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Journal of Systems Engineering and Electronics Vol. 22, No. 3, June 2011, pp.500–506 Available online at www.jseepub.com

Adaptive neural control for a class of uncertain stochastic nonlinear systems with dead-zone Zhaoxu Yu∗ and Hongbin Du Department of Automation, East China University of Science and Technology, Shanghai 200237, P. R. China

Abstract: The problem of adaptive stabilization is addressed for a class of uncertain stochastic nonlinear strict-feedback systems with both unknown dead-zone and unknown gain functions. By using the backstepping method and neural network (NN) parameterization, a novel adaptive neural control scheme which contains fewer learning parameters is developed to solve the stabilization problem of such systems. Meanwhile, stability analysis is presented to guarantee that all the error variables are semi-globally uniformly ultimately bounded with desired probability in a compact set. The effectiveness of the proposed design is illustrated by simulation results.

Keywords: adaptive control, neural network (NN), backstepping, stochastic nonlinear system.

DOI: 10.3969/j.issn.1004-4132.2011.03.020

1. Introduction The topic of designing controllers for stochastic nonlinear systems has been an intensive area of research in recent years. Some nonlinear control design methods such as Lyapunov function method, backstepping techniques and nonlinear optimality were extended to the case of stochastic nonlinear systems [1−6]. The main obstacle in the Lyapunov design for stochastic systems is that the Itˆo stochastic differentiation involves not only the gradient but also the higher order Hessian term [7]. In addition, the combined problem of the control of stochastic nonlinear systems with simultaneously nonlinear uncertainties is still cumbersome issue. Adaptive neural network (NN) control method and adaptive fuzzy system control method have been applied to some classes of uncertain nonlinear systems [8−10], and so on. However, there are only a few papers dealing with uncertain stochastic nonlinear systems using NN or fuzzy systems [11−13]. Reference Manuscript received January 18, 2010. *Corresponding author. This work was supported by the National Natural Science Foundation of China (60704013) and the Special Foundation of East China University of Science and Technology for Youth Teacher (YH0157134).

[11] developed an adaptive neural control schemes to solve the output tracking control problem of uncertain nonlinear systems disturbed by unknown covariance noise. Based on Takagi-Sugeno (T-S) fuzzy model, [13] proposed a fuzzy adaptive control design scheme for a class of uncertain stochastic strict-feedback systems, in which the virtual control gain function sign is unknown. A common weakness of these control methods is that the number of adaptation laws depends on the number of the NN nodes or fuzzy rules. With increase of NN nodes or fuzzy rules, the parameters to be estimated will increase significantly. As a result, the on-line learning time becomes prohibitively large. Nonsmooth nonlinear characteristics such as dead-zone, backlash, hysteresis are common in actuators and sensors, such as mechanical connections, hydraulic actuators and electric servomotors. Dead-zone which can severely limit system performances is one of the most important nonsmooth nonlinearities in many industrial processes. Therefore, the study of dead-zone has been drawing much interest in the control community for a long time [14−16]. However, the stabilization of the stochastic nonlinear system with dead-zone input is dealt with by little paper. Reference [17] presented an adaptive fuzzy tracking control scheme for a class of stochastic nonaffine uncertain nonlinear systems with unknown dead-zone. In this paper, combining the ideas of backstepping design, NN parameterization, and NN control, a novel adaptive NN control scheme is presented, which can guarantee that all the error variables are semi-globally uniformly ultimately bounded with desired probability in a compact set. The proposed method extends the class of uncertain stochastic nonlinear strict-feedback systems with deadzone that can be handled using adaptive NN control techniques. In addition, the NN parameterization technique [18] is firstly applied to the design of adaptive control for uncertain stochastic nonlinear systems. In this way, param-

Zhaoxu Yu et al.: Adaptive neural control for a class of uncertain stochastic nonlinear systems with dead-zone

eters to be estimated will be greatly decreased. As a result, the on-line learning time will be dramatically decreased.

