2315ijcsitce02

International Journal of Computational Science, Information Technology and Control Engineering (IJCSITCE) Vol.2, No.3, July 2015

19

CHAOS CONTROL VIA ADAPTIVE INTERVAL TYPE-2

FUZZY NONSINGULAR TERMINAL SLIDING MODE

CONTROL

Rim Hendel1, Farid Khaber1 and Najib Essounbouli2

1 QUERE Laboratory, Engineering Faculty, University of Setif 1, 19000 Setif, Algeria 2 CReSTIC of Reims Champagne-Ardenne University, IUT de Troyes, France

ABSTRACT In this paper, a novel robust adaptive type-2 fuzzy nonsingular sliding mode controller is proposed to stabilize the unstable periodic orbits of uncertain perturbed chaotic system with internal parameter uncertainties and external disturbances. This letter is assumed to have an affine form with unknown mathematical model, the type-2 fuzzy system is used to overcome this constraint. A global nonsingular terminal sliding mode manifold is proposed to eliminate the singularity problem associated with normal terminal sliding mode control. The proposed control law can drive system tracking error to converge to zero in finite time. The adaptive type-2 fuzzy system used to model the unknown dynamic of system is adjusted on-line by adaptation law deduced from the stability analysis in Lyapunov sense. Simulation results show the good tracking performances, and the efficiently of the proposed approach.

KEYWORDS Chaotic System, Type-2 Fuzzy Logic System, Nonsingular Terminal Sliding Mode Control, Lyapunov Stability.

1. INTRODUCTION Chaos is a particular case of nonlinear dynamics that has some specific characteristics such as extraordinary sensitivity to initial conditions and system parameter variations. The study of chaos can be introduced in several applications as: medical field, fractal theory, electrical circuits and secure communication [1]. Nowadays, the scientific community has identified two problems in chaos control: suppression and synchronization. The chaos suppression problem can be defined as the stabilization of unstable periodic orbits (UPO's) of a chaotic attractor in equilibrium points or periodic orbits with period n embedded into the chaotic attractor [2]. Many nonlinear control techniques have been applied for chaos elimination and chaos synchronization such as linear and nonlinear control techniques based on feedback [3-6], variable structure control [7-8], nonlinear control [9-11], active control [12-14], backstepping design [15-17], fuzzy logic control [18-19], and adaptive control [20-21]. Unfortunately, most of the above approaches mentioned have not considered the uncertainties and unknown parameters of the chaotic system, internal and external disturbances. Then, a useful and effective control scheme to deal with uncertainties, time varying properties, nonlinearities and bounded externals disturbances is the sliding mode control (SMC). Since then, different controllers based on sliding mode control schemes have been proposed to control chaotic systems [22-23] However, its major drawback in practical applications is the chattering problem. A lot of works have proceeded to solve this problem by using adaptive control [24-26], intelligent approach [27-29], and higher order sliding mode control [30]. In general, the sliding surface is designed as a


20

linear dynamic equation s e ce= +& . However, the linear sliding surface can only guarantee the asymptotic error convergence in the sliding mode, i.e., the output error cannot converge to zero in finite time. The terminal sliding mode TSM has a nonlinear surface pqees += & , while reaching the terminal sliding mode, the system tracking error can be converged to zero in finite time. Furthermore, TSM controller design methods have a singularity problem. Moreover, the known bounds of uncertainties is required. Based on TSM, some nonsingular terminal sliding mode (NTSM) control systems have been proposed to avoid the singularity in TSM [31-33].

The objective of this paper is to force the n-dimensional chaotic system to a desired state even if it has uncertainties system, external and internal disturbances, by incorporation the fuzzy type-2 approach and nonsingular terminal sliding mode (NTSM) control. We introduced an adaptive type-2 fuzzy system for model the unknown dynamic of system, and we use boundary layer method to avoid a chattering phenomenon.

The organization of this paper is as follows. After a description of system and problem formulation in section II, the adaptive type-2 fuzzy nonsingular terminal sliding mode control scheme is presented in section III. Simulation example demonstrate the efficiently of the proposed approach in section IV. Finally, section V gives the conclusions of the advocated design methodology.

