chaotic synchronization and information transmission ... - Springer Link

C IRCUITS S YSTEMS S IGNAL P ROCESSING VOL . 26, N O . 4, 2007, P P. 427–449

c Birkhäuser Boston (2007) DOI: 10.1007/s00034-007-4001-0

C HAOTIC S YNCHRONIZATION AND I NFORMATION T RANSMISSION E XPERIMENTS : A F UZZY R ELAXED H∞ C ONTROL A PPROACH * Leonardo A. Mozelli,1 Cláudio D. Campos,2 Reinaldo M. Palhares,1 Leonardo A. B. Tôrres,1 and Eduardo M. A. M. Mendes1 Abstract. This paper addresses the problem of the synchronization of a class of chaotic oscillators using techniques borrowed from a recently proposed control design based on Takagi-Sugeno (TS) fuzzy modeling. In order to attain better synchronization, this TS fuzzy modeling is combined with the robust H∞ observer theory based on linear matrix inequalities. A laboratory setup based on Chua’s oscillator circuit is used to demonstrate the main ideas of the paper. Information transmission experiments are performed as an index 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 oscillators under some assumptions. In that sense, synchronization can be achieved by coupling a nonlinear chaotic system with its partially duplicated counterpart and driving them by a common signal produced by the unduplicated part. Because a chaotic signal may be characterized by a noise-like behavior, many researchers have 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 the

Brazilian agencies CNPq and FAPEMIG. 1 Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antônio Carlos

6627 – 31270-010, Belo Horizonte, MG, Brazil. E-mail for Mozelli: [email protected]; E-mail for Palhares: [email protected] (corresponding author); E-mail for Tôres: [email protected]; E-mail for Mendes: [email protected] 2 Department of Electronics and Telecommunications Engineering, Pontifical Catholic University of Minas Gerais, Av. Dom José Gaspar 500 – 30535-610, Belo Horizonte, MG, Brazil. E-mail: [email protected]

428

ˆ M OZELLI , C AMPOS , PALHARES , T ORRES , AND M ENDES

instance, [9], [12], [13], and [31] and the references therein). As pointed out in [10], the spread spectrum of a chaotic signal can also be beneficial in many other aspects. For instance, high signal attenuation can occur in narrow frequency band communication in indoor environments because of multipath propagation or narrowband interference. The reader can find an interesting discussion on the topic 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 elegant flexibility when dealing with issues related to nonlinear systems. In particular, the stability analysis and stabilization of nonlinear systems in a TS setting combined with the Lyapunov theory have played an important role since the 1990s [27], [35]. This is so because the methodologies based on Lyapunov functions provide an easy way to describe stabilization and regulator design issues by means of solutions of linear matrix inequalities (LMIs). This represents an interesting aspect because LMIs can be efficiently solved by convex optimization algorithms. Besides, the flexibility introduced by LMI approaches allows one to incorporate a range of extra constraints over input/output signals and control matrices [29] or to 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 conservative, and in such a manner that no feasible solution can be found, even for a stable system. This motivated the pioneer work in [25], where the authors proposed more relaxed conditions for system stability and controller design. Later, significant contributions in this scenario were discussed in [28], [29], providing less conservative results. Concurrently with those contributions, some alternatives to avoid conservatism have been developed such as piecewise Lyapunov functions [8] or those based on nonquadratic stabilization criteria, using a multiple Lyapunov function approach [14], [23], [39]. All those aspects are suitable for the nonlinear synchronization problem, especially chaotic oscillators. Because of the presence of noise in real circuit applications, as well interferences on the transmission channel, an H∞ control strategy ensures a more robust synchronization, reducing disturbance effects [12], [16]. In such a situation where some analysis/synthesis approaches present conservatism, relaxed stability conditions could provide better performances, or yet a solution, when other methods seems 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 results borrowed from robust control and observer theory. However, whereas the previous works are suitable for piecewise linear systems, the method proposed here is far more general as it is based on TS fuzzy modeling. Another aspect that distinguishes the proposed method from other synchronization schemes that also rely

