: adaptive chattering free variable structure control for a class

7/29/2019 : Adaptive chattering free variable structure control for a class

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Physics Letters A 335 (2005) 274281

www.elsevier.com/locate/pla

Adaptive chattering free variable structure control for a classof chaotic systems with unknown bounded uncertainties

Jun-Juh Yan a,, Wei-Der Chang b, Jui-Sheng Lin a, Kuo-Kai Shyu c

a Department of Electrical Engineering, Far-East College, No. 49, Jung-Hwa Road, Hsin-Shih Town, Tainan 744, Taiwan, ROCb Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan, ROC

c Department of Electrical Engineering, National Central University, Chung-Li 320, Taiwan, ROC

Received 27 February 2004; received in revised form 12 December 2004; accepted 14 December 2004

Available online 27 December 2004

Communicated by A.P. Fordy

Abstract

A new adaptive control scheme is developed for a class of chaotic systems with unknown bounded uncertainties. It is im-

plemented by using variable structure control. The concept of extended systems is used such that continuous control input is

obtained to avoid chattering phenomenon as frequently in the conventional variable structure systems. Furthermore, it is worthy

of note that the proposed adaptive control scheme does not involve any information about the bounds of uncertainties. Thus, thelimitation of knowing the bounds of uncertainties in advance is certainly released. A numerical simulation is included to verify

the validity of the developed adaptive chattering free variable structure control.

2004 Elsevier B.V. All rights reserved.

Keywords: Chaotic system; Adaptive control; Variable structure control; Extended systems; Chattering

1. Introduction

Nowadays more and more chaotic phenomena are

being found in many engineering systems. Chaotic

system is a very complex dynamical nonlinear systemand its response possesses some characteristics, such

as excessive sensitivity to initial conditions, broad

Supported by the National Science Council of Republic of

China under contract NSC-92-2213-E-269-001.* Corresponding author.

E-mail address: [email protected] (J.-J. Yan).

spectra of Fourier transform, and fractal properties

of the motion in phase space [1]. Due to its pow-

erful applications in chemical reactions, power con-

verters, biological systems, information processing,

secure communications, etc., controlling these com-plex chaotic dynamics for engineering applications

has emerged as a new and attractive field and has de-

veloped many profound theories and methodologies

to date. In 1990, Ott et al. [2] showed that a chaotic

attractor could be converted to any one of a large

number of possible attracting time-periodic motions

by making only small time-dependent parameter per-

0375-9601/$ see front matter 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.physleta.2004.12.028
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J.-J. Yan et al. / Physics Letters A 335 (2005) 274281 275

turbations. Chen and Dong [3] showed that the con-

ventional linear state feedback control is still an ef-

fective tool for chaotic systems. Vincent and Yu [4]

used a bangbang controller to cope with the controlproblem of chaotic system. In addition, several con-

trol methods have also been successfully applied to

chaotic systems, e.g., robust control [5], adaptive con-

trol [69], H control [10], digital redesign control

[11], etc. In the above-mentioned studies, the con-

trollers were synthesized based on the full knowledge

of the chaotic model and the upper-bounds of un-

certainties and disturbance. However, in real physical

systems or experimental situations, this may not be

easy obtained in practice and the control schemes ob-

tained using these upper-bounds of uncertainties and

disturbances yields over-conservative (high) feedbackgains [12]. At present, little attention has been given to

adaptive controlling uncertain chaotic systems without

the limitation of knowing the bounds of uncertain-

ties.

For designing a robust control, variable structure

control (VSC) is frequently adopted due to its inher-

ent advantages of easy realization, fast response, good

transient performance and insensitive to variation in

plant parameters or external disturbances. Recently,

Tsai et al. [13], Yang et al. [14], Yau et al. [1] have

successfully applied the concept of variable structurecontrol to cope with the control problem for uncer-

tain chaotic systems. However, in the above works,

the control schemes are all still under the limitation

of knowing the bounds of uncertainties and chattering

of the control input occurred in Tsai et al. [13]. For the

above reasons, it is highly desirable to propose a new

adaptive chattering free controller for chaotic systems

to not only preserve the advantages of variable struc-

ture control but also release the limitation of knowing

the bounds of uncertainties.

In this Letter, the tracking problem for uncertain

chaotic systems with unknown bounded uncertain-

ties is considered. An adaptive variable structure con-

trol (VSC) is designed to guarantee the existence of

the sliding mode for the tracking error dynamics. In

particular, the chaotic systems can be driven to ar-

bitrary trajectory, even when the desired trajectories

are not located on the embedded orbits of a chaotic

system. Since the continuous control input is used in

this Letter, chattering phenomenon is removed. Mean-

while, the limitation of knowing the bounds of un-

certainties is also removed due to the adaptive mech-

anism. Finally, a numerical example is illustrated to

demonstrate the validity of the derived adaptive VSC.