2. Preliminary results and problem formulation 2.1 Preliminary results Consider the following system: dx = f (t, x)dt + h(t, x)dω

(1)

where x ∈ Rn is the state, ω is r-dimensional standard Brownian motion, f and h are vector-value or matrix-value function with appropriate dimensions. We define the operator L known as infinitesimal generator for C 2 function V (t, x) as follows: ∂V ∂V 1 ∂2V L V (t, x) = + f + tr{hT 2 h} ∂t ∂x 2 ∂x

for the matrix A, A∞ = max |aij | denotes the infini,j

ity norm of matrix A ∈ Rm×n . In addition, throughout the letter, θˆi (i = 1, . . . , n) denotes the estimate for 1 θi , and define θ˜i := the unknown constant parameter bm 1 θˆi − θi ; λi and σi (i = 1, . . . , n) are positive conbm stants. μi (i = 1, . . . , n) are positive constants which will be specified later. 2.2 Problem formulation Consider the following uncertain stochastic nonlinear systems: Plant: xi ) + gi (¯ xi )xi+1 )dt + ψiT (¯ xi )dω, dxi = (fi (¯

(2)

1in−1 dxn = (fn (¯ xn ) + gn (¯ xn )u)dt + ψnT (¯ xn )dω,

where tr(A) is the trace of A. Definition 1 The trivial solution x(t) of system (1) is semi-globally bounded in probability, if for some ε: 0 < ε < 1, there exists d = d(t0 , x0 ) > 0 such that for all initial values of x0 ∈ Υ 0 (Υ0 is some compact set containing the origin), the trivial solution satisfies lim inf P {x(t)  d}  1 − ε.

t→∞

(3)

If d is independent of x0 , and for some T  0 such that inf P {x(t)  d}  1 − ε, ∀ t  t0 + T , then x(t) is semi-globally uniformly ultimately bounded in probability. Lemma 1 [11] Consider the stochastic nonlinear system (1). If there exists a positive definite, radially unbounded function V : Rn → R, and constants Ci > 0 (i = 1, 2), satisfying the following inequality L V (t, x)  −C1 V (t, x) + C2 ,

(4)

then (i) the system has a unique solution surely and (ii) the system is bounded in probability. Lemma 2

Consider the following inequality ˙ θˆ  −λθˆ

where λ is a positive constant. If the initial condition ˆ  0 holds, then θ(t) ˆ  0 for all t  0. θ(0) Remark 1 In fact, it is always reasonable to choose ˆ θ(0)  0 in practical situation, as θˆ is an estimate of the unknown positive constant θ. The conclusion will be used in each design step. For a clearer explanation, some notations will be simplified, and the argument of some function √ is omitted. For example, fi represents fi (·), A = AT A denotes the Euclidean norm for vector or the induced Euclidean norm

501

(5)

Dead-zones:

⎧ ⎨ pr (v), v  br u = D(v) = 0, bl < v < br ⎩ pl (v), v  bl

(6)

where xi ∈ R (i = 1, . . . , n) is the state of the system, and x ¯i = [x1 , . . . , xi ]T , x = [x1 , . . . , xn ]T ; ω is r-dimensional standard Brownian motion defined on the complete probability space (Ω , F, P ) with Ω being a sample space, F being a σ-field, and P being the probability measure. fi (·), gi (·) : Ri → R, ψiT (·) : Ri → Rr for i = 1, . . . , n are unknown nonlinear smooth functions with fi (0) = 0, ψiT (0) = 0. u ∈ R is the output of the dead-zone, v ∈ R is the input of the dead-zone, bl and br are the unknown parameters of the unknown dead-zone. The control objective is to design an adaptive state feedback control law v for system (5) such that for all initial ˆ 0 ) in some compact set, all the sigconditions x(t0 ), θ(t nals in the closed-loop system are semi-globally uniformly ultimately bounded in probability as t → ∞. Assumption 1 For i = 1, . . . , n, nonlinear functions xi ) are unknown, but their signs are known, and there gi (¯ exist positive constants bm and bM such that 0 < bm  xi )|  bM < ∞, ∀ x¯i ∈ Ri . Without loss of general|gi (¯ ity, it is further assumed that 0 < bm  gi (¯ xi )  bM < ∞ , for i = 1, . . . , n. Remark 2 bm and bM are only used for stability analysis, and will not be used for designing the adaptive NN control law. Thus, bm and bM can be unknown constants. Assumption 2 For 1  i  n, there exist nonnegative unknown smooth functions φi such that ψi (¯ xi )  i  |xk |φi (¯ xi ).

k=1

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Journal of Systems Engineering and Electronics Vol. 22, No. 3, June 2011

Remark 3 It seems that Assumption 2 is a restrictive growth condition. However, Assumption 2 does not require smooth functions φi to be known. For this reason, Assumption 2 can be seen as a generalization of the feedback linearizable condition. Assumption 3 able.