2. DESCRIPTION OF SYSTEM AND PROBLEM FORMULATION

Consider n-order uncertain chaotic system which has an affine form:

+++==

+

,)()(),(),(,11,1

tutdtxftxfxnixx

n

ii

&

&

(1)

where 1 2[ ( ) ( ) ( )] nnx x t x t x t= is the measurable state vector, ( , )f x t is unknown nonlinear continuous and bounded function, ( )u t is control input of the system, ( )d t is the external bounded disturbance, and ( , )f x tD represents the uncertainties,

df tdtxfFtxf < )(,),(,),( (2)

where fF , and d are positive constants.

The control problem is to get the system to track an n- dimensional desired vector ( )dy t which belong to a class of continuous functions on 0[ , ]t . Lets the tracking error as;

)](...)()([])()()()()()([

)()()(

)1(

)1()1(

tetete

tytxtytxtytx

tytxte

n

n

dn

dd

d

=

=

=

&

K&& (3)

Therefore, the dynamic errors of system can be obtained as;


21

+++=

=

=

)()(),()(),(

,

)(

32

21

tutdtxftytxfe

ee

ee

n

dn&

M

&

&

(4)

The control goal considered is that;

lim ( ) lim ( ) ( ) 0,dt t

e t x t y t

= (5)

2.1. Terminal Sliding Mode Control

We consider a second order nonlinear system (1), the conventional TSM is described by the following first order terminal sliding variable;

q ps e e= +& (6)

where 0 > is a design constant, and ( ), ,p q p q> are positive odd integers. The sufficient condition to ensure the transition trajectory of the tracking error from approaching phase to the sliding one is:

),(),(),(),(21 2 testestestes

dtd

= & (7)

where 0 > is a constant.

If ( , )f x t is known and free of uncertainties and external disturbances, and when the system (1) is restricted to the ( , ) 0s e t = , it will be governed by an equivalent control equ obtained by:

+=

eepqytxfu pqdeq &&& 1),( (8)

The global control is composed of the equivalent control and discontinuous term, such that; sgn( )dis su k s= (9)

where ( 0)s sk k > is switching gain, by adding this term to (8), we obtain the global control:

++=

)(),( 1 ssignkeepqytxfu spqdTSM &&& (10)

which ensures that TSM occurs. Then, we can choose switching gain as follows: sk D= + (11)

Where fdD += . If (0) 0s , its clear that the tracking errors will reach the sliding mode ( 0s = ) within the finite time

rt , which satisfies;

(0)r

st

(12)


22

Suppose the attaining time is st from ( ) 0re t to ( ) 0s re t t+ = . In this phase, the sliding mode

( )0s = is reached, i.e., the system dynamics is determined by the following nonlinear differential equation:

0=+ pqee & (13)

By integrating the differential equation pqee =& , we have:

( ) pq

rs teqpp

t

=1)( (14)

From TSM control (10), the term containing ee pq &1 may cause a singular problem.

2.2. Non Singular Terminal Sliding Mode Control

In order to overcome the singularity problem in the conventional TSM systems, the proposed NTSM model is described as follows:

qpees &

1+= (15)

where ,q and ( )1 2p p q< < have been defined in (6). For system (1) with the nonsingular sliding mode manifold (15), the control is designed as;

++=

)()(),( 2 ssignkepq

tytxfu sqpdNTSM &&& (16) Thus to satisfy the transition condition (7), the time derivative of s is:

( )11 ( , ) ( ) ( ) ( , ) ( )pq

dp

s e e f x t y t u t f x t d tq

= + + + +& & & && (17)

Using control law (16),

( )11 sgn( ) ( , ) ( )pq

s

ps e k s f x t d t

q

= + +& &

After some manipulations, we obtain:

( )11

1sgn( )

1

pq

s f d

pq

pss e k s s

q

pe s

q

+ +

& &

&

(18)

