A novel adaptive control method for a class of stochastic switched pure feedback systems

Neurocomputing 367 (2019) 337–345

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

A novel adaptive control method for a class of stochastic switched pure feedback systemsR Yumei Sun a,∗, Bing Chen b, Fang Wang a, Shaowei Zhou a, Honghong Wang b a b

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

a r t i c l e

i n f o

Article history: Received 2 April 2019 Revised 16 May 2019 Accepted 20 June 2019 Available online 23 August 2019 Communicated by Shaocheng Tong

a b s t r a c t This paper proposes a novel adaptive control approach for a class of stochastic switched nonlinear system with pure feedback structure. An important lemma is developed to overcome the design difficulty from the non-affine pure feedback structure. Combining backstepping technique with neural network approximation, a state feedback adaptive controller is given. And this controller can ensure that all of the signals in the closed-loop system are bounded, and the tracking error converges to a small enough neighborhood of the zero. A simulation example is used to verify the effectiveness of our results.

Keywords: Adaptive neural network control Stochastic switched nonlinear systems Backstepping Non-affine pure feedback systems

1. Introduction Over the past decades, approximate-based adaptive control design of unknown nonlinear systems has attracted lots of attention. By combing universal function approximators, i.e., neural networks or fuzzy logic systems, with backstepping technique, many interesting results have been obtained. Adaptive backstepping design method was presented in [1–9] for strict-feedback systems, and was further extended to stochastic cases in [10–14]. On the other hand non-affine pure feedback systems, that is more general than lower-triangular systems, have attracted a great deal of attention due to their widely existence in real engineering, for instance, see [15–21]. Adaptive neural output-feedback control for a class of nonlinear pure-feedback systems has been studied in [17,18]. However, stochastic disturbance is not taken into account in these literatures. In [20,21], adaptive control have been considered for stochastic systems, where diffusion terms ψ ( · )s were supposed to be the function of the previous variable x¯i or x¯i+1 . However, when they are the nonlinear function of the whole state variables, how to design the virtual control signal α i , which is independent of the state variables x j , j = i + 1, · · · , n, becomes difficult. In [15], the assumption that ψ ( · )s must be less than or equal to increasing functions has been proposed to over the difficulty. R ∗

Communicated by Shaocheng Tong Corresponding author. E-mail address: [email protected] (Y. Sun).

https://doi.org/10.1016/j.neucom.2019.06.061 0925-2312/© 2019 Elsevier B.V. All rights reserved.

© 2019 Elsevier B.V. All rights reserved.

Such an assumption is then not only very difficult to be tested in real engineering, but also leads to a very complex control design procedure. In addition, switched systems have caused great attention, owing to its extended application in engineering practice such as circuit and power systems, aircraft control systems, robot manipulators and multi-systems [22–26]. Switched systems present changing-over a series of subsystems in light of environmental changes. In [27], a common Lyapunov function (CLF) is used to the stability analysis of switched systems under arbitrary switching. Then, some notable results have been achieved in the control of nonlinear switched systems (see [28–34]). Meanwhile, by combing backstepping technique with approximation-based adaptive control method, many notable results have been achieved for unknown nonlinear strict-feedback systems under arbitrary switching (see [35–38]). As we have seen, there is little research for switched stochastic nonlinear systems. Despite some progress has been made in [39], the switched stochastic systems must be in triangular form. Those methods will be invalid for switched stochastic pure-feeback systems. Motivated by the above observation, we will propose the adaptive neural control method for a class of switched stochastic purefeedback nonlinear systems. It will been proven that the proposed controller ensures that all the closed-loop signals remain bounded and the tracking error converges to a small neighborhood around the zero in the sense of mean quadratic value. The main contributions of our work are listed as: 1) the restrictions on the system

