Exponential stabilization of chaotic systems with delay by periodically intermittentcontrolChuandong Li, Xiaofeng Liao, and Tingwen Huang Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 17, 013103 (2007); doi: 10.1063/1.2430394 View online: http://dx.doi.org/10.1063/1.2430394 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/17/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Design of coupling for synchronization in time-delayed systems Chaos 22, 033111 (2012); 10.1063/1.4731797 Exponential synchronization of stochastic neural networks with leakage delay and reaction-diffusion terms viaperiodically intermittent control Chaos 22, 013124 (2012); 10.1063/1.3685523 Synchronization of chaotic systems with delay using intermittent linear state feedback Chaos 18, 033122 (2008); 10.1063/1.2967848 Comment on “Exponential stabilization of chaotic systems with delay by periodically intermittent control”[Chaos17, 013103 (2007)] Chaos 18, 018101 (2008); 10.1063/1.2896089 Robust anti-synchronization of a class of delayed chaotic neural networks Chaos 17, 023113 (2007); 10.1063/1.2731306

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Exponential stabilization of chaotic systems with delay by periodicallyintermittent control

Chuandong Lia�

School of Computer, Hangzhou Dianzi University, Hangzhou 310018, Chinaand College of Computer Science, Chongqing University, Chongqing 400030, China

Xiaofeng LiaoCollege of Computer Science, Chongqing University, Chongqing 400030, China

Tingwen HuangTexas A&M University at Qatar, c/o Qatar Foundation, P.O. Box 5825, Doha, Qatar

�Received 3 September 2006; accepted 7 December 2006; published online 25 January 2007�

This paper studies the exponential stabilization problem for a class of chaotic systems with delay bymeans of periodically intermittent control. A unified exponential stability criterion, together with itssimplified versions, is established by using Lyapunov function and differential inequality tech-niques. A suboptimal intermittent controller is designed with respect to the general cost functionunder the assumption that the control period is fixed. Numerical simulations on two chaotic oscil-lators are presented to verify the theoretical results. © 2007 American Institute of Physics.�DOI: 10.1063/1.2430394�

Chaos stabilization is important and useful in appliedphysics, electronic engineering, and biological sciences.Recently, several new control schemes including impul-sive control and switch control have been developed tostabilize chaotic systems. In this paper, the so-called pe-riodically intermittent control is introduced to stabilize aclass of delayed chaotic systems. In this new study, anexponential stabilization condition together with a subop-timal controller design is derived by means of rigorousmathematical analysis on periodically switched delayeddifferential equations. Two numerical examples will bepresented, to show the effectiveness of the theoreticalresults.

I. INTRODUCTION

A large number of monographs and papers on the stabi-lization of the chaotic systems without or with delays havebeen published1–5 since the seminal paper of Ott, Grebogi,and Yorke.6 In these publications, many control techniquesincluding continuous feedback and discontinuous feedbackhave been developed. Recently, discontinuous feedback con-trols, such as occasional bang-bang control �OBBC�,7 occa-sional proportional feedback �OPF�,8–10 impulsive control,3,4

switch control,5 and intermittent control,11 have been attract-ing much attention. The main idea of OBBC and OPF is touse repeated application of discrete pulses �with nonzero du-ration� of an available system parameter to stabilize the sys-tem. Switch control is naturally suitable to switched systems,while impulsive control and intermittent control can be ap-plied to most dynamical systems, particularly on sampling-data systems. The common intermittent control is different

from the impulsive control because impulsive control is ac-tivated only at some isolated moments �namely, it is of zeroduration�; the common intermittent control has a nonzerocontrol width. Generally, discontinuous control dynamicalsystems are subject to the more complex mathematical mod-els with more complex dynamical behaviors. For example,an impulsive control system is modeled by impulsive differ-ential equation; a system with intermittent control is de-scribed by time-switched dynamical equations mathemati-cally.

Intermittent control is a straightforward engineering ap-proach to process control of any type—an approach that hasbeen used for a variety of purposes in such engineering fieldsas manufacturing, transportation, and communication. Re-cently, intermittent control has been introduced to controlnonlinear dynamical systems.11 However, to the best of ourknowledge, there is little theoretical analysis of intermittentcontrol dynamical systems in the literature. In Ref. 11, theauthors investigated numerically chaos synchronization un-der the condition that the interacting systems �i.e., masterand slave systems� are coupled intermittently. In recentwork,12 the stabilization problem of chaotic systems de-scribed by ordinary differential equations �ODE� was ad-dressed by means of periodically intermittent control, wherean exponential stabilization condition for a class of periodi-cally intermittent control systems was derived in virtue of theLyapunov function and differential inequality techniques. Inthis paper, a further step is taken to formulate and analyzethe intermittent control systems with time delay. In this newstudy, a general and effective sufficient condition, togetherwith its simplified versions, for exponential stability of aclass of periodically intermittent dynamical systems with de-lay, is derived by means of a rigorous mathematical analysison the time-switched functional differential equations. A sub-optimal stabilization algorithm is then suggested based on

a�Author to whom correspondence should be addressed. Electronic mail:licd@cqu.edu.cn

