Adaptive interval type 2 fuzzy sliding mode control based on feedback linearization

Scientific Journal of Control Engineering October 2014, Volume 4, Issue 5, PP.138-149

Adaptive Interval Type-2 Fuzzy Sliding Mode Control Based on Feedback Linearization Jing Hua, Yang Cai, Yimin Li# Faculty of Science, Jiangsu University, Zhenjiang Jiangsu 212013, China #

Email: llym@ujs.edu.cn

Abstract Basing on feedback linearization, adaptive interval type-2 fuzzy sliding mode control (SMC) is proposed for nonlinear systems in the paper. Feedback linearization, which is usually used to realize nonlinear system transformation, can effectively simplify the controller design. So simple feature of feedback linearization is applied to design SMC. Moreover, we use type-2 fuzzy system instead of type-1 fuzzy system to approximate the unknown functions. With type-reduction, the type-2 fuzzy system is replaced by the average of two type-1 fuzzy systems. Compared with existing type-1 fuzzy sliding mode control (FSMC), as the advantage of handling numerical and linguistic uncertainties, type-2 FSMC have the potential to produce a better performance than the type1 fuzzy logic control (FLC) in many respects, such as robustness and the property of resistance to disturbance. Ultimately, simulation results illustrate the effectiveness of the proposed method.

Keywords: Sliding Mode Control; Type-2 Fuzzy Systems; Feedback Linearization

1 INTRODUCTION Sliding mode control (SMC) techniques [1-5] have been widely applied to the field of robust control, especially in nonlinear systems. The SMC provides discontinuous control laws to drive the system states to a specified sliding surface and to keep them on the sliding surface. The dynamic performance of the SMC has been adopted as an effective robust control approach for the problems of system uncertainties and external disturbances. However, its major drawback in practical applications is the chattering problem. Numerous techniques have been proposed to eliminate this phenomenon in SMC [6,7]. The fuzzy sliding mode control (FSMC) [8-11], a hybrid of the SMC and fuzzy logic control (FLC), gives a simple way to design the controller systematically and provides the asymptotical stability of the system. In general, the FSMC can also reduce the rule number in the FLC and still possess robustness in the face of model uncertainties and external disturbances.

So far, lots of important results on FSMC have been reported. The paper[12] is concerned with a framework that unifies SMC and FLC. In paper[13], a FLC is used to replace the discontinuous sign function of the reaching law in traditional SMC, and hence a control input without chattering is obtained in the chaotic systems with uncertainties.

The paper [14] propose two fuzzy systems to be used as reaching and equivalent parts of the SMC. In this way, it make use of the fuzzy logic to handle uncertainty or disturbance in the design of the equivalent part and provide a chattering free control for the design of the reaching part, so a novel Adaptive Fuzzy Sliding Mode Control (AFSMC) methodology is proposed based on the integration of SMC and Adaptive Fuzzy Control (AFC). In paper[15], a hybrid adaptive fuzzy control (HAFC) design methodology is presented, where the nonlinear system is controlled by a state feedback controller and an adaptive fuzzy controller. In the adaptive fuzzy controller, an adaptive law is developed to tune a robust gain of the sliding-mode controller (SMC) so as to cope with the uncertainties and model errors.

However in the FSMC, uncertainties bounds may not be easily obtained due to the complexity of the structure uncertainties as using precise type-1 fuzzy sets. In general, the rule uncertainty will be existed in the following three possible ways [16-20]: (i) the words that are used in antecedents and consequents of rules can mean different things - 138 http://www.sj-ce.org

to different people; (ii) consequents obtained by polling a group of experts will often be different for the same rule because the experts will not necessary be in agreement; and (iii) noisy training data. Therefore, antecedent or consequent uncertainties translate into uncertain antecedent or consequent membership functions. So type-1 FLSs are unable to handle rule uncertainties directly. To resolve this problem, many methods of type-2 fuzzy sliding mode control (T2FSMC) is presented.

