gi α i − L′α i − 1 + gi z i + z i φi − 4 4l i ⎜ ⎝ 4

i

+

∑ j =1

i −1

∑ j =1

∂α i − 1 φ ∂x j j

⎞ 4/3 ̇ 3n ⎛ ∂α i − 1 ⎞ W⎟ 1 3 2 b ⎟⎟ + i ⎟ + gi z i4+ 1 + l i − rVQ i − m θ˜i θ^i. z i ⎜⎜ 4 ⎝ ∂x j ⎠ 4 z i3 ⎟ 4 ri ⎠

(18)

The differential operator L of Vzi is given by Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

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4

Remark 4. The term Wi/zi3 is not well defined at zi ¼ 0, that is, lim zi → 0 Wi/zi3 = ∞. So, RBF NNs cannot be used to approximate this term. By Lemma 3, the term 4 tanh4 (zi /νi ) Wi/zi3 will be introduced to overcome this difficulty. Since lim zi → 0 tanh4 (zi /νi ) Wi/zi3 = 0, the unknown term 4 tanh4 (zi /νi ) Wi/zi3 can be approximated by RBF NNs. Thus, (18) can be rewritten as

1 LVi ≤ zi3 (f¯i + gi αi ) − rVQ i + 1 − 4 tanh4 (zi /νi ) Wi + gi zi4+ 1 4 ̇ 1 3 b + li2 − m θ˜i θ^i − gi − 1zi , 4 4 ri

(

)

i

+

∑ j=1

i−1

∑ j=1

Step n: In the final step, the actual control law u will be derived. Similar to the step i, one can get

⎛ + ⎜⎜ φn − ⎝

(19)

n− 1 j=1

+

Remark 5. It is obvious that f¯i is a function of the whole state variable x. So, the exiting backstepping-based adaptive neural control method cannot be used for systems (6). In [38–40], some adaptive neural/fuzzy control designs were proposed for nonstrict-feedback systems. Then, those control design schemes require that the bounding functions of f¯i must be monotonic increasing and the function gi (·) cannot contain the state variable xi + 1, … , xn . Therefore, the method proposed in [38–40] cannot be utilized to systems (6). The following discussion shows that the structural characteristic of RBF NNs, shown in Lemma 4, can help us extend adaptive neural backstepping to system (6) easily without the assumptions in [38–40]. For any given εi > 0, there exists a RBF NN Wi⁎ T Si (Zn ) such that

(20)

≤

with θi =

Wi⁎ bm

3 2 zn 2

n− 1

φn −

∑ j=1

2

n− 1

∂αn − 1 φ ∂xj j

− z n3

∂αn − 1 q (x (τj )) ∂xj j

∑ j=1

(27)

n − 1 ∂αn − 1 qj (x (τj )) ∂xj

with L′αn − 1 = Lαn − 1 + ∑ j = 1

∂αn − 1 ∂xn

and

= − 1.

By using Young's inequality and Remark 3, the following inequalities hold: 3 2 z n φn − 2

n

∑

n −1

∑ j =1

2

∂α n − 1 φ ∂x j j

∂α n − 1 q (x (τj )) ≤ ∂x j j

n −1

3

z 4 φn − 2 n

≤

4l n

n

∑ j =1

4

∂α n − 1 φ ∂x j j

+

3 2 ln 4

(28)

n

∑ ∑ |zn3 ∂α n − 1 |ρjk (xk (τj )) ∂x j

j =1 k=1

4/3 3n 4 ⎛ ∂α n − 1 ⎞ ⎟⎟ + z n ⎜⎜ 4 ⎝ ∂x j ⎠

∑

n

n

∑∑ k=1 j =1

1 4 ρ (xk (τj )). 4 jk

(21)

(22)

(29)

Then it follows immediately from substituting (28) and (29) into (27) that

⎛ 3 ⎜ LVzn ≤ z n3 ⎜ fn + gn u − L′αn − 1 + z n φn − ⎜ 4l n ⎝

εi4

n

+

αi and the adaptive law as

∑ j=1

4/3⎞ 3n ⎛ ∂αn − 1 ⎞ ⎟ ⎟ ⎟+ zn ⎜ 4 ⎝ ∂xj ⎠ ⎟ ⎠

n

n

∑∑ j=1 k=1

n− 1

∑ j=1

4

∂αn − 1 φ ∂xj j

1 4 3 ρ (xk (τj )) + ln2 4 jk 4

(30)