CHAOS 17, 013103 �2007�

1054-1500/2007/17�1�/013103/7/$23.00 © 2007 American Institute of Physics17, 013103-1

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the exponential stability conditions. Numerical simulationsare reported to show their good agreement with the theoret-ical results.

The rest of this paper is organized as follows: In the nextsection, we formulate the problem of intermittent control ofnonlinear systems with delays, and introduce some necessarypreliminaries. We then establish, in Sec. III, a general expo-nential stability criterion, together with its simplified ver-sions, for periodically intermittent control systems with de-lay. The simplified criterion is then used in Sec. IV for thedesign of a suboptimal controller provided that control pe-riod is known. As an application, intermittent control of twochaotic systems is discussed in Sec. V. Finally, some conclu-sions and remarks are drawn in Sec. VI.

II. PROBLEM FORMULATION AND PRELIMINARIES

Consider a class of delayed nonlinear systems describedas

�x�t� = Ax�t� + f�x�t�� + g�x�t − ��� + u�t� ,

x��� = ����, � � �− �,0� ,� �1�

where x�Rn presents the state vector, f :Rn→Rn andg :Rn→Rn are both continuous functions satisfying f�0�=0and g�0�=0, and positive constants exist � ,� such that�f�x�����x� and �g�x�����x� for x�Rn. u�t� denotes theexternal input of system �1�, and ��0 is the time delay. It isknown that many chaotic systems such as, e.g., the Mackey-Glass equation �see Ref. 13�, Ikeda equation �see Ref. 14�,and chaotic delayed neural network models reported byGilli15 and Lu,16 are system �1� with u�t�=0.

In order to stabilize the origin of the system �1� bymeans of periodically intermittent feedback control, we as-sume that the control imposed on system �1� is of the form

u�t� = �− kx�t� , nT � t � nT + 0.5T;

0, nT + 0.5T � t � �n + 1�T ,� �2�

where k�0 is the control strength, T�0 denotes the controlperiod. In the sequel, our objective is to design suitable T andk such that system �1� can be stabilized at the origin. Withcontrol law �2�, the system �1� can be rewritten as

�x�t� = �A − kI�x�t� + f�x�t�� + g�x�t − ��� , nT � t � nT + 0.5T;

x�t� = Ax�t� + f�x�t�� + g�x�t − ��� , nT + 0.5T � t � �n + 1�T ,� �3�

where I denotes the identity matrix. Equation �3� represents aclass of periodically switched systems where the switchingrule only depends on the time. However, to the best of ourknowledge, the �exponential� stability analysis on system �3�has received little attention so far. In the present paper, wewill establish a unified exponential stability criterion for sys-tem �3� by the Lyapunov function and differential inequalitytechniques.

To make the paper self-contained, the following defini-tion and lemmas are necessary.

Definition 1: The zero solution of Eq. �1� is said to beglobally exponentially stable if two constants exist, M������0, �0 such that

�x�t�� � M�����exp− t, t � 0,

where ���=sup−����0������.Lemma 1 (Halanay Inequality17): Let : ��−� ,��

→ �0,�� be a continuous function such that

�t� � − a�t� + b max t

is satisfied for t �. If a�b�0, then

�t� � �max ��exp− �t − ��, t � ,

where max t=supt−����t���, and �0 is the smallest realroot of the equation

a − − b exp� = 0.

Lemma 2: Let : ��−� ,��→ �0,�� be a continuousfunction such that

�t� � a�t� + b max t

is satisfied for t �. If a�0,b�0, then

�t� � max t � ���exp�a + b��t − ��, t � ,

where max t=supt−����t���.Proof: Note that

�t� � �a + b�max t.

Integrating both sides of this inequality from � to t, we have

�t� � ��� + ��

t

�a + b�max �d� ,

which implies

�t� � max t � ��� + ��

t

�a + b�max �d� .

From the Gronwall-Bellman inequality,18 we then derive

�t� � max t � ���exp�a + b��t − ��, t � .