C HAOTIC S YNCHRONIZATION AND F UZZY M ODELING

429

on TS fuzzy modeling and LMIs (such as [11], [12], [26], [36]) is the relaxation imposed on the LMI conditions. The proposed method ensures relaxed LMIs conditions and optimal disturbance rejection in an H∞ -norm sense, and it takes the 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 the main contributions of this paper is to present experimental results to address the problem of information transmission under a robust H∞ synchronization setting. The information transmission experiments are performed in a laboratory setup with an inductorless implementation of a controlled Chua’s oscillator circuit [32], subjected to noise, interference, parameter mismatch, and another aspects inherent from practical applications. The information transmission principle proposed in [31] is used to establish communication between the master and the slave by the synchronization of coupled Chua’s chaotic oscillators. In this work, transmission is used as a performance index to measure the quality of the synchronization attained with the approach developed herein because, as stated in [31], synchronization is a sufficient condition to guarantee information recovery as the control signal used to keep systems synchronized. In this context, high-quality quasiidentical synchronization is very desirable. This paper is organized as follows. In Section 2 the master-slave synchronization scheme and information transmission are presented. A brief review of the Takagi-Sugeno fuzzy model is given in Section 3. Stability conditions and H∞ performance are discussed in Section 4. The fuzzy H∞ synchronization control design is given in Section 5. The experimental results of the proposed methodology are discussed in Section 6. Our conclusions are presented in Section 7. The following notation is used throughout the text: ri< j . For instance, if r = 3 it means: 3 αi α j = α1 α2 + α1 α3 + α2 α3 . (1) i< j

2. Master-slave synchronization scheme and information transmission Consider the following synchronization scheme of nonlinear discrete-time systems with sampling time t:

x(k + 1) Master: y(k) x(k ˆ + 1) Slave: yˆ (k)

= f x(k) = h x(k) = f x(k) ˆ + u(k) , = h x(k) ˆ

(2)

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

430


respectively; the state transition is denoted by the map f : Rn → Rn ; y(k), yˆ (k) ∈ Rm are the measured outputs given by the map h : Rn → Rm , and u(k) ∈ Rn denotes the synchronization control vector. To avoid clutter, assume k + 1 = t + t. By synchronization one may define the condition achieved when the master and the slave describe a common trajectory on the state space simultaneously. What we call robust synchronization is obtained when the trajectories become close to each other as imposed by ; a high-quality quasi-identical synchronization, that is, lim x(k) − x(k) ˆ

≤ .

k→∞

(3)

A possible strategy to attain robust synchronization for general nonlinear systems is to drive one of the systems, the slave, into the same trajectory as the other one, the master, using an appropriate control law proportional to the systems output difference (see below for the expression) added to the slave system u(k) = g y(k) − yˆ (k) . (4) This control law is then applied to minimize the synchronization error defined by e(k) = x(k) − x(k). ˆ Thus, the goal of robust synchronization is to find a control law that ensures asymptotic stability for the following dynamical error system derived from (2) and with (4) operating in the slave system: e(k + 1) = f x(k) − f x(k) ˆ − u(k) (5) e(k + 1) = f x(k) − f x(k) ˆ − g h x(k) − h x(k) ˆ . Injecting an information signal into the master system as an additive perturbation, a pair of dynamical equations is established: Transmitter: x(k + 1) = f x(k) + i(k) (6) Receiver: x(k ˆ + 1) = f x(k) ˆ + u(k). The information signal, i(k), can be recovered in the slave system using a coherent 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 unfeasible in real applications, mainly because of the presence of circuit noise, channel interference, or parametric uncertainties. In this situation, H∞ performance turns out to be relevant as it establishes a reliable synchronization and consequently a better communication performance.