The dead-zone output, u, is not avail-

Assumption 4 The dead-zone parameters br and bl are unknown bounded constants, but their signs are known, i.e. br > 0 and bl < 0. Assumption 5 The functions pl (v) and pr (v) are smooth, and there exist unknown positive constants ql0 , ql1 , qr0 and qr1 such that 0 < ql0  pl (v)  ql1 , ∀ v ∈ (−∞, bl ] 0 < qr0  pr (v)  qr1 , ∀ v ∈ [br , +∞ ) and β0  min{ql0 , qr0 } is a known positive constant,   dpl (s)  dpr (s)    where pl (v) = and pr (v) = . ds s=v ds s=v Based on Assumption 5, the dead-zone (6) can be written as follows (it is also shown in [16]): u = D(v) = Θ T (t)Φ(t)v + d(v)

si (Z) = exp[

where μi = [μi1 , . . . , μin ]T is the center of the receptive field and νi is the width of the Gaussian function. By choosing enough nodes, NN can approximate any continuous function over a compact region ΥNN ⊂ Rn with arbitrary accuracy, namely, f (Z) = W ∗T Φ(Z) + δ,

Φ(t) = [ϕr (t), ϕl (t)]T , Θ (t) = [qr (v), ql (v)]T 

 1, v(t) > bl 1, v(t) < br , ϕl (t) = 0, v(t)  bl 0, v(t)  br  0, v  bl qr (v) = pr (ξr (v)), bl < v < +∞   pl (ξl (v)), −∞ < v < br ql (v) = 0, v  br ⎧ ⎨ −pr (ξr (v))br d(v) = −[pl (ξl (v)) + pr (ξr (v))]v ⎩ −pl (ξl (v))

ϕr (t) =

and |d(v)|  p∗ , p∗ is an unknown positive constant with p∗ = (qr1 + ql1 ) max{br , −bl }. 2.3 NN and its parameterization Radial basis function (RBF) NN is often used in practical control engineering due to its simple structure and nice approximation properties. In this letter, a continuous function f (·) : Rn → R will be approximated by the Gaussian ˆ T Φ(Z), where Z ∈ ΥNN ⊂ RBF NN. Namely, fˆ = W n ˆ = [w1 , . . . , wl ]T ∈ Rl R is the input vector, W is the weight vector and the kernel vector is Φ(Z) = [s1 (Z), s2 (Z), . . . , sl (Z)]T with active function si (Z) being chosen as the commonly used Gaussian functions:

∀Z ∈ ΥNN .

(8)

The ideal weight vector W ∗ is an “artificial” quantity re∗ quired for analytical  purposes. It is defined as W :=   arg min { sup f (Z) − fˆ(Z)}. ˆ ∈Rn Z∈ΥNN W

Assumption 6 For ∀Z ∈ ΥNN , there exists an ideal constant weight vector W ∗ such that W ∗ ∞  wmax and |δ|  δmax with bounds wmax , δmax > 0. From (8), it is easy to obtain   W ∗T Φ(Z) + δ  W ∗T Φ(Z) + |δ|  l 

|sm (Z)|wmax + δmax  θβ(Z)

m=1



(7)

where

− Z − μi 2 ], i = 1, 2, . . . , l νi

where β(Z) =

(l + 1)(

l 

m=1

(9)

s2m (Z) + 1), and θ =

max{δmax , wmax }. Remark 4 Based on the technique used in [18], an upper bound for the NN parameterization can be obtained, such that there is only one positive parameter to be adapted.

3. Controller design and stability analysis 3.1 Adaptive controller design In this section, we use the backstepping method to design an adaptive state feedback controller. To introduce the design of controller, we need to make the following coordinate transformation: z1 = x1 ,

zi = xi − αi−1 ,

i = 2, . . . , n

where αi−1 is the virtual control signal, which will be specified later. Step1

It is easy to be obtained that dz1 = (f1 + g1 x2 )dt + ψ1T dω.