In paper [21], a novel direct adaptive interval type-2 fuzzy-neural tracking control equipped with sliding mode and Lyapunov synthesis approach is proposed to handle the training data corrupted by noise or rule uncertainties for nonlinear single-input single-output (SISO) nonlinear systems involving external disturbances. By employing adaptive fuzzy-neural control theory, the update laws will be derived for approximating the uncertain nonlinear dynamical system. In the meantime, the sliding mode control method and the Lyapunov stability criterion are incorporated into the adaptive fuzzy-neural control scheme such that the derived controller is robust with respect to unmodeled dynamics, external disturbance and approximation errors. Robust adaptive tracking control of multivariable nonlinear systems based on interval type-2 fuzzy approach is developed in paper [22]. The paper [23] propose an interval type-2 fuzzy sliding mode control (IT2FSMC) for linear and nonlinear systems. The proposed IT2FSMC is a combination of the interval type-2 fuzzy logic control (IT2FLC) and the SMC, which inherits the benefits of these two methods. Paper [24] proposed a method to reduce chattering by replacing the reaching component of SMC by an the interval type-2 fuzzy logic system (IT2FLS) when taking into consideration possible system uncertainties. Using this method, IT2FLS enhances fuzzy system performance by incorporating uncertainties in rule base.

For affine nonlinear systems of inaccurate model, a novel adaptive interval type-2 fuzzy sliding mode control based on feedback linearization is proposed in this paper. In order to give a new control method with simple structure, the design of this control combines the simplification of the feedback linearization and the intelligence of type-2 fuzzy logic with the efficiency of SMC. The designed control scheme ensures stronger robustness of the IT2FSMC system, attenuates the chattering phenomena emerging in the sliding-mode control system, and obtains a smooth control effect. The simulation result shows the validity of the control scheme proposed in this paper. It also shows this control scheme guarantee the higher tracking performance and resistance to external disturbances than traditional type-1 FLC method.

The rest of this paper is organized as follows: The method of feedback linearization is presented in Section 2. In section 3, sliding mode control design is presented. In Section 4, a brief introduction of the interval type-2 fuzzy logic system is presented. Indirect adaptive fuzzy sliding mode control design using interval type-2 fuzzy logic system is presented in Section 5. In Section 6 and 7, a simulation example is provided to illustrate the validity of the proposed control scheme. In Section 8, we conclude the work of the paper.

2 THE METHOD OF FEEDBACK LINEARIZATION An easy way to comply with the journal paper formatting requirements is to use this document as a template and simply type your text into it.

Consider SISO system can be expressed as: x  f 0  g0 ( x)u y  h( x )

(1)

where x  Rn is the state vector, f 0 , g 0 and h belong to: Rn  Rn space. Respectively, f 0 and h satisfy the follow conditions: f0 (0)  0 , h(0)  0 .

So from (1), y is obtained by: y

def h h h x f 0 ( x)  g0 ( x)u  f1 ( x)  g1 ( x)u x x x

assuming g1 ( x)  0 , linearized feedback control law is designed as: - 139 http://www.sj-ce.org

(2)

u  g1 ( x)1[ R  f1 ( x)]

then linear system could be achieved by(2) defined as: yR ( n 1) T d

Let Yd   yd , yd ,..., y

T

  Rn be the desired trajectory with Y   y, y,..., y ( n1)   R n , define R as follows: R  yd  a( y  yd )

(3)

where a  0 , when e  y  yd .

From(3), we can get error dynamic equation as follows: e  ae  0

the resulting state response of the tracking error vector satisfies the following condition: lim e(t ) t   0

when e(0)  e(0)  0 , e(t ) 's always been 0 .

3 SLIDING MODE CONTROL DESIGN Consider nonlinear SISO systems whose dynamical can be expressed in the canonical form: x( n)  f ( x, t )  g ( x, t )u

(4)

Where x  Rn is the state vector, f ( x, t ) and g ( x, t ) are the unknown functions.

Assuming xd is the the desired trajectory. Define the tracking error as follows: e  x  xd  e, e,..., e( n 1) 

T

The control goal considered in this paper is that for any given target xd .

