Robust neural tracking control for switched nonaffine stochastic nonlinear systems with unknown control directions and backlash-like hysteresis

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Robust Neural Tracking Control for Switched Nonaffine Stochastic Nonlinear Systems with Unknown Control Directions and Backlash-like Hysteresis Yanjun Shu , Yanhui Tong , Chaogang Yu PII: DOI: Reference:

S0016-0032(19)30898-1 https://doi.org/10.1016/j.jfranklin.2019.12.011 FI 4328

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

21 June 2019 15 October 2019 4 December 2019

Please cite this article as: Yanjun Shu , Yanhui Tong , Chaogang Yu , Robust Neural Tracking Control for Switched Nonaffine Stochastic Nonlinear Systems with Unknown Control Directions and Backlash-like Hysteresis, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.12.011

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Robust Neural Tracking Control for Switched Nonaffine Stochastic Nonlinear Systems with Unknown Control Directions and Backlash-like Hysteresis Yanjun Shu*, Yanhui Tong, Chaogang Yu, Shanghai University of Engineering and Science, 201620, Shanghai [email protected]

Abstract This paper is concerned with the tracking control problem for a class of switched stochastic nonlinear systems in nonaffine form with both unknown control directions and unknown backlash-like hysteresis, and a novel neural tracking control scheme is proposed based on backstepping technique and Nussbaum function. Dynamic surface control(DSC) is adopted to overcome the problem of complexity explosion of the traditional backstepping design. High-order neural networks (HONNs) are utilized to approximate the lumped unknown nonlinear functions, and only one adaptive parameter needs to be updated. Stability analysis shows all closed-loop error signals are semi-globally uniformly ultimately bounded in the fourth-moment(or mean square), and the system tracking error is ensured to converge to a small neighborhood of zero. Finally, simulation results illustrate the effectiveness of the proposed scheme.

Keyword Neural tracking control; Switched nonaffine stochastic systems; Backlash-like hysteresis; DCS; Nussbaum function

1. Introduction During the past decades, because stochastic disturbance commonly exists in engineering practice, e.g., aircraft, biology, and many kinds of process control systems, stochastic nonlinear systems have drawn considerable attention and a series of prominent results have been obtained. The early research results usually require parametric linearization of the systems[1]-[7], such that they cannot tackle the stochastic nonlinear systems with completely unknown functions which are more common in practice. Thus, robust control methods based on neural networks(NNs)[8]-[12] or fuzzy logical systems(FLS)[13]-[20] are intensively developed. The overwhelming majority of the FLS/NNs for stochastic nonlinear systems are based on the backstepping technique, which has presented systematic approaches for the control synthesis of general nonlinear systems with mismatched uncertainties. Unfortunately, a main disadvantage of the backstepping design is the problem of complexity explosion arising from the repeated differentiation of virtual control laws. To eliminate this disadvantage of backstepping, DSC technique was first proposed in [21] for a class of strict-feedback nonlinear systems. By incorporating the DSC into backstepping framework, some adaptive neural or fuzzy controllers for stochastic nonlinear systems have been developed[22]-[28]. In [25]-[27], the DSC technique was employed to achieve the neural or fuzzy output-feedback control for stochastic nonlinear systems. It is worth mentioning that these control schemes are designed for strict-feedback or pure-feedback stochastic nonlinear systems in non-switching model. So far, to be sure, there is no report on the application of DSC for the tracking control of switched stochastic nonaffine nonlinear systems. The nonaffine systems have a more general representation than the other nonlinear systems. Compared with the strict-feedback or nonstrict-feedback systems, the nonaffine systems have no appearance of the variables being used as virtual controls or actual control, and compared with the