Lemma 3 (Sanchez and Perez19): Given any real matri-ces �1 ,�2 ,�3 of appropriate dimensions and a scalar ��0such that 0��3=�3

T, then the following inequality holds:

�1T�2 + �2

T�1 � ��1T�3�1 + �−1�2

T�3−1�2.

013103-2 Li, Liao, and Huang Chaos 17, 013103 �2007�

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Throughout this paper, we use PT ,�M�m��P� to denote thetranspose and the maximum �minimum� eigenvalue of asquare matrix P, respectively. The vector �or matrix� norm istaken to be Euclidian, denoted by �·�. We use P�0��0,�0, 0� to denote a symmetrical positive �negative, semin-egative, semipositive� definite matrix P.

III. EXPONENTIAL STABILITY ANALYSIS

This section addresses the exponential stability problemof the delayed system �3� by means of the aforementionedlemmas. The main results are stated as follows:

Theorem 1: Suppose that a continuous and positive-definite function exists V :Rn→R+ satisfying the followingconditions:

�i� c1�x�r�V�x�����x��, where � :R+→R+ is a continu-ous and strictly monotonic increasing function with��0�=0; c1 and r are some positive constants.

�ii� positive constants exist gi and hi�i=1,2� such that thederivative of V along the trajectories of system �3�satisfies, for n=0,1 ,2 , . . .,

�V�x�t�� � − g1V�x�t�� + h1V�x�t − ��� ,

nT � t � nT + 0.5T ,

V�x�t�� � g2V�x�t�� + h2V�x�t − ��� ,

nT + 0.5T � t � �n + 1�T .

�iii� g1�h1 and �g2+h2, where is the smallest realroot of the equation

g1 − − h1 exp� = 0.

Then, the origin of the system �3� is globally expo-nentially stable, and moreover,

V�x�t�� � �max−����0 V�x�����

�exp− 12 � − g2 − h2�t, ∀ t 0.

Proof: Let �t�=V�x�t��. When nT� t�nT+0.5T, bycondition �ii�, we have

�t� � − g1�t� + h1�t − ��

� − g1�t� + h1 maxt−����t ��� .

It follows from Lemma 1 that

�t� � �maxnT−����nT ����exp− �t − nT� , �4�

where is the smallest real root of the equation g1−−h1 exp�=0.

When nT+0.5T� t� �n+1�T, we have from condition�ii�

�t� � g2�t� + h2�t − ��

� g2�t� + h2 maxt−����t ��� .

Then, applying Lemma 2 yields

�t� � maxt−����t ���

� �maxnT+0.5T−����nT+0.5T ����

�exp�g2 + h2��t − nT − 0.5T� . �5�

Now, we estimate �t� based on Eqs. �4� and �5�.

�a� For 0� t�0.5T,

�t� � �max−����0 ����exp− t .

�b� For 0.5T� t�T,

�t� � maxt−����t ���

� �0.5T�exp�g2 + h2��t − 0.5T�

� �max−����0 ����

�exp�g2 + h2��t − 0.5T� − 0.5T .

�c� For T� t�T+0.5T,

�t� � �maxT−����T ����exp− �t − T�

� �max−���� ����

�exp0.5�g2 + h2�T − 0.5T − �t − T�

= �max−���� ����

�exp0.5�g2 + h2�T − �t − 0.5T� .

�d� For T+0.5T� t�2T,

�t� � maxt−����t ���

� �T + 0.5T�exp�g2 + h2��t − T − 0.5T�

� �max−����0 ����exp�g2 + h2��t − T� − T .

By induction, we have the following estimate of �t�for any integer n.

�e� For nT� t�nT+0.5T,

�t� � �max−����0 ����

�exp− �t − 0.5nT� + 0.5�g2 + h2�nT .

Note that nT� t in this case, we obtain

�t� � �max−����0 ����exp− 0.5� − g2 − h2�t .

�f� For nT+0.5T� t� �n+1�T,

�t� � �max−����0 ����

�exp− 0.5�n + 1�T

+ �g2 + h2��t − 0.5�n + 1�T� .

Note that t� �n+1�T in this case, we also obtain

�t� � �max−����0 ����exp− 0.5� − g2 − h2�t .

Therefore, for any t 0,

V�x�t�� � �max−����0 V�x�����exp− 12 � − g2 − h2�t .

Based on Definition 1 and the first condition in thetheorem, we conclude the proof.

Remark 1: g2 is a positive number because the systemconsidered in this paper is assumed to be chaotic. Also, for asuccessful stabilization, k should be taken to be large enoughso that the first subsystem is exponentially stable, which canguarantee the existence of a positive number g1.

Remark 2: In the theorem, the parameter implies theestimated rate of exponential convergence of the controlledsystem, which depends on the control parameters k and �.