In the traditional SMC, a switching surface representing a desired system dynamics is constructed. n 1

s  en   ci ei

(5)

i 1

The switching surface parameters  ci , i  1, 2,..., n  1 are chosen based on two criteria. First, the values are chosen such that the system during sliding mode is stable, Routh-Hurwitz is used to determine the range of coefficients ci that produce stable dynamics that is, all the roots of the characteristic polynomial describing the sliding surface: p( )   n1  cn1 n2  ...  c2   c1

Second, The values are chosen such that the system has fast and smooth response.

According to the Linearization feedback technology, the SMC control law could be designed as: u  g 1 ( x, t )[ R  fˆ ( x, t )]

(6)

where 1 ( x, t )  xd  c1e , R  1 ( x, t )  k sgn(s) , k  0 According to the Lyapunov stability theory, a Lyapunov function is define as: V

1 2 s 2

For second order systems: s  c1e  e , The time-derivative of V is V  ss  s(e  c1e)  s( x  xd  c1e)  s( f ( x, t )  g ( x, t )u  xd  c1e)

Putting (6) substituted into (7), we can get - 140 http://www.sj-ce.org

(7)

V  s[k sgn(s)]  k s

Therefore V  0 .

To obtain the sliding mode control (6), the system functions f ( x, t ) and g ( x, t ) must be known in advance.

However, f and g are unknown in our problem, it is impossible to obtain the control. The purpose of this paper is to approximate the unknown functions f and g by interval type-2 fuzzy logic system(FLS). Furthermore, the adaptive laws used to adjust parameters will be derived.

4 INTERVAL TYPE-2 FUZZY LOGIC SYSTEM In this section, the interval type-2 fuzzy set and the inference of the type-2 fuzzy logic systems will be presented.

The architecture of an IT2FLC controlled system contains the following five components: fuzzifier, rule base, fuzzy inference engine, type-reducer, and defuzzifier [18-24]. The concept of type-2 fuzzy set is the expansion of type-1 fuzzy set. A type-2 fuzzy set is denote by A which is characterized by a type-2 membership function  A ( x, u) .

The  A ( x, u) is a type-1 fuzzy set in  0,1 .

A  {(( x, u),  A ( x, u)) | x  X

u  J x  [0,1]}

(8)

in which 0   A ( x, u)  1 . Uncertainty in the primary memberships of a type-2 fuzzy set A , consists of a bounded region that is called the Footprint of Uncertainty(FOU). It is the union of all primary memberships: FOU ( A) 

Jx  xX

 x, u  : u   ( x),  ( x) A

A

The distinction between type-1 and type-2 rules is associated with the nature of the membership functions: the structure of the rules remains exactly the same in the type-2 case, but all the sets involved are type-2 now. We can consider a type-2 FLS having n inputs x1  X1 ,..., xn  X n and one output y  Y , assuming there are M rules, the i th rule of type-2 FLS has this form: R j : IF x1 is F1 j and x2 is F2j and ...and xn is Fnj , THEN y is F j , j  1,...M

In paper [25], the type reduced set using the center of sets type-reduction can be expressed as follows: Ycos Y ,...., Y , F ,...., F 1

M

1

M

   y , y    ...  ... l

r

y1

yM

f1

 

M

fM

1

i 1 M

f i yi

i 1

fi

i and f i  F i   f , f i  . In the meantime, an interval type-2 FLS with singleton fuzzification and meet under

i

minimum or product t-norm f and f i can be obtained as f   F i ( x1 )      F i ( xn ) i

1

and

n

f   F i ( x1 )      F i ( xn ) i

1

(9)

n

This interval set is determined by its two end points, yl and yr ,which corresponds to the centroid of the type-2 interval consequent set G i .

 

N

CGi   i J y1 ...  N J yN 1 /

i 1 N

yii

 i 1 i

  yli , yri 

(10)

Before the computation of Ycos ( x) , we must evaluate (10) and its two end points, yl and yr .

For illustrative purpose, the Karnik-Mendel iterative procedure is to compute yl and yr [21]. Therefore, the yl can be rewritten as

 f   L

yl

i 1

i

yli   i  L 1 f yli M

f   i  L 1 f i 1 L

i

M

i

i

L

  ql yli  i 1

i

y  i i Q Ql   l    T  q y   l l  l   l  l l i  L 1 y  M

- 141 http://www.sj-ce.org

(11)

where ql  f / Dl , ql  f / Dl and Dl  ( i 1 f   i  L 1 f ) . In the meantime, we have i

i

i

i

L

i

M

i

l L 1 L2 M l l 1 2 L Ql  ql , ql ,..., ql  , Q  ql , ql ,..., ql  , lT  Ql Q  and Tl   y l y  .      

