[IEEE Multiconference on "Computational Engineering in Systems Applications - Beijing, China (2006.10.4-2006.10.6)] The Proceedings of the Multiconference on "Computational Engineering

IMACS Multiconference on "Computational Engineering in Systems Applications"(CESA), October 4-6, 2006, Beijing, China.

Output Feedback Control for Chaotic System via GeneralizedFuzzy Hyperbolic Model

Dongsheng Yang and Huaguang Zhang$

Abstract

This paper studies the output feedback control forchaotic continuous-time systems whose dynamics may notbe exactly known. A chaotic system is modelled basedon the generalizedfuzzy hyperbolic model (GFHM). A dy-namic outputfeedback controller is designed such that theclosed-loop system is asymptotic stable. All the results arepresented based on the solvability of a set of linear ma-trix inequalities (LM7s). The effectiveness of the proposedmethod is shown by a simulation example.

1. INTRODUCTION

Chaos being a ubiquitous phenomenon has been drawnmore and more attentions within the latest thirty years. Fur-thermore, how to utilize chaos and how to control chaos hasbecame a hot point in recent years. Since Ott,Grebogi andYorke proposed their famous OGY method [1], the controlproblem of chaotic system has been studied by many re-searchers, but much ofthe research work makes an assump-tion that exact model of the chaotic system is known[2-3].In practice, it is usually difficult to obtain an explicit modelfor a chaotic system because of the intrinsical complexityof the system and affection of unknown factors of environ-ment.

Fuzzy control methods have been widely used in con-trolling chaotic systems. Successful applications have beendemonstrated, in particular, situations where the dynamicsof systems are so complex that it is impossible to constructan accurate model[5]. So fuzzy control provides an effec-tive approach to handle chaotic systems, especially in thepresence of incomplete knowledge of the real plant or thesituation where precise control action is unavailable[6-9].

*This work was supported by the National Natural Science Foundationof China 60534010, 60572070, 60521003, 6032531 land the Program forChangjiang Scholars and Innovative Research Team in University.

tD.S Yang is with School of Information Science and Engineer-ing Northeastern University, Shenyang, Liaoning 110004, P.R. [email protected]

tH.G Zhang is with School of Information Science and Engineer-ing Northeastern University, Shenyang, Liaoning 110004, P.R. [email protected]

Recently, Zhang and his co-authers proposed the fuzzyhyperbolic model (FHM) and the generalized fuzzy hyper-bolic model (GFHM)[10-11]. It was proved that the gen-eralized fuzzy hyperbolic model is a universal approxima-tor. Compared with the T-S fuzzy model, the GFHM hasmany merits. For example, there is no need to identifypremise structure when modeling a plant by GFHM, there-fore, there is much less computation expense than that ofusing T-S fuzzy model, especially when a lot of fuzzy rulesare needed to approximate nonlinear complex systems. Inthis paper, a fuzzy hyperbolic controller is designed for aclass of chaotic systems with modelling uncertainties.

The main purpose ofthis paper is to develop a fuzzy out-put feedback controller for a class of uncertain chaotic sys-tems. The chaotic control problem is converted into a fea-sible problem of a set of linear matrix inequalities (LMIs).The presented method can also be extended to a wide classof nonlinear systems in which not all states are available.

The rest of the paper is organized as follows: In Sec. 2,the basis of fuzzy representation of a nonlinear system inthe GFHM format is first revisited. In Sec. 3, the GFHM isused to model a chaotic system, and a fuzzy dynamic out-put feedback controller is designed. In Sec. 4, a simulationexample verifies the effectiveness of the proposed method.In Sec. 5, conclusions are made.

2. Preliminaries

In this section we review some necessary preliminaries forthe GFHM.Definition 1 [10]: Given a plant with n input variablesx [X1 (t), ...,xn(t)]T and one output variable y(t). we de-fine the generalized input variables as xi = x -dzi, my'n wi are the numbers of generalized input variables,w,(z= 1,...,n) are the numbers to be transformed aboutx:, dzi(z=l ,..., n, i=l ,...,w.) are constants where x: are trans-formed. We define the fuzzy rule based on the generalizedfuzzy hyperbolic rule base if the following conditions aresatisfied:

The fuzzy rules have the following form:

1. IF (x -d1i) is Fxll and ... and (xl -dlwl) is Fxl,

1973


and (x2- d21) is FI2 and ... and (xn -dpl) is F,,1 and... and (xn-dnwn ) is Fxnw THEN

y CFII+ + CFi.1 + CF2i + + CFni + + CFnwn

(1)

where Fx,z are fuzzy sets of x:- dj, which include Px(Positive) and NX (Negative) subsets.