Since 0 > , p and q are positive odd integers ( )1 2p q< < , we have 1 0p qe >& (when 0e & ), then;

1

( ),1( ) 0

pq

ss e s

pe e

q

= >

& &

& & for 0e & (19)

Therefore, the condition for Lyapunov stability is satisfied when 0e & , and the tracking errors can reach the sliding mode s=0 within finite time. Substituting the control (16) into system (4) yields;


23

2( , ) ( ) sgn( )

pq

s

qe e f x t d t k s

p = + + && &

When 0e =& , we obtain ,

( , ) ( ) sgn( )e f x t d t D s= + +&& , 00

e for se for s

> there exists a vicinity of 0e =& , such that e


24

),()t,( *f

xfxfw

=

We can write,

f

fT

f

MF

xxfw

)()t,( *

By using = fMF , it can be easily concluded that w is bounded ,w (i.e. w L ).

To study the closed loop stability and to find the adaptation law of adjustable parameter, we consider the following Lyapunov function:

fT

fffssV

+=

~~2

121)~,( 2 (22)

where ,~ *fff

= and f is arbitrary positive constant, so the time derivative of (22) is:

fT

ffssV

+=

&&&~1

(23)

Using the control law (21), and (20), the time derivative of the NTSM manifold (15) becomes:

))(),()sgn(),(),(),(),(()(

)(),()sgn(),(),()(** tdtxfskxfxfxftxfe

tdtxfskxftxfes

sfff

sf

+++=

++=

&

&&

))(),()sgn()()(()( * tdtxfskxwes sfff ++= && (24)

such that 11( )

pqpe e

q

=& & .

The substitution of (24) in (23) will be: ( )

+++=

)()(~1)(),()sgn()( xestdtxfskwesV fffffs

&&&& (25)

By choosing the following adaptation law: )()( xes

fff

= && (26)

where ff

= &&~ , therefore, we obtain:

( )( )( )

( ) sgn( ) ( , ) ( )( ) ( , ) ( )( )

s

s

s

V s e w k s f x t d te sw k s s f x t sd te w k D s

= + += + +

= +

& &

&

&


25

Then,

( )( )

( )( )

V e w s

e s

& &

& (27)

From the universal approximation theorem, it is expected that will be very small (if not equal to zero) in the adaptive fuzzy system, and ( ) 0e >& . So, we have 0.V


26

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

time(s)

Stat

e re

sponse

x1 x2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-3

-2

-1

0

1

2

3

x1

x2

Figure 2. Time response (x1, x2) and typical chaotic behavior of duffing oscillator

In order to force the states system ( ), 1,2ix t i = to track the reference trajectories ( )dy t and ( )dy t& in finite time, such as ( )( ) ( / 30) sin( ) 0.3sin(3 )dy t t tpi= + , the adaptive interval type-2

fuzzy nonsingular terminal sliding mode control ( )u t is added into the system as follows: 1 2

32 2 1 1

,

0.4 1.1 2.1cos(1.8 ) ( , ) ( ) ( )x x

x x x x t f x t d t u t=

= + + +

&

& (29)

We choose 15, 3, 5f q p = = = and 1 = , the TSM and NTSM manifolds are selected as, /q p

TSMs e e= +& and ( ) /1/ p qNTSMs e e= + & , respectively. To design the fuzzy system ( , )ff x , we define seven type-2 Gaussian membership functions depending ( ), 1,2ix t i = selected as

, 1,...,7liF l = are shown in table. 1, with variance 0.5 = and initial values 2 7(0)f = .