The computation procedure for yr is similar to compute yl . So yr can also be rewritten as

 f  R

yr 

i

i 1

R

i 1

yri   i  R 1 f i yri M

f i   i  R 1 f i M

where qri  f i Dr , qri  f i Dr and Dr 



R i 1

y  R M   i 1 qri yri   i  R 1 qri yri = Q r Qr   r   rT r  yr 



f i   i  R 1 f i . In the meantime, we have M

Qr   qr1 , qr2 ,..., qrR  , Qr  qr1 , qr2 , , qrR  rT  Qr Qr  and Tr   y r yr  .

We defuzzify the interval set by using the average of yl and yr , hence, the defuzzified crisp output becomes Y fuzz 2 ( x) 



Where

yl  yr 1 T = l l  rT r  2 2

1 T T l  l r      T  2  r 

1 T T l r    T and Tl Tr   T .

2

5 INTERVAL TYPE-2 INDIRECT ADAPTIVE FUZZY SLIDING MODE CONTROL DESIGN In the part, we use type-2 logic system to approximate the unknown function f ( x, t ) and g ( x, t ) , then the i th rule for the unknown function f ( x, t ) and g ( x, t ) can be written as: R fˆj : IF x1 is A1j and x2 is A2j and ...and xn is Anj ,THEN fˆ ( x  f ) is F j Rgˆj : IF x1 is A1j and x2 is A2j and ...and xn is Anj , THEN gˆ ( x  g ) is G j

where Anj s are antecedent type-2 sets and F j s, G j s are consequent type-2 sets. j  1, 2,..., m is the total number of the fuzzy rules for each fuzzy model and Ai j (i  1, 2,..., n) denote the fuzzy sets associated with xi (i  1, 2,..., n) .

F j and G j are fuzzy singletons for fˆ ( x  f ) and gˆ ( x  g ) . Ai j is the label of the fuzzy set characterized by a generalized Gaussian-type membership function:   x  m 2  u A j ( x)  exp      , m   m1 , m2  i      Corresponding to each value of m , we will get a different membership curve,

1 1 fˆ  ( fˆl  fˆr ) , gˆ  ( gˆ l  gˆ r ) 2 2 i f  ifl   i  L 1 f  ifl  fl  L M  i i 1 ˆ   i 1 ql ifl   i  L 1 ql ifl  Ql Ql     lT  fl where fl  i L M i  fl   i 1 f   i  L1 f L

i

M

i

i f  ifr   i  R 1 f  ifr  fr  R M i ˆf   i 1   i 1 q r ifr   i  R 1 q r ifr  Ql Ql     lT  fr r i R M i  fr   i 1 f   i  R1 f R

 f     f  L

gˆ l

i

M

i

i

i 1

L

i 1

i gl

i

M i  L 1 M i  L 1

i

f  gli f

i

i  gl  L M i   i 1 ql gli   i  L 1 ql gli  Ql Ql     lT  gl  gl 

- 142 http://www.sj-ce.org

gˆ r

 

R i 1



f  gri   i  R 1 f  gri M

i

R i 1

f   i  R 1 f i

M

i i

i  gr  R M i   i 1 q r gri   i  R 1 q r gri  Ql Ql     lT  gr  gr 

So we can replace f and g by the interval type-2 system fˆ ( x  f ) and gˆ ( x  g ) as:  fr  1 fˆ ( x  f )  rT lT      ( x)T  f 2  fl 

(12)

1 T T  gr  r l      ( x)T  g 2  gl 

(13)

gˆ( x  g ) 

So we can replace (6) by the following equation: u(t )  gˆ 1 ( x  g )[ R  fˆ ( x  f )]

(14)

6 STABILITY ANALYSIS Theorem 1. Consider the nonlinear SISO system(4) , if the fuzzy based adaptive laws are chosen as:

 fr  r2 sr ( x)

(15)

 fr  r2 sr ( x)

(16)

 gl  r3 sl ( x)u

(17)

 gr  r4 sr ( x)u

(18)

Then, the overall adaptive scheme guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded and the tracking error will converge to zero asymptotically.