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of the diffusion terms ψ ( · )s are relaxed, 2) the proposed control scheme can be applied to the non-affine in virtual control switched stochastic pure-feedback systems. The remainder of this paper is organized as follows. The preliminaries and problem description are given in Section 2. A novel adaptive tracking control design scheme is presented in Section 3. The simulation example is given in Section 4. Finally, the conclusion is drawn in Section 5. 2. Preliminaries and problem description 2.1. Stochastic system stability

(1)

where x = [x1 , x2 , · · · , xn ]T ∈ R, w denotes an r-dimensional standard Brownian motion defined on the complete probability space (, F, P) with  being a sample space, F being a σ -field, and P being a probability measure. f( · ) and h( · ) are locally Lipschitz functions in x and satisfy f (0 ) = 0 and h(0 ) = 0. Definition 1. For any given V(x) ∈ C2 , associated with the stochastic differential equation (1), we define the differential operator L as follows:



LV =



∂V 1 ∂ 2V f + T r hT h , ∂x 2 ∂ x2

(2)

where Tr(A) is the trace of matrix A. Definition 2 [40]. The solution process {x(t), t ≥ 0} of stochastic system (2) is said to be bounded in probability, if limc→∞ sup0≤t≤∞ P (||x(t )|| > c ) = 0, where P(B) denotes the probability of event B. Lemma 1 [41]. Consider the stochastic system (3). If there exists a positive definite. radially unbounded, twice continuously differentiable Lyapunov function V: Rn → R, and constants a0 > 0, b0 > 0 such that

LV (x ) ≤ −a0V (x ) + b0 , then i) the system has a unique solution almost surely, and ii) the system is bounded in probability. Lemma 2 [42]. For any (x, y) ∈ R2 , the following inequality holds:

xy ≤

εp p

|x| p +

∀x ∈ Rn

Assumption 2 means that the sign of gi,k ( · ) is strict positive or strict negative. Without loss of generality, we can further assume that 0 < bm ≤ gi,k ( · ) ≤ bM . Remark 1. In [20,21], it is required that the diffusion terms ψ ( · )s must not contain state variable x¯k for k ≥ i + 1. So, the system (3) is a more general than the one in case [20,21].

fi (x¯i , xi+1 ) = fi (x¯i , 0 ) + gθi (x¯i , xθi )xi+1 ,

fn (x¯n , u ) = fn (x¯n , 0 ) + gθn (x¯n , θn u )u,

where,

1 ≤ i ≤ n,

for

g θi ( · ) =

∂ fi (x¯i ,xi+1 ) ∂ xi+1 |xn+1 =xθi ,

xθi = θi xi+1 ,

0 < θ i < 1 and xn+1 = u. Notice that gθi is the function of x¯i+1 = [x¯i , xi+1 ], and the virtual control gain function gi ( · ) in systems (3) can be the function of x¯i+1 , therefore the scheme in this paper can be applied to the case of nonaffine pure-feedback. 2.3. Function approximation with neural networks In this paper, we use radial basis function (RBF) neural network to approximate unknown functions. Universal approximation result indicate that, given a desired accuracy ε , approximation to that level of accuracy can be guaranteed by making l sufficiently large [43]. Thus the neural network W∗T S(Z) can approximate any continuous function

fnn (Z ) = W ∗T S(Z ) + ε (Z ),

∀ Z ∈ Z ⊂ R q ,

where W∗ is the ideal constant weight vector and defined as

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

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) is chosen as Guassian function, namely,



si (Z ) = exp −

 ( Z − μi ) T ( Z − μi ) , i = 1, · · · , l ηi2

(4)

where μi = [μi1 , · · · , μiq ]T is the center of the receptive field and ηi is the width of Gaussian function. Based on the structure of the Guassion function in (4), an important lemma is introduced first as following.

