c synchronization i experiments af r h a - ufmg · ∗ received july 13, 2006; revised february 21,...

CIRCUITS SYSTEMS SIGNAL PROCESSING c© Birkhauser Boston (2007)VOL. 26, NO. 4, 2007, PP. 427–449 DOI: 10.1007/s00034-007-4001-0

CHAOTIC SYNCHRONIZATION

AND INFORMATION

TRANSMISSION EXPERIMENTS:A FUZZY RELAXED H∞CONTROL APPROACH*Leonardo A. Mozelli,1 Claudio D. Campos,2

Reinaldo M. Palhares,1 Leonardo A. B. Torres,1 andEduardo M. A. M. Mendes1

Abstract. This paper addresses the problem of the synchronization of a class of chaoticoscillators using techniques borrowed from a recently proposed control design based onTakagi-Sugeno (TS) fuzzy modeling. In order to attain better synchronization, this TSfuzzy modeling is combined with the robust H∞ observer theory based on linear matrixinequalities. A laboratory setup based on Chua’s oscillator circuit is used to demonstratethe main ideas of the paper. Information transmission experiments are performed as anindex to measure the effectiveness of the proposed approach.Key words: Chaotic systems, chaos synchronization, robust control, Takagi-Sugeno (TS)fuzzy model, linear matrix inequalities (LMIs), Chua’s circuit, H∞ performance.

1. Introduction

Over the last decades, nonlinear dynamics systems, including the chaotic ones,have been extensively studied. In this context, the seminal papers [3] and [20]were very important in demonstrating the possibility of synchronizing chaotic os-cillators under some assumptions. In that sense, synchronization can be achievedby coupling a nonlinear chaotic system with its partially duplicated counterpartand driving them by a common signal produced by the unduplicated part. Becausea chaotic signal may be characterized by a noise-like behavior, many researchershave considered its application in secure communication in the past (see, for

∗ Received July 13, 2006; revised February 21, 2007; This work has been supported in part by theBrazilian agencies CNPq and FAPEMIG.

1 Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antonio Carlos6627 – 31270-010, Belo Horizonte, MG, Brazil. E-mail for Mozelli: [email protected];E-mail for Palhares: [email protected] (corresponding author); E-mail for Tores: [email protected]; E-mail for Mendes: [email protected]

2 Department of Electronics and Telecommunications Engineering, Pontifical Catholic Universityof Minas Gerais, Av. Dom Jose Gaspar 500 – 30535-610, Belo Horizonte, MG, Brazil. E-mail:[email protected]

428 MOZELLI, CAMPOS, PALHARES, TORRES, AND MENDES

instance, [9], [12], [13], and [31] and the references therein). As pointed outin [10], the spread spectrum of a chaotic signal can also be beneficial in manyother aspects. For instance, high signal attenuation can occur in narrow frequencyband communication in indoor environments because of multipath propagationor narrowband interference. The reader can find an interesting discussion on thetopic in the special issue on application of chaos in communications [22], [33],[37], [38].

As far as modeling of nonlinear systems is concerned, the Takagi-Sugeno (TS)fuzzy methodology [21] has been attracting increasing attention for its elegantflexibility when dealing with issues related to nonlinear systems. In particular, thestability analysis and stabilization of nonlinear systems in a TS setting combinedwith the Lyapunov theory have played an important role since the 1990s [27],[35]. This is so because the methodologies based on Lyapunov functions providean easy way to describe stabilization and regulator design issues by means ofsolutions of linear matrix inequalities (LMIs). This represents an interesting as-pect because LMIs can be efficiently solved by convex optimization algorithms.Besides, the flexibility introduced by LMI approaches allows one to incorporatea range of extra constraints over input/output signals and control matrices [29] orto consider index performances as, e.g., the H∞ or H2 norms, as in [4], [6], [7],[24] in the optimization problem.

However, LMI descriptions for certain nonlinear systems tend to be very con-servative, and in such a manner that no feasible solution can be found, evenfor a stable system. This motivated the pioneer work in [25], where the authorsproposed more relaxed conditions for system stability and controller design. Later,significant contributions in this scenario were discussed in [28], [29], providingless conservative results. Concurrently with those contributions, some alternativesto avoid conservatism have been developed such as piecewise Lyapunov func-tions [8] or those based on nonquadratic stabilization criteria, using a multipleLyapunov function approach [14], [23], [39]. All those aspects are suitable for thenonlinear synchronization problem, especially chaotic oscillators.