Proof. Define the optimal parameters vector:    f  arg min  sup fˆ ( X  f )  f ( X , t )   f  f

 xRQ



  xRQ

 

 g  arg min  sup gˆ ( X  g )  g ( X , t )   g  g

Where  f ,  g are constraint sets for  f ,  g respectively.

Assumption 1 Define the minimum approximation error as: w   fˆ ( x, t )  fˆ ( x  f )   gˆ ( x, t )  gˆ ( x  g ) u 

  

 m 

 f   f  RQ  f  m f g

g

 RQ  g

g

Hence, according to two-order systems s  c1e  e  c1e  x  xd  c1e  f ( x, t )  g ( x, t )u  xd  f ( x, t )  g ( x, t )u  c1e  xd

 f ( x, t )  g ( x, t )u  1 ( x, t ) - 143 http://www.sj-ce.org

Putting u into above formula s  f ( x, t )   g ( x, t )  gˆ ( x, t )  gˆ ( x, t )u  1 ( x, t )  f ( x, t )   g ( x, t )  gˆ ( x, t ) u  gˆ ( x, t )  gˆ 1 ( x, t )( f ( x, t )  R)   1 ( x, t )  f ( x, t )  fˆ ( x, t )   g ( x, t )  gˆ ( x, t )  u  R  1 ( x, t )

 f ( x, t )  fˆ ( x, t )   g ( x, t )  gˆ ( x, t ) u  1 ( x, t )  k sgn(s)  1 ( x, t )  f ( x, t )  fˆ ( x, t )   g ( x, t )  gˆ ( x, t )u  k sgn(s)

Putting w into s , s  fˆ ( x  f )  fˆ ( x, t )  ( gˆ ( x  g )  gˆ ( x, t ))u  k sgn(s)  w 1 1  ( f T  (t )   f T  (t ))  ( gT  (t )   gT  (t ))u  k sgn( s)  w 2 2

1 1  ( Tfl l   Tfrr )  ( glT l   grT  r )u  k sgn(s)  w 2 2

where  fl   fl   fl ,  fr   fr   fr ,  gl   gl   gl ,  gr   gr   gr Now consider the Lyapunov function: 1 1 1 T 1 1 T V  (s 2   Tfl  fl   fr fr   glT  gl   gr gr ) 2 2r1 2r2 2r3 2r4

The time derivative of V along the error trajectory is: V  ss 

1 T 1 T 1 1 T  fl  fl   fr fr   glT  gl   gr gr 2r1 2r2 2r3 2r4

1 1 1 1 T 1 1 T  fr fr   glT  gl   gr gr  s[k sgn(s)  ( Tfl l   Tfr r )  ( glT l   grT r )u  w]   Tfl  fl  2r1 2r2 2r3 2r4 2 2



1 T 1 1 1 T 1 1 1 1 T s fl l ( x)   Tfl  fl  s Tfrr ( x)   fr fr  s glT l ( x)u   glT  gl  s grT r ( x)u   gr gr  k s  sw 2 2r1 2 2r2 2 2r3 2 2r4



1 1 T 1 T 1  fl (r1sl ( x)   fl )   fr (r2 sr ( x)   fr )   glT (r3 sl ( x)u   gl )   grT (r4 sr ( x)u   gr )  k s  sw 2r4 2r1 2r2 2r3

where  fl  r1sl ( x) ,  fr  r2 sl ( x) ,  gl  r3 sl ( x)u ,  gr  r4 sr ( x)u , V  k s  sw Because of the minimum approximation error w is very small, so V  0 .

7 SIMULATION In this section, the performance of the proposed approach is evaluated for the circle of Willis system. We give the simulation results of our proposed adaptive fuzzy controller in the circle of Willis problem.