To compensate for the nonlinear time-delay terms in (30), construct a Lyapunov–Krasovskii function VQ n as follows:

(23)

n

VQ n =

n

∑ ∑ e − r (τ j ) ∫ j=1 k=1

̇ r θ^i = i 2 zi6 SiT (Zi ) Si (Zi ) − σi θ^i 2a i

p, q = 1

∂ 2αn − 1 T φ φ ∂xp ∂x q p q

yd(j + 1) ,

LVzn = z n3 (gn u + fn + qn (x (τn )) − L′αn − 1) +

j =1

. Furthermore, substituting (21) into (19) gives

⎛ 3⎞ 1 3^ T αi = − ⎜ k i + ⎟ z i − zi θi Si (Zi ) Si (Zi ), 4⎠ 2ai2 ⎝

∂yd(j)

j=0

∑

The differential operator L of Vzn is given by

≤

2

Now, take the virtual control signals follows:

∂αn − 1

∑

j

εi4

⎛ ⎛ z ⎞⎞ 1 1 + + + gi z i4+ 1 + ⎜⎜ 1 − 4 tanh4 ⎜ i ⎟ ⎟⎟ Wi − gi − 1z i . ⎝ νi ⎠ ⎠ 2 4 4 4 ⎝

n− 1

∂αn − 1 ^ ̇ θj + ∂θ^

n

⎛ z3 ̇ 3b m ⎞ 3 2 b LVi ≤ z i3 ⎜⎜ i b m θ i SiT (Z i ) Si (Z i ) + gi α i + z i ⎟⎟ − rVQ i − m θ˜i θ^i + l i 2 2 4 4 a ri ⎝ i ⎠ a i2

n− 1

1

j =1

+ εi )

1 6 3 z b m θ i Si (Z i )T Si (Z i ) + + z i4 + 2a i2 i 2 4 4

(26)

Vzn = 4 z n4 .

−z n3

a i2

j=1

⎞T ∂αn − 1 ⎟ φj ⎟ dw, ∂xj ⎠

Consider a Lyapunov function candidate as

where δ i (Zn ) denotes the approximation error and satisfies |δ i (Zn )| < εi . Applying Lemma 4 and Young's inequality to (20) yields: Si (Z i )

∑ j=1

1

+ δ i (Z n )) ≤ |z i3 |( Wi⁎ T

∑

∂αn − 1 1 (gj xj + 1 + f j + qj (x (τj ))) + ∂xj 2

n− 1

The term of 4 gi − 1zi is from step i − 1 and the term of 4 gi zi4+ 1 in (17) will be handled in step i + 1.

z i3 f¯i ≤ |z i3 |( Wi⁎ T Si (Z n )

∑

Lαn − 1 =

4/3 3n ⎛ ∂αi − 1 ⎞ W ⎟ + 4 tanh4 (zi /νi ) 3i . zi ⎜ 4 ⎝ ∂xj ⎠ zi

f¯i = Wi⁎ T Si (Z n ) + δi (Z n )

n− 1

where

4

∂αi − 1 φ ∂xj j

1

(25)

dz n = dxn + dαn − 1 = (gn u + fn + qn (x (τn )) − Lαn − 1) dt

where

3 1 3 f¯i = fi − L′αi − 1 + gi zi + gi − 1zi + 2 zi φi − 4 4 4l i

ε4 a2 σi 3 1 bm θ˜i θ^i + li2 + i + i + gi zi4+ 1 ri 4 2 4 4 ⎛ ⎛ z ⎞⎞ 1 + ⎜ 1 − 4 tanh4 ⎜ i ⎟ ⎟ Wi − gi − 1zi . ⎝ νi ⎠ ⎠ 4 ⎝

LVi ≤ − ki bm zi4 − rVQ i −

(24)

with ki being the positive design parameter. And then by taking (22), (23) and (24) into account, one has

t

e rσ

4 ρjk (xk (σ ))

4

t − τj

dσ .