Because of the presence of noise in real circuit applications, as well interfer-ences on the transmission channel, an H∞ control strategy ensures a more robustsynchronization, reducing disturbance effects [12], [16]. In such a situation wheresome analysis/synthesis approaches present conservatism, relaxed stability con-ditions could provide better performances, or yet a solution, when other methodsseems to fail, for the same problems.

In this paper, an alternative methodology for the design of discrete-time H∞fuzzy control applied to the synchronization of coupled oscillators is proposed.As in [2], [13], [16], [17], the proposed methodology consists in adapted resultsborrowed from robust control and observer theory. However, whereas the previousworks are suitable for piecewise linear systems, the method proposed here is farmore general as it is based on TS fuzzy modeling. Another aspect that distin-guishes the proposed method from other synchronization schemes that also rely

CHAOTIC SYNCHRONIZATION AND FUZZY MODELING 429

on TS fuzzy modeling and LMIs (such as [11], [12], [26], [36]) is the relaxationimposed on the LMI conditions. The proposed method ensures relaxed LMIsconditions and optimal disturbance rejection in an H∞-norm sense, and it takesthe recent results presented in [29] as the starting point in the LMI formulation.In order to demonstrate the effectiveness of the proposed approach, one of themain contributions of this paper is to present experimental results to address theproblem of information transmission under a robust H∞ synchronization setting.The information transmission experiments are performed in a laboratory setupwith an inductorless implementation of a controlled Chua’s oscillator circuit [32],subjected to noise, interference, parameter mismatch, and another aspects inherentfrom practical applications. The information transmission principle proposed in[31] is used to establish communication between the master and the slave by thesynchronization of coupled Chua’s chaotic oscillators. In this work, transmissionis used as a performance index to measure the quality of the synchronizationattained with the approach developed herein because, as stated in [31], synchro-nization is a sufficient condition to guarantee information recovery as the controlsignal used to keep systems synchronized. In this context, high-quality quasi-identical synchronization is very desirable.

This paper is organized as follows. In Section 2 the master-slave synchroniza-tion scheme and information transmission are presented. A brief review of theTakagi-Sugeno fuzzy model is given in Section 3. Stability conditions and H∞performance are discussed in Section 4. The fuzzy H∞ synchronization controldesign is given in Section 5. The experimental results of the proposed methodol-ogy are discussed in Section 6. Our conclusions are presented in Section 7.

The following notation is used throughout the text:∑r

i< j . For instance, if r = 3it means:

3∑i< j

αiα j = α1α2 + α1α3 + α2α3. (1)

2. Master-slave synchronization scheme and informationtransmission

Consider the following synchronization scheme of nonlinear discrete-time sys-tems with sampling time �t :

Master:

{x(k + 1) = f

[x(k)

]y(k) = h

[x(k)

]Slave:

{x(k + 1) = f

[x(k)

] + u(k)

y(k) = h[x(k)

] ,

(2)

where x(k), x(k) ∈ Rn are the state vectors of the master and slave systems


respectively; the state transition is denoted by the map f : Rn → R

n ; y(k),y(k) ∈ R

m are the measured outputs given by the map h : Rn → R

m , andu(k) ∈ R

n denotes the synchronization control vector. To avoid clutter, assumek + 1 = t + �t .

By synchronization one may define the condition achieved when the master andthe slave describe a common trajectory on the state space simultaneously. Whatwe call robust synchronization is obtained when the trajectories become close toeach other as imposed by ε; a high-quality quasi-identical synchronization, thatis,

limk→∞ ‖ x(k) − x(k) ‖ ≤ ε. (3)

A possible strategy to attain robust synchronization for general nonlinear sys-tems is to drive one of the systems, the slave, into the same trajectory as theother one, the master, using an appropriate control law proportional to the systemsoutput difference (see below for the expression) added to the slave system

u(k) = g[y(k) − y(k)

]. (4)

This control law is then applied to minimize the synchronization error definedby e(k) = x(k) − x(k). Thus, the goal of robust synchronization is to find acontrol law that ensures asymptotic stability for the following dynamical errorsystem derived from (2) and with (4) operating in the slave system:

e(k + 1) = f[x(k)

] − f[x(k)

] − u(k)

e(k + 1) = f[x(k)

] − f[x(k)

] − g{h[x(k)

] − h[x(k)

]}.