Throughout this Letter, it is noted that (W) de-notes an eigenvalue of W and max(W ) represents the

max[i (W )], i = 1, . . . , n. |w| represents the absolute

value of w and W represents the Euclidean norm

when W is a vector or the induced norm when W is

a matrix. sign(s) is the sign function of s, if s > 0,

sign(s) = 1; ifs = 0, sign(s) = 0; if s < 0, sign(s) =

1. In represents the identity matrix ofn n.

2. System definition for uncertain chaotic systems

In this Letter, we consider a class of uncertain

chaotic systems with unknown bounded uncertainties

described as

xi (t) = xi+1(t ), i = 1, . . . , n 1,

(1)xn(t) = f(X,t) + f(X,t) + (t) + u(t),

where

X(t) =

x1(t),x2( t ) , . . . , xn(t)

= x(t), x ( t ) , . . . , x(n1)(t) Rnis the state vector, f(X,t) R is a given nonlinear

function of X and t, u(t) R is the control input,

f (X, t) is the unknown parameter uncertainty ap-

plied to the system, and (t) denotes the unknown

external disturbance. The superscript n denotes the or-

der of differentiation. In many previous reports, the

upper-bounds of parameter uncertainties and external

disturbance must be obtained in advance. However, in

this study, this limitation of knowing the upper-bounds

of uncertainties will be released.

Remark 1. (1) is not restrictive. Several nonlinear

chaotic systems can be transformed into the control-

lable canonical form (1) with some state transfor-

mation [15,16]. For example, Rssler systems [15],

Lure-like system [16] and DuffingHolmes system

[22] all belong to the class defined by (1).

The control problem is to get the system to track an

n-dimensional desired vector Xd(t) (i.e., the original

nth-order tracking problem of state xd(t) as discussed


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276 J.-J. Yan et al. / Physics Letters A 335 (2005) 274281

in [13]),

Xd(t) =

xd1(t),xd2( t ) , . . . , xdn (t)

= xd(t), xd( t ) , . . . , x(n1)d ,which belong to a class of C function on [t0, ]. Let

us define the tracking error as

E(t) = X(t) Xd(t)

=

x(t) xd(t), x(t) xd(t),. . .,

x(n1)(t) x(n1)d (t)

=

e(t), e(t),. . .,e(n1)(t)

(2)=

e1(t),e2( t ) , . . . , en(t)

.

The control goal considered in this Letter is that

for any given target orbit Xd(t), a VSC is designed in

spite of the unknown bounded uncertainties, such that

the resulting tracking error vector satisfies

(3)limt

E(t) = limt

X(t) Xd(t) 0.Before proceeding with the main result of this Let-

ter, the following assumption is made:

Assumption 1. For the nonlinear function f(X,t),

parameter uncertainty f (X, t), the disturbance (t)

and xd(t), there exist a positive constant large

enough such that ddtf(X,t) + f (X, t) + (t) x(n+1)d (t)

(4) < .

Remark 2. Generally speaking, Assumption 1 is rea-

sonable and not restrictive. Since the constant is only

introduced to prove Theorem 1 later and does not ap-

pear in our proposed controller, we can suppose that

is a constant large enough (i.e., ) such that

Assumption 1 is always satisfied.

Using the concept of extended systems, the stan-

dardized state-space equations of the error states can

be obtained as

ei (t) = ei+1(t), 1 i n 1,

en(t) = xn(t) xd n(t)

= f(X,t) + f (X, t) + (t) + u(t) x(n)d (t)

= en+1,

(5)

en+1(t) =d

dt

f(X,t) + f (X, t) + (t)

x

(n+1)d (t) + u(t).

Furthermore, we arrange system (5) in matrix equa-tion form as

(6)E(t) = AE(t) + BF + Bu(t),

where

E(t) = [ e1 e2 . . . en+1 ]T R(n+1)1

is the extended error vector. Consequently, we have

u(t) = u(t),

A =

0 1 0 0

0 0 1 0 0......

.... . .

. . ....

0 0 0

(n+1)(n+1)

,

B =

0...

0

1

(n+1)1

,

F =d

dt

f(X,t) + f(X,t) + (t)

x

(n+1)d (t).

3. Sliding surface design

In consequence, for the adaptive chattering free

VSC design for uncertain chaotic systems with un-

known bounded uncertainties, there exist two major

phases. First, we need to select an appropriate switch-

ing surface such that the sliding motion on the mani-

fold has the desired properties. Second, we need to de-

termine an adaptive continuous control law such that

the existence of the sliding mode can be guaranteed

even without knowing the upper-bounds of uncertain-

ties.Now we select a switching function s(t) corre-

sponding to E(t) in the extended error space as fol-

lows:

(7)s(t) = E(t)

t0

( A + B K )E()d,

where s(t) R; and K R1(n+1) are constant ma-

trices to be designed. = [1 2 . . . n+1] is chosen


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such that B is nonzero. K is the control feedback

gain matrix to be determined later so that the error

state can fit the performance of control in the sliding

mode. When the system operates in the sliding mode,it satisfies the equations [17,18]

(8)s(t) = 0 and s(t) = 0.