Table 1. Interval Type-2 Fuzzy Membership Functions For ( 1,2).ix i =

Mean

Mean m1 m2 m1 m2

1 ( )i

iFx

-3.5 -2.5 5 ( )i

iFx

0.5 1.5 2 ( )

iiF

x -2.5 -1.5 6 ( )

iiF

x 1.5 2.5 3 ( )

iiF

x -1.5 -0.5 7 ( )

iiF

x 2.5 3.5 4 ( )

iiF

x -0.5 0.5

In this section, two control laws are adopted, adaptive type-2 fuzzy nonsingular terminal sliding mode control (AT- 2FNTSM) described in (21), and adaptive type-2 fuzzy terminal sliding mode control (AT-2FTSM), which is designed as follow;

++=

)(),( 1 ssignkeepqyxfu spqdfTSM &&& (1)


27

The simulation results are presented in the presence of uncertainties ( ) ( )1 2( , ) sin 2 ( ) sin 3 ( )f x t x t x tpi pi = , external disturbance ( ) sin( )d t t= , and white Gaussian

noise is applied to the measured signal ( ), 1,2ix t i = with Signal to Noise Ratios (SNR=20dB). A boundary layer method is used to eliminate chattering.

4.1. Adaptive Interval Type-2 Fuzzy Terminal Sliding Mode Control (AT-2FTSM) The tracking performance of states ( )x t is shown in Figure 3. The control input ( )u t and the phase-plane trajectories of system are represented in Figures 4-5.

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

x1yd

0 2 4 6 8 10 12 14 16 18 20

-0.2

0

0.2

time(s)

x2yd'

Figure 3. The output trajectories of (x1, x2).

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

e1

e2

ideal sliding modepractical trajectory

-0.1 -0.05 0 0.05 0.1-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

x1,x2yd,yd'

Figure 4. Phase-plane of tracking error and typical chaotic behavior of duffing oscillator


28

0 2 4 6 8 10 12 14 16 18 20-10

-5

0

5

10

time(s)

u

Figure5. Control input u(t)

4.2. Adaptive Interval Type-2 Fuzzy Non-singular Terminal Sliding Mode Control (AT-2FNTSM)

0 2 4 6 8 10 12 14 16 18 20

-0.1

-0.05

0

0.05

0.1

x1yd

0 2 4 6 8 10 12 14 16 18 20

-0.2

-0.1

0

0.1

0.2

0.3

time(s)

x2yd'

Figure6. The output trajectories of (x1, x2)

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

x1

x2

ideal sliding modepractical trajectory

-0.1 -0.05 0 0.05 0.1-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

x1,x2yd,yd'

Figure7. Phase-plane of tracking error and typical chaotic behavior of duffing oscillator


29

0 2 4 6 8 10 12 14 16 18 20-6

-4

-2

0

2

4

time(s)

u

Figure8. Control input u(t)

According to the above simulation results, we can see that both controller provide a good tracking of outputs system 1 2( , )x x to their trajectories in finite time. Furthermore, a singularity problem occurs in the case of AT-2FTSM control as shown in Figure 5. The proposed approach allows obtaining a smooth control signal (Figure 8), then, the NTSM manifold (15) can eliminate the singularity problem associated with conventional TSM manifold.

5. CONCLUSION

In this paper, the problem of stabilization orbit of nonlinear uncertain chaotic system in the presence of external, internal disturbances and disturbances is solved by incorporation of interval type-2 fuzzy approach and non-singular terminal sliding mode control. In order to eliminate the chattering phenomenon efficiently, a boundary layer method is used, and an adaptive interval type-2 fuzzy system is introduced to approximate the unknown part of system. Based on the Laypunov stability criterion, the adaptation law of adjustable parameters of the type-2 fuzzy system and the stability of closed loop system are ensured. A simulation example has been presented to illustrate the effectiveness and the robustness of the proposed approach.

REFERENCES [1] E. Tlelo-Cuautle, (2011) Chaotic Systems, Janeza Trdine 9, 51000 Rijeka, Croatia. [2] I. Zelinka, S. Celikovsky, H. Richter & G. Chen, (2010) Evolutionary Algorithms and Chaotic

Systems, Studies in Computational Intelligence, vol. 267, Springer-Verlag Berlin Heidelberg. [3] R. Femat, G. Solis-Perales, (2008), Robust Synchronization of Chaotic Systems via Feedback,

Springer-Verlag, Berlin Heidelberg. [4] X. Xiao, L. Zhou & Z. Zhang, (2014) Synchronization of chaotic Lure systems with quantized

sampled-data controller, Commun Nonlinear Sci Numer Simulat, vol. 19, pp. 20392047. [5] L-X. Yang, Y-D. Chu, J.G. Zhang, X-F. Li & Y-X. Chang, (2009) Chaos synchronization in

autonomous chaotic system via hybrid feedback control, Chaos, Solitons and Fractals, vol. 41, pp. 214223.