pure-feedback systems, the nonaffine systems are of nonlower triangular structure, which make the control design for nonaffine systems more challenging and meaningful. However, the present results are restricted to deterministic nonaffine systems or stochastic pure-feedback systems [29]-[36], and there is few achievement concerned with the switched stochastic nonaffine systems. Very recently, Wang [37] extended the fuzzy-based backstepping tracking control method to nonaffine stochastic nonlinear switched systems. However, the above control designs did not take the problem of complexity explosion into consideration, and did not involve the unknown control directions or unknown backlash-like hysteresis, which are this paper attempt to solve. On the other hand, as a typical non-smooth nonlinear input, backlash-like hysteresis widely exists in physical systems such as actuator and sensor, which cause system performance deteriorates severely and even instability[38]. Therefore, the control problem of nonlinear systems involving backlash-like hysteresis has attracted much attention for a long time. A few adaptive controllers have been developed for various deterministic nonlinear systems with unknown backlash-like hysteresis[39]-[41]. However, the adaptive control design for stochastic nonlinear systems with hysteresis nonlinearity has received relatively less attention[42]-[45]. Adaptive neural control schemes for nonstrict-feedback stochastic nonlinear systems with backlash-like hysteresis are proposed in [42][43]. An adaptive tracking control for switched stochastic nonlinear pure-feedback systems with backlash-like hysteresis is developed in [44]. Unfortunately, in the aforementioned results, the complexity explosion and many adaptive parameters occurred as the order of the system increases. An adaptive neural DSC control scheme is proposed for a class of pure-feedback stochastic nonlinear systems with backlash-like hysteresis in [45]. Nevertheless, the model of [45] is non-switching, and don‘t involve the problem of unknown control directions. Moreover, the control directions, defined as the signs of the control gains, usually cannot be known in a prior in many practical systems, which make the control design more difficult since we cannot decide the direction along which the control operates. It is also a major source of instability. The Nussbaum gain function is frequently utilized to overcome the problem of unknown control direction, and plentiful results have been available for deterministic nonlinear systems [46]-[51]. However, there are relatively less results on stochastic nonlinear systems with unknown control directions[43][53]-[60]. Some adaptive neural output feedback control schemes have been developed in [43][53]-[57]. It is worth to mention that these output feedback control method are designed only for stabilization problem of non-switching stochastic nonlinear systems, and unable to deal with the tracking control problem of switched stochastic nonlinear systems. Few results are available on the state feedback control design for the tracking control problem of stochastic systems with unknown control directions[58]-[60]. An adaptive fuzzy control scheme is studied for a class of stochastic strict-feedback nonlinear systems with unknown control directions in [58]. In [59][60], some neural tracking control methods have been developed for stochastic pure-feedback nonlinear systems with unknown control directions. However, the above results do not involve the unknown backlash-like hysteresis. To the best of author‘s knowledge, Up to now, there has been no any result for stochastic nonaffine nonlinear switched systems with both unknown control directions and backlash-like hysteresis, which motivates us to carry out the present work. Inspired by the preceding observation, this paper investigates the robust neural tracking control problem for a class of switched stochastic nonaffine nonlinear systems. Compared with the published results, the main contributions of this paper can be summarized as follows: (1) The

tracking control problem of switched stochastic nonaffine nonlinear systems with both unknown control directions and backlash-like hysteresis is first investigated and solved by combining neural networks, backstepping design and Nussbaum gain function. (2) The problem of complexity explosion is first addressed for the considered system by DSC technique. Compared with [36][58]-[60], the process of controller design is significantly simplified and no more parameters occur. (3) Compared with the stochastic pure-feedback systems in non-switching model[45], the considered system is a more general form in completely nonaffine form and nonlower triangular structure, and additionally both the unknown backlash-like hysteresis and unknown control directions are solved. Both stability analysis and simulation example show that the proposed control scheme guarantees all the error signals of the considered system are semi-globally uniformly ultimately bounded(SGUUB) in the sense of 4th-moment(or mean square), and the tracking error converges to an arbitrarily small neighborhood of origin by choosing the parameters appropriately. The organization of the paper is as follows. The problem formulation and preliminaries are stated in section 2. Section 3 presents the main results, and section 4 demonstrates an illustrative example. Finally, the conclusions are given in section 5 . Notation: In this paper, the notation used is quite standard. TrX  denotes the trace of square matrix X . The notation



refers to the Euclidean vector norm. R

n

stands for the

n-dimensional Euclidean space. The superscript ‗T‘ denotes matrix (vector) transposition. C i stands for the set of all functions with continuous ith partial derivative. E[V ] represents the expectation of stochastic variable V .

2. Problem formulation and preliminaries 2.1 Problem formulation Consider the following stochastic nonaffine nonlinear switched system:

 dxi  f i , (t ) ( xi , xi 1 )dt  iT, (t ) ( x)dw,   1  i  n  1,  (1)   T dx  f ( x , u (  )) dt   ( x ) dw ,  n n , ( t ) n  (t ) n , ( t ) y  x , 1    T i n where xi  [ x1 , x2 ,, xi ]  R , i  1,2,..., n with x  xn  R is the system state vector, y(k )  R is the system output; w is an r-dimensional independent standard Brownian motion defined on a complete probability space[36][45];  (t ) : [0, )    {1,, m}, m  1 denotes switching signal, which is a piecewise constant function;    and i 1 n r 1  i  n , f i , () : R  R and i , () : R  R are unknown nonlinear functions which satisfy locally Lipschitz condition. u ( )  R is the control input of system(1) and meanwhile the output of the unknown backlash-like hysteresis given as

du d d (2)    (c  u )  B  , dt dt dt where  and B are designed constants, and c  0 is the slope satisfying c  B , and   is the input signal of the backlash-like hysteresis .