(5)

Injecting an information signal into the master system as an additive perturba-tion, a pair of dynamical equations is established:

Transmitter: x(k + 1) = f[x(k)

] + i(k)

Receiver: x(k + 1) = f[x(k)

] + u(k).(6)

The information signal, i(k), can be recovered in the slave system using acoherent demodulation technique if robust synchronization is established. In [30]it was proved that, if condition (3) is satisfied for the discrete-time systems (6),then u(k) ≈ i(k). Perfect information recovery, u(k) ≡ i(k), becomes unfeasiblein real applications, mainly because of the presence of circuit noise, channelinterference, or parametric uncertainties. In this situation, H∞ performance turnsout to be relevant as it establishes a reliable synchronization and consequently abetter communication performance.

3. Takagi-Sugeno fuzzy model

The TS fuzzy model is now briefly reviewed. Consider the master-slave schemein (2), with a control signal acting over the slave oscillator and a noisy signal


disturbing the master dynamics. The TS fuzzy model is given by the followingIF-THEN rules:

Rule i:

IF: q1(k) is Mi1 and q2(k) is Mi

2 and . . . qs(k) is Mis

THEN:

Master:

{x(k + 1) = Ai x(k) + Eiw(k)

y(k) = Ci x(k) + Diw(k)

Slave:

{x(k + 1) = Ai x(k) + u(k)

y(k) = Ci x(k),

(7)

where i = 1, 2, . . . , r denotes the number of rules; for the i th rule, Mij ( j =

1, 2, . . . , s) denotes the fuzzy sets; q j (k), j = 1, 2, . . . , s, denotes the premisevariables, which may be measured; x(k) ∈ R

n is the master state vector; y(k) ∈R

m is the output vector; w(k) ∈ Rp is the exogenous input; the control signal is

represented by u(k) ∈ Rn , and Ai : R

n → Rn, Ei : R

p → Rn, Ci : R

n → Rm ,

and Di : Rp → R

m are the matrices related to the local linear model for the rulei .

Let µij [q j (k)] be the membership function of the fuzzy set Mi

j , q(k) =[q1(k) q2(k) · · · qs(k)] and mi [q(k)] = ∏s

j=1 µij [q j (k)].

Considering that

αi [q(k)] = mi [q(k)]∑ri=1 mi [q(k)] ,

r∑i=1

αi [q(k)] = 1,

αi [q(k)] ≥ 0 (i = 1, 2, . . . , r),

(8)

the master-slave scheme in (2) can be rewritten as a result of the TS modelingdescribed in (7):

Master:

{x(k + 1) = ∑r

i=1 αi [q(k)]{Ai x(k) + Eiw(k)}y(k) = ∑r

i=1 αi [q(k)]{Ci x(k) + Diw(k)}

Slave:

{x(k + 1) = ∑r

i=1 αi [q(k)]Ai x(k) + u(k)

y(k) = ∑ri=1 αi [q(k)]Ci x(k)

.

(9)

The control signal, u(k), is given by the parallel distributed compensation(PDC) principle [25], [27] and shares the same fuzzy sets and premise variablesas the system models. The control law is proportional to the difference betweenthe systems outputs, so it follows that

u(k) =r∑

i=1

αi [q(k)]Li [y(k) − y(k)], (10)

where Li : Rm → R

n is the local gain for the i th rule.


4. Stability conditions and H∞ performance

Based on the TS fuzzy model in (9), one can obtain a fuzzy description for thesynchronization error dynamics as follows:

e(k + 1) =r∑

i=1

αi [q(k)]{Ai e(k) + Eiw(k)} − u(k), (11)

where e(k) = x(k) − x(k).Taking equation (9) into account, the control law in (10) can be rewritten as

u(k) =r∑

i=1

r∑j=1

αi [q(k)]α j [q(k)]Li{C j e(k) + D jw(k)

}. (12)

Finally, combining equations (11) and (12), the synchronization error dynamicsis given by

e(k + 1) =r∑

i=1

r∑j=1

αi [q(k)]α j [q(k)]{[Ai − Li C j ]e(k) + [Ei − Li D j ]w(k)}.