1 |y|q , qε q

where ε > 0, p > 1, q > 1 and ( p − 1 )(q − 1 ) = 1. 2.2. System description Consider a class of switched stochastic pure-feedback systems as follows:

dxi = (xi+1 gi,τ (t ) (x¯i , xi+1 ) + fi,τ (t ) (x¯i , xi+1 ))dt + ψi,Tτ (t ) (x )dw,

x¯q = [x1 , · · · , xq ]T Lemma 3 [44–46]. Let and S(x¯q ) = T [S1 (x¯q ), · · · , Sl (x¯q )] be the basis function vector of a RBF NN. Then, for all positive integers q ≤ p, the following formula is established:

||S(x¯ p )||2 ≤ ||S(x¯q )||2 . Remark 3. This Lemma provides a simple but useful characteristic of RBF NN. According to this lemma,we can apply adaptive neural backstepping design method to the system (3) easily.

1≤i≤n−1 T dxn = (ugn,τ (t ) (x ) + fn,τ (t ) (x ))dt + ψn, τ (t ) (x )dw,

y = x1 ,

0 < bm ≤ |gi,k (· )| ≤ bM < ∞,

Remark 2. According to the mean value theorem, the nonaffine function fi (x¯i , xi+1 ) in [15,17,20] can be rewritten as

Consider the following stochastic system

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

Assumption 2. The sign of gi,k ( · ) is known and there exist constants bm and bM such that

(3) Rn ,

where x¯i = [x1 , · · · , xi ] ∈ u ∈ R and y ∈ R are state variable, system input and system output, respectively, w is an r-standard Brownian motion defined on the complete probability space. τ (t ) : [0, +∞ ) → M  {1, 2, · · · , m} denotes a piecewise continuous switched signal, τ (t ) = k (k ∈ M ) implies that the kth subsystem is active. fi,k (· ) : Ri+1 → R, gi,k (· ) : Ri+1 → R, and ψ i,k ( · ): Rn → Rr are all unknown smooth nonlinear functions. Assumption 1. The reference signal yd (t) and its time derivatives up to the n−th order are continuous and bounded.

3. Adaptive tracking control design In this section we will contribute to develop a backsteppingbased adaptive neural control design procedure. The design procedure of the controller can be visualized using the block diagram shown in Fig. 1. Usually, a backstepping design procedure is implemented by the following coordinate transformation.





¯ zi = xi − αi−1 x¯i−1 , θˆi−1 , y¯ d(i−1) , 1 ≤ i ≤ n

(5)

Y. Sun, B. Chen and F. Wang et al. / Neurocomputing 367 (2019) 337–345

339

with li represents a positive design parameter. Substituting (8) and (9) into (7) yields

LVi ≤ zi3 ( f¯i,k + gi,k αi ) +

1 3 2 bm ˜ ˆ˙ 1 l − θi θi + gi,k zi4+1 − gi−1,k zi4 , (10) 4 i ri 4 4

where

3 3zi 1 f¯i,k = fi,k − Lαi−1 + gi,k zi − gi−1,k zi + 2 ||ψi,k 4 4 4li −

i−1 j=1

Remark 4. Since f¯i,k is a function of whole state variable x, the existing approximate-based backstepping control schemes can not be utilized to the system (3). In [15], an adaptive fuzzy control strategy was presented for stochastic pure-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 [15].

Fig. 1. The block diagram of controlled system.

with α0 = yd . By (5) and It oˆ formula, the zi subsystem is described by



dzi = (xi+1 gi,k + fi,k − Lαi−1 )dt +

ψi,k −

i−1 j=1

∂αi−1 ψ ∂ x j j,k

T

dw, (6)

where

Lαi−1 =

i−1 j=1

+

j=1

∗T f¯i,k = Wi,k Si,k (Zi ) + δi,k (Zi )

with Zi = [x1 , · · · , xn , θˆ1 , · · · , θˆi , yd , · · · , yd(i ) ] and δ i,k (Zi ) ≤ ε i,k being the approximation error. Furthermore, applying Lemma 2 and Lemma 3 to the term zi3 f¯i,k gives ∗T zi3 f¯i,k = zi3 Wi,k Si,k (Zi ) + δi,k (Zi )

i−1 ∂αi−1 ˆ˙ ∂αi−1 ( j+1) θj + y , ( j) d ˆ ∂θj j=0 ∂ yd

∂α0 ∂α0 ∂ xk = 0 and ∂θk = 0, xn+1 = u.