Fig.1. Backlash-like hysteresis curve

According to the analysis of [61], Eq.(2) can be solved into u (t )  c (t )  d ( ) , where d ( )  [u 0  c 0 ]e and

 (  0 ) sgn

(3)

 s

 e  sgn  [ B  c]e  sgn d with u 0  0  0

 0  0 are the initial value of u and   , respectively. d ( ) is bounded by an unknown 

constant D , that is

d ( )  D ,

(4)

Fig.1 depicts the curve generated by Eq.(2), where the parameters are chosen as   1 , c  3.1635 , B  0.345 and the input signal  (t )  6.5 sin(2.3t ) . The design objective of this paper is to develop a robust neural controller for the considered system(1) such that the system output y can track the reference signal y r , and all error signals of the closed-loop systems remain SGUUB. According to mean value theorem [64], it yields

where

f i , ()

   f i , ( xi , xi 1 )  f i , ( xi , xi01 )  g i , ( xi 1  xi01 ),  1  i  n  1,   f ( x , u )  f ( x , u 0 )  g (u  u 0 ), n , n  n ,    n , n   f i , ( xi , xi 1 ) is described between and

(5)

 f i , ( xi , xi01 )

,

xi ,  i , xi 1  (1  i , ) xi01 , i , is a certain constant satisfying 0  i ,  1,   gi , : gi , ( xi , xi , )  f i , ( xi , xi 1 ) xi 1  x  x , 1  i  n , xn1  u . i 1

i ,

Furthermore, by substituting (5) into (1), we have

 dxi  ( g i , xi 1  f i , ( xi ))dt  iT, ( x)dw,  1  i  n  1,    T dxn  ( g n , u  f n , ( xn ))dt  n , ( x)dw, y  x , 1 

Substituting (3) into (6), it yields

(6)

 dxi  ( g i , xi 1  f i , ( xi ))dt  iT, ( x)dw,  1  i  n  1,    T dxn  ( g n , c  g n , d ( )  f n , ( xn ))dt  n , ( x)dw, y  x , 1 

(7)

2.2 preliminaries For the ease of control design, the following assumptions and lemmas are presented. Assumption 1[59]: Assume gi ,  0, i  1,2,..., n  1 , and the sign of g n , is unknown, and there exist unknown scalars bl and br such that, for 1  i  n

 0  bl  gi ,  br   , xi 1  Ri 1 ,

(8)

Assumption 2[45]: The reference signal yr (t ) and its time derivatives up to the n-th order are continuous



and

bounded,

yr , y r , yr T  d

and



with

known

compact

set

d  yr , y r , yr  : yr2  y r2  yr2  B0  R3 , where the size B0 is a known positive T

constant. To raise some important preliminaries, we consider the following stochastic nonlinear system (9) dx  f ( x)dt  h( x)dw, x  R n , where x  R is system state, w is defined as (1), and f (x) and h(x) are locally Lipschitz functions with f (0)  h(0)  0 . n

Definition 1 [62]: Let V ( x)  C : R  R 2

n



be a positive function with respect to the system

(9). A differential operator  is defined as

V  where the term

 V 1   2V f ( x)  TrhT ( x) 2 h( x) , x 2  x 

(10)

 1  T  2V Trh ( x) 2 h( x) is called Itοˆ formula. 2  x 

Definition 2 [63]: If for some compact set   R

n

and any initial state x0  x(t0 ) , there exist

a constant   0 and a time constant T  T ( , x0 ) such that E[ x(t ) ]   , t  t0  T , p

then the state trajectory

x(t ), x  0

of system (9) is called SGUUB in pth moment. Especially,

when p  2 , it is said to be SGUUB in mean square. Definition 3 [43]: If any even differentiable function N ( ) : R  R has the following properties, then it is said to be a Nussbaum-type function :

1 s N ( )d   , s  s 0 1 s (2) lim inf  N ( )d   , s  s 0 (1) lim sup

Many even functions such as function. Lemma 1 [43]: Let

  2 cos( ) , exp( 2 ) cos(  ) can serve as a Nussbaum gain 2

 (t ), t [0, t f ) be a smooth function and N ( )   2 cos( ) be a

special Nussbaum-type function. Consider the stochastic nonlinear system (9), if there exists a positive definite, radially unbounded function V (t , x)  C 2 ([0, t f ), R n ) and constants ci  0 (i = 1, 2), satisfying the following inequality:

(13) V (t , x)  c1V (t , x)  ( g  N ( )  1)  c2 ,  where g is a nonzero constant, and then, E (V (t , x)) is bounded on [0, t f ) with t f   . Lemma 2[59]:   R and   0 , the inequality 0    tanh(  )  0.2785 holds. Lemma 3[62]: ( x, y)  R , the inequality 2

xy 

  0, p  1, q  1, and ( p  1)(q  1)  1 .

p l

| x |p 

1 | y |q holds, where q m

2.3 High-order neural networks(HONNs) The structure of HONN is expressed as: Γ W , z   W T S ( z ) , where W is the adjustable

synaptic weight vector and S ( z )  s1 ( z ), s2 ( z ),, si ( z )  R with T



si ( z )   s( z j )



where z  z1 , z2 ,..., zq

I , I ,..., I  1

2



 

jI i

T

z



d j (i )