(13)

At this point, it is necessary to introduce an auxiliary output variable, z(k),which evaluates and weights the synchronization error and the exogenous inputin the system:

z(k) =r∑

i=1

αi [q(k)]{�i e(k) + iw(k)}, (14)

where z(k) ∈ Rl , i ∈ R

l×p, and �i ∈ Rl×n . i and �i are weighting matrices

to be chosen.The dynamics system to be considered in the analysis study for the H∞ per-

formance is given by the combination of (13) and (14), i.e., with input w(k), statevector e(k), and output z(k), or:

e(k + 1) =r∑

i=1

r∑j=1

αi [q(k)]α j [q(k)]{[Ai − Li C j ]e(k) + [Ei − Li D j ]w(k)}

z(k) =r∑

i=1

αi [q(k)]{�i e(k) + iw(k)}.(15)

Therefore, the controller must fulfill two goals: (i) to maintain the synchroniza-tion error kept to a minimum level, i.e., to ensure equation (3); (ii) to minimizethe presence of the exogenous entries, w(k), into the weighted error, z(k).

There are several ways to quantify the effect of w(k) on z(k). In robust control


theory, one way to perform this is by means of the H∞-norm, which for the signalsz(k) and w(k) and considering stability for the system in (15), is given by

sup0<‖w‖2<∞

‖ w ‖2

‖ z ‖2, (16)

where ‖ · ‖2 indicates the L2-norm. In other words, the H∞-norm computes thegreatest ratio between the weighted error signal energy and the exogenous inputenergy. In addition to guaranteeing robust synchronization for the system (15),the control signal must minimize, ∀z �= 0, the H∞ disturbance attenuation levelgiven by γ :

sup0<‖w‖2<∞

‖ w ‖2

‖ z ‖2< γ. (17)

Remark 1. The matrices �i and i are “ad hoc” choices. One should choosethem according to the system properties, taking into account, for instance, whichstate variable is more contaminated by noise or which state variable is morerelevant to the control strategy.

The main result is stated in the following theorem which presents sufficientconditions that guarantee asymptotical stability for the error system with an H∞level of performance for the robust synchronization of the master-slave schemein (9).

Theorem 1. Let matrices Li , i = 1, . . . , r, be given. The fuzzy error systemin (15) is asymptotically stable with minimum H∞ disturbance attenuation level,given by γ �

√δ, if the following optimization problem is feasible:

minP,Ti jh ,Ri j ,Si jh ,δ

δ

s.t. Kh ≺ 0, P � 0, Ti jh � 0,

(i, j, h = 1, 2, . . . , r, i < j),(18)

where

Kh �

Q1 − Z1h Q12 + N12h · · · Q1r + N1rh

Q12 + N21h Q2 − Z2h · · · Q2r + N2rh...

.... . .

...

Q1r + Nr1h Q2r + Nr2h · · · Qr − Zrh

,

Qi �

−P GT

ii P 0 �Ti

PGii −P P Mii 00 MT

ii P −δ I Ti

�i 0 i −I

,


Qi j �

−P H T

i j P 0 �Ti j

P Hi j −P P Oi j 00 OT

i j P −δ I ϒTi j

�i j 0 ϒi j −I

, (19)

Gi j � Ai − Li C j , Mi j � Ei − Li D j , Hi j � Gi j + G ji

2,

Oi j � Mi j + M ji

2, �i j � �i + � j

2, ϒi j � i + j

2(20)

so that

Ni jh ={

Ti jh + Wi jh + Si jh − STi jh if i < j

Tjih + W jih − S jih + STjih if i > j

(21)

and

Wi jh ={

Ri j , if i = h or j = h

0, if i �= h and j �= h(22)

Zih =

Rih, if i < h

Rhi , if i > h

0, if i = h

. (23)

Proof. First, consider the following notation:

r∑i=1

αi [q(k)]Ni = N (24)

which is extended to the other matrices. Using this notation, rewrite (15) as fol-lows: {

e(k + 1) = Ge(k) + Mw(k)

z(k) = �e(k) + w(k), (25)

where G = A − LC and M = E − L D.In order to demonstrate that the optimization problem guarantees the minimum

H∞ performance and asymptotic stability, select the following Lyapunov functioncandidate:

V [e(k)] = e(k)T Pe(k) (26)

and its finite difference defined as

�V [e(k)] � e(k + 1)T Pe(k + 1) − e(k)T Pe(k). (27)


The I∞ performance can be given by (see, e.g., [18], [19]):

I∞ =∞∑

k=0

z(k)T z(k) − γ 2w(k)T w(k) ≺ 0. (28)

Taking the initial conditions equal to zero, equation (28) can be rewritten as:

�V [e(k)] + zT (k)z(k) − γ 2wT (k)w(k) ≺ 0. (29)

Substituting equations (25) and (27) into (29), the following augmented systemcan be obtained:

e(k)T � e(k) ≺ 0, (30)

where e(k) � [e(k) w(k)]T , and

� �[

GT PG − P + �T � GT P M + �T

∗ −γ 2 I + MT P M + T

]. (31)