2. The constant CF1 (z1.ni=... w.) in the "THEN"part correspond to Fxz,in the "IF" part, that is, if thereis Fxz in the "IF" part, cF,,must appear in the "THEN"part. Otherwise, cF,,does not appear in the "THEN"part. Denoting CF by cp, and CF by CN,.

3. There are 2m(m = 1I w,) fuzzy rules for the out-put variable in the rule base, that is, all the possiblePx and Nx combinations of input variables in the "IF"part and all the linear combinations of constants in the"THEN" part.

From Definition 1, if we set P, and NX negative to eachother, we can obtain a homogeneous GFHM:

x = BT tanh(Kx) (5)

In this paper, we will model chaotic system and design afuzzy control scheme based on the GFHM.Lemma 3 [16] : For a symmetric matrix S

E s1 22 ]2 where SII is r x r dimension. There areS21 S22

three equivalent conditions:

1. S<O;

2. Sii <0 , S22 S42SI11512 <0;

3. S22 <0,S11 -S12S22$12<°

3. Main Results

In the FHM, there are two types of fuzzy sets, includingPositive (Ps) and Negative (Nx). The membership functionsof P, and Nx are defined as:

kz)2giv (Xz) = e-2 (x:+ k:) (2)

where kz > 0. We can see that only two fuzzy sets are usedto represent the input variables. If we transform the inputvariable x:, the fuzzy sets may cover the whole input spaceifw is large enough.Lemma 1 [10] : For a plant with n input variables and

an output variable y(t), if we define the generalized fuzzyhyperbolic rule base and generalized input variables as def-inition 1, and define the membership functions of the gen-eralized input variables P, and NX as (2), then we can derivethe following model:

M cpi ekixi + CN, e-kX,ii= 1 ekixi + e-kixi

=A +BT tanh(KxD,

m m ekixi e-kixiY, ai +I bi -kx e-tXek-,xi+ e-k(,x)

(3)

According to Lemma 1 , the real chaotic system canbe represented by a GFHM as follows, which is a kind ofglobal description.

x(t) = Atanh(Kx7 + H1Aow (,y(t) = Ctanh(Kx7 + H2Aw

"')

where x [xI,x2, . ,x,*X,Xm. T e Rm, y E Rl . A,C,H1and H2 are known matrices with appropriate dimensions.K = diag [kl, ..., km]. A1 is an unknown matrix denoting theapproximation error between the real plant and the modelrepresented by the GFHM.

In this section, the main idea is to design a controlleru(t) which is an output feedback controller such that thesystem (6) can be stabilized to the zero equilibrium point.

Find a controller u(t) in the following form:

(t) =Aktanh(Kx) +Bkyu(t) = Cktanh(Kxe +Dky (7)

CP- CN- CPi CNi iwhere ai = 2 , 2, A Y, ai,

i=1B= [bi, ,bm]T, tanh(Kx) is defined by tanh(Kx=[tanh(k1x )...* tanh(kmxmi)]TT, K = diag [k1, ..., km]. Wecall (3) the generalized fuzzy hyperbolic model (GFHM).Lemma 2 [10] For any given real continuous g(x) on

the compact set U c R' and arbitrary £ > 0, there existsF(x) C Y such that

sup g(x) - F(x) < c.xcU

where x C R' is the state of the controller, Ak, Bk,,Ck andDk are undeterminate parameters matrices with appropriatedimensions. The system (6) with the controller (7) can berewritten as following:

x(t) = Atanh(Kx7 + HlIAo+Cktanh(Kx) +Dky (8)

y(t) = Ctanh(Kx7 + H2Ao1.

Now combining (7) and (8), we can obtain the new form ofthe system (6) as following

(4)(9)

1974

-I(X,pp', (x,) = e 2

. (t) = Atanh(K.) + HAco (t)


where

4 (t) [ xT(t) x (t) ]T, A =[ A+DkC CkBkC Akl

H HIi+Dk-H2] K~0kH = Bk_+DH2 K- 0 KO

Assumption 1: Given a positive definite matrix (1 to satisfyfollowing inequality for any 4.