Its time derivative is given by n

VQ̇ n = − rVQ n + Wn −

n

∑∑ j=1 k=1

1 4 ρ (xk (τj )), 4 jk

(31)

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

Y. Sun et al. / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎ n

n

1

4 with Wn = ∑ j = 1 ∑k = 1 4 erτj ρjk (xk (t )). Choose a Lyapunov function

candidate as

(32)

One has

⎛ 3 ⎜ LVn ≤ z n3 ⎜ fn + gn u − L′αn − 1 + z n φn − ⎜ 4l n ⎝ n

+

∑ j=1

n− 1

∑ j=1

4

∂αn − 1 φ ∂xj j

⎞ 4/3 ̇ W ⎟ 3 b 3n ⎛ ∂αn − 1 ⎞ ⎟ + 3n ⎟ + ln2 − rVQ n − m θ˜n θ^n ≤ z n3 (f¯n zn ⎜ rn 4 ⎝ ∂xj ⎠ zn ⎟ 4 ⎠

+ gn u) − rVQ n + (1 − 4 tanh4 (z n/νn ) ) Wn + −

3.1. Stability analysis At the present stage, we can summarize the main result in the following theorem.

b 2 + m θ˜n 2rn

Vn = Vzn + VQ n

3 2 ln 4

Theorem 1. Consider the closed-loop system consisting of plant (6), the controller (37) together with the virtual control signals (23) and adaptive law (24) under Assumptions 1 and 2. Suppose that for 1 ≤ i ≤ n, the packaged unknown functions f¯i (Zn ) can be approximated by neural network WiT Si (Zn ) in the sense that the approximate error δi is bounded, then for bounded initial conditions, all the signals in the closed-loop system remain bounded in probability, and the tracking errors can be arbitrarily small by choosing the suitable design parameters as t → ∞. Proof. Consider the Lyapunov function candidate

1 bm ˜ ^ ̇ θn θn − g n − 1 z n , 4 rn

(33)

n

V=

1 3 f¯n = fn − L′αn − 1 + gn − 1z n + 2 z n 4 4l n n

∑ j=1

n− 1

φn −

∑ j=1

By (25) for 1 ≤ i ≤ n, it can be verified that

4

∂αn − 1 φ ∂xj j

LV ≤ −

4/3 3n ⎛ ∂αn − 1 ⎞ W ⎟ + 4 tanh4 (z n/νn ) 3n . zn ⎜ 4 ⎝ ∂xj ⎠ zn

bm

. Now, taking (33) together with (35) into account,

ri

⎛ ⎛ ⎞⎞ ∑ ⎜⎜ 1 − 4 tanh4 ⎜ zi ⎟ ⎟⎟ Wi. ⎝ νi ⎠ ⎠ i=1 ⎝ n

(40)

⎛

bm σi ˜ 2 ⎞⎟ θi ⎟ 2ri ⎠

∑ ⎜⎜ ki bm zi4 + rVQ i + ⎝

⎛3 ε4⎞ a2 b σ + ∑ ⎜⎜ li2 + i + m i θi2 + i ⎟⎟ 4 2 2ri 4⎠ i=1 ⎝ ⎞⎞

n

⎛

⎛

i=1

⎝

⎝ νi ⎠ ⎠

∑ ⎜⎜ 1 − 4 tanh4 ⎜ zi ⎟ ⎟⎟ Wi.

(41)

⎛ Let a0 = min {4ki bm , σi, 1 ≤ i ≤ n, r} and D = ∑ni = 1 ⎜ 43 li2 + ⎝

a2 i 2

+

bm σ i 2ri

θi2 +

ε4⎞ i ⎟. 4 ⎠

It follows that

⎛ z3 ⎞ ̇ b 3 n bm θn SnT (Z n ) Sn (Z n ) + gn u⎟ − rVQ n − m θ˜n θ^n + ln2 rn 4 ⎝ 2a n2 ⎠