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


431

disturbing the master dynamics. The TS fuzzy model is given by the following IF-THEN rules: Rule i: q1 (k) is M1i and q2 (k) is M2i and . . . qs (k) is Msi

IF: THEN:

Master: Slave:

x(k + 1) = Ai x(k) + E i w(k) y(k) = Ci x(k) + Di w(k)

(7)

x(k ˆ + 1) = Ai x(k) ˆ + u(k) , yˆ (k) = Ci x(k) ˆ

where i = 1, 2, . . . , r denotes the number of rules; for the ith rule, M ij ( j = 1, 2, . . . , s) denotes the fuzzy sets; q j (k), j = 1, 2, . . . , s, denotes the premise variables, which may be measured; x(k) ∈ Rn is the master state vector; y(k) ∈ Rm is the output vector; w(k) ∈ R p is the exogenous input; the control signal is represented by u(k) ∈ Rn , and Ai : Rn → Rn , E i : R p → Rn , Ci : Rn → Rm , and Di : R p → Rm are the matrices related to the local linear model for the rule i. Let µij [q j (k)] be the membership function of the fuzzy set M ij , q(k) = [q1 (k) q2 (k) · · · qs (k)] and m i [q(k)] = sj=1 µij [q j (k)]. Considering that r m i [q(k)] , αi [q(k)] = 1, αi [q(k)] = r i (8) i=1 m [q(k)] i=1 αi [q(k)] ≥ 0 (i = 1, 2, . . . , r ), the master-slave scheme in (2) described in (7): x(k + 1) Master: y(k) x(k ˆ + 1) Slave: yˆ (k)

can be rewritten as a result of the TS modeling = ri=1 αi [q(k)]{Ai x(k) + E i w(k)} r = i=1 αi [q(k)]{Ci x(k) + Di w(k)} = ri=1 αi [q(k)]Ai x(k) ˆ + u(k) . r = i=1 αi [q(k)]Ci x(k) ˆ

(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 variables as the system models. The control law is proportional to the difference between the systems outputs, so it follows that r αi [q(k)]L i [y(k) − yˆ (k)], (10) u(k) = i=1

where L i :

Rm

→

Rn

is the local gain for the ith rule.

432


4. Stability conditions and H∞ performance Based on the TS fuzzy model in (9), one can obtain a fuzzy description for the synchronization error dynamics as follows: e(k + 1) =

r

αi [q(k)]{Ai e(k) + E i w(k)} − u(k),

(11)

i=1

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

r r

αi [q(k)]α j [q(k)]L i C j e(k) + D j w(k) .

(12)

i=1 j=1

Finally, combining equations (11) and (12), the synchronization error dynamics is given by e(k + 1) =

r r

αi [q(k)]α j [q(k)]{[Ai − L i C j ]e(k) + [E i − L i D j ]w(k)}.

i=1 j=1

(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 input in the system: z(k) =

r

αi [q(k)]{ i e(k) + i w(k)},

(14)

i=1

where z(k) ∈ Rl , i ∈ Rl× 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∞ performance is given by the combination of (13) and (14), i.e., with input w(k), state vector e(k), and output z(k), or:  r r    e(k + 1) = αi [q(k)]α j [q(k)]{[Ai − L i C j ]e(k) + [E i − L i D j ]w(k)}   i=1 j=1

    z(k)

=

r

αi [q(k)]{ i e(k) + i w(k)}.

i=1

(15) Therefore, the controller must fulfill two goals: (i) to maintain the synchronization error kept to a minimum level, i.e., to ensure equation (3); (ii) to minimize the 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


433

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

0< w 2 h . if i = h

(20)

(21)

(22)

(23)

Proof. First, consider the following notation: r

αi [q(k)]Ni = N¯

(24)

i=1

which is extended to the other matrices. Using this notation, rewrite (15) as follows: ¯ ¯ e(k + 1) = Ge(k) + Mw(k) , (25) ¯ ¯ z(k) = e(k) + w(k) ¯ where G¯ = A¯ − L¯ C¯ 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 function candidate: 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)


435

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

∞

z(k)T z(k) − γ 2 w(k)T w(k) ≺ 0.

(28)

k=0

Taking the initial conditions equal to zero, equation (28) can be rewritten as: V [e(k)] + z T (k)z(k) − γ 2 w T (k)w(k) ≺ 0.