Notice that (30) can be put in the equivalent form e(k)T �e(k) ≺ 0, where

� �

−P GT P 0 �T

PG −P P M 00 MT P −γ 2 I T

� 0 −I

. (32)

Using the notation in (20) and according to (8):

� =

θ(−P) θ(GT

i j P) 0 θ(�Ti )

θ(PGi j ) θ(−P) θ(P Mi j ) 00 θ(MT

i j P) θ(−γ 2 I ) θ( Ti )

θ(�i ) 0 θ( i ) θ(−I )

, (33)

where θ �∑r

i=1∑r

j=1 αi [q(k)]α j [q(k)].Because all the summations in (33) have the same indexes, � can be rewritten

as a summation of matrices:

� =r∑

i=1

r∑j=1

αi [q(k)]α j [q(k)]

−P GT

i j P 0 �Ti

PGi j −P P Mi j 00 MT

i j P −γ 2 I Ti

�i 0 i −I

. (34)

Finally, using (1) and the definitions introduced in (19) yields

� =r∑

i=1

αi [q(k)]2 Qi +r∑

i< j

αi [q(k)]α j [q(k)]Qi j . (35)

At this point, we introduce a new LMI relaxation for TS fuzzy systems based onthe results presented in [29]. Following Appendix B from [29], consider Qi , Qi j ,


Ri j , Si jh , and Ti jh as the same matrices presented in this paper. Then, replacingx(t) by e(k), and αi [z(t)] by αi [q(k)] and applying similar steps as shown in [29],the following inequality is obtained:

e(k)T �e(k) < e(k)T κ e(k), (36)

where κ = [α1 I · · · αr I ]∑r

h=1 Kh [α1 I · · · αr I ]T .Therefore, if Kh ≺ 0, (h = 1, 2, . . . , r) (defined in (19)), as established in

the LMI constraints proposed in (18), it follows that e(k)T κ e(k) ≺ 0. Thus,according to the inequality (36), the H∞ performance as well as the asymptoticstability in (30) is also guaranteed. Because the optimization problem is convex,the feasibility ensures that the minimum H∞ disturbance attenuation level isattained. ✷

Remark 2. In Theorem 1 the matrices P , Ti jh , and Ri j are symmetric whereasSi jh are skew matrices. Moreover, P ∈ R

n×n , and the other matrices have thesame dimension as Qi j .

5. Fuzzy H∞ synchronization control design

Based on the results of the last section, the next theorem establishes the LMImachinery to obtain the synchronization gains Li ensuring an H∞ level of per-formance for the error system dynamics in (15).

Theorem 2. Consider the error system in (15). If the following optimization prob-lem is feasible:

minP,Ti jh ,Ri j ,Si jh ,Xi ,δ

δ

s.t. Kh ≺ 0, P � 0, Ti jh � 0,

(i, j, h = 1, 2, . . . , r, i < j),(37)

where

Kh �

V1 − Z1h V12 + N12h · · · V1r + N1rh

V12 + N21h V2 − Z2h · · · V2r + N2rh...

.... . .

...

V1r + Nr1h V2r + Nr2h · · · Vr − Zrh

,

Vi �

−P Y T

ii 0 �Ti

Yii −P Jii 00 J T

ii −δ I Ti

�i 0 i −I

, Vi j �

−P BT

i j 0 �Ti j

Bi j −P Ui j 00 U T

i j −δ I ϒTi j

�i j 0 ϒi j −I

,

(38)


R

C2C1Gid(t)

vc1(t) vc2

(t)

+ +

+-

Lu1(t) u2(t)

u3(t)

Figure 1. Chua’s circuit schematic diagram.

Yi j � P Ai − Xi C j , Ji j � P Ei − Xi D j ,

Bi j � Yi j + Y ji

2, and Ui j � Ji j + J ji

2, (39)

then the synchronization gains that guarantee the minimum H∞ level perfor-mance are given by Li � P−1 Xi .

Remark 3. In this theorem, the matrices �i j , ϒi j , Zi j , and Ni jh are the same asthe ones in Theorem 1.

Remark 4. The proof of Theorem 2 is omitted as it follows directly from thederivation of Theorem 1, just carrying out the linearization change of variablesgiven by Xi � P Li .