[6] M.T. Yassen, (2005) Controlling chaos and synchronization for new chaotic system using linear feedback control, Chaos, Solitons and Fractals, vol. 26, pp. 913920.

[7] S. Vaidyanathan, (2012) Hybrid synchronization of hyperchaotic liu systems via sliding mode control, International Journal of Chaos, Control, Modelling and Simulation (IJCCMS), Vol.1, no.1.

[8] F. Farivar, M. A. Shoorehdeli, M. A. Nekoui & M. Teshnehlabc, (2012) Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control, Chaos, Solitons & Fractals, vol. 45, pp. 8097.

[9] J. H. Park, (2005) Chaos synchronization of a chaotic system via nonlinear control, Chaos, Solitons and Fractals, Vol. 25, pp. 579584.

[10] Q. Zhang & J-a. Lu, (2008) Chaos synchronization of a new chaotic system via nonlinear control, Chaos, Solitons and Fractals, vol. 37, pp. 175 179.

[11] J. H. Park, (2006) Chaos synchronization between two different chaotic dynamical system, Chaos, Solitons and Fractals, vol. 27, pp. 549554.


30

[12] U. E. Vincent, (2008) Chaos synchronization using active control and backstepping control: a comparative analysis, Nonlinear Analysis, Modelling and Control, Vol. 13, No. 2, pp. 253261.

[13] Y. Wu, X. Zhou, J. Chen & B. Hui, (2009) Chaos synchronization of a new 3D chaotic system, Chaos, Solitons and Fractals, vol. 42, pp. 18121819.

[14] M.T. Yassen, (2005) Chaos synchronization between two different chaotic systems using active control, Chaos, Solitons and Fractals, vol. 23, pp. 131140.

[15] B. A. Idowu, U. E. Vincent & A. N. Njah, (2008) Control and synchronization of chaos in nonlinear gyros via backstepping design, International Journal of Nonlinear Science, Vol. 5, No.1 , pp. 11-19.

[16] J.A. Laoye, U.E. Vincent & S.O. Kareem, (2009) Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller, Chaos, Solitons and Fractals, vol. 39, pp. 356362.

[17] P. Roy, S. Ray & S. Bhattacharya, (2014) Control of Chaos in Brushless DC Motor, Design of Adaptive Controller Following Back-stepping Method, International Conference on Control, Instrumentation, Energy & Communication (CIEC), pp. 41 -45.

[18] H.-T. Yau & C.-S. Shieh, (2008) Chaos synchronization using fuzzy logic controller, Nonlinear Analysis: Real World Applications, Vol. 9, pp. 1800 1810.

[19] M. A. khanesar, M. Teshne hlab & O. Kaynak, (2012) control and synchronization of chaotic systems using a novel indirect model reference fuzzy controller, Soft Comput, vol. 16, pp. 1253-1265.

[20] S. Vaidyanathan, (2011) hybrid synchronization of liu and l chaotic systems via adaptive control, International Journal of Advanced Information Technology (IJAIT), Vol. 1, No. 6.

[21] X. Zhang, H. Zhu & H. Yao, (2012) Analysis and adaptive synchronization for a new chaotic system, Journal of Dynamical and Control Systems, Vol. 18, No. 4, pp. 467-477.

[22] N. Vasegh & A. K. Sedigh, (2009) Chaos control in delayed chaotic systems via sliding mode based delayed feedback, Chaos, Solitons and Fractals, vol. 40, pp. 159165.

[23] S. Vaidyanathan & S. Sampath, (2012) Hybrid synchronization of hyperchaotic Chen systems via sliding mode control, Second International Conference, CCSIT Proceedings, Part II, India, pp. 257-266.