Consequently, the equivalent control ueq(t) in the slid-

ing manifold is obtained by differentiating (7) with

respect to time and substituting from (6)

s(t) =

E(t) AE(t) BK E(t)

(9)= Bueq(t) BKE(t) + BF = 0.

Therefore, the equivalent control ueq(t) in the slidingmode is given by

(10)ueq(t) = KE(t) F.

Substituting ueq(t) into (6), the following sliding

mode equation is obtained as

(11)

E(t) = (A + BK)E(t).

It is obvious that

RankAn1B..

. A

n2

B

..

.

..

. A

1

B

..

. B= Rank[I(n+1)] = n + 1.

Thus, (A,B) is controllable. It do exist a parameter

vector K such that the maximum real part eigenvalue

of (A + BK ) is negative, that is, max(A + BK) < 0.

Furthermore, we can easily assign the system perfor-

mance in the sliding mode just by selecting an appro-

priate matrix K using any pole assignment method.

4. Adaptive sliding mode control design

Once a proper switching plane has been decided

with appropriate matrices and K. It followed by de-

signing an adaptive continuous variable structure con-

troller to not only derive the system trajectories onto

the sliding surface without chattering, but also remove

the limitation of knowing the bounds of the uncertain-

ties in advance.

Before proceeding to the adaptive sliding mode

control design, the Barbalat lemma is provided.

Lemma 1 (Barbalat lemma) [19]. If f : R R is a

uniformly continuous function fort 0 and if the limit

of the integral

(12)limt

t0

f ()dexists and is finite, then

(13)limt

f(t) = 0.

To ensure the occurrence of the sliding motion, an

adaptive scheme is proposed as

u(t) = u(t) = KE (B)1| B| sign(s),

(14)u(0) = u0,

where > 1 and u0 is the bounded initial value of

u(t). The adaptive law is

(15)

= q1| B|s(t), (0) = 0,

where 0 is the bounded initial value of . q is positive

constant specified by the designer. The adaptive con-

trol scheme can be also rewritten in the integral form

as

(16)

u(t) =

t

0

K E (B)1| B| sign(s)dt + u0and

(17) = q1t

0

| B||s|

dt + 0.

Remark 3. In the conventional variable structure con-

trol, the control scheme is often discontinuous and the

feedback gains need to switch with infinite switch-

ing frequency. However, infinite switching frequency

cannot be implemented because of the existence of in-

herent delay and other problems [20]. This will cause

chattering in the sliding mode. Chattering is highly

undesirable because it may excite high-frequency un-

modelled plant dynamics, which probably leads to

unforeseen instability. However, the adaptive control

scheme proposed as (14) or (16) is continuous. There-

fore, it does not need to switch as infinite switching

frequency and the chattering in the sliding mode will

be removed.


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In the following, the proposed adaptive scheme (14)

will be proved to be able to derive the extended error

system onto the sliding mode s(t) = 0.

Theorem 1. Consider the extended error system (6)

with unknown bounded uncertainties, this system is

controlled by the adaptive controller u(t) in (14) with

adaptation law (15). Then the system trajectory con-

verges to the sliding surface s(t) = 0.

Proof. Consider the following Lyapunov function

candidate

(18)V(t) =1

2

s2 + qe2

.

Taking the derivative of V(t) with respect to time t,one has

V(t) = ss + qee

= s(BF BK E + Bu) + qee

|s|| B||F| sBKE + s Bu + qee

(19) |s|| B| sBKE + s Bu + qee.

Now define e = denote the adaptation error.

Since is constant, thus the following expression

holds:

(20)e =

=

.

Inserting (14) and (15) into the right-hand side of in-

equality (19) this yields

V(t) |s|| B| sBKE + s Bu + qee

= |s|| B|

e ( ) + | B||s|

| B||s| + q

e

(21)= (1 )| B||s|.

By Eq. (21) and > 1, one can obtain

(22)V(t)( 1)| B||s| = w(t) 0,

where w(t) = ( 1)| B||s|. Integrating the above

equation from zero to t, it yields

(23)V (0) V(t) +

t0

w()d

t0

w()d.

As t goes infinite, the above integral is always less than

or equal to V (0). However, V (0) is positive and finite,

thus according to Barbalat lemma (see Lemma 1), we

obtain

(24)limt

w(t) = limt

( 1)| B||s| = 0.

