Adaptive synchronization of coupled self-sustained electrical systems Samuel Bowonga , Ren´ e Yamapib , Paul Woafob,1 a

Laboratoire de Math´ematiques Appliqu´ees, D´epartement de Math´ematiques et Informa-

tique, Facult´e des Sciences Universit´e de Douala, B.P. 24157 Douala (Cameroun) b

Laboratoire de M´ecanique, Facult´e des Sciences, Universit´e de Yaound´e I, B.P. 812 Yaound´e (Cameroon), Email: {sbowong, ryamapi, pwoafo}@uycdc.uninet.cm Abstract This study addresses the adaptive synchronization of coupled self-sustained electrical systems described by the Rayleigh-Duffing equations. We show that the synchronization of two coupled such systems can be achieved by means of nonlinear feedback coupling. We use the Lyapunov direct method to study the asymptotic stability of the solutions of the synchronization error system. Numerical simulations are given to explain the effectiveness of the proposed control scheme. Keywords: Adaptive synchronization, Self-sustained system, Lyapunov direct method.

1

Introduction

The phenomenon of synchronization in dynamical and, in particular, mechanical and electrical systems has been the focus of a growing literature during the last past decade [1]. It has been demonstrated that two or more dynamical systems which can exhibit either chaotic or non chaotic behavior can synchronize by linking them with mutual coupling or with a common signal or signals [1-6]. In the case of linking a set of identical chaotic systems (the same set of ordinary differential equations (ODEs) and values of the system parameters) ideal or complete synchronization can be obtained. The ideal synchronization takes place when all trajectories converge to the same value and remain in step with each other during further evolution (i.e., lim |x(t) − y(t)| = 0 for two arbitrarily chosen trajectories x(t) and y(t)). In