6. Experimental results

In this section, the proposed methodology is applied to the robust synchronizationof a coupled nonlinear system. The practical implementation used in this work isa chaotic Chua’s oscillator [5]. Basically, this circuit is composed of a resonanttank (parallel capacitor, C2, and inductor, L) linked by a linear resistor R to acapacitor C1, as illustrated in Figure 1. In parallel with this capacitor, C1, thereis a nonlinear resistor, G, called Chua’s diode. Figure 1 also illustrates Chua’soscillator with extra voltage and current sources.

The oscillator dynamics can be described as

dvc1(t)

dt= 1

C1

{1

R[vc2(t) − vc1(t)] − G[vc1(t)]

}dvc2(t)

dt= 1

C2

{1

R[vc1(t) − vc2(t)] + il(t)

}dil(t)

dt= 1

L[−vc2(t) − R0il(t)], (40)

where the state vector is given by x � [vc1 vc2 il ]T .The nonlinear dynamic arises from the fact that the diode resistance is voltage


id(t)

vc1(t)

GaGb

E–E–d

d

Figure 2. Chua’s diode conductance feature.

dependent, i.e., G[vc1(t)], which is piecewise linear. When the voltage is withinthe range ±E , the diode conductance is equal to Ga , and outside this sector itis equal to Gb. This behavior is depicted in Figure 2. The slope for the current-voltage curve between ±E represents the conductance Ga ; for the rest of thefigure the slope corresponds to Gb.

Therefore, it is straightforward to formulate two fuzzy rules that describe com-pletely the behavior of this Chua’s circuit. The first linear model assumes that theconductance is Ga for low voltages. This is represented in Figure 2 by the straightline crossing the horizontal axis at the origin.

Assuming that the voltage over the capacitor C1 is bounded, vc1 ∈ [−d, d],the other linear model can be built. This second linear model captures the localdynamic for higher voltage levels, representing the dynamics exactly at ±d andapproximately in the vicinity. In Figure 2, this is represented by the other linecrossing the diode curve at ±d and passing through the origin. For this model,the slope in the graphics, or physically the conductance, is described by Gg =(Gb + (Ga−Gb)E

d ). More details about this fuzzy modeling can be found in [34].In [11] the reader can find an overview on chaotic systems TS fuzzy modeling,including Chua’s circuit.

Therefore, the fuzzy rules are

Rule 1:

IF: vc1(t) is M1 (nearby zero)

THEN: x(t) = A1x(t) + E1w(t)y(t) = C1x(t) + D1w(t)z(t) = �1x(t) + 1w(t)

Rule 2:

IF: vc1(t) is M2 (nearby ± d)

THEN: x(t) = A2x(t) + E2w(t)y(t) = C2x(t) + D2w(t)z(t) = �2x(t) + 2w(t)

.


–6 –4 –2 0 2 4 6

0

0.2

0.4

0.6

0.8

1

Figure 3. Membership functions.

–6 –4 –2 0 2 4 6–3

–2

–1

0

1

2

3x 10

– 3

vc1

i d

Figure 4. Comparison between the diode conductance curve and the curve obtained with the fuzzymodel.

The membership functions may be chosen as in [34] and are depicted in Fig-ure 3. These inference functions combining the two fuzzy rules can represent theexact diode characteristic. This is illustrated in Figure 4, where the line repre-sents the actual diode characteristic and the points marked with × are the diodecharacteristic according to the chosen fuzzy model.

Because the methodology deals with discrete-time systems it is necessary toconsider a discretization approach to Chua’s circuits as, for instance, the onepresented in [15] which is shown to preserve the fixed points of the originalsystem. This approach is chosen to be used in this work as it guarantees thereconstruction of the original dynamics even in a large range of values of theincrement time.

The actual circuit parameters are given in Table 1 and were obtained using aparameter estimation scheme called the unscented Kalman filter method. See [1]for details on the estimation.

All the experiments were performed in a laboratory setup, called PCCHUA,developed and implemented in [32], in which constructive and operational detailscan be found.

Consider the information transmission scheme exhibited in Figure 5. This


Table 1. Circuit and model parameters

Parameters Values

C1 30.14 µFC2 185.6 µFL 52.28 HR 1673 �

R0 0 �

Ga −0.801 mSGb −0.365 mSE 1.74 Vd 6 V

R R

C2 C2C1 C1

id(t) id(t)

i(t)

η(t)

u1(t)

u2(t)

u3(t)

y(t) y (t)

+

+ -

++ +

+-

LL

ControllerTransmitter Receiver

Figure 5. Unidirectional communication system.

scheme shows the Chua’s oscillator circuits on a unidirectional coupling, wherey(t) is the master output and y(t) is the slave output.