First, we introduce the circle of Willis system. Besides heart attacks and diverse types of cancers, cerebrovascular diseases are reported to be one of the most common reasons of death. Among such illness one can distinguish intracranial aneurysm, defined as a pouching or a dilatation of the arterial wall, according to the analyses presented in “management of cerebral aneurysms-how can current management be improved”[25] and “Unruptured intracranial aneurysms: a review” its incidence varies between 0.5 and 0.05 of the human population. It has been observed that the existing dilatation of the arterial wall can transform into the pouching due to stresses having an - 144 http://www.sj-ce.org

effect on the impaired aneurysm walls. Sizes of the pouching are related to oscillatory influence of hem dynamical forces. The resultant increased risk of its rupture can lead to serious neurophysiological complications. The points of bifurcation of large arteries within the circle of Willis are particularly to occurrence of the aneurysm.

The state equations of the circle of Willis system can be expressed by [26]:  x1  x2  2 3  x2  3x1  0.1x2  2 x1  x1  u y  x 1 

Where f  3x12  0.1x2  2 x13  x1 , g  1 , x1 is indicated the blood velocity at aneurysm, x2 represents the change rate of blood velocity at aneurysm. The objective is to generate an appropriate actuator force u to control the blood velocity in aneurysm.

The sliding surface is designed as: s  c1e  e , c1  5 . In the simulation, we require that x1  2 and x2  1 . The initial conditions values are selected as: x(0)  [0.5 1]T .

The SMC design algorithm for type-2 adaptive fuzzy control is proposed as follows.

Step 1: define the type-2 membership functions as:

 A  x1   e

 x  0.51 0.1    1  10  

2

,  A2  x1   e

1 1

 A  x2   e

 x 1.01 0.1    1  10  

2

1

 x  0.5*1 0.1    2  10  

2

,  A2  x2   e

1 2

,  A3  x1   e

 x 1.51 0.1    1  10  

1

 x 1.0*1 0.1    2  10  

2

2

,  A3  x2   e

 x 1.5*1 0.1    2  10  

1 1



1 2

A

1 1

,

2 2

A

,

3 2

1 1

A

,

2

2

  A *  A  A *  A  A * A  A *  A  A *  A  A * A

l  

2

2 1

1 2

,

A

2 1

2 2

A

3 2

2 1

,

A

A *  A A *  A A * A  3 1

,

1 2

A

3 1

,

2 2

A

3 1

,

2

1

2

2

1

1

2

1

2

2

1

1

1

2

    A3 *  A3 A

A   A1 *  A1   A1 *  A2   A1 *  A3   A2 *  A1   A2 *  A2   A2 *  A3   A3 *  A1   A3 *  A2 1

3 2

2

1

2

  A1 *  A1  A1 *  A2  A11 *  A3  A2 *  A1  A2 *  A2  A12 *  A3  A3 *  A21  A3 *  A22  A3 *  A3  1 2 2 2 2 2 2 2 , 1 , , 1 , 1 , , 1 , 1 , 1  B B B B B B B B B   B   A1 *  A1   A1 *  A2   A1 *  A3   A2 *  A1   A2 *  A2   A2 *  A3   A3 *  A1   A3 *  A2   A3 *  A3

r  

1

2

1

2

1

9 T l

2

1

2

1

2

1

2

1

2

9 T r

1

2

1

l   ,  ,...,   r   ,  ,...,   Step 2: we assume that there exist some language rules of unknown functions 1, 2 and 3, respectively.

1 l

2 l

1 r

2 r

1 1 fˆ ( x  f )   fˆl ( x  f )  fˆr ( x  f )  , gˆ ( x  g )   gˆ l ( x  g )  gˆ r ( x  g )  2 2 T T fˆl  lT  fl  l1 , l2 ,..., l9   fl , fˆr  rT  fr  r1 , r2 ,..., r9   fr T

T

gˆ l  lT  gl  l1 , l2 ,..., l9   gl , gˆ r  rT  gr  r1 , r2 ,..., r9   gr Step 3 : specify positive design parameters as follows: k  0.5 , r1  r2  r3  r4  1