(29)

Substituting equations (25) and (27) into (29), the following augmented system can be obtained: ˜ e(k) e(k) ˜ T ˜ ≺ 0, where e(k) ˜ [e(k) w(k)]T , and T ¯ T ¯ G¯ P G¯ − P + ˜ ∗

¯ T ¯ G¯ T P M¯ + ¯ . ¯ T −γ 2 I + M¯ T P M¯ +

(30)

(31)

˜ ≺ 0, where Notice that (30) can be put in the equivalent form e(k) ˜ T e(k)   T T ¯ −P G¯ P 0  P G¯ ¯ −P P M 0  .  (32) T 2  0 ¯ ¯ M P −γ I T  ¯ ¯ 0

−I Using the notation in (20) and according to (8):   0 θ( iT ) θ(−P) θ(G iTj P)   0  θ (P G i j ) θ(−P) θ (P Mi j ) (33) = , 0 θ (MiTj P) θ (−γ 2 I ) θ( iT )  θ ( i ) 0 θ ( i ) θ (−I ) r r where θ i=1 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:   0 iT −P G iTj P r r   −P P Mi j 0   P Gi j = (34) αi [q(k)]α j [q(k)]  . T T 2 Mi j P −γ I i   0 i=1 j=1 i 0

i −I Finally, using (1) and the definitions introduced in (19) yields =

r i=1

αi [q(k)]2 Q i +

r

αi [q(k)]α j [q(k)]Q i j .

(35)

i< j

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

436


Ri j , Si j h , and Ti j h as the same matrices presented in this paper. Then, replacing x(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) r where κ = [α1 I · · · αr I ] h=1 K h [α1 I · · · αr I ]T . Therefore, if K h ≺ 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 asymptotic stability in (30) is also guaranteed. Because the optimization problem is convex, the feasibility ensures that the minimum H∞ disturbance attenuation level is attained. ✷ Remark 2. In Theorem 1 the matrices P, Ti j h , and Ri j are symmetric whereas Si j h are skew matrices. Moreover, P ∈ Rn×n , and the other matrices have the same dimension as Q i j . 5. Fuzzy H∞ synchronization control design Based on the results of the last section, the next theorem establishes the LMI machinery to obtain the synchronization gains L i ensuring an H∞ level of performance for the error system dynamics in (15). Theorem 2. Consider the error system in (15). If the following optimization problem is feasible: min

δ

s.t.

h ≺ 0, K

P,Ti j h ,Ri j ,Si j h ,X i ,δ

P 0, Ti j h 0, (i, j, h = 1, 2, . . . , r, i < j), (37)

where



V1 − Z 1h V12 + N21h h  K  ..  . V1r + Nr 1h 

−P  Yii Vi   0 i

YiiT −P JiiT 0

0 Jii −δ I

i

V12 + N12h V2 − Z 2h .. . V2r + Nr 2h

 iT 0  ,

iT  −I

··· ··· .. . ··· 

 V1r + N1r h V2r + N2r h   , ..  . Vr − Z r h

−P  B Vi j  i j  0 i j

BiTj −P UiTj 0

0 Ui j −δ I ϒi j

 iTj  0  , ϒiTj  −I (38)


vc1(t) id(t) u1(t)

G

+

C1

R

vc2(t)

u2(t)

+

L

C2 u3(t)

437

+

-

Figure 1. Chua’s circuit schematic diagram.

Yi j P Ai − X i C j ,

Ji j P E i − X i D j ,

Yi j + Y ji Ji j + J ji , and Ui j , (39) 2 2 then the synchronization gains that guarantee the minimum H∞ level performance are given by L i P −1 X i . Bi j

Remark 3. In this theorem, the matrices i j , ϒi j , Z i j , and Ni j h are the same as the ones in Theorem 1. Remark 4. The proof of Theorem 2 is omitted as it follows directly from the derivation of Theorem 1, just carrying out the linearization change of variables given by X i P L i .

6. Experimental results In this section, the proposed methodology is applied to the robust synchronization of a coupled nonlinear system. The practical implementation used in this work is a chaotic Chua’s oscillator [5]. Basically, this circuit is composed of a resonant tank (parallel capacitor, C2 , and inductor, L) linked by a linear resistor R to a capacitor C1 , as illustrated in Figure 1. In parallel with this capacitor, C1 , there is a nonlinear resistor, G, called Chua’s diode. Figure 1 also illustrates Chua’s oscillator with extra voltage and current sources. The oscillator dynamics can be described as 1 1 dvc1 (t) = [vc (t) − vc1 (t)] − G[vc1 (t)] dt C1 R 2 dvc2 (t) 1 1 = [vc (t) − vc2 (t)] + il (t) dt C2 R 1 dil (t) 1 (40) = [−vc2 (t) − R0 il (t)], dt L 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