LV ≤ − a 0 V + D +

z n3 ⎜

⎛ ⎛ z ⎞⎞ 1 + + + ⎜ 1 − 4 tanh4 ⎜ n ⎟ ⎟ Wn − gn − 1z n4 . ⎝ νn ⎠ ⎠ 2 4 4 ⎝ εn4

(36)

̇ Now, take the actual control signals u and the adaptive law θ^n as follows:

u = − kn zn −

i=1

n

(35)

one has

a n2

i=1

i=1

+

2

Wn⁎

i=1

n

LV ≤ −

z n3 f¯n ≤ |z n3 |(∥Wn⁎ T Sn (Z n )∥ + δ n (Z n )) ≤ |z n3 |(∥Wn⁎ T ∥∥Sn (Z n )∥ + εn )

with θn =

n

1 1 2 By using the inequality θ˜i θ^i ≤ 2 θi2 − 2 θ˜i , i = 1, … , n, (40) can be further expressed as

(34)

where δn (Zn ) denotes the approximation error and satisfies |δn (Zn )| < εn . Applying Young's inequality, the following inequality is obtained:

a2 ε4 1 6 3 ≤ z b θ S (Z )T Sn (Z n ) + n + z n4 + n 2 n m n n n 2 4 4 2a n

n

⎛3 a2 ε4⎞ + ∑ ⎜⎜ li2 + i + i ⎟⎟ + 4 2 4⎠ i=1 ⎝

n

f¯n = Wn⁎ T Sn (Z n ) + δ n (Z n ),

n

∑ ki bm zi4 − r ∑ VQ i + ∑ bm σi θ˜i θ^i n

Furthermore, the neural network Wn⁎ T Sn (Zn ) is utilized to approximate the unknown function f¯ such that for any given εn > 0

LVn ≤

∑ Vi i=1

where

+

5

1 3^ T z n θn Sn (Z n ) Sn (Z n ), 2a n2

n

⎛

i=1

⎝

⎛ zi ⎞ ⎞ ⎟ ⎟ Wi. νi ⎠ ⎟⎠

∑ ⎜⎜ 1 − 4 tanh4 ⎜⎝

For the last term in (42), we define a set Ω zi as Ω zi≔{zi zi | < 0.8814νi } with νi is a positive parameter. It is obvious that the last item can be rewritten as follows: n

⎛

i=1

⎝

⎞

⎛

⎞

∑ ⎜⎜ 1 − 4 tanh4 ⎜ zi ⎟ ⎟⎟ Wi = ∑ ⎝ νi ⎠

⎠

zi ∈ Ω z i

⎞ ⎛ ⎜ 1 − 4 tanh4 ⎛⎜ zi ⎞⎟ ⎟ W i ⎜ ⎝ νi ⎠ ⎟ ⎠ ⎝

(37)

+

∑ zi ∈¯ Ω z i

̇ r θ^n = n 2 z n6 SnT (Z n ) Sn (Z n ) − σn θ^n 2a n

(38)

with ki being the positive design parameter. Then, substituting (37) and (38) into (36) shows that

ε4 a2 σn 3 bm θ˜n θ^n + ln2 + n + n 4 2 4 rn ⎛ ⎛ zn ⎞ ⎞ 1 + ⎜ 1 − 4 tanh4 ⎜ ⎟ ⎟ Wn − gn − 1z n4 . ⎝ νn ⎠ ⎠ 4 ⎝

LVn ≤ − k n bm z n4 − rVQ n +

⎛ ⎝

where qI = ∑ z ∈ Ω z ⎜ 1 − 4 tanh4 i

i

() zi νi

⎞ ⎛ ⎜ 1 − 4 tanh4 ⎛⎜ zi ⎞⎟ ⎟ W = q + q i I J ⎜ ⎝ νi ⎠ ⎟ ⎠ ⎝

⎞ ⎛ 4 ⎟ Wi , q = ∑ ⎜ J ¯ Ω z ⎝ 1 − 4 tanh zi ∈ ⎠ i

() zi νi

⎞ ⎟ Wi . ⎠

¯ Ω zi , from Lemma 4 and the fact that Wi ≥ 0, it can be For zi ∈ obtained that qJ ≤ 0. On the other hand, for zi ∈ Ω zi , zi is bounded, and qi is also bounded. Furthermore, there exist positive parameter b0, such that |D + qJ | < b0 . Thus, we can further rewrite (42) as

LV ≤ − a 0 V + b0 . (39)

(42)

(43)

Obviously, (43) means that

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

Y. Sun et al. / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

Fig. 4. u.

