Stabilizing a memristor-based chaotic systemby switching control and fuzzy modeling

MA Fuling, XING Dahong

(Department of Public Courses,Zhongshan Huoju Polytechnic,Zhongshan 528436,China)

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Stabilizing a memristor-based chaotic systemby switching control and fuzzy modeling

MA Fuling, XING Dahong

(Department of Public Courses,Zhongshan Huoju Polytechnic,Zhongshan 528436,China)

Abstract:We investigate the stabilization of a memristor-based chaotic system.Based on the system,a T-S model is established and a switching controller is designed.Then,by Lyapunov stability theory,one can acquire a criterion to guarantee that the conclusion holds.Finally,numerical results are demonstrated to verify the effectiveness of our method.

Key words:chaotic system; memristor; fuzzy modeling; switching control

CLC number: TM 132Document code: AArticle ID: 1000-5137(2016)01-0001-06

1Introduction

During the recent decades,memristor has been focused on by many researchers.It was postulated as the missing fourth passive circuit element by Chua in 1971[1].In the following almost 40 years,scientists had aimed at inventing a practical device.In 2008,a large development that a passive two terminal physical implementation was immediately related to memristor theory was achieved by Hewlett-Packard Labs[2].Many researchers show great interest in its physical properties and applications on computer[3-17].For instance,Saeed Afshar presented a memristor-based neuromorphic competitive (mNCC) circuit which utilized a single sensor and could control the output of N actuators delivering optimal scalable performance,and immunity from device variation and environmental noise[3].Adam Rak and Gyorgy Cserey introduced a new simulation program with integrated circuit emphasis macromodel of the recently physically implemented memristor[4].Sung Hyun Jo experimentally displayed a nanoscale silicon-based memristor device and showed a hybrid system composed of complementary metal-oxide semiconductor neurons[5].

In addition,numerous authors have investigated the dynamical properties and synchronization of memristor-based systems[18-23].Wu analyzed the dynamic behaviors for a class of memristor-based Hopfield networks and supplied several sufficient conditions to guarantee a novel memristive neural network for realizing winner-take-all behavior[18-19].Some others considered the problem of fuzzy modeling and impulsive control chaotic system and memristorbased chaotic system[24-28]and Tanaka K provided approaches to fuzzy modeling and design[29].In Ref.[28],Huang researched into the problem of intermittent control of a memristor-based Chua′s oscillator and presented the oscillator as the T-S fuzzy model system.Nevertheless,in this paper we will design a switching controller which is different from the controller in letter[28]and construct a Lyapunov function for stabilizing a fourth-order memristor-based chaotic system[30].We also utilize fuzzy modeling and analysis for this system.

The rest of this paper is presented as follows.In Section 2,a memristor-based system is introduced and a T-S fuzzy model is built according to the system.Furthermore,in Section 3,a switching controller is given to stabilize the memristor-based chaotic system and a criterion obtained by Lyapunov function theory is provided.Numerical simulation examples are given to demonstrate the scheme in Section 4. And finally,conclusions are drawn in Section 5.

2Preliminaries

Several years ago,Itoh and Chua[30]put forward a memristor-based chaotic system by replacing the Chua′s diode.Fig.1 displays a Canonical Chua′s oscillator with a flux-controlled memristor,which is depicted by

Figure 1　Canonical Chua′s oscillator with aflux-controlled memristor[30]

(1)

wecanchangesystem(1)into

(2)

where

(3)

(4)

Whenα=4,β=1,γ=0.65,a=0.2 andb=10,the system (2) has a chaotic attractor as shown in Fig.2.

Figure 2　The chaotic attractor of the memristor oscillator

In the following,we will establish the T-S fuzzy model for the memristorbased chaotic system and gain its stabilization by switching control under the Lyapunov function sense.

Through Ref.[30],we can construct the fuzzy model for the system (2) as follows.

wherex(t)=(x1(t),x2(t),x3(t),x4(t))T,Miisfuzzysetfori=1,2,

With a center-average defuzzifier,the overall fuzzy system is described as

(5)

where the grade of the membership function

We add a switching control to the memristor-based chaotic system (2) for its stabilization as follows.

(6)

whereBis a symmetric matrix andu(t) is the switching feedback control gain given that

(7)

whereki(i=1,2) is a scalar denoting control strength.

In the following,we will give the unique theorem of this paper,which expresses our main idea.

3Main Result

In this section,one utilizes the Lyapunov function method to obtain the stability of the trivial solution of the system (6).The main result is presented as follows.

Theorem 1Assume that there exists a matrixBmentioned in Section 2 such that

(8)

Then the trivial solution of the system (6) is asymptotically stable,i.e.,the memristor-based chaotic system (2) is stabilized.

ProofConsider the Lyapunov function candidate

(9)

The derivative of (9) with respect to t along the trajectories of the system (6) is calculated as follows.

(10)

It is easy to deduce that if there exists a symmetric matrixBsuch that

(11)

Thatis,thetrivialsolutionofthesystem(6)isstable,andthememristor-basedchaoticsystem(2)isstabilized.Theproofiscompleted.

4Numericalsimulations

Toillustratetheproposedcontrolscheme,numericalsimulationsarecarriedoutbyusingMATLAB.Selectingtheparametersα=4,β=1,γ=0.65,a=0.2,b=10andlettingk1=1,k2=2,weachievethat

and

Via calculation,we get a

Let the initial condition bex=[2,5,9,15]T.The stability of the trivial solution of the memristor-based chaotic system under switching control is shown in Fig.3,which demonstrates the effectiveness of our method.

Figure 3　Memristor-based chaotic system under switching control

5Conclusions

In this paper,the control problem and fuzzy modeling of memristor-based chaotic system are considered.Furthermore,a switching controller is designed to stabilize the original system.Finally,numerical results are acquired to illustrate the effectiveness of our method.

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(責任編輯:馮珍珍,包震宇)

通過切換控制和模糊建模鎮定基于憶阻的混沌系統

馬芙玲, 刑大紅

(中山火炬職業技術學院 公共課部,中山 528436)

摘要:研究了以憶阻器為基礎的混沌系統的穩定性.基于該系統建立了一個T-S模型并設計了一個開關控制器.通過李雅普諾夫穩定性理論,取得了一個以保證結論成立的準則.最后用數值結果驗證了本研究方法的有效性.

關鍵詞:混沌系統; 憶阻器; 模糊建模; 切換控制

Corresponding author:MA Fuling,Department of Public Courses,Zhongshan Huoju Polytechnic,No.60,Zhongshan harbor Rd.,Zhongshan 528436,China,E-mail:flima@sina.com

Received date: 2015-11-24