(10)

Consider the following Lyapunov function: V1 =

1 4 1 ˜2 z + bm λ−1 1 θ1 . 4 1 2

(11)

From (2), we obtain 1 T ˜ ˜˙ L V1 = z13 (f1 + g1 x2 ) + bm λ−1 1 θ1 θ 1 + ψ1 ψ1 . (12) 2

Zhaoxu Yu et al.: Adaptive neural control for a class of uncertain stochastic nonlinear systems with dead-zone

503

From Assumption 2, the following inequality can be obtained: 3 3 2 T z1 ψ1 ψ1  z14 φ21 . (13) 2 2 Then, by (13), we have

From (19)−(22), and using the inequality θ˜θˆ  12 θ˜2 − 1 2 2 θ , it is easy to show that

˜ ˜˙ 3 4 2 L V1  z13 (f1 +g1 (z2 +α1 ))+bm λ−1 1 θ1 θ 1 + z1 φ1 . (14) 2 3 Define a new function f¯1 as f¯1 = f1 + z1 φ21 . Since 2 f1 and φ1 are unknown, f¯1 can not be directly implemented to construct the virtual controller α1 . Thus according to the RBF NN approximation property, f¯1 can be approximated by a RBF NN on the compact set ΥZ1 , f¯1 = W1∗T S1 (x1 ) + δ1 (x1 ), where W1∗T S1 (x1 ) represents the ‘ideal’ NN approximation of f¯1 , and δ1 (x1 ) denotes the approximation error.   W1∗T S1 (x1 ) + δ1 (x1 )  W1∗T S1 (x1 ) + |δ1 (x1 )| 

where c1 := μ1 − 3/4 > 0 and κ1 := 0.278 5θ1 σ1 +

l1 

|s1m (x1 )| w1 max + δ1 max  θ1 β1 (·)

m=1



where β1 (x1 ) =



(l1 + 1)

l1 

m=1

s21m (x1 )

1

1

1

1

1 1

2

Choose the virtual control as

1



α1 = −μ1 z1 − θˆ1 β1 (·) tanh The adaptive law is chosen as ˙ θˆ1 = −λ1 θˆ1 + λ1 z13 β1 (·) tanh

β1 (·)z13 σ1



+ 1 , and

m 1

1 1

(16)

(17)

.

.

(18)

Under the virtual control law (17) and parameter update law (18), using Lemma 2 yields

β1 (·)z13 3 4 3 ˆ g1 z1 α1 = −μ1 g1 z1 − bm θ1 z1 β1 (·) tanh σ1 (19)

3 β (·)z 1 ˙ 1 −1 ˜ ˜ 3 bm λ θ1 θ 1 = −bm θ˜1 θˆ1 +bm θ˜1 z1 β1 (·) tanh . σ1 (20) And using the lemma provided by [19], we get

3  3 z1  θ1 β1 (·)−θ1 z13 β1 (·) tanh β1 (·)z1  0.278 5θ1 σ1 . σ1 (21) By applying Young’s inequality, it can be shown that 3 1 g1 z13 z2  g1 z14 + g1 z24 . 4 4

bm ˜2 1 θ1 + κ1 + g1 z24 2 4

(23) θ12 . 2bm

According to Itˆo formula, we

i−1  ∂αi−1 dzi = (fi + gi xi+1 − Lαi−1 )dt + (ψi − ψj )T dω ∂x j j=1

where Lαi−1 = i−1 

2

i−1  l=1

(24) i−1  ∂αi−1 ˆ˙ ∂αi−1 (fl + gl xl+1 )+ θ + ˆl l ∂xl l=1 ∂ θ

∂ αi−1 T 1 ψ ψq . 2 p,q=1 ∂xp ∂xq p Define the following Lyapunov function: 1 1 ˜2 Vi = Vi−1 + zi4 + bm λ−1 i θi . 4 2

(25)

From (2), we can obtain ˜ ˜˙ L Vi = L Vi−1 +zi3 (fi + gi xi+1 −Lαi−1 )+bm λ−1 i θi θ i + i−1 i−1   ∂αi−1 ∂αi−1 3 2 zi [ψi − ψj ]T [ψi − ψj ]. 2 ∂x ∂xj j j=1 j=1

