Controlling chaos in a memristor-based Chua's circuit

Jian ZHANG, Hongbin ZHANG and Guofeng ZHANG

where

Fig. I. Chua 's circuit

II. THE MEMRISTOR-BASED CHUA'S CIRCUIT

A typical implementation of the Chua's circuit is shownin Fig.l.

(1)

Chua' sdiode

q(e/» = be/> + 0.5 (a- b) ( Ie/> + I I+ Ie/> - II)

W(e/» =dq(e/» /de/> = { a, le/>I < Ib, le/>I > 1.

It turns out that if the Chua diode is replaced by amemristor(with the appropriate nonlinearity), we can stillobtain chaos[8] . A function representation of the circuitcontains a piece-wise linear function.

The aim of this paper is to introduce a simple, switchedcontroller for resolving the control problem of the memristor­based Chua's circuit system. A common quadratic Lyapunovfunction is introduced to design a switched controller and thefeedback gains are obtained by solving a set of linear matrixinequalities . The rest paper is organized as follows. Section IIdescribes the memristor-based Chua's circuit along with sim­ulation results . Next, a linear switched feedback controlleris designed for stabilizing memristor-based Chua's circuit insection III. Furthermore, the corresponding simulation resultsare also provided and these results illustrate the effectivenessof the proposed method. Finally, a conclusion is drawn inSection IV.

If replacing Chua's diode with an active two-terminalcircuit consisting of a conductance and a flux-controlledmemristor, then we obtain the following circuit in Fig.2.The dynamics of the above circuit can be represented bythe following equations

Abstract- In this paper, we study the problem of controllingchaos in a memristor-based Chua's circuit, which can berepresented as a linear switched system. A linear switchedcontroller is obtained by solving a set of LMIs based on acommon Lyapunov function. Finally, a numerical simulation isprovided to illustrate the effectiveness of the proposed method.

Chaotic behavior occurs in many mechanical and elec­tronic oscillators, in chemical reactions, in laser cavities,etc. However, these irregular and complex phenomena areoften undesirable. In practical applications , it is necessaryto control a chaotic system to a periodic orbit or a steadystate, but particularly the last one, in order to improve systemperformance or avoid fatigue failures of mechanical systems.There has been some increasing interest in recent years inthe study of controlling chaotic systems . In the literatures,there are two basic approaches used to the control of chaoticsystem: non-feedback control and feedback control. In thefirst method, it is demonstrated that a small parametricperturbation can tame chaos [I]. For the second method,various control methodologies have been developed by manyresearchers from a point of view of dynamic system theoryand traditional feedback control. These control algorithmsinclude differential geometric method [2],linear state spacefeedback [3], and output feedback control [4], etc.

In this paper, we consider the feedback control of amemristor-based Chua's circuit. The Chua's circuit is anextensively studied and well-understood chaotic system [5].Chua's circuit is perhaps the simplest autonomous physicaldevice that exhibits very rich and complex nonlinear dynam­ics such as bifurcations and chaos, and hence has becomesa prototype of experimental dynamics generator.

The mernristor was postulated as the forth circuit elementby Leon. O. Chua in 1971 [6]. In May 2008, HP scientists an­nounced the discovery of the missing circuit element memris­tor, acronym for memory resistor[7]. The memsristor conceptgives a simple explanation for many puzzling voltage-currentcharacteristics in nanoscale electronics . With the reportedadvance, HP is promising prototypes of ultradense memorychips in 2009.

I. INTRODUCTION

This work was supported by National Natural Science Foundation ofP. R. China (No. 60502009)

Jian ZHANG is with School of Automation Engineering , University ofElectronic Science and Technology of China, Sichuan, Chengdu, 610054,P. R. China. zhangj@ues tc.edu.cn

Hongbin ZHANG and Guofeng ZHANG are with School ofElectronic Engineering, University of Electronic Science andTechnology of China, Sichuan, Chengdu, 610054, P. R. China.zhanghb@uestc .edu .cn ;guo fengua@gma i l .com

978-1-4244-4888-3/09/$25.00 ©2009 IEEE 961

Communications, Circuits and Systems, 2009. ICCCAS 2009. International Conference on

Flux-controlledmemristor

Fig. 2. Chua 's oscillator with a flux-controlled memristor and a negativeconductance .

IfwesetL =0.08,r =0.025,R = I,C] = 0. I, C2= 0.6,G=1.5 and choose the initial conditions i (0) = 7, V2 (0) =5, l/J (0) = 1.6, VI (0) = 8,a = 0.3 and b = 0.8, then thesimulation shows the equation(l) can generate a chaoticattractor shown in Fig.3.

where x = [ i V2 l/J v] r ,B= [ 0 0 0 I r

AI ~ [

- ri L IlL 0 0

]- t / C2 - II RC2 0 l l RC20 0 0 I0 IIRC] 0 GIC] - I IRC]- aIC]

and

[ - ' IL IlL 0 0

]A, ~ -If' - I IRC2 0 II RC20 0 I

I IRC] 0 GlC] - II RC]- bIC]

Next, we design a linear switched state feedback controller

{ U] = K]x , IX31 < I , (4)U2 = K2X, IX31 > I

to stabilize the chaotic system(l), where K] and K2 aredesigned feedback gains.