Furthermore, (t) > 0 for all t > 0 and > 1 is cho-

sen. Thus Eq. (24) implies s(t) 0 as t . Hence

the proof is achieved completely. 2

Remark 4. Since from theoretical point of view, s

will not be exactly equal to zero in finite time, thus

the adaptive parameter will increase (even if s is a

very small number) until s = 0. A simple way for over-

coming this disadvantage is to modify the adaptive law

(15) by dead-zone technique [21] as

(25)

(t) =

q1| B||s|, |s| ,

0, |s| < ,

where is a small positive constant.

5. An illustrative example: chaos control to

arbitrary desired trajectories

In this section, a numerical experiment to demon-

strate the effectiveness of the proposed adaptive con-

trol scheme. Fourth-order RungeKutta method isused to integrate the differential equations with the

step 0.01. The system interested here is an uncertain

DuffingHolmes system with unknown bounded un-

certainties, which can be described by

x1(t) = x2(t),

(26)

x2(t) = p1x1(t) p2x2(t) x31 (t) + q cos(w1t)

+ f (X, t) + (t) + u(t).

We will show that, by using the proposed adaptive

VSC (14) with adaptation law (15), one can control a

chaotic system to arbitrary trajectories even under theinfluence of parameter uncertainty and external distur-

bance. Obviously, the corresponding nominal system

of(26) is as follows:

x1(t) = x2(t),

(27)

x2(t) = p1x1(t) p2x2(t) x31 (t) + q cos(w1t).

The parameters p1, p2, q and w1 are chosen p1 = 1,

p2 = 0.25, q = 0.3 and w1 = 1.0, respectively, in this


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simulation to ensure the existence of chaos in the ab-

sence of control. Furthermore, there exists a bounded

region R2 containing the whole attractor such that

every orbit of system (27) never leaves it [22].The control objective is to drive the uncertain

chaotic system (26) to the following trajectory:

(28)xd(t) = A sin(w2t).

Obviously, the desired trajectory xd(t) with A = 1,

w2 = 1.1 does not belong to the embedded orbits of

the strange attractor. According to (6), the extended

error system can be described as

(29)E(t) = 0 1 0

0 0 1

0 0 0 A

E(t) +0

0

1B

F +0

0

1u(t),

where

F =d

dt

p1x1 p2x2 x

31 + q cos(w1t)

+ f(X,t) + (t)

x(3)d (t).

The uncertainties f(X,t) = 0.1x2(t) and bound-

ed (t) = 0.2sin(t) will be used for simulation.

Since the every orbit of (27), (1)(t) and x(3)d (t)

are bounded, (4) is always easy to be satisfied with

a very large constant < . Thus according to(7), we select = [1 1 1] such that B = 1 = 0

and K = [6 11 6] such that max(A + BK ) =

1 < 0. Therefore, we have a stable sliding mode and

the switching surface equation is

s(t) = [ 1 1 1 ]E(t)

(30)

t0

[ 6 10 5 ]E()d.

From (15) and (16), the control input is determinated

as

(31)u(t) =

t0

[ 6 11 6 ]E sign(s)

dt,

where = 1.1 > 1 and u0 = 0.

The adaptive law is

(32) =

t0

|s| dt, where q = 1 and 0 = 2.

Fig. 1. The time response for the switching function s(t) of the con-trolled system.

Fig. 2. The phase plane of x1(t) versus x1(t).

The simulation results with initial value x0 =

[1 2.5]T are shown in Figs. 16. Figs. 1 and 2 show,

respectively, the corresponding s(t) and phase plane

(x1(t) versus x1(t)) of controlled system under the

proposed adaptive VSC control. Fig. 3 shows the state

responses for the controlled DuffingHolmes system.

The extended error state time responses and control

input are shown in Figs. 4 and 5, respectively. The

adaptation parameter (t) is shown in Fig. 6. From the

simulation, it shows that the proposed adaptive VSC

works well for the uncertain DuffingHolmes system.

In particular, it is worthy of note that, no information

of upper-bounds of uncertainties is used in our con-


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Fig. 3. The state responses of the controlled DuffingHolmes sys-

tem.

Fig. 4. The time responses of error states.

trol design. Also it shows that the chattering does not

appear due to the continuous control.

6. Conclusions

In this Letter, a new robust controller for uncertain

chaotic has been proposed. Based on the Lyapunov

stability theory and Barbalat lemma, an adaptive vari-

able structure controller is designed for the tracking

problem of the state vector to a desired vector in the

Fig. 5. Time response of the control input.

Fig. 6. Time response of adaptation parameter .

state space. Compared with existing other chaos con-

trol laws, the proposed chattering free variable struc-

ture control can be also achieved through adaptive

control without knowing the upper-bounds of the un-

certainties in advance. Finally, a numerical example isgiven to verify the validity of the developed controller.

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