(26)

Firstly, we estimate the last term in (26). By Assumption 2, there exists nonnegative unknown smooth function φ˜i such that   i−1 i−1      j i ψi − ∂αi−1 ψj  |xj | φi + ∂αi−1 |xk | φj =  ∂xj  ∂xj j=1 j=1 j=1 k=1

3

β1 (·)z1 σ1

Step i (2  i  n − 1) have

(15)

θ1 = max{δ1 max , w1 max }. Moreover, using RBF NN and its parameterization (15), we have   L V  z 3  θT β (x ) + g z 3 (z + α ) + b λ−1 θ˜ θ˜˙ . 1

L V1  −bm c1 z14 −

(22)

   ∂αi−1  φk  |xj |  ∂xk  j=1 j=1 k=j ⎧ ⎫  i i−1  i ⎨    ∂αi−1  ⎬    φk  |xj | φi + |xj | φi   ∂xk  ⎭ ⎩ i 

|xj | φi +

j=1

i−1  i−1 

j=1

k=j

i  j=1

|zj + αj−1 | φi 

i 

|zj | φ˜i .

(27)

j=1

From (27), we have 2 i−1 i  ∂αi−1 3 2  3 2 zi ψi − z ψj  ( |zj | φ˜i )2  i 2 ∂x 2 j j=1 j=1 i−1 3i 4 ˜2 3i 2 ˜2  zi φi + zi φi |zj |2  2 2 j=1

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Journal of Systems Engineering and Electronics Vol. 22, No. 3, June 2011 i−1 3i 4 ˜2 3 3 2 4 ˜4  2 zi φi + + i zi φi ( |zj | )2 . 2 4 4 j=1

(28)

Define a new function f¯i as i−1  gi−1 3i 3 2 f¯i = fi −Lαi−1 + zi + zi φ˜2i + i2 zi φ˜4i ( |zj | )2 . 4gi 2 4 j=1

f¯i can be approximated by an RBF NN on the compact set ΥZi , and can be parameterized   Wi∗T Si (·) + δi (xi )  Wi∗T Si (·) + |δi (xi )|  li 

|sim (·)| wi max + δi max  θi βi (·)

¯ where βi (¯ xi , θˆi−1 ) =

(li + 1)(

li 

m=1

(29)

¯ s2im (¯ xi , θˆi−1 ) + 1)

and θi = max{δi max , wi max }. Using RBF NN and its parameterization (29), it is easy to obtain   ¯ L Vi  L Vi−1 + zi3  θiT βi (¯ xi , θˆi−1 ) + gi zi3 (zi+1 + αi )− 3 1 ˜ ˜˙ gi−1 zi4 + bm λ−1 i θi θ i + . 4 4 Define the virtual control as

(30) 3

βi (·)zi αi = −μi zi − θˆi βi (·) tanh( ). σi The adaptive law is chosen as ˙ θˆi = −λi θˆi + λi zi3 βi (·) tanh



βi (·)zi3 σi

(31)

.

(32)

Under the virtual control law (31) and parameter adaptive law (32), using Lemma 2 yields gi zi3 αi = −μi gi zi4 − bm θˆi zi3 βi (·) tanh(

βi (·)zi3 ) (33) σi

3 1 4 gi zi3 zi+1  gi zi4 + gi zi+1 . 4 4 Substituting (33) and (34) into (30), we have i  k=1

bm ˜2 1 4 θk + κi + gi zi+1 −bm ck zk4 − 2 4

(34)

(35)

where ci := μi − 3/4 > 0 and κi := 3(i − 1)/4 + i  θ2 ( k + 0.278 5θk σk ). k=1 2bm Step n