By combing backstepping with adaptive neural control, we will give the controller design procedure as follows: Step i: For 1 ≤ i ≤ n − 1, take the Lyapunov function candidate as

Vi =

For the unknown f¯i,k (Zi ), there exists a neural network ∗ S (Z ) such that for given ε > 0, Wi,k i,k i,k i



i−1 ∂αi−1 1 ∂ 2 αi−1 T (x j+1 g j,k + f j,k ) + ψ ψ ∂xj 2 ∂ x p ∂ xq p,k q,k p,q=1

i−1

1 4 bm θ˜i2 z + , 4 i 2ri



ψi,k −

i−1 j=1

∂αi−1 ψ ∂ x j j,k

T  ψi,k −

i−1 j=1

∂αi−1 ψ ∂ x j j,k



||W ∗ ||2

3 2 z 2 i ≤

ψi,k −

i−1 j=1

3 4 z ||ψi,k − 4li2 i

∂αi−1 ψ ∂ x j j,k i−1 j=1

ψi,k −

i−1 j=1

∂αi−1 3 ψ ||4 + li2 ∂ x j j,k 4

∂αi−1 ψ ∂ x j j,k

2a2i,min

bm θi SiT (Xi )Si (Xi ) +

3 zi + αi gi,k 4



+

a2i,max 2

bm ˜ ˆ˙ 1 3 2 εi,k 1 l + − θi θi + gi,k zi4+1 − gi−1,k zi4 . 4 i 4 ri 4 4

  3 αi = − ki + zi −

(8)

T 

zi3

4

+

4bm

(7)

3 1 g z4 + gi,k zi4+1 4 i,k i 4

(11)

(12)

˙ Choose the virtual control signal α i and the adaptive laws θˆi as follows

Furthermore, it follows from Young inequality and Assumption 2 that



4 a2i,k bm 6 T 3 4 εi,k z θ S ( X ) S ( X ) + + z + i i i i i 2 4 i 4 2a2i,k i

sign parameter and θi = max{ bi,k : k ∈ M}. Let ai,min = min{ai,k : m k ∈ M} and ai,max = max{ai,k : k ∈ M}. Thus, it follows immediately from substituting (11) into (10) that



bm ˙ − θ˜i ˆθi. ri

gi,k zi3 zi+1 ≤

≤

where Xi = [x1 , · · · , xi , θˆ1 , · · · , θˆi , yd , · · · , yd(i ) ], ai,k is a positive de-

where ri > 0 is a design parameter and θ˜i = θi − θˆi . By (3), (5) and (6) one has

3 + zi2 2

4 a2i,k εi,k bm 6 T 3 z i θi S i ( Z i ) S i ( Z i ) + + zi4 + 2 2 4 4 2ai,k



LVi = zi fi,k + gi,k αi + gi,k zi+1 − Lαi−1



≤

LVi ≤ zi3

 3

∂αi−1 ψ ||4 , g0,k = 0. ∂ x j j,k

θˆ˙ i =

1 z3 θˆi SiT (Xi )Si (Xi ), 2a2i,min i

ri z6 ST (Xi )Si (Xi ) − σi θˆi , θˆi (0 ) ≥ 0, 2a2i,min i i

(13)

(14)

where ki and σ i are positive design parameters. Remark 5. It is apparent that (14) means that for any initial condi˙ tion θˆi (t0 ) ≥ 0, the solution θˆi (t ) holds for t ≥ t0 . Thus, throughout this paper, it is assumed that θˆi (t ) ≥ 0.