013103-3 Stabilization of delayed systems Chaos 17, 013103 �2007�

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Remark 3: The proof of Theorem 1 provides a unifiedapproach to analyze the exponential stability of system �3�.In practice, we can select an appropriate positive-definitefunction V to derive a sufficient condition guaranteeingexponential stability for a given system, as shown inTheorem 2.

Remark 4: The exponential stability criterion describedin Theorem 1 is independent of the control period T.

Theorem 2: Suppose that positive constants exist ri ,gi�i=1,2� such that

�i� A+AT+ �−2k+2�+g1+r1−1�I�0;

�ii� A+AT+ �2�−g2+r2−1�I�0;

�iii� g1�r1�2, and �g2+r2�2, where is the smallestreal root of the equation

g1 − − r1�2 exp� = 0.

Then, the origin of system �3� is globally exponen-tially stable, and moreover,

�x�t�� � ���exp− 14 � − g2 − r2�2�t, t 0.

Proof: We construct the following positive-definite func-tion candidate

V�x� = �x�2. �6�

Obviously, condition �i� in Theorem 1 holds with c=1, r=2,and ���x��= �x�2. In the sequel, we will prove that conditions�ii� and �iii� in Theorem 1 are also satisfied. For this purpose,we calculate the derivative of Eq. �6� with respect to time talong the trajectories of system �3� as follows.

When nT� t�nT+0.5T, for n=0,1 ,2 , . . .,

V�x�t�� = 2xT�t�x�t�

= 2xT�t���A − kI�x�t� + f�x�t�� + g�x�t − ����

= xT�t���A − kI� + �A − kI�T�x�t� + 2xT�t�f�x�t��

+ 2xT�t�g�x�t − ��� .

Note that by virtue of Lemma 2 we have

2xT�t�f�x�t�� � 2�xT�t�x�t� ,

and

2xT�t�g�x�t − ��� � r1−1xT�t�x�t� + r1�2xT�t − ��x�t − �� .

Therefore, we obtain

V�x�t�� � xT�t���A − kI� + �A − kI�T

+ �2� + r1−1�I�x�t� + r1�2V�x�t − ���

= − g1V�x�t�� + xT�t���A − kI� + �A − kI�T

+ �2� + r1−1 + g1�I�x�t� + r1�2V�x�t − ���

� − g1V�x�t�� + r1�2V�x�t − ��� .

Similarly, when nT+0.5� t� �n+1�T, we have

V�x�t�� � g2V�x�t�� + r2�2V�x�t − ��� .

Letting h1=r1�2 and h2=r2�2. The remaining proof is trivialand omitted.

Remark 5: If we replace the first two conditions in Theo-

rem 2 by the scalar equalities g1=2k−�M�A+AT�−2�−r1−1

and g2=�M�A+AT�+2�+r2−1, then, Theorem 2 also holds.

Particularly, if we further choose r1=r2=1/�, Theorem 2will reduce the following corollary.

Corollary 1: Let be the smallest real root of the equa-tion

2k − �M�A + AT� − 2� − � − − � exp� = 0.

If

= − �M�A + AT� − 2� − 2� � 0,

then, the origin of system �3� is globally exponentially stable,and moreover,

�x�t�� � ���exp− 14 t, t 0.

IV. CONTROLLER DESIGN WITH OPTIMIZATIONCONSIDERATION

This section addresses the design of controller �2� withoptimization consideration. We first notice that the proposedcontroller �2� contains two parameters, namely, controlstrength k and control period T. However, we know fromRemark 4 that the proposed stability condition is indepen-dent of time T. Therefore, it is assumed that the period T isfixed and known in the sequel. In addition, in order to main-tain the convergent speed of the controlled system, a smallestexponential convergence degree 0 of the controlled systemis predefined.

Consider the cost function of the form

R = �0

�

�u�t��dt . �7�

Our objective is then to find the suitable control strength ksuch that R is minimized, provided that control period T andsmallest exponential convergence degree 0 are both known.Based on Corollary 1, the following dynamical optimizationproblem is immediate,

�P1��min R = �

0

+�

�u�t��dt

s.t.,

= * − �M�A + AT� − 2� − 2� 0,

* = inf � R+:2k − �M�A + AT� − 2� − � − − � exp��� = 0 .

Note that

R = �0

+�

�u�t��dt

= �n=0

+� �nT

nT+0.5T

�− kx�t��dt

= k�n=0

+� �nT

nT+0.5T

�x�t��dt

013103-4 Li, Liao, and Huang Chaos 17, 013103 �2007�

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� k����n=0

+� �nT

nT+0.5T

exp�−1

4t�dt

= k���4

�1 − exp�−

1

8T���

n=0

+�

exp�−1

4nT�

=4���k

�1 + exp− 18 T� .