The adaptive laws are designed as:

 fl  sl ( x)  (ql1 , ql2 , ql3 , ql4 , ql5 , ql6 , ql7 , ql8 , ql9 )  (5e  e)lT

 gl  sl ( x)u  ( pl1 , pl2 , pl3 , pl4 , pl5 , pl6 , pl7 , pl8 , pl9 )  (5e  e)lT u

 gr  sr ( x)u  ( p1r , pr2 , pr3 , pr4 , pr5 , pr6 , pr7 , pr8 , pr9 )  (5e  e)rT u Linearized feedback control law can be written as: 1 1 u (t )  ( gl1  gr1 )[ xd  5e  k sgn(s)  ( fl 1  f r1 )] 2 2 - 145 http://www.sj-ce.org

2

The simulation results are shown in Figs 3-9.

Fig. 1 and Fig. 2 show the state variable output with type-2 FLC and type-1 FLC respectively. We can obtain that the type-2FLC has a higher performance in terms of response speed than type-1FLC.

FIG. 1 STATE VARIABLE OUTPUTS WITH TYPE-2 FLC.

FIG. 2 STATE VARIABLE OUTPUTS WITH TYPE-1FLC.

Fig. 3 show the trajectories of controller u , respectively. From the observation of the simulation results, the adaptive fuzzy control method proposed above guarantee the good tracking performance.

FIG. 3 TRAJECTORIES OF CONTROLLER WITH TYPE-2 FLC. - 146 http://www.sj-ce.org

In order to show the property of resistance to disturbances, training data is corrupted by a random noise 0.5x , i.e., x is replaced by ( 1  random(0.5)) x . Figs 4, 6 show the responses of state variables with disturbances based on type-2 FLC. Figs 5, 7 are on type-1 FLC. It is obvious that, the system based on type-2 FLC has the better property of resistance to disturbances.

FIG. 4 TRAJECTORIES OF THE STATE VARIABLE x1 WITH EXTERNAL DISTURBANCES WITH TYPE-2FLC.

FIG. 5 TRAJECTORIES OF THE STATE VARIABLE x1 WITH EXTERNAL DISTURBANCES WITH TYPE-1FLC.

FIG. 6 TRAJECTORIES OF THE STATE VARIABLE x2 WITH EXTERNAL DISTURBANCES WITH TYPE-2FLC. - 147 http://www.sj-ce.org

FIG. 7 TRAJECTORIES OF THE STATE VARIABLE x2 WITH EXTERNAL DISTURBANCES WITH TYPE-1FLC.

From all the outputs of simulation, we can obtain that the adaptive type-2 fuzzy sliding mode control method based on Feedback Linearization proposed in this paper guarantee the higher tracking performance and resistance to external disturbances than traditional type-1FLC method.

8 CONCLUSION Based on feedback linearization for a class of nonlinear systems, we have set up a design scheme for interval fuzzy sliding mode control in this paper. Compared with existing adaptive type-1 fuzzy sliding mode controller, type-2 FLC is recommended in this approach. From the simulation results, the main conclusions can be drawn: (1) The sliding mode control method based on type-2 FLS guarantees the outputs of the closed-loop system follow the reference signal, and all the signals in the closed-loop system are uniform ultimately bounded.

(2) Compared by traditional sliding mode control method based on type-1 FLS, the type-2 FLC has higher performance in terms of robustness, response speed and resistance to external disturbances.

ACKNOWLEDGMENT This work was supported by national science foundation (11072090) and project of advanced talents of Jiangsu University (10JDG093).

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AUTHORS 1

2

Jiangsu province, China. She obtained his

University, Zhenjiang, China.

master degree in Applied Mathematics

3

Jing Hua born in 1979, in Zhenjiang,

from Jiangsu University in 2006.

Yang Cai is a Graduate Student in Faculty of Science, Jiangsu

Yimin Li is a Professor and Postgraduate Supervisor in Faculty

of Science, Jiangsu University, Zhenjiang, China.

Now, she is a PhD candidate in Systems Engineering

at

Jiangsu

University,

Zhenjiang, China. She is currently engaged in the following research fields: (1) Modeling of complex ecosystem and study of features of biological system; (2) Type-2 fuzzy control methods.

- 149 http://www.sj-ce.org