Based on Assumption 2, the following inequality holds:

(9)

  b 3 αi gi,k ≤ − ki bm + zi − 2m zi3 θˆi SiT (Xi )Si (Xi ). 4

2ai,min

(15)

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Y. Sun, B. Chen and F. Wang et al. / Neurocomputing 367 (2019) 337–345

Define εi,max = max{εi,k : k ∈ M}, substituting (14) and (15) into (12) yields

LVi ≤ −ki bm zi4 + −

4 εi,max

a2 bm ˜ ˆ 3 σi θi θi + li2 + i,max + ri 4 2

4

+

1 g z4 4 i,k i+1

1 g z4 . 4 i−1,k i

Step n: In this step, we will give the real controller. Take stochastic Lyapunov function as follows

(17)





3 + zn2 2

ψn,k −

n−1 j=1

∂αn−1 ψ ∂ x j j,k

T  ψn,k −

n−1 j=1

∂αn−1 ψ ∂ x j j,k

ψn,k −

n−1 j=1



(18)

Using similar as (9)-(12) in step i, we have



∂αn−1 ψ ∂ x j j,k n−1

3 ≤ 2 zn4 ψn,k − 4ln j=1

T 

ψn,k −

n−1 j=1

4

∂αn−1 3

ψ j,k + ln2 ∂xj 4

∂αn−1 ψ ∂ x j j,k



(19)

1 3 2 bm ˜ ˆ˙ l − θn θn − gn−1,k zn4 , 4 n rn 4

(20)

∗T f¯n,k = Wn,k Sn,k (Zn ) + δn,k (Zn ),

4 a2n,k εn,k bm 6 3 zn θn SnT (Zn )Sn (Zn ) + + zn4 + , 2 2 4 4 2an,k

zn3 bm θn SnT (Xn )Sn (Xn ) + ugn,k 2a2n,min

4 εn,k

bm ˜ ˆ˙ 1 + − θn θn − gn−1,k zn4 , 4 rn 4

−

1 g z4 . 4 n−1,k n (26)

Theorem. Consider the nonlinear stochastic system (3) under Assumption 1–2. For bounded initial conditions, under the actual controller u = αn , associated with the virtual signals (13) for 1 ≤ i ≤ n − 1 and the adaptive laws (14), all the signals in the closedloop system are bounded in probability and there exist T1 such that tracking error is in the set 1 for all t > T1 ,

with a0 = min{4ki bm , 4 εi,max

4

bm σi θi2 2ri

+

n

bm ri σi |1

≤ i ≤ n} and b0 =

n

i=1 (

a2i,max 2

+ 34 li2 +

)

Proof. Define V =

n

i=1 Vi ,

it follows from (16) and (22) that

n n bm ˜ ˆ σ i θi θi + ri

ki bm zi4 +

i=1

i=1



a2i,max 2

+

3 2 l + 4 i

4 εi,max

4

 .

Since

1 1 θ˜i θˆi = θ˜i (θi − θ˜i ) ≤ − θ˜i2 + θi2 , 2

LV ≤ −

(21)

1 zn3 θˆn SnT (Xn )Sn (Xn ), 2a2n,min

n i=1

+

n

ki bm zi4 −



2



n bm σi θ˜i2 2ri i=1

a2i,max 2

i=1

≤ −a0

+

 +

a2n,max 3 + ln2 2 4

(23)

4 εi,max

4

+

bm σi θi2 2ri



+ b0

i=1

= −a0V + b0 ,

(22)

3 2 l + 4 i

n n 1 4 bm θ˜i2 zi + 4 2ri i=1

with an,min = min{an,k : k ∈ M} and an,max = max{an,k : k ∈ M}. ˙ Take the real control signal u and the adaptive laws θˆn as follows

u = −kn zn −

4

and

At present stage, we will summarize our main conclusions in the following theorem.



where θ n is defined in step i. Substituting (21) into (20), we get

LVn ≤

4 εn,max

(24)