N

is the basis function vector

, i  1,2,...,  ,

(14)

 R q is the input vector;   1 is the NN node number;

is a collection of  not-ordered subsets of

1,2,..., m

and d j ( j ) are

nonnegative integers; s ( z j ) is a hyperbolic tangent function z

z

s( z j )  (e j  e j ) (e j  e j ) , (15) For any unknown smooth function  (z ) , there exists an ideal NN such that  (z ) can be z

z

approximated with arbitrary accuracy as

 ( z)  W T S ( z )   ( z ) ,

(16)

where  (z ) is called the NN approximation error, and the ideal constant weight vector W is 

  q expressed as W   arg min sup  ( z )  W T S ( z )  ,  z  R . W R  z  z  Assumption 3[65]: There exist constants max  0 and

W



 max and

 max  0 , such that

 ( z )   max ,  z  R q . From (16), it is easily obtained that 

W T S ( z )   ( z )  W T S ( z )   ( z )   si ( Z )max   max   ( Z ) . i 1



where  ( Z )  (  1)



s

 i 1

2 i

 ( Z )  1 and   maxmax ,  max . 

3. Main result 3.1 controller design In this section, a neural tracking control scheme based on the DSC technique is proposed, where the intermediate control functions, the parameters adaptive laws, and the actual controller will be given in the following n-step backstepping design, respectively. The main results are based on the coordinate change:

z1  x1  yr , zi  xi   if , i  2,3,, n , where  if will be specified later. Define an adaptive parameter as

(17)

   : max  i , i  1,2,, n ,  bl

(18)



where i will be specified during the backstepping procedure. Let ˆ denotes the estimation of

~  , and     ˆ is the estimate error.





For the sake of clarity, we denote f i , ( xi ) , g1, ( xi ) and

i , ( x) by f i , , g1, and i ,

respectively in this section. Step 1: Based on z1  x1  yr , its differential is

dz1  ( g1, x2  f1,  y r )dt  1T, dw ,

(19)

Consider the following stochastic Lyapunov function candidate:

Vz1 

1 4 bl ~ 2 z1   , 4 2

(20)

It can be obtained from (10), (19) and (20) that

LVz1  z13 ( g1, x2  f1,  y r ) 

3 2 T ~ z1 1,1,  bl ˆ , 2

(21)

From Lemma 3, it yields

3 2 T 3 z1 1,1,  z12 1, 2 2

2



3 4 z1 1, 4l12

4

3  l12 , 4

(22)

Substituting (22) into (21) results in

3 3 ~ Vzi  z13 ( g1, x2  F1, ( Z1 ))  z14  l12  bl ˆ , 2 4 4 3 3 T T where F1, ( Z1 )  f1,  y r  2 z1 1,  z1 , with Z1  [ x , yr , y r ] . 4l 2

(23)

It is easy to get that

z13 F1, (Z1 )  z13 F1 (Z1 ) ,    ,

(24)

where F1 ( Z1 )  max  {| F1, ( Z1 ) |} . Since the function F1, ( Z1 ) contains the unknown functions f1, ( x1 ) and

1, , the prior

knowledge of F1 ( Z1 ) can not be available as well. Hence, there is a neural network T

W1 S1 ( Z1 ) satisfying F1 (Z1 )  W1 S1 (Z1 )  1 (Z1 ) , T

where 1 ( Z1 ) is the approximation error. Thus, using Assumption 3, Eq.(18) and Lemma 2, we obtain



z13 F1 ( Z1 )  z13 W1 S1 ( Z1 )   1 ( Z1 ) T

(25)



T  z13  W1 S1 ( Z1 )   1 ( Z1 )   

 z13 11 ( Z1 )

(26)

 z13 1    0.2785bla1 ,  z b 1 tanh a  1  where a1 is a positive parameter, and 1  max1max , 1max  with 1max and  1max being 3 1 l



define as the bound of W1



and

 1 respectively. As a result, only one adaptive parameter is

required to be updated, no matter how many neural networks are used and the order of the system. Substituting (26) into (23), we have

Vz1  z13 ( g1, z2  g1, 2 f  g1,1 )  z13 g1,1  z 3  ~ 3  z13bl1 tanh 1 1   bl ˆ  z14  1 , 2  a1  3 2 where 1  0.2785bla1  l1 . 4 Construct

(27)

1 as

 z13 1  ˆ  , 1  k1 z1  1 tanh a  1 

(28)

where k1 is positive constant.

 z13 1    0 , substitute (28) into (27), it obtains  a1 

Note that z1 ˆ1 tanh 3

Vz1  z13 ( g1, z2  g1, 2 f  g1,1 )  k1 g1, z14  z 3   3 ~  bl ( z13 1 tanh 1 1   ˆ)  z14  1 , 2  a1 

(29)

1 is passed through a first-order filter with a time constant  2 to get a new state variable  2 f , such that the repeatedly differentiating 1 is avoided. The operation is as  2 2 f   2 f  1 ,  2 f (0)  1 (0) , (30) In this paper,