438

ˆ M OZELLI , C AMPOS , PALHARES , T ORRES , AND M ENDES Gb

id(t)

Ga

E –d

–E

d vc1(t)

Figure 2. Chua’s diode conductance feature.

dependent, i.e., G[vc1 (t)], which is piecewise linear. When the voltage is within the range ±E, the diode conductance is equal to G a , and outside this sector it is equal to G b . This behavior is depicted in Figure 2. The slope for the currentvoltage curve between ±E represents the conductance G a ; for the rest of the figure the slope corresponds to G b . Therefore, it is straightforward to formulate two fuzzy rules that describe completely the behavior of this Chua’s circuit. The first linear model assumes that the conductance is G a for low voltages. This is represented in Figure 2 by the straight line 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 local dynamic for higher voltage levels, representing the dynamics exactly at ±d and approximately in the vicinity. In Figure 2, this is represented by the other line crossing 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 G g = b )E ). More details about this fuzzy modeling can be found in [34]. (G b + (G a −G d In [11] the reader can find an overview on chaotic systems TS fuzzy modeling, including Chua’s circuit. Therefore, the fuzzy rules are  IF: vc1 (t) is M1 (nearby zero)      THEN: x(t) ˙ = A1 x(t) + E 1 w(t) Rule 1:   y(t) = C1 x(t) + D1 w(t)    z(t) = 1 x(t) + 1 w(t)  IF: vc1 (t) is M2 (nearby ± d)      THEN: x(t) ˙ = A2 x(t) + E 2 w(t) Rule 2: .   y(t) = C x(t) + D w(t)  2 2   z(t) = 2 x(t) + 2 w(t)


439

1 0.8 0.6 0.4 0.2 0 –6

–4

–2

0

2

4

6

4

6

Figure 3. Membership functions.

x 10

–3

3 2

id

1 0 –1 –2 –3 –6

–4

–2

0

2

vc1

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

The membership functions may be chosen as in [34] and are depicted in Figure 3. These inference functions combining the two fuzzy rules can represent the exact diode characteristic. This is illustrated in Figure 4, where the line represents the actual diode characteristic and the points marked with × are the diode characteristic according to the chosen fuzzy model. Because the methodology deals with discrete-time systems it is necessary to consider a discretization approach to Chua’s circuits as, for instance, the one presented in [15] which is shown to preserve the fixed points of the original system. This approach is chosen to be used in this work as it guarantees the reconstruction of the original dynamics even in a large range of values of the increment time. The actual circuit parameters are given in Table 1 and were obtained using a parameter 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 details can be found. Consider the information transmission scheme exhibited in Figure 5. This

440


Table 1. Circuit and model parameters

η(t)

Parameters

Values

C1 C2 L R R0 Ga Gb E d

30.14 µF 185.6 µF 52.28 H 1673 0 −0.801 mS −0.365 mS 1.74 V 6V

y(t)

+

-

R +

C1

R

Controller

id(t) Transmitter i(t)

^y (t)

+

C2

id(t)

Receiver +

L

u1(t) u2(t) u3(t)

C1

+

L C2 +

-

Figure 5. Unidirectional communication system.

scheme shows the Chua’s oscillator circuits on a unidirectional coupling, where y(t) is the master output and yˆ (t) is the slave output. The information to be transmitted i(t) is injected on the master Chua’s chaotic oscillator (transmitter) as a perturbation only in the direction of vc1 (t), as illustrated in Figure 5. Mathematically this perturbation corresponds to a modification on the differential equations that govern the motion of the oscillator. Then a scalar signal y(t) is used to carry the information (from the master) and to serve as a reference signal to allow the synchronization of the slave oscillator (receiver). This signal may be corrupted by noise interferences η(t) on the transmission channel. In this framework, the same circuit operates as the master and the slave. When configured as the master, it oscillates freely, and the signal to be transmitted is 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 random initial conditions. At some instant in time the system has to synchronize with the master reference that was previously recorded. Because the elapsed time between data acquisition and the synchronization experiment is short, the assumption that master and slave systems have the same dynamics is plausible, and consequently there is no time for significant changes in the circuit parameters.