So, (27) can be expressed as

where Zn = Xn = [x1 , · · · , xn , θˆ1 , · · · , θˆn , yd , · · · , yd(n ) ] and δ n (Zn ) represents the approximate error and satisfies |δ n (Zn )| < ε n . Applying Lemma 2 and Lemma 3 to the term zn3 f¯n,k gives



substituting

(27)

n,k

zn3

(25)

a2 bm 3 σn θ˜n θˆn + ln2 + n,max + rn 4 2

i=1

∗T S (Z ) is applied to approximate Similar to step i, the network Wn,k n n ¯ the unknown function f such that for all ε n > 0

∗T zn3 f¯n,k = zn3 Wn,k Sn,k (Zn ) + δn,k (Zn )

Using

bm zn3 θˆn SnT (Xn )Sn (Xn ). 2a2n,min

Define εn,max = max{εn,k : k ∈ M}, (25) into (22) yields

LV ≤ −

4

n−1 3 z ∂α

n n −1 f¯n,k = fn,k − Lαn−1 + zn + 2 ψn,k − ψ j,k . ∂ xj 4ln

j=1

≤

parameters.

a0

where



design

  8b 1 = y(t ) ∈ R|E[|y − yd |4 ] ≤ 0 , ∀t > T1

with ln representing a positive design parameter. Substituting the inequalities (19)into (18), we can get

LVn ≤ zn3 ( f¯n,k + ugn,k ) +

positive

(24)



bm ˜ ˆ˙ − θn θn. rn

3 2 z 2 n

ugn,k ≤ −kn bm zn −

LVn ≤ −kn bm zn4 +

with rn being a positive design constant. On the basis of (6), we have

LVn = zn3 fn,k + ugn,k − Lαn−1

rn zn6 SnT (Xn )Sn (Xn ) − σn θˆn , θˆn (0 ) ≥ 0, 2a2n,min

where kn and σ n are Assumption 2, we get

(16)

1 bm θ˜n2 Vn = zn4 + 4 2rn

θˆ˙ n =

(28)

which indicates that all the signals in the closed-loop system are bounded.  Remark 6. From Theorem, the tracking error can be bounded by b a constant 8 a0 . However, this constant is unknown because b0 de0 pends on the unknown constants bm and θ i . Although an explicit b estimation of 8 a0 is impossible, it is clear that reducing the design 0 parameters ai , li , ε i and σ i , and increasing ri at the meantime will b lead to a smaller 8 a0 . 0

Y. Sun, B. Chen and F. Wang et al. / Neurocomputing 367 (2019) 337–345

341

1 x1

0.8

yd1

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

0

10

20

30 Time(sec)

40

50

60

Fig. 2. x1 and yd1 (t).

2 x1 yd2

1.5 1 0.5 0 −0.5 −1 −1.5 −2

0

10

20

30 Time(sec)

40

50

60

Fig. 3. x1 and yd2 (t).

4. Simulation example In this section, one example is used to verify the effectiveness of the presented method. Example. Consider the third-order switched nonlinear system as follows:

dx1 = (x2 g1,τ (t ) (x¯2 ) + f1,τ (t ) (x¯2 ))dt + ψ1T,τ (t ) (x )dw, dx2 = (x3 g2,τ (t ) (x¯3 ) + f2,τ (t ) (x¯3 ))dt + ψ2T,τ (t ) (x )dw,

dx3 = (ug3,τ (t ) (x ) + f3,τ (t ) (x ))dt + ψ3T,τ (t ) (x )dw, y = x1 ,

(29)

To make all the signals are bounded and y follow a given signal under arbitrary switchings, select the virtual controller (13), the controller (14) and the adaptive law (11). In simulation, the two reference signals yd1 = 0.5(sin(t ) + sin(0.5t )); and yd2 = 1.5(sin(t )) are given, the switched law is τ (t ) : [0, ∞ ) → N = {1, 2, 3} and systems gi,τ (t) , fi,τ (t) , ψ i,τ (t) , 1 ≤ i ≤ 3 are switched among the following three different equation sets:

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20 x2 x3

15

10

5

0

−5

−10

−15

0

10

20

30 Time(sec)

40

50

60

Fig. 4. State variable x2 and x3 .