The information to be transmitted i(t) is injected on the master Chua’s chaoticoscillator (transmitter) as a perturbation only in the direction of vc1(t), as illus-trated in Figure 5. Mathematically this perturbation corresponds to a modificationon the differential equations that govern the motion of the oscillator. Then a scalarsignal y(t) is used to carry the information (from the master) and to serve as areference signal to allow the synchronization of the slave oscillator (receiver).This signal may be corrupted by noise interferences η(t) on the transmissionchannel.

In this framework, the same circuit operates as the master and the slave. Whenconfigured as the master, it oscillates freely, and the signal to be transmittedis added by the actuators. The generated time series is then recorded in a file.Afterward, the circuit is reconfigured to operate as the slave starting from randominitial conditions. At some instant in time the system has to synchronize with themaster reference that was previously recorded.

Because the elapsed time between data acquisition and the synchronizationexperiment is short, the assumption that master and slave systems have the samedynamics is plausible, and consequently there is no time for significant changesin the circuit parameters.


Table 2. Synchronization gains provided by Theorem 2

L1 L2

u1(k) 1.0668 × 102 9.4710 × 101

u2(k) 2.9186 2.9188u3(k) 5.4422 × 10−5 5.4084 × 10−5

It is assumed that the disturbance input acts only over the state variable vc1 ,as information is added only into this state variable. Under the assumptions thatthe master and slave oscillators share the same dynamics and considering that theweighted error z(t) is simply the summation of all state variables, the fuzzy modelmatrices are given by

A1 =1 − T/(RC1) − T Ga/C1 T/(RC1) 0

T/(RC2) 1 − T/(RC2) T/C20 −T/L 1 − T R0/L

,

A2 =1 − T/(RC1) − T Gg/C1 T/(RC1) 0

T/(RC2) 1 − T/(RC2) T/C20 −T/L 1 − T R0/L

,

E(1,2) =0.001

00

, C(1,2) =1

00

T

, �(1,2) =1 0 0

0 1 00 0 1

,

(1,2) = [0 0 0

]T, D(1,2) = [0.0001],

(41)

where T is the sampling time.

6.1. Results

The information transmission problem is investigated in the light of the informa-tion transmission via control (ITVC) principle developed in [31]. The informationtransmission test is used as a performance index to validate the proposed approachfor robust synchronization.

The ITVC principle states that any controller that guarantees an identical, orquasi-identical, master-slave synchronization can actually perform as a demodu-lator and thus recover the transmitted information. In this way, the control signalu1(t) corresponds to the demodulated information signal i(t).

This section presents the synchronization experimental results with the gainsobtained by applying Theorem 2 to the discretized TS fuzzy model in (41). Usingthe LMI Control Toolbox for MATLAB, the synchronization gains were calcu-lated (see Table 2). These gains guarantee an H∞ level of 9.061 × 10−4, whenconsidering a sampling time of T = 10 ms.

The PCCHUA setup [32] allows the user to use a wide variety of signals to


0 20 40 60 80 100 120–6

–4

–2

0

2

4

6

time (s)

volta

ge (

V)

0 20 40 60 80 100 120–6

–4

–2

0

2

4

6

time (s)

volta

ge (

V)

Figure 6. The master-slave times series for vc1 in a Chua’s circuit.

input into the circuit, such as sinusoid, square, and sawtooth. Some relevant testsresults are described next.

6.1.1. First experiment: Sinusoidal signal

During data acquisition, the master oscillator runs for 90 s. From 15 s to 75 s asinusoid voltage with a frequency of 0.5 Hz and 0.12 mV of amplitude is addedto vc1 . The circuit is restarted in the slave configuration and runs freely for 15 s,when control action begins. Information transmission effectively begins after 30 sand ends at 90 s. The control signal persists for 15 more seconds and the controlexperiment finishes after 120 s.

Figure 6 illustrates the master and the slave time series for the state variable vc1 .As can be seen in Figure 7, the error is very small, less than 1% of the maximumerror, which clearly shows the synchronization.

The transmitted and recovered signals (control action signal) are depicted inFigure 8. Notice that after applying a second-order Butterworth low-pass digitalfilter, with cutting frequency fc = 1 Hz, the final received signal is obtained asdepicted in Figure 9.

It is worth mentioning that the small time delay between the original and therecovered signals, in Figure 9, is due to the filter phase lag.