AWTA < tanhT (K ) 1)tanh(K ). (10)

Theorem 1 : Consider the system (9) with a controller ofform (7), suppose that there exists a positive diagonal ma-trix P c R2mx2m such that the following matrix inequalityholds, then the control system is asymptotic stable.

PA+ATp+q PH<<° (I11)

Proof: Choose the following Lyapunov functional candi-date for system (9)

2xmV(t) =2 Ik ln(coshkii) (12)

i= I

where is the ith element of 4 (t) and ki is the ith diagonalelement ofK. Here, ki > 0 and Pi > 0. Because

o hki + ek-icosh(kjCi) = 2 1

we know that V(t) > 0 for all Xi(t) and when 00V(t) t=o = 0, otherwise Xi > 0, V(t) > 0.

V(t) = 2 Y pitanh(kiXi) ii=l _ __

= 2tanhT(K )P; = 2tanhT(K4)P[Atanh(K4) +HAo)(t)]= tanhT (KO9 (PA + ATP) tanh(KOJ + 2 tanhT (KOPHAo\= tanhT (K; ) (PA + ATP) tanh(K ) + 2 tanhT (K; )PHAw+AWTAA AWTA

<tanhT(K4)(PA + ATP+ 1)) tanh(K ) +2tanhT(K4)PHAw

[ tah(4K) ifPA+ATP+ PH tanh(K4) 1AoL HTP -I Aw)=nTz,(

(13)

where = [ tanh(K4) Ao)]T,PA+ATp+ ( PHH[TP -I if z < 0, then V(t) < 0

for r1 7t 0. P = diag{Pl P2,...Pm} is a positive diagonalmatrix.This completes the proof.It is noted that in the matrix inequalities 1, the controllerparameters are unknown, the controller parameters can besolved by LMIs technique.Theorem 2: There exists a controller for the chaoticsystem (6), if there exist positive diagonal matrices X =

diag{pl,p2,...pm} > 0,Y= diag{pm+1,Pm+2...P2m} >Osuch that the following LMIs holds:

Al + (TC)T XH1 +VH2 I

+ 6T +'1)2 PH2 0 < 0* -I 0* * 7(-1)1

(14)T

with 1A=1 AX+XA + nc+(17C)TProof: First, partition P and Q as

P= X 0 D1i0

0(2[

where X = diag{p1,p2,...pm} and Y =

diag{pm+1 Pm+2, .. P2m } are positive diagonal matri-ces, (DI and (2 are chosen diagonal matrices too.

According to Theorem 1, because I < 0, the formulacan be obtained:

[A2 XCk + (YBkC)TYAk+ (YAk)T+12L**

I <0.

(15)

XH1 +XDkH2YBkH2-I

TwithA2=AX+XA +XDkC+(XDkC)T+ (lthen by Schur complement, (15) is equivalent to following

[A3 XCk + (YBkC)T* YAk+ (YAk)T+12* *

* *

-XDkH2 I3kH2 0 < O.-I 0* ¢1)1

(16)

XH1 4Y4

T

with A3 =AX+XA +XDkC+ (XDkC)T

1975

2 2> (ek-i.i)' (ek-i.i)'

r77


Defined

n = XDIG = XCkW = YBk^E = YAk

(16) can be rewrite

Here, we choose membership functions of Pj1 and Nj1 asfollows:

gpp (x:) = e 2 k )2 N (X ) Xe(x+ k)2

[3 E + (TC)T* E +ET+q2*

XH1 +HH2 IPH2 0 < 0. (17)-I 0* ¢1l

GFHM can be seen as a neural network model, so we canlearn the model parameters by back-propagation (BP) al-gorithm. We can obtain parameters as followingK = diag [0.2018,0.3352,0.1266,2.683,2.683]

withA3 =AX+XA +nC+(HC<The stability analysis and controller design are expressedas an LMI problem.

Dk =X-1H Ck =XG'()Bk= Y1'T Ak =y 1E

This completes the proof.

4. Simulation Study

In this section, we shall present an example to demon-strate the effectiveness and applicability of the proposedmethod.