2 1

1.8

2

1.6

3

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

10

20

30 Time(sec)

40

50

60

Fig. 5. Adaptive law θˆ1 , θˆ2 and θˆ3 .

dx1 = [0.5x2 sin(x2 ) + (1 + sin(x2 ) sin(x2 ))x2 ]dt

dx2 = [x22 sin(x1 ) cos(x3 ) + (2 + x23 /(1 + x22 + x23 ))x3 ]dt

+ 0.25x2 sin(x3 )dw;

+ 0.5x1 cos(x1 )dw;

dx2 = [x2 sin(x1 ) cos(x3 )x2 + (2 + x23 /(1 + x22 + x23 ))x3 ]dt

dx3 = [x23 sin(x1 − x2 ) + (2 + exp(x1 x3 ))u]dt

+ 0.25x1 cos(x3 )dw; dx3 =

[x23

+ 0.5x1 sin(−x22 )dw;

sin(x1 − x2 ) + (2 + cos(x1 x3 ))u]dt

+ 0.15x1 exp(−x22 )dw;

(30)

(31)

dx1 = [0.5x22 sin(x1 ) + (1 + sin(x2 )2 + 0.5x1 sin(x1 )dw; dx2 = [x22 sin(x1 ) cos(x3 ) + (2 + x23 /(1 + x22 + x23 ))x3 ]dt +0.5x1 cos(x1 )dw;

dx1 = [0.5x22 sin(x3 ) + (1 + cos(x2 ) sin(x2 ))x2 ]dt + 0.5x1 sin(x1 )dw;

dx3 = [x23 sin(x1 − x2 ) + (1 + exp(x1 x3 ))u]dt +0.5x1 cos(−x21 )dw;

(32)

Y. Sun, B. Chen and F. Wang et al. / Neurocomputing 367 (2019) 337–345

343

150 u 100 50 0 −50 −100 −150 −200 −250 −300

0

10

20

30 Time(sec)

40

50

60

Fig. 6. The controller u.

4 switched signal 3.5 3 2.5 2 1.5 1 0.5 0

0

10

20

30 Time(sec)

40

50

60

Fig. 7. The switched signal τ (t).

Select the design parameters as a1,min = a2,min = a3,min = 1, r1 = r2 = r3 = 7.5,k1 = k2 = k3 = 10, σ1 = σ2 = σ3 = 0.5. The simulation is performed under the initial conditions [x1 (0 ), x2 (0 ), x3 (0 )]T = [0.25, 0, 0]T , and [θˆ1 (0 ), θˆ2 (0 ), θˆ3 (0 )]T = [0, 0, 0]T .

Figs. 2–7 demonstrate the corresponding simulation results. Figs. 2–3 show the tracking two different signal under switching. Figs. 4–7 display that all the signals are bounded.

Remark 6: Figs. 2 and 3 show that the method proposed in this paper has good tracking performance for different signals. 5. Conclusion In this article, we have presented a new neural control method for a class of switched stochastic nonlinear systems in form of pure-feedback. The given adaptive neural tracking controller ensures that all the closed-loop signals are bounded in probabil-