Fig. 1. x1 and yd .

dE [V (t )] ≤ − a 0 E [V (t )] + b0 . dt Thus, one has

⎡ b ⎤ b 0 ≤ E [V (t )] ≤ ⎢ V (0) − 0 ⎥ e−a0 t + 0 ⎣ a0 ⎦ a0 which implies that

0 ≤ E [V (t )] ≤ V (0) +

b0 , a0

∀ t > 0.

(44)

Therefore, based on the inequality (44) and the definition of V, all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in the sense of four-moment. □

4. Simulation Example

Fig. 2. x2 .

In this section, a numerical example is used to test the proposed method. Consider the following second-order stochastic nonlinear non-strict-feedback systems with time delays: ⎛ ⎛ x12 ⎞ 0.5x1 (τ1) x2 (τ1) ⎞⎟ ⎟x + dx1 = ⎜⎜ x12 x22/3 + ln ⎜⎜ 3 + ⎟ dt + 0.25dw , 2⎟ 2 1 + x2 ⎠ 1 + x12 (τ1) + x22 (τ1) ⎠ ⎝ ⎝ ⎛ ⎞ dx2 = ⎜⎜ x23 x12 + x1 cos (x1x2 ) + 2e x1x 2 u + 0.5x12 (τ 2 ) sin (x2 (τ 2 )) ⎟⎟ dt + 0.15dw , ⎝ ⎠

(45)

y = x1.

x12 x22/3

It should be pointed out that in this system, f1 = , ⎛ x12 ⎞ 0.5x1(τ1) x2 (τ1) ⎟ and q1 = , they are the functions of g1 = ln ⎜ 3 + 1 +x12 (τ1) +x22 (τ1) 1 +x22 ⎠ ⎝ all state variables. So, it is a non-strict-feedback system. Though there have been a few control strategies to be reported on adaptive fuzzy/neural control for non-strict-feedback systems [39,40], these existing methods require that the function g1 cannot contain the state variable x2. So, these existing methods cannot be used to control this system. In addition, as a 2-order system, (45) is a pure⎛ ⎛ x2 ⎞ ⎞ feedback system, and f1 = x12 x22/3 + ⎜ ln ⎜ 3 + 1 2 ⎟ ⎟ x2. Furthermore, 1 +x2 ⎠ ⎠ ⎝ ⎝ ⎛ x2 ⎞ 1 +x22 −1 ∂f 2 . Since we one has ∂x1 = 3 x12 x2 3 + ln ⎜ 3 + 1 2 ⎟ − 2x12 x22 1 +x2 ⎠ 3 +x12 +3x22 2 ⎝ cannot prove Fig. 3. θ^1 and θ^2.

∂f1 ∂x2

> 0 (or <0), the existing control approaches for

pure-feedback systems are infeasible for this system. However, the

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

Y. Sun et al. / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

proposed control strategy in the paper can be used for control design of the system (45). According to Theorem 1, the virtual control signal and the true control law are chosen as the following:

⎛ 3⎞ 1 α1 = − ⎜ k1 + ⎟ (x1 − yd ) − (x1 − yd )3θ^1S1T (Z1) S1 (Z1), 4⎠ 2a12 ⎝

[10]

[11]

(46) [12]

u = − k2 (x2 − α1) −

1 (x2 − α1)3θ^2 S2T (Z2 ) S2 (Z2 ) 2a22

(47)

[13]

The time delays are selected as τ1 = 2.5, τ 2 = 2.5 with the adaptive law as follows:

[14]

̇ r θ^i = i zi6 SiT (Zi ) Si (Zi ) − σi θ^i, 2a i

[15]

i = 1, 2.

(48)

In this simulation the design parameters are adopted, respectively, as k1 = 8, k2 = 8, a1 = 0.1, a2 = 0.1, σ1 = 0.005, σ 2 = 0.005, r1 = 6, r2 = 6, and the reference signal yd = 0.5 sin (t ) + 0.25 sin (0.5t ) and the initial conditions are given by x (0) = [0.2, 0.05] , θ^ = [0, 0].