Let

 2   2 f  1 be the output error of this filter, and we have  2 f  

1

2

 2 , thus

 1   d 2  d ( 2 f  1 )     2  B2 ( zn ,  2 ,ˆ, yr , y r , yr ) dt  C2 ( z1 ,ˆ, yr , y r , yr )dw , (31)  2  2 ˆ 3z  1 3  where B2 ()  1 2 ( f1,  y r )  ( z13ˆ  3z1ˆ1T,1, ) , C2 ()  2 z12ˆ1T, . 2 2a1 2a1 2a1 Then, it follows that

 z 3   3 ~ Vz1  z13 g1, z2  z13 g1, 2  k1 g1, z14  bl ( z13 1 tanh 1 1   ˆ)  z14  1 , 2  a1 

(32)

From Lemma 3 and Assumption 1, it yields

3 4 1 4 4 z1  br z2 , 4 4 3 1 z13 g1, 2  z14  br4 24 , 4 4 z13 g1, z2 

(33) (34)

Substituting (33) and (34) into (32), it obtains

 z 3   1 1 ~ Vz1  br4 z24  br4 24  k1 g1, z14  bl ( z13 1 tanh 1 1   ˆ)  1 , 4 4  a1  Step i ( 2  i  n  1 ): According to zi  xi   if and Itοˆ formula, we obtain

(35)

T

dzi  ( g i , xi 1  f i ,

i 1      if )dt   iT,   i 1  Tj,  dw ,   j 1 x j  

Consider the following stochastic Lyapunov function candidate:

(36)

Vz i 

1 4 zi , 4

(37)

It can be obtained easily from (10) and (36) that

Vzi  z ( g i , xi 1  f i , 3 i

i 1   3    if )  zi2  i ,   i 1  j ,   2  j 1 x j 

T

i 1    i ,    i 1  j ,  ,   j 1 x j  

(38)

From Lemma 3, it yields i 1   3 2  zi i ,   i 1  j ,   2  j 1 x j 



T

 3 4 z i ,   i 1  j , 2 i 4li j 1 x j i 1

i 1    i ,    i 1  j ,    j 1 x j  

(39)

4

3  li2 , 4

Substituting (39) into (38) results in

3 3 Vzi  zi3 ( gi , xi 1  Fi , ( Z i ))  zi4  li2 , 2 4 where Fi , ( Z i )  f i ,

i 1  3   if  2 zi iT,   i 1  Tj, 4l j 1 x j

4



(40)

3  zi , with Z i  [ xT , yr(i ) ,ˆ]T , 2

 yr(i )  [ yr , y r ,..., yr(i ) ] . It is obvious that

zi3 Fi , (Z i )  zi3 Fi (Z i ) ,    , where Fi ( Z i )  max  {| Fi , ( Z i ) |} .

(41)



Since the function Fi , ( Z i ) contains the unknown functions f i , ( xi ) and

i , , the prior

knowledge of Fi ( Z i ) can not be available as well. Hence, there is a neural network T

Wi  Si ( Z i ) satisfying Fi (Z i )  Wi  Si (Z i )   i (Z i ) , T

where  i ( Z i ) is the approximation error. Considering Assumption 3, Lemma 2 and (18) results in



zi3 Fi ( Z i )  zi3 Wi  Si ( Z i )   i ( Z i ) T

(42)



T  zi3  Wi  Si ( Z i )   i ( Z i )   

 zi3 i  i ( Z i )

(43)

 z 3   zi3bl i tanh i i   0.2785blai ,  ai  where ai is a design parameter, and i  maxi max ,  i max  with i max and  i max being define as the bound of Wi

 

and

 i respectively.

Substituting (43) into (40), it yields

 z 3  3 Vi  zi3 ( gi , zi 1  gi , i 1 f  g i , i )  zi3 gi , i  zi3bl i tanh i i   zi4  i , (44)  ai  2 3 2 where  i  0.2785blai  li . 4

Construct  i as

 zi3  i   ,  ai 

 i  ki zi  ˆ i tanh

(45)

 zi3  i  3 ˆ    0 , substitute (45) into (44), it obtains z   tanh Note that i i a  i   z 3  3 ~ Vzi  zi3 g i , zi 1  zi3 gi , i 1 f  zi3 g i , i  ki gi , zi4  zi3bl  i tanh i i   zi4  i , (46)  ai  2 where k i is positive constant. Similarly to step 1, let  i pass through a first-order filter with time constant  i 1 to get a new filtered virtual control law  i 1 f as

 i 1 i 1 f   i 1 f   i ,  i 1 f (0)   i (0) , Define output error

(47)

 i 1   i 1 f   i ,

Combining (47) and (48), we have  i 1 f  

1

 i 1

(48)

 i 1 , thus

d i 1  d ( i 1 f   i ) (49)   1        i 1  Bi 1 ( zn ,  i 1 ,ˆ, yr , y r , yr ) dt  Ci 1 ( zi ,  i 1 ,ˆ, yr , y r , yr )dw,   i 1  2 ˆ 3z  1 3  where Bi 1 ()  i 2 ( f1,   if )  ( zi3ˆ  3ziˆiT,i , ) , Ci 1 ()  2 zi2ˆiT, . 2 2ai 2ai 2ai Then, it follows that

 zi3  i  3 4   zi  i , Vzi  z g i , zi 1  z gi , i 1  ki g z  z b   i tanh a  i  2 3 i