Consider the following globally quadratic Lyapunov func­tion candidate

V = xTpx, (5)10

Fig. 3. Chaotic attractor of Chua's circuit in Fig.2.

(8)

(7)

0.1579 ]0.59930.3621 '0.9174,

0.28300.26590.93710.3621

[

0.6817 0.1422P = 0.1422 4.5776

0.2830 0.26590.1579 0.5993

{A]L +LAf + BE ]+ ETBT < 0A2L +LAI + BE2+ ETBT < O.

In the following, we still choose the same parameters andinitial conditions as the ones in section II. By using MatlabLMI Control Toolbox to solve LMIs (7), we can get a feasiblesolution of Lyapunov function

we obtain

which is obviously a continuous function. It is easy to deducethat if there exist a positive matrix matrix P = pT such thatthe following inequalities hold

{PAl + A f p+ PBK]+ KTBTp < 0 (6)PA2 + AI p+ PBK2+ KTBTp < 0,

then every trajectory of (l) tends to zero exponentially.It is noted that the condition (6) is not still linear matrix

inequalities and can not be implemented by using standardnumerical software due to the terms PBKi (i E 1,2). Anappropriate state feedback controller gain K, (i E 1,2) can notbe found by using a convex optimization algorithm yet.

Pre- and post-multiply p~] respectively, and make thefollowing matrix change defined by

p -] = L,K]P- ] = E],K2P- ] = E2,

(3)

5

{

I = (V2 - ri) ILV2 = (v] - V2) IRC2 - ilC2 (2)~ = V]v] = (v] - v2) IRC]+ GIC]V]- W (l/J )V]+ U.

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III. CONTROLLING CHAOS IN MEMRISTOR-BASED

CHUA'S CIRCUIT

We firstly add a control input into the last state, thecontrolled system becomes

As an alternative representation, the above formula (2)can be represented in the following form of piecewise linearsystems

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and the feedback gains

s; = [ 0.3825 - 23.0222 - 1.7511 - 404823] (9)K2 = [ 0.3825 - 23.0222 - 1.7511 0.5177 ] .

Finally, substituting (9) into (4), then we obtain the linearswitched state feedback controller.

In order to illustrate the effectiveness of the proposedcontroller (4), a numerical simulation has been carried outby Matlab. The simulation results are shown in FigA .

1 5 ,------ - - - - - - - - - - - - - -----,

1 0

5

° v

REFERENCES

[I] Y. Braiman and I. Goldhirsch, "Taming chaotic dynamics with weakperoidic perturbations" , Phys. Rew. Lett., vol.66, pp.2545-2548, 199I.

[2] C.e. Fuh and P.e. Tung, "Controlling chaos using differential geo­metric method", Phys. Rew. Lett., vol.75, pp.2952-2955,1995.

[3] G.R. Chen and X.N. Dong,"On feedback control of chaoticcontinuous-time systems" IEEE Trans. circuits Syst. I, vol.40, pp.591­601, 1993.

[4] J. Zhang, c.o. Li, H.B. Zhang and J.B. YU,"Chaos synchronizationusing single variable feedback based on backstepping method" Chaos,Solitions and Fractals, vol.21, pp.1183-1193, 2004.

[5] R.N. Madan(Ed.),"Chua's circuit: A paradigm for chaos" , WorldScientific, Singapore, 1993.

[6] L.O. Chua, "Mernristor-The missing circuit element" , IEEE Trans.Circuit Syst. i, vo1.18, pp.507-519, 1971.

[7] D. Strukov, G. Snider,G. Stewart, and R. Williams,"The missingmemristor found", Nature, vol.453, pp.80-83, 2008.

[8] M. Itoh and L.O. Chua,"Memristor oscillators", int. J. Bifurcation andChaos, vo1.18, pp.3183-3205, 2008.

6,------ - - - - - - - - - - - - -----,

4

2

o ~llr~--------------

-2

8 06 040t

2 0- 4

0=-- - ----=-=----- - -----:'-=----- - ----:::::-- - ----:!

6,------ - - - - - - - - - - - - --,5

4 ~3

2

8 06 040t

2 0

o- 1 L- ~ ~ ~_____'

o

8 ,---------------------,

6

4

>- 2

°v-2

8 06 040t

2 0- 4

0=-- - ----=-=----- - -----:'-=----- - ----:::::-- - ----:!

Fig. 4. Simulation results.

It is obviously seen from FigA that the switched controller(4) can drive a chaotic attractor to a steady state.

IY. CONCLUSiON

In this paper, a Lyapunov-based approach has been pro­posed for controlling memristor-based Chua's chaotic sys­tem. The linear switched state feedback controller has beenobtained by solving a set of LMls. It has also been shownthat the presented method can control chaotic motion to anequilibrium point by numerical simulation .

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