441

Table 2. Synchronization gains provided by Theorem 2

u 1 (k) u 2 (k) u 3 (k)

L1

L2

1.0668 × 102 2.9186 5.4422 × 10−5

9.4710 × 101 2.9188 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 that the master and slave oscillators share the same dynamics and considering that the weighted error z(t) is simply the summation of all state variables, the fuzzy model matrices are given by   T /(RC1 ) 0 1 − T /(RC1 ) − T G a /C1 T /(RC2 ) 1 − T /(RC2 ) T /C2  , A1 =  0 −T /L 1 − T R0 /L   1 − T /(RC1 ) − T G g /C1 T /(RC1 ) 0 1 − T /(RC2 ) T /C2  , T /(RC2 ) A2 =  (41) 0 −T /L 1 − T R0 /L    T   0.001 1 1 0 0 E (1,2) =  0  , C(1,2) = 0 , (1,2) = 0 1 0 , 0 0 0 0 1 T D(1,2) = [0.0001],

(1,2) = 0 0 0 , where T is the sampling time. 6.1. Results The information transmission problem is investigated in the light of the information transmission via control (ITVC) principle developed in [31]. The information transmission test is used as a performance index to validate the proposed approach for robust synchronization. The ITVC principle states that any controller that guarantees an identical, or quasi-identical, master-slave synchronization can actually perform as a demodulator and thus recover the transmitted information. In this way, the control signal u 1 (t) corresponds to the demodulated information signal i(t). This section presents the synchronization experimental results with the gains obtained by applying Theorem 2 to the discretized TS fuzzy model in (41). Using the LMI Control Toolbox for MATLAB, the synchronization gains were calculated (see Table 2). These gains guarantee an H∞ level of 9.061 × 10−4 , when considering a sampling time of T = 10 ms. The PCCHUA setup [32] allows the user to use a wide variety of signals to

442

ˆ M OZELLI , C AMPOS , PALHARES , T ORRES , AND M ENDES 6

voltage (V)

4 2 0 –2 –4 –6

0

20

40

60

80

100

120

80

100

120

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voltage (V)

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0

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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 tests results 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 a sinusoid voltage with a frequency of 0.5 Hz and 0.12 mV of amplitude is added to 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 s and ends at 90 s. The control signal persists for 15 more seconds and the control experiment 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 maximum error, which clearly shows the synchronization. The transmitted and recovered signals (control action signal) are depicted in Figure 8. Notice that after applying a second-order Butterworth low-pass digital filter, with cutting frequency f c = 1 Hz, the final received signal is obtained as depicted in Figure 9. It is worth mentioning that the small time delay between the original and the recovered signals, in Figure 9, is due to the filter phase lag.


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Figure 7. Error in the state variable vc1 for the first experiment. x 10

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

6.1.2. Second experiment: Low-frequency signal combined with higher-frequency signals Here the transmitted input signal is a combination of three signals: a square wave with 0.5 Hz and 0.16 mV; a sinusoid with 2 Hz and 0.08 mV; and a sawtooth with 4 Hz and 0.04 mV. The experimental procedure is the same as the one

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

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

described previously. Once more, the synchronization has been achieved with a synchronization 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 f c = 10 Hz was used. 6.1.3. Third experiment: Signals with close frequencies and amplitudes For the last experiment, the transmitted signal consists of a combination of three signals with close frequencies and amplitudes. They are: sinusoid with 1.2 Hz and 0.08 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 error magnitude was a little higher than in the other cases, but still very small, limited by almost 4%, not impairing the robust synchronization. As shown in Figures 14 and 15, the transmitted signal can be recovered, thus illustrating the efficiency of the proposed method.

C HAOTIC S YNCHRONIZATION AND F UZZY M ODELING 4

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

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

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

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

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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 been given to check the synchronization of coupled nonlinear systems in practice. The H∞ synchronization approach has performed successfully when the goal is the problem of signal transmission through a chaotic channel.

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

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