Let

R1 =k

�1 + exp− 18 T� .

Then, a suboptimization problem associated with �P1� canbe formulated as

�P2��min R1 =k

�1 + exp− 18 T� ,

s.t.,

= * − �M�A + AT� − 2� − 2� 0,

2k − �M�A + AT� − 2� − � − * − � exp�*�� = 0.

This is a constrained nonlinear minimization problem thatcan be solved numerically by the program FMINCON inMatlab.

V. NUMERICAL EXAMPLE

In this section, we study numerically the stabilization ofthe Mackey-Glass oscillator13 and Ikeda-type oscillator byapplying the presented results. The programs DDE23 andFMINCON in Matlab are used to integrate the delay differ-ential equations and to solve numerically optimization prob-lem �P2�, respectively.

Example 1: Consider the Mackey-Glass oscillator whichis described by

FIG. 1. The chaotic attractor of the Mackey-Glass oscillator described byEq. �8� with time interval �100, 1000� and initial value x���=0.5, �� �−17,0�.

FIG. 2. The time response curve of the Mackey-Glass oscillator with theperiodically intermittent control u�t�=−8.2546x�t� when nT� t�nT+0.5T;and u�t�=0 when nT+0.5T� t� �n+1�T, n=0,1 ,2 , . . ..

FIG. 3. The control signal under the control law described in the caption ofFig. 2.

FIG. 4. The chaotic attractor of the Ikeda-type oscillator described by Eq.�9� with time interval �100, 2000� and initial value x���=1, �� �−2,0�.

013103-5 Stabilization of delayed systems Chaos 17, 013103 �2007�

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x�t� = − ax�t� +b · x�t − ��

1 + xc�t − ��. �8�

When a=0.1, b=0.2, c=10, and �=17, system �8� exhibitschaotic behavior,13 as shown in Fig. 1. It is easy to calculatethe parameters required in �P2�:

� = 0, � = 0.2, �M�A + AT� = − 0.2.

For numerical simulation, we choose the control period Tand the smallest degree of exponential convergence 0 as T=4 and 0=0.1. Solving �P2�, we obtain the suboptimal con-trol strength k=8.2546 with min R1=71.3673. The time re-sponse curves of the controlled oscillator with two differentinitial conditions are shown in Fig. 2, and the correspondingcontrol signals are shown in Fig. 3.

Example 2: Consider the delay system described by

x�t� = − ax�t� + b sin�x�t − ��� . �9�

When a=0.9,b=1.5,�=2, system �8� exhibits chaotic behav-ior, as shown in Fig. 4. Note that the parameters requiredin �P2�:

� = 0, � = 1.5, �M�A + AT� = − 1.8.

In this case, we choose the control period T and smallestdegree of exponential convergence 0 as T=2 and 0

=0.005. Based on Corollary 1, we take the smallest controlstrength k=8.8. The time response curves of the controlledoscillator with two different initial conditions are shown inFig. 5, and the corresponding control signals are shown inFig. 6.

VI. CONCLUSIONS AND REMARKS

This paper has dealt with the exponential stabilizationproblem for a class of chaotic systems with delay by meansof periodically intermittent control. A general exponentialstability criterion, together with its simplified versions hasbeen obtained. Provided that the control period is known, wehave presented a suboptimal intermittent control law thatwould minimize the general cost function. It is worth notingthat the proposed method can be applied to the general non-linear systems. The future work in this endeavor will focuson the following issues: �i� analysis and design of the generalintermittent controller. The present controller is periodicallyintermittent with constant control strength, and fixed controlperiod and control width, which might be inadequate in thepractical application; �ii� controllability and then intermittentcontroller design when only the scalar output is available forfeedback.

ACKNOWLEDGMENTS

The work described in this paper was partially supportedby the National Natural Science Foundation of China �GrantNo. 60574024, 60573047, 60604007� and Natural ScienceFoundation Project of CQ �Grant No. 2006BB2228,2006BB2230�, Program for New Century Excellent Talentsin University.

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FIG. 5. The time response curve of system �9� with the periodically inter-mittent control u�t�=−8.8x�t� when nT� t�nT+0.5T; and u�t�=0 whennT+0.5T� t� �n+1�T, n=0,1 ,2 , . . ..

FIG. 6. The control signal under the control law described in the caption ofFig. 5.

013103-6 Li, Liao, and Huang Chaos 17, 013103 �2007�

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