3 i

4 i , i

3 i l

~

(50)

From Lemma 3 and Assumption 1, it yields

3 4 1 4 4 zi  br zi 1 , 4 4 3 1 zi3 g i , i 1  zi4  br4 i41 , 4 4 zi3 g i , zi 1 

(51) (52)

Substitute (51) and (52) into (50), it obtains

 zi3  i  1 4 4 1 4 4 4 3 ~   i , Vzi  br zi 1  br  i 1  ki gi , zi  zi bl  i tanh 4 4 a  i 

(53)

Step n: The actual control law is constructed in this final step. According to zn  xn   nf and

Itοˆ formula, we obtain T

n 1     dzn  ( g n, c  g n, d ( )  f n, ( xi )   nf )dt   nT,   n1  Tj,  dw , j 1 xn  

(54)

Consider the following stochastic Lyapunov function candidate:

Vz n 

1 4 zn , 4

It can be obtained easily from (10) and (54) that

(55)

Vzn  zn3 ( g n , c  g n , d ( )  f n ,   nf ) n 1   3   zn2  n ,   n 1  j ,   2  j 1 x j 

T

n 1    n ,    n 1  j , ,   j 1 x j  

(56)

From Lemma 3, it yields n 1   3 2  z n n ,   n 1  j ,   2  j 1 x j 



T

 3 4 z n ,   n 1  j , 2 n 4ln j 1 x j n 1

n 1    n ,    n 1  j ,    j 1 x j  

(57)

4

3  ln2 , 4

Substituting (57) into (56) results in

3 Vzn  zn3 ( g n, c  g n, d ( )  Fn, ( Z n ))  ln2 , 4

(58)

4

where Fn , ( Z n )  f n ,   n , f

 yr( n)  [ yr , y r ,..., yr( n) ] .

n 1  3   2 zn n,   n1  j , , with Z n  [ xT , yr( n ) ,ˆ]T , 4l j 1 x j

It is easy to get that

zn3 Fn, (Z n )  zn3 Fn ( Z n ) ,    ,





(59)

where Fn ( Z n )  max  Fn, ( Z n ) . Since the function Fn, ( Z n ) contains the unknown

 functions f n , ( xn ) and n , , the prior knowledge of Fn ( Z n ) can not be available as well. T

Hence, there is a neural network Wn S n ( Z n ) satisfying

Fn (Z n )  Wn S n (Z n )   n (Z n ) , T

where  n ( Z n ) is the approximation error. Considering Assumption 3, Lemma 2 and (18) results in



z n3 Fn ( Z n )  z n3 Wn S n ( Z n )   n ( Z n ) T

(60)



T  z n3  Wn S n ( Z n )   n ( Z n )   

 z n3 n  n ( Z n )

(61)

 z 3   z n3bl n tanh n n   0.2785blan ,  an  where an is a design parameter, and n  maxn max ,  n max  with n max and  n max being 

define as the bound of Wn



and

 n respectively.

Substituting (61) into (58), it yields

 z 3  3 Vzn  zn3 ( g n, c  g n, d ( ))  zn3bl n tanh n n   0.2785blan  ln2 , 4  an 

(62)

Due to

zn3 g n, d ( )  Hence, construct the actual control law as

3 4 1 4 4 zn  br D , 4 4

(63)



1 c

 zn3  n     ,  a  n 

  N ( ) kn zn  ˆ n tanh 

(64)

 zn3  n  4 3 ˆ   ,   kn zn  zn n t a n h a  n 

(65)

where  is a positive parameter to adjust the Nussbaum gain N ( ) , and k n is positive constant. Then we have

  z 3   zn3 g n, c  g n, N ( ) kn zn4  zn3ˆ n tanh n n    g n, N ( ) ,  an   

(66)

Substituting (63) and (66) into (62), it obtains

 zn3  n  3 4 3~     n , Vzn  (kn  ) zn  g n, N ( )  1  zn  n tanh 4 a  n  where

n 

(67)

1 4 4 3 br D  0.2785an  ln2 . 4c 4

3.2 Stability Analysis Theorem 1: Consider the closed-loop system including the system (1) under the Assumptions 1 and 2, neural networks under the Assumption 3, backlash-like hysteresis (2), the actual control law (64), the virtual control law (45) together with the adaptive law (72), then, V (0)  p , the actual control law (64) can guarantees all the error signals of the closed-loop system are SGUUB in the 4th-moment(or mean square). Proof: Define the Lyapunov function candidate of the closed-loop system as follows: n