Consider the Van der Pol oscillator with parameters as

follows [18]

[

X.1

X.2

x3 I

a

0

m

1

b

01

-1

-cJ [

X1

X2

X3 I +[

(TX2I

0 [

X1

X2

X3

Suppose that we have the following fuzzy rule base:RI: IF x1 is P,10 andx2 is P120 andx3 is P130 and (xi -

is PX11 and (xl + 1) is Px12Thenx1 c=CX + CX2 + CX3 + CXII + CX12R2: IF x1 is Px10 andx2 is PX20 andx3 is PX30 and (xi -

is Pxll and (xl + 1) is NxI2Thenx1 c=CX + CX2 + CX3 + CXI- CX12R3: IF x1 is Px10 andx2 is PX20 andX3 is PX30 and (xl -

is NxlI and (xl + 1) is NxI2Then x2 = Cxl + CX2 +C3 CX Cx12

I

Then, using Matlab LMI Control Toolbox to solve theLMI, the solution can be obtained as follows:

- 0.56- 0.01

Ak = 0

- 0.01- 0.02

0.10- 0.01

Bk= 0.00

0.000.00

- 217.9914.35

Ck = - 1.363.893.89

- 984.15- 178.40

Dk = - 25.21- 46.57- 46.56

1)

1)

1)

0,

0

0,0,

0,- C

.01 00.56 - 0.18.18 0.18.06 0.02 0

).01 0.000.08 - 0.03).03 0.09.00 0.00).01 0.00

12.15-220.1337.02 -i12.6512.65

- 176.41 -

- 879.58 -

-476.22 -

- 153.09153.09

0.01- 0.060.18- 0.56

0

0.00- 0.01

0.000.100.00

0.9641.27241.27-1.30-1.30

25.26481.68846.26-20.53-20.53

0.02- 0.02

0

0.03- 0.56

0.000.01

0.000.000.10

4.1213.791.05

-249.9812.33

-46.60154.86-20.56691.7590.47

4.1213.79- 1.05- 12.33

- 249.98

- 46.60- 154.86- 20.5690.47

- 691.75

. . .. ....

R25: IF x1 is NX10 and X2 is N 20 and X3 is PX30 and (xl1) is Nx11 and (xi+ 1) is NxI2 Then x3 =-cX -CX2-CX3cxI -cx12

X = .Oe -003 x diag [0.54, 0.54,0.54,0.54,0.54]Y =diag [0.93,0.93,0.93,0.93,0.93]

The result of simulation is illustrated in Fig. 1-Fig.8.

1976

A=

1755.000175175

300-3.00900300300

0-8.00-1.60000

-99.9100

- 99.91- 99.91

-99.9100

- 99.91- 99.91


3.

20,

1-.

-1

-2-

-3,i0.2

0.5

x2 -0.2 -1 xl

Fig. 1. Chaotic attractor of the Van der Pol oscillatorbased on GFHM.

0..

0.2

0.15

<9j 0.1c:

-Z 0.05cs

0

-0.05

1.8xl

Fig. 2. The state figure ofx1 and x2 based on GFHM.

0.t

0

-0.1 - /__ (t)

-0.2 -_ .(t)

-0.3

-0.4

-0.5

-0.60 20 40 60 80 100

time (sec)

Fig. 3.The curve of the state xl and x1.

-0.10 20 40 60 80 1

time (sec)

Fig. 4.The curve of the state x2 and x2.

o

0 20 40 60 80 1 Ctime (sec)

Fig. 5. The curve of the state X3 and X3.

-0.410 20 40 60 80 100

time (sec)

Fig. 6. The error curve of the state error xl -x1I.

1977

0.2)

0.2

0.1

0

-0.1

-0.2

-0.3

x (t)- - - ;6 (t)-

x

0

0

-0

-0

1.15

0.8 02

-0.8 -0. -04 -. .2 04 06 0

9-h;

3Q


0.

0.2

0.15

<< 0.1

0.05

0

-0.05

-0.10 20 40 60

time (sec)80 100

Fig. 7. The error curve of the state error=x2-x2.

20 40 60time (sec)

80 100

Fig. 8. The error curve of the state error=X3-X3.

5. CONCLUSIONS AND FUTURE WORKS

This paper studies the global stabilization by a outputfeedback controller for uncertain chaotic systems whosedynamic may not exactly known. The generalized fuzzyhyperbolic model (GFHM) is used to modelling a chaoticsystem.The dynamic output feedback controller is designedsuch that the closed-loop system is asymptotic stable.Theeffectiveness ofthe proposed method is presented by a sim-ulation example. There exits some work to derive estimatesfor the GFHM model parameters through an identificationprocedure using neural network.

6. ACKNOWLEDGMENTS

The authors gratefully acknowledge the contribution ofNational Research Organization and reviewers' comments.