3 1 ˜ ˜˙ gn−1 zn4 + bm λ−1 (36) n θn θ n + 4 4 ln  ¯ ¯ xn , θˆn−1 ) = (ln +1)( s2nm (¯ xn , θˆn−1 )+1) where βn (¯ m=1

and θn = max{δn max , wn max }. Choose the actual control v as follows: v=−

1 βn (·)zn3 (μn zn + θˆn βn (·) tanh( )), β0 σn

(37)

and the adaptive law is chosen as

m=1

L Vi 

  ¯ xn , θˆn−1 ) + gn zn3 u− L Vn  L Vn−1 + zn3  θnT βn (¯

Define the Lyapunov function as 1 1 ˜2 Vn = Vn−1 + zn4 + bm λ−1 n θn . 4 2

Similar to the design in Step i (2  i  n − 1), it is easy to obtain

3

βn (·)zn ˙ θˆn = −λn θˆn + zn3 βn (·) tanh( ). σn

(38)

Since Θ T (t)Φ(t)  β0 , we have gn zn3 u = gn zn3 (Θ T (t)Φ(t)(−

1 (μn zn + θˆn βn (·)· β0

βn (·)zn3 ))) + d(v))  σn βn zn3 3 bM ∗4 p . −gn μn zn4 − bm θˆn βn zn3 tanh( ) + gn zn4 + σn 4 4 (39) From (39), it holds true that tanh(

n 

bm ˜2 θ ) + κn  −ϑVn + κn 2 i i=1 (40) where cn := μn − 3/4 > 0, ςi := min{4bmci , λi }, n  θ2 ( i + 0.278 5θi σi ) + ϑ = min {ςi } and κn := 1in i=1 2bm bM ∗4 3 (n − 1) + p . 4 4 L Vn 

(−bm ci zi4 −

3.1 Stability analysis In this section, the stability analysis for the closed-loop stochastic system under the proposed control scheme will be derived. Theorem 1 For the uncertain stochastic nonlinear systems (5)−(6) satisfying Assumptions 1−5, if the control law and parameter adaptive adjusted law are respectively designed in (37) and (38), then for initial conditions xi (t0 ), zi (t0 ) and θˆi (t0 ) (i = 1, 2, . . . , n) starting in some compact set Υ0 , the following results hold: (i) The error signals zi and θ˜i remain in a compact set Υzi and Υθ˜i respectively , which are defined by 4

Υzi := { zi ∈ R| E[|zi | ]  4Ω }, i = 1, 2, . . . , n (41)     2λ   i ˜ Υθ˜i := { θi ∈ R θ˜i   Ω }, i = 1, 2, . . . , n bm (42)

Zhaoxu Yu et al.: Adaptive neural control for a class of uncertain stochastic nonlinear systems with dead-zone

κn . ϑ (ii) Moreover, the error signals zi and θ˜i converge to the following steady state compact sets respectively, as t → ∞, κn 4 Υzi∞ := { zi ∈ R| E[|zi | ]  4 }, i = 1, 2, . . . , n ϑ (43)     2λ κ    i n }, i = 1, 2, . . . , n. Υθ˜i∞ := { θ˜i ∈ R θ˜i   bm ϑ (44) (iii) All the signals in the closed-loop systems are semiglobally uniformly ultimately bounded in probability.

where Ω := Vn (t0 ) +

Proof Consider the Lyapunov function candidate V = Vn , and define a compact set  ¯  4 Υz := { [¯ zn , θ˜n ] E(|zi | )  4Ω ,    2λ ˜  i Ω , i = 1, 2, . . . , n}. θi   bm Let τε be the first exit time of the state variables’ trajectories from Υz . From (40) and Lemma 1, it is easy to get that the system is bounded in probability in Υz . Namely ∀ ε > 0, inf P {τε = ∞}  1 − ε. Using Dynkin’s formula [20], we have  τε (t) E[V (τε (t))] − V (t0 ) = E{ L V (s)ds} (45) t0

505

It is easy to get β0 = 0.5. By Theorem 1, the virtual control law, the actual control law and the adaptive laws are chosen, respectively, as α1 = −μ1 x1 − θˆ1 β1 (·) tanh(β1 (·)x31 /σ1 ) v=−