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[31] B. Niu, L. Li, Adaptive neural network tracking control for a class of switched strict-feedback nonlinear systems with input delay, Neurocomputing 173 (2016) 2121–2128. [32] Q.T. Yin, M. Wang, X.L. Li, G.H. Sun, Neural network adaptive tracking control for a class of uncertain switched nonlinear systems, Neurocomputing 301 (2018) 1–10. [33] W.L. Zhou, B. Niu, X.J. Xie, F.E. Alsaadi, Adaptive neural-network-based tracking control strategy of nonlinear switched non-lower triangular systems with unmodeled dynamics, Neurocomputing 322 (2018) 1–12. [34] L.J. Long, J. Zhao, Switched-observer-based adaptive neural control of MIMO switched nonlinear systems with unknown control gains, IEEE Trans. Neural Netw. Learning Syst. 28 (7) (2017) 1669–1709. [35] X. Zhao, X. Zheng, B. Niu, L. Liu, Adaptive tracking control for a class of uncertain switched nonlinear systems, Automatica 52 (C) (2015) 185–191. [36] Y. Li, S. Tong, T. Li, Adaptive fuzzy backstepping control design for a class of pure-feedback switched nonlinear systems, Nonlinear Anal. Hybrid Syst. 16 (2015) 72–80. [37] X. Zheng, X. Zhao, R. Li, Y. Yin, Adaptive neural tracking control for a class of switched uncertain nonlinear systems, Neurocomputing 168 (C) (2015) 320–326. [38] F. Wang, B. Chen, Y.M. Sun, C. Lin, Finite time control of switched stochastic nonlinear systems, Fuzzy Sets Syst. 365 (2019) 140C152. [39] M. Hou, F. Fu, G. Duan, Global stabilization of switched stochastic nonlinear systems in strict-feedback form under arbitrary switchings, Automatica 49 (2013) 2571–2575. [40] R.Z. Khas’minskii, Stochstic stability of differential equations, Kluwer Academic Publishers, Norwell, MA, 1980. [41] Z.J. Wu, X.J. Xie, S.Y. Zhang, Adaptive backstepping controller design using stochastic small-gain theorem, Automatica 43 (2007) 608–620. [42] H. Deng, M. Krstic, Stochastic nonlinear stabilization, part i: a backstepping design, Syst. Control Lett. (1997) 143–150. [43] R.M. Sanner, J.E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Netw. 3 (1992) 837–863. [44] Y.M. Sun, B. Chen, C. Lin, H.H. Wang, Adaptive neural control for a class of stochastic nonlinear systems by backstepping approach, Inf. Sci. 369 (2016) 748–764. [45] Y.M. Sun, B. Chen, C. Lin, H.H. Wang, Adaptive neural control for a class of stochastic non-strict-feedback nonlinear systems with time-delay, Neurocomputing, 214 (19) (2016) 750–757. [46] Y.M. Sun, B. Chen, C. Lin, H.H. Wang, Finite-time adaptive control for a class of nonlinear systems with nonstrict feedback structure, IEEE Transactions on Cybernetics, 48 (10) (2018) 2774–2782. Yumei Sun received the B.Sc. degree in mathematics from Shandong University, Jinan, China, in 2002, the M.Sc. degree in mathematics from Sun Yat-sen University, Guangzhou, China, in 2005, and Ph.D. degrees from Qingdao University Qingdao University, Qingdao, China in 2018. Her current research interests include adaptive neural or fuzzy control and stochastic control of nonlinear systems.

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.

Fang Wang received the B.S. degree from the Qufu Normal University, Qufu, China, the M.S. degree from Shandong Normal University, Jinan, China, and the Ph.D. degree from Guangdong University of Technology, Guangzhou, China, in 1997, 2004, and 2015, respectively. Since 2005, she has been at the Shandong University of Science and Technology, Qingdao, China. Her current research interests include stochastic nonlinear control systems, quantized control, backstepping control, and adaptive fuzzy control.

Y. Sun, B. Chen and F. Wang et al. / Neurocomputing 367 (2019) 337–345 Shaowei Zhou received the B.S. degree from Shandong Normal University, China, the M.S. and Ph.D. degrees from Shandong University of Science and Technology, China, in 20 0 0, 20 06 and 2012, respectively. Since 20 0 0, she has been at Shandong University of Science and Technology, Qingdao, China. Her current research interests include stochastic system control theory, and fuzzy adaptive control.

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Honghong Wang received her B.S. degree M.S. degree from Shandong University and Ph.D. degrees from Qingdao University in 20 01 20 04 and 2018 respectively. Currently she is a Teacher of the School of Automation Engineering, Qingdao University, Qingdao, PR China. Her current research interests are mainly in systems analysis and control, neural networks and fuzzy control theory.