References are important to the reader; therefore, eachcitation must be complete and correct. If at all possible,references should be commonly available publications.

References

[1] E. Ott, C. Grebogi & J. A. Yorke, "Controlling chaos. Phys.Rev. Lett", 64 1990, pp. 1196-1199.

[2] Sanchez E N. Perez. JP, Martinez. M & Chen G, "Chaosstablilization: An inverse optimal control approach", LatinAmer. Appl.Res. Int.J.32 2002, pp. 111-114.

[3] Z. H. Guan, R. Q. Liao, F. Zhou & H. 0. Wang, "On im-pulsive control and its application to chen's chaotic system", International Journal of Bifurcation and Chaos, 12 5 2002,pp. 1191-1197.

[4] Chen. G. & Lai. D, "Making a dynamical system chaotic:feedback control of Lyapunov exponents for discrete-timedynamical systems", IEEE Trans. Circuits Syst.-I: Fund. Th.Appl. 44 1997, pp. 250-253.

[5] Chen. L & Chen. G, "Fuzzy predictive control of uncertainchaotic systems using time series", Int. J. Bifurcation andChaos 9 1999, pp. 757-767.

[6] Yen. J &Langari.R, "Fuzzy Logic, intelligence, control andinformation prentice-hall", Upper Saddle River, NJ, 1999.

[7] J. Hun, K. C. Park, E. Kim & M. Park, "Fuzzy adaptivesynchronization of uncertain chaotic systems", Physics Let-ters, A. 334 2005, pp. 295-305.

[8] J. G. Barajas-ramiirez, G. R. Chen & Leng S. Shieh, "Fuzzychaos synchronization via sampled driving signals Interna-tional Journal of Bifurcation and Chaos", 8 2004, pp. 2721-2733

[9] H. G. Zhang, Z. L. Wang & D. R. Liu, "Chaotifying fuzzyhyperbolic model based on adaptive inverse optimal controlapproach", Int.J. Bifurcation And Chaos. 12 2003, pp. 32-43.

[10] H. G. Zhang, Z. L. Wang, M. Li, et al, "Generalized fuzzyhyperbolic model: a universal approximator", ACTA AU-TOMATIC ASINICA.30 2004, pp. 416-422.

[11] H. G. Zhang & Y B. Quan, "Modelling, identification andcontrol of a class of non-linear system", IEEE Trans. onFuzzy Systems. 9 2001, pp. 349-354.

[12] Corless M, "Guaranteed rates of exponential convergencefor uncertain systems", Journal of Optimization and Appli-cation. 64 1999, pp. 48 - 494.

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2bz0 F,


[13] R. Ordones & K. M. Passino, "Stable multi-output adap-tive fuzzy/neural control", IEEE Trans. on Fuzzy Systems.7 1999, pp. 345-353.

[14] S. Lun, H. Zhang & D. Liu, "Fuzzy hyperbolic H°° fil-ter design for a class of nonlinear continuous-time dynamicsystems", Proc. of the 43th IEEE Int. Conf. on Decision andControl, Atlantis, Paradise Island, Bahamas. 2004, pp. 225-230.

[15] M. Margaliot & G. Langholz, "Hyperbolic optimal controland fuzzy control", IEEE Transactions on Systems, Man,and Cybernetics. 29 1999, pp. 1-9.

[16] S. Boyd, L. E. Ghaoui, E. Feron & V. Balakrishan, "Linearmatrix inequalities in system and control theory", Philadel-phia. PA: SIAM, 1994.

[17] S. S.Chang. L. Peng, & T. K. C, "Adaptive guaranteed costcontrol of systems with uncertain parameters. IEEE Trans-actions on Automatic Control", 17 1972, pp. 474-483.

[18] H.B. Fotsin & P. Woafo, "Adaptive synchronization of amodified and uncertain chaotic Van der Pol-Duffing oscilla-tor based on parameter identification", Chaos, Solitons andFractals, 24 2005, pp. 1363-1371.

[19] L.U, J, Zhou, T. & Zhang, "S. Chaos synchronizationbetween linearly coupled chaotic systems", Chaos Solit.Fract,2002, pp. 14 529-541.

1979

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[IEEE Multiconference on "Computational Engineering in Systems Applications - Beijing, China (2006.10.4-2006.10.6)] The Proceedings of the Multiconference on "Computational Engineering