[16]

Simulation results are shown in Figs. 1–4. Fig. 1 shows the systems output and the reference signal. Figs. 2–4 show that all the closedloop signals are bounded. This accords with Theorem 1 and sufficiently verifies the effectiveness of the proposed control scheme.

[18]

[17]

[19]

[20]

Remark 6. From Theorem 1, the tracking error can be bounded by b a constant 8 a0 . However, this constant is unknown because b0 0 depends on the unknown constants bm and θ0. Although an exb plicit estimation of 8 a0 is unable to achieve, it is clear that reducing 0 the design parameters ai, li, εi and σi, meanwhile increasing ri, will d0 lead to a smaller 2 a .

[21] [22] [23]

0

[24] [25]

5. Conclusion In this paper, a tracking control problem has been investigated for a class of stochastic systems with time delays. A novel adaptive NN control scheme was proposed for nonlinear stochastic nonstrict-feedback systems. And the Lyapunov–Krasovskii function is presented to compensate the time delay terms. It has been shown that the proposed controller can guarantee that the closed-loop system is SGUUB and the tracking error is convergent to a small region of zero.

[26] [27] [28]

[29] [30]

[31] [32]

References

[33] [34]

[1] Y.C. Wang, H.G. Zhang, Y.Z. Wang, Fuzzy adaptive control of stochastic nonlinear systems with unknown virtual control gain function, Acta Autom. Sin. 32 (2006) 170–178. [2] Y.J. Liu, S.C. Tong, W. Wang, Adaptive fuzzy output tracking control for a class of uncertain nonlinear systems, Fuzzy Sets Syst. 160 (October (19)) (2009) 2727–2754. [3] W.S. Chen, Z.Q. Zhang, Globally stable adaptive backstepping fuzzy control for output-feedback systems with unknown high-frequency gain sign, Fuzzy Sets Syst. 161 (March (6)) (2010) 821–836. [4] Z. Liu, G.Y. Lai, Y. Zhang, C.L. Philip Chen, Adaptive fuzzy tracking control of nonlinear time-delay systems with dead-zone output mechanism based on a novel smooth model, IEEE Trans. Fuzzy Syst. http://dx.doi.org/10.1109/TFUZZ. 2015.2396075. [5] S.C. Tong, S. Sui, Y.M. Li, Adaptive fuzzy output feedback dynamic surface control of interconnected nonlinear pure-feedback systems, IEEE Trans. Cybern. 45 (1) (2015) 138–149. [6] F. Wang, Z. Liu, Y. Zhang, C.L. Philip Chen, Adaptive quantized controller design via backstepping and stochastic small-gain approach, IEEE Trans. Fuzzy Syst. 24 (2) (2016) 330–343. [7] F. Wang, Zhi. Liu, G.Y. Lai, Fuzzy adaptive control of nonlinear uncertain plants with unknown dead zone output, Fuzzy Sets Syst. 263 (1) (2015) 27–48. [8] F. Wang, B. Chen, C. Lin, G. Li, Y.M. Sun, Adaptive quantized control of switched stochastic nonlinear systems, Neurocomputing. http://dx.doi.org/10.1016/j. neucom.2016.05.024. [9] Z. Liu, F. Wang, Y. Zhang, X. Chen, C.L. Philip Chen, Adaptive tracking control