V  Vzi  i 1

1 n 4 i , 4 i 2

(68)

Then it can be obtained that n 1 n 1 n 1 1 1 V   ki g i , zi4   br4 zi 1   br4 i 1 i 1 i 1 4 i 1 4 3  (k n  ) z n4  g n , N ( )  1 4 n  z 3    bl ( z  i tanh i i   ˆ)    i i 1 i 1  ai  n n 1 3   (  i4   i3 Bi ())    i2 Tr CiT ()Ci () , 2 i 2 i 2  i

~

(69)

n

3 i





Since B0  0 and p  0 , the set r  yr , y r , yr



T





: yr2  y r2  yr2  B0  R3 and

1 n 1 n 1~       ( zn ,  n ,ˆ) :  zi4   g i4, i4   2  p   R 2 n 2 are compact sets, thus, 4 i 1 4 i 2 2    d   is also a compact set. Therefore, M i and N i denote the maximum of Bi () and





Tr CiT ()Ci () on  d   , respectively. From Lemma 3, it follows that n

  i3 Bi ()  i 2

3 n 4 1 n M i4 , i  4  4 i 2 i 2

(70)





3 n 2 3 n 4 3 n 2 T  Tr C (  ) C (  )  Ni ,  i i  4  i i 2 i 2 4 i 2 i 2

(71)

Substituting (70) and (71) into (69), we obtain n 1 n n 1 1 3 V   ki g i , zi4  br4  zi4  br4   i4  (k n  ) z n4 4 i 2 4 i 2 4 i 1 n  z 3   ~ n  g n , N ( )  1  bl ( zi3  i tanh i i   ˆ)    i i 1 i 1  ai  n n n n n 1 3 1 3 3    i4    i4   M i4    i4   N i2 4 i 2 4 i 2 4 i 2 4 i 2 i 2  i n 1 1 3 1 4 4 n 1 1 4 3 4   k1 g1, z14   (ki g i ,  br4 ) zi4  (k n   br ) z n   (  br  ) i 4 4 4 4 2 i 2 i 2  i n  z 3   1 n 3 n ~ n  g n , N ( )  1  bl ( zi3  i tanh i i   ˆ)    i   M i4   N i2 , 4 i 2 4 i 2 i 1 i 1  ai 

 According to (72), the adaptive law ˆ can be designed as  z 3   n ˆ   zi3  i tanh i i   ˆ , i 1  ai  where  is a designed parameter.

(72)

(73)

Substituting (73) into (72), and with the fact that

1~ 2

~

1 2

 ˆ    2   2 ,

(74)

It yields n 1 1 3 1 4 4 n 1 1 4 3 4 V  k1 g1, z14   (ki g i ,  br4 ) zi4  (k n   br ) zn   (  br  ) i 4 4 4 4 2 i 2 i 2  i n bl ~ 2 1 n 3 n 2 bl 4     g n , N ( )  1    i   M i   N i   2 , 2 4 i 2 4 i 2 2 i 1

(75)

Define



1 4

 z  min k1 g1, , ki gi ,  br4 , kn  

1

3 1 4  br , i  2,3,..., n  1 , 4 4 

3

1

   min   br4  , i  2,3,4,..., n , 2  i 4

(76)

(77)

Based on (75)~(77) and assumption 1, we have n

n

V   z zi4     i4  i 1



i 2



bl ~ 2   br N ( )  1  c2 2

(78)

 c1V  g N ( )  1   c2 . 

where c1  min 4 z ,4  , , g  br  , c2  

n

 i  i 1

1 n 3 n 2 bl 4 M  N i   2 .  i 4 4 i 2 2 i 2

Thus, according to Lemma 1, it‘s obvious that the closed-loop error signals including zi and

~  i 1 (i  1,2,..., n) are SGUUB in the sense of 4th-moment for all t  0 , and  is SGUUB in mean square. Remark 1: According to (78) and the proof of Lemma 1[43], it can be derived that

EV (t )   V (0) 

c2  A0  D [43], though choosing the proper initial condition, V (0) and c1

A0 can be made enough small, such that the bound of EV  is also small. Additionally, if the

design parameters k i , ai ,  i ,  and  are appropriately chosen such that the value of c1 is increased or the value of c2 is decreased to make c2 c1 enough small. Accordingly, by designing the appropriate initial condition and the suitable parameters, the closed-loop error signals zi ,

~

 i 1 (i  1,2,..., n) and  can be made enough small.