20 30 40 50 60 70 80 90 100

–0. 6

–0. 4

–0. 2

0

0.2

0.4

0.6

time (s)

volta

ge (

V)

Figure 7. Error in the state variable vc1 for the first experiment.

40 41 42 43 44 45 46 47 48 49 50

–2

–1

0

1

2

x 10– 4

time (s)

volta

ge (

V)

40 41 42 43 44 45 46 47 48 49 50

–2

–1

0

1

2

x 10– 4

time (s)

volta

ge (

V)

Figure 8. Transmitted and recovered signals for the first experiment.

6.1.2. Second experiment: Low-frequency signal combinedwith higher-frequency signals

Here the transmitted input signal is a combination of three signals: a square wavewith 0.5 Hz and 0.16 mV; a sinusoid with 2 Hz and 0.08 mV; and a sawtoothwith 4 Hz and 0.04 mV. The experimental procedure is the same as the one


40 41 42 43 44 45 46 47 48 49 50

–2

–1

0

1

2

x 10– 4

time (s)

volta

ge (

V)

Figure 9. Transmitted (- -) and filtered (-) signals for the first experiment.

30 40 50 60 70 80 90 100 110 120

–0. 6

–0. 4

–0. 2

0

0.2

0.4

0.6

time (s)

volta

ge (

V)

Figure 10. Error in the state variable vc1 for the second experiment.

described previously. Once more, the synchronization has been achieved with asynchronization error limited by ±1.5% (see Figure 10).

To filter the control signal, Figure 11, and to obtain the input signal, Figure 12,a first-order Butterworth low-pass digital filter with fc = 10 Hz was used.

6.1.3. Third experiment: Signals with close frequencies andamplitudes

For the last experiment, the transmitted signal consists of a combination of threesignals with close frequencies and amplitudes. They are: sinusoid with 1.2 Hz and0.08 mV; sawtooth with 1.3 Hz and 0.12 mV; square with 2.1 Hz and 0.08 mV.The synchronization error is depicted in Figure 13. As can be verified, the errormagnitude was a little higher than in the other cases, but still very small, limitedby almost 4%, not impairing the robust synchronization.

As shown in Figures 14 and 15, the transmitted signal can be recovered, thusillustrating the efficiency of the proposed method.


36 36. 2 36. 4 36. 6 36. 8 37 37. 2 37. 4 37. 6 37. 8 38–4

–2

0

2

4x 10

– 4

time (s)

volta

ge (

V)

36 36. 2 36. 4 36. 6 36. 8 37 37. 2 37. 4 37. 6 37. 8 38–4

–2

0

2

4x 10

– 4

time (s)

volta

ge (

V)

Figure 11. Transmitted and recovered signals for the second experiment.

36 36. 2 36. 4 36. 6 36. 8 37 37. 2 37. 4 37. 6 37. 8 38–4

–2

0

2

4x 10

– 4

time (s)

volta

ge(V

)

Figure 12. Transmitted (- -) and filtered (-) signals for the second experiment.

7. Conclusion

In this paper, a new LMI methodology for the synchronization of coupleddiscrete-time systems in an H∞ fuzzy setting has been proposed. The methodis based on a recent relaxation technique for TS fuzzy systems presented in


30 40 50 60 70 80 90 100 110 120

–0. 6

–0. 4

–0. 2

0

0.2

0.4

0.6

time (s)

volta

ge (

V)

Figure 13. Error in state variable vc1 for the third experiment.

35 35. 5 36 36. 5 37 37. 5 38 38. 5 39 39. 5 40

–2

–1

0

1

2

x 10– 4

time (s)

volta

ge (

V)

35 35. 5 36 36. 5 37 37. 5 38 38. 5 39 39. 5 40

–2

–1

0

1

2

x 10– 4

time (s)

volta

ge (

V)

Figure 14. Transmitted and recovered signals for the third experiment.

the literature as well as on the robust characteristic of the H∞ performance.Several experimental results using a laboratory setup for Chua’s circuit have beengiven to check the synchronization of coupled nonlinear systems in practice. TheH∞ synchronization approach has performed successfully when the goal is theproblem of signal transmission through a chaotic channel.


35 35. 5 36 36. 5 37 37. 5 38 38. 5 39 39. 5 40

–2

–1

0

1

2

x 10– 4

time (s)

volta

ge (

V)

Figure 15. Transmitted (- -) and filtered (-) signals for the third experiment.

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c synchronization i experiments af r h a - ufmg · ∗ received july 13, 2006; revised february 21,...

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