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

7

for a class of nonlinear systems with fuzzy dead-zone input, IEEE Trans. Fuzzy Syst. 23 (1) (2015) 193–204. Y.X. Hou, S.C. Tong, Adaptive fuzzy output-feedback control for a class of nonlinear switched systems with unmodeled dynamics, Neurocomputing 168 (2015) 200–209. C. Kwan, F. Lewis, Robust backstepping control of nonlinear systems using neural networks, IEEE Trans Syst. Man Cybern. A: Syst. Hum. 30 (November (6)) (2000) 753–766. S.S. Ge, J. Wang, Robust adaptive neural control for a class of perturbed strict feedback nonlinear systems, IEEE Trans. Neural Netw. 13 (November (6)) (2002) 1409–1419. Y.H. Li, S. Qiang, X.Y. Zhuang, O. Kaynak, Robust and adaptive backstepping control for nonlinear system using RBF neural networks, IEEE Trans. Neural Netw. 15 (May (3)) (2004) 693–701. W.S. Chen, Adaptive backstepping dynamic surface control for systems with periodic disturbances using neural networks, IET Control Theory Appl. 3 (October (10)) (2009) 1383–1394. J. Li, W.S. Chen, J.M. Li, Adaptive NN output-feedback decentralized stabilization for a class of large-scale stochastic nonlinear strict-feedback systems, Int. J. Robust Nonlinear Control 21 (March (4)) (2011) 452–472. H.H. Wang, B. Chen, C. Lin, Y.M. Sun, Observer-based adaptive neural control for a class of nonlinear pure-feedback systems, Neurocomputing 171 (2016) 1517–1523. L. Yua, S. Fei, F. Long, M. Zhang, J. Yu, Multilayer neural networks-based direct adaptive control for switched nonlinear systems, Neurocomputing 74 (2010) 481–486. S.S. Ge, C. Wang, Adaptive neural control of uncertain MIMO nonlinear systems, IEEE Trans. Neural Netw. 15 (May (3)) (2004) 674–692. Y.M. Li, S.C. Tong, T.S. Li, Observer-based adaptive fuzzy tracking control of MIMO stochastic nonlinear systems with unknown control direction and unknown dead-zones, IEEE Trans. Fuzzy Syst. 23 (4) (2015) 1228–1241. S.C. Tong, S. Sui, Y.M. Li, Fuzzy adaptive output feedback control of MIMO nonlinear systems with partial tracking errors constrained, IEEE Trans. Fuzzy Syst. 23 (4) (2015) 729–742. K.Q. Gu, V.L. Karitonov, J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, 2003. M. Krstić, H. Deng, Stabilization of Uncertain Nonlinear Systems, Springer, New York, 1998. W. Michiels, S.I. Niculescu, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach, SIAM, Singapore, 2007. S.I. Niculescu, Delay Effects on Stability, Springer, New York, 2001. J.P. Richard, Time-delay systems: an overview of some recent advances and open problem, Automatica 39 (2004) 1667–1694. Q.C. Zhong, Robust Control of Time-Delay System, Springer, New York, 2006. S.S. Ge, K.P. Tee, Approximation-based control of nonlinear MIMO time-delay systems, Automatica 43 (2007) 31–43. Y. Li, S. Tong, T. Li, Adaptive fuzzy output feedback control of MIMO nonlinear uncertain systems with time-varying delays and unknown backlash-like hysteresis, Neurocomputing 93 (2) (2012) 56–66. Y.S. Fu, Z.H. Tian, S.J. Shi, Output-feedback stabilization for a class of stochastic time-delay nonlinear systems, IEEE Trans. Autom. Control 50 (2005) 847–850. Y.G. Liu, J.F. Zhang, Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics, SIAM J. Control Optim. 45 (2006) 885–926. C.Y. Lu, T.J. Su, J.S.H. Tsai, On robust stabilisation of uncertain stochastic timedelay systems an LMI-based approach, J. Frankl. Inst. 342 (2005) 473–487. S.L. Xie, L.H. Xie, Stabilization of a class of uncertain large-scale stochastic systems with time delays, Automatica 36 (2000) 161–167. L. Liu, X.J. Xie, Output-feedback stabilization for stochastic high-order nonlinear systems with time-varying delay, Automatica 47 (2011) 2772–2779. S.C. Tong, T. Wang, Y.M. Li, B. Chen, A combined backstepping and stochastic small-gain approach to robust adaptive fuzzy output feedback control, IEEE Trans. Fuzzy Syst. 21 (2) (2013) 314–327. Y.M. Li, S. Sui, S.C Tong, Adaptive fuzzy control design for stochastic nonlinear switched systems with arbitrary switchings and unmodeled dynamics, IEEE Trans. Cybern. doi:CYB-E-2015-07-0990. Z. Liu, F. Wang, Y. Zhang, C.L. Philip Chen, Fuzzy adaptive quantized control for a class of stochastic nonlinear uncertain systems, IEEE Trans. Cybern. 46 (2) (2016) 524–534. F. Wang, Z. Liu, Y. Zhang, C.L. Philip Chen, Adaptive fuzzy control for a class of stochastic pure-feedback nonlinear systems with unknown hysteresis, IEEE Trans. Fuzzy Syst. 24 (1) (2016) 140–152. B. Chen, X.P. Liu, S.S. Ge, C. Lin, Adaptive fuzzy control of a class of nonlinear systems by fuzzy approximation approach, IEEE Trans. Fuzzy Syst. 20 (2012) 1012–1021. B. Chen, X.P. Liu, K.F. Liu, C. Lin, Fuzzy approximation-based adaptive control of nonlinear delayed-time systems with unknown dead zone, IEEE Trans. Fuzzy Syst. 22 (2) (2014) 237–248. H.Q. Wang, B. Chen, K.F. Liu, X.P. Liu, C. Lin, Adaptive neural tracking control for a class of non-strict-feedback stochastic nonlinear systems with unknown backlash-like hysteresis, IEEE Trans. Neural Netw. Learn. Syst. 25 (2014) 947–958. S.J. Liu, J.F. Zhang, Z.P. Jiang, Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems, Automatica 43 (2) (2007) 238–251. H.B. Ji, H.S. Xi, Adaptive output-feedback tracking of stochastic nonlinear