[24] M. Pourmahmood, S. Khanmohammadi & G. Alizadeh, (2011) Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller, Commun Nonlinear Sci Numer Simulat, Vol. 16, pp. 28532868.

[25] X. Zhang, X. Liu & Q. Zhu, (2014) Adaptive chatter free sliding mode control for a class of uncertain chaotic systems, Applied Mathematics and Computation, vol. 232, pp. 431 435.

[26] C. Yin, S. Dadras, S-m. Zhong & Y. Q. Chen, (2013) Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach, Applied Mathematical Modelling, vol. 37, pp. 24692483.

[27] D. Chen, R. Zhang, J. C. Sprott & X. Ma, (2012) Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control, Nonlinear Dyn, vol. 70, pp. 15491561.

[28] R. Hendel & F. Khaber, (2013) Stabilizing Periodic Orbits of Chaotic System Using Adaptive Type-2 Fuzzy Sliding Mode Control, Proceedings of The first International Conference on Nanoelectronics, Communications and Renewable Energy ICNCRE13 , Jijel, Algeria, p. 488-493.

[29] T-C. Lin, M-C. Chen, M. Roopaei & B. R. Sahraei, (2010) Adaptive Type-2 Fuzzy Sliding Mode Control for Chaos Synchronization of Uncertain Chaotic Systems, IEEE International Conference on Fuzzy Systems (FUZZ), Barcelona, pp. 1-8.

[30] R. Hendel & F. Khaber, (2013) Stabilizing periodic orbits of chaotic system adaptive interval type-2 fuzzy second order sliding mode control, International Conference on Electrical Engineering and Automatic Control ICEECAC13, Stif, Algeria.

[31] H. Wang, Z-Z. Han, Q-Y. Xie & W. Zhang, (2009) Finite-time chaos control via nonsingular terminal sliding mode control, Commun Nonlinear Sci Numer Simulat, vol. 14, pp. 27282733.

[32] Q. Zhang, H. Yu & X. Wang, (2013) Integral terminal sliding mode control for a class of nonaffine nonlinear systems with uncertainty, Mathematical Problems in Engineering, pp. 111.

[33] C. S. Chiu, (2012) Derivative and integral terminal sliding mode control for a class of MIMO nonlinear systems, Automatica, vol. 48, pp.316326.

[34] N. N. Karnik, J. M. Mendel & Q. Liang, Type-2 fuzzy logic systems, IEEE Trans. Fuzzy Syst, Vol. 7, 1999, pp. 643-658.


31

Authors Rim Hendel received here engineering and Master degrees in Automatic from Setif University (Setif 1), Algeria, in 2009 and 2012 respectively. From November 2012, she is Ph.D. student in the Engineering Faculty with the QUERE laboratory at the University of Setif 1. Here research interests are higher order sliding mode control, fuzzy type-1 and type-2 systems, nonlinear systems

Farid khaber received his D.E.A in 1990 and his Master in 1992 degrees in industrial control, and his PhD in 2006 from Setif University (Setif 1), Algeria, in automatic control. He is currently a Professor in the Engineering Faculty from the same university. His research interests include multivariable adaptive control, LMI control and type-2 fuzzy control of renewable energy systems

Najib Essounbouli received his Maitrise from the University of Sciences and Technology of Marrakech (FSTG) in Morocco, his D.E.A. in 2000, his Ph.D. in 2004, and its Habilitation from Reims University of Champagne- Ardenne, all in Electrical Engineering. From September 2005 to 2010, he has been an Assistant Professor with IUT of Troyes, Reims Champagne Ardenne University. He is a currently a Professor and Head of the Mechanical Engineering Department of IUT at Troyes, Reims University. His current research interests are in the areas of fuzzy logic control, robust adaptive control, renewable energy and control drive.

Date post:	01-Sep-2015
Category:	Documents
Upload:	ijcsitcejournal
View:	3 times
Download:	1 times

2315ijcsitce02

Documents