4. Simulation example In this section, a numerical example is presented to demonstrate the correctness and feasibility of the proposed control scheme in the previous sections. Consider the following 2-order switched stochastic nonlinear systems containing two subsystems:

dx1  f1,1 ( x1 , x2 )dt  1T,1 ( x1 , x2 )dw,   T dx2  f 2,1 ( x2 , u1 )dt  2,1 ( x1 , x2 )dw, y  x ,  1

dx1  f1, 2 ( x1 , x2 )dt  1T, 2 ( x1 , x2 )dw,   T dx2  f 2, 2 ( x2 , u2 )dt  2, 2 ( x1 , x2 )dw, y  x ,  1

where

f1,1 ( x1 , x2 )  (0.3  0.8x1 ) x2  sin x23 , 1T,1 ( x1 , x2 )  0.1x2 sin x1  f 2,1 ( x2 , u1 ( ))  x2  0.2 sin u1  g 2,1u1 , 2T,1 ( x1 , x2 )  0.5x2 cos x1 f1, 2 ( x1 , x2 )  0.8(1  x12 ) x2  x23 , 1T, 2 ( x1 , x2 )  x1 x22 2  f 2, 2 ( x2 , u2 ( ))  x2e0.5 x1  g 2，2u2  0.5u23 , 2T, 2 ( x1 , x2 )  0.6 x2 sin x1 where

g 2,1  1 and

g 2，2  1 are the control gains representing the control directions.

u ( ),   1,2 is the output of backlash-like hysteresis of Eq.(2), the parameters of which are designed as   1 , c  3.1635 , B  0.345 . The reference signal is given as yr (t )  sin(t )  0.5 sin(0.25t ) satisfying Assumption 2. The stochastic disturbance w is chosen as the 1-dimensional Gaussian white noise. According to Theorem 1, the virtual control law

1 , the actual control input   , and adaptive

 laws ˆ are chosen respectively as  z13 1   ,  a1 

1  k1 z1  ˆ1 tanh

 zi3  i    ˆ , z  i tanh  a i 1  i    z 3   1   N ( ) k2 z2  ˆ 2 tanh 2 2   ,   1,2 , c  a2     z 3    k2 z24  z23ˆ 2 tanh 2 2  ,  a2   ˆ 

(79)

2

3 i

where z1  x1  yr , z2  x2   2 f . In the simulation, the design parameters are chosen as

(80)

(81)

(82)

  0.05 , a1  a2  0.1 ,   1 , k1  k2  4.5 and   0.1 . The time constant of the low-pass filter is  2  0.1 . In this simulation, the node number of HONN is chosen as   81 to acquire enough generalization capability, and the initial system conditions are set as x1 (0), x2 (0)T  0.4,0T , ˆ(0)  0 and  (0)  0 . The simulation results are shown in Figs. 2-8. Fig. 2 gives the reference signal y r and the response of system output y , and the subplot shows the satisfactory transient phase. Fig. 3 displays the curve tracking error z1 . Fig. 2 and Fig. 3 shows that the proposed control scheme obtains fairly good tracking performance. Fig. 4 illustrates the trajectory of adaptive law of the NN weight ˆ . Fig. 5 shows the backlash-like hysteresis output signal u  and Fig. 6 depicts the

actual control input signal   . Fig. 7 shows the Nussbaum gain function. An arbitrary given switching signal is presented in Fig. 8.

Fig. 2. Reference signal

yr and system output y

Fig. 3. Tracking error

z1

Fig. 4. Adaptive law

ˆ

Fig. 5. Hysteresis output

u

Fig. 6. Actual control input



Fig. 7. Nussbaum gain function: N ( ) and



Fig. 8. Switching signal  (t )

Furthermore, in order to demonstrate the effectiveness of the proposed design, consider the different control direction case. Let the control gains g 2,1  1 and g 2，2  1 . Using the same control scheme as the preceding case, Fig. 9 and Fig. 10 are obtained, which show that proposed method remains effective when the control directions are changed. Based on the simulation results, it can be concluded that the proposed control scheme has been achieved the design objective of the paper.

Fig. 9. Reference signal

yr and system output y

Fig. 10. Nussbaum gain function: N ( ) and



5. Conclusions This paper investigates the tracking control problem for a class of switched nonaffine stochastic nonlinear systems by combining backstepping technique, HONN and Nussbaum gain function. Dynamic surface control(DSC) technique is introduced into the backstepping procedure to overcome the problem of complexity explosion, such that the process of controller design becomes much simpler. Nussbaum gain function is adopted to deal with the problem of unknown control directions. High-order neural networks are employed to approximate the lumped unknown nonlinear functions, and only one parameter adaptive law is required. It is shown the SGUUB in the sense of 4th-moment(or mean square) of all closed-loop error signals is guaranteed. Finally, a simulation example is presented to verify the correctness and effectiveness of the proposed scheme. In our future research works, the finite-time command filtered backstepping technique will be adopted for the tracking control of switched nonaffine stochastic nonlinear systems.

Acknowledgement This work was supported by the Scientific Research Fund of Shanghai University of Engineering Science, and partly by National Natural Science Foundation of China under Grant No. 11602144. The authors gratefully acknowledge these supports.

Conflict of interest statement We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or

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