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i

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Y. Sun et al. / Neurocomputing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

systems, IEEE Trans. Autom. Control. 51 (2006) 355–360. [43] S.C. Tong, Y. Li, Y.M. Li, Y.J. Liu, Observer-based adaptive fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems, IEEE Trans. Syst. Man Cybern. B: Cybern 41 (2011) 1693–1704. [44] R.M. Sanner, J.E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Netw. 3 (1992) 837–863. [45] W. Lin, C.J. Qian, Adaptive control of nonlinear MIMO time-delay systems, Automatica 43 (1) (2007) 304–311. [46] S.S. Ge, K.P. Tee, Approximation-based control of nonlinear MIMO time-delay systems, Automatica 43 (1) (2007) 31–43.

Yumei Sun received the B.Sc. degree in mathematics from Shangdong University, Jinan, China, in 2002 and the M.Sc. degree in mathematics from Sun Yat-sen University, Guangzhou, China, in 2005. She is currently a Ph.D. candidate of Qingdao University, Qingdao, China. Her current research interests include adaptive fuzzy control and stochastic nonlinear systems.

Chong Lin received the B.Sc. and M.Sc. degrees in applied mathematics from Northeastern University, Shenyang, China, in 1989 and 1992, respectively, and the Ph.D. degree in electrical and electronic engineering from Nanyang Technological University, Singapore, in 1999. He was a research associate with the Department of Mechanical Engineering, University of Hong Kong, Hong Kong, in 1999. From 2000 to 2006, he was a research fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Since 2006, he has been a professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. He has published more than 60 research papers and co-authored two monographs. His current research interests include systems analysis and control, robust control, and fuzzy control.

Honghong Wang received her B.S. degree and M.S. degree at Shandong University in 2001 and 2004, respectively. Currently she is a teacher of the School of Automation Engineering, QingDao University, Qingdao, PR China. Now she is pursuing her Ph.D. degree in system theory from the Institute of Complexity Science, Qingdao University. Her current research interests are mainly in systems analysis and control, neural networks and fuzzy control theory.

Bing Chen received the B.A. degree in mathematics from Liaoning University, Liaoning, China, the M.A. degree in mathematics from the Harbin Institute of Technology, Heilongjiang, China, and the Ph.D. degree in electrical engineering from Northeastern University, Shenyang, China, in 1982, 1991, and 1998, respectively. He is currently a professor with the Institute of Complexity Science, Qingdao University, Qingdao, China. His current research interests include nonlinear control systems, robust control, and adaptive fuzzy control.

Please cite this article as: Y. Sun, et al., Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing (2016), http://dx.doi.org/10.1016/j.neucom.2016.06.060i