1 (μ2 z2 + θˆ2 β2 (·) tanh(β2 (·)z23 /σ2 )) β0

˙ θˆi = −λi θˆi + λi zi3 βi (·) tanh(zi3 βi (·)/σi ),

i = 1, 2

where z1 = x1 , z2 = x2 − α1 , β1 (·) = β1 (x1 ) and β2 (·) = β2 (x1 , x2 , θˆ1 ). In this work, one RBF NN is taken for each unknown function. In particular, W1∗T S1 (x1 ) contains 3 nodes with centers spaced in [−5, 5]; W2∗T S2 (x1 , x2 , θˆ1 ) contains 27 nodes with centers evenly spaced in [−5, 5] × [−5, 5] × [0, 10]; and all the widths are chosen as vi = 1 (i = 1, 2, 3). The initial conditions are given by [x1 (0), x2 (0)]T = [−0.2, 0.5]T and [θˆ1 (0), θˆ2 (0)]T = [3, 5]T . Design parameters are taken as follows: μ1 = 10, μ2 = 15, λ1 = λ2 = 10 and σ1 = σ2 = 1. Simulation results are shown in Figs. 1−3. Fig. 1 verifies the effectiveness of the proposed adaptive NN controller. For the selected set of parameters, the control input is shown in Fig. 2. And Fig. 3 exhibits adaptive parameters curves under the control action of our method. From Figs. 1−3, it can clearly verify the bounded performance and the effectiveness of the proposed design.

where τε (t) = min{t, τε }, and applying the mean value operator on (45), we get t E[V (t)]  e−ϑ(t−t0 ) V (t0 ) + κn e−ϑ(t−τ ) dτ  t0

κn e−ϑ(t−t0 ) V (t0 ) + , ∀ t ∈ [t0 , τε ] . (46) ϑ Furthermore, it is easy to see that as t → ∞, we have κn 4 E[|zi | ]  4E[V (t)]  4 ϑ 2λ 2λ i i κn . E[θ˜i2 ]  E[V (t)]  bm bm ϑ From the above discussion, we can conclude that all the closed-loop signals are semi-globally uniformly ultimately bounded in probability.

Fig. 1

States of closed-loop system x1 (t) and x2 (t)

4. Simulation examples In order to demonstrate the effectiveness of our result, consider the following stochastic nonlinear system with deadzone:  dx1 = (x1 e0.5x1 + (1 + x21 )x2 )dt + x31 dω dx2 = (x1 x22 + (3 + cos(x1 ))u)dt + (x1 + x2 ex2 )dω (47) ⎧ ⎨ 0.5(v − 2.5), v  2.5 where u = D(v) = 0, −1.5 < v < 2.5 . ⎩ 0.8(v + 1.5), v  −1.5

Fig. 2

Control input v(t)

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Fig. 3

ˆ1 and θ ˆ2 Adaptive parameters θ

5. Conclusion Adaptive NN controller design is presented to guarantee the boundedness in probability for the solution of the uncertain stochastic nonlinear system with both unknown dead-zone and unknown gain function. The proposed controller has the following advantages: (i) the controller can become simpler than the backstepping controller which requires the repeated differentiations of virtual controllers; (ii) few tuning parameters are required due to minimal learning parameterization; (iii) the steady-state errors can be made arbitrarily small by adjusting the design parameters; and (iv) the mathematical model of the system which considers the unknown dead-zone as nonlinear input is not required to be known accurately, such that the controller design is less dependent on the system model. Hence, the application of the controller is extensive in the many industrial processes. The given theoretical and simulation results clearly verify the effectiveness of the proposed design.

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Biographies Zhaoxu Yu received the B.S. degree in mathematics from Jiangxi Normal University in 1998, the M.S. degree in applied mathematics from Tongji University in 2001 and the Ph.D. degree in control science and engineering from Shanghai Jiaotong University in 2005. He is currently a lecturer in the Department of Automation in East China University of Science and Technology. His research interest includes nonlinear control, adaptive control and stochastic system. E-mail: [email protected] Hongbin Du received the B.S., M.S., and Ph.D. degrees from Dalian University of Technology, China, in 1996, 1999, and 2002, respectively. He was with the Department of Automation of Shanghai Jiaotong University as a postdoctor from 2002 to 2004, and with the Department of Electrical and Computer Engineering, National University of Singapore as a research fellow in 2005 and 2006. Since 2005, he joined in the Department of Automation, East China University of Science and Technology, Shanghai as an associate professor. His research interests include process modelling, intelligent nonlinear control and embedded systems application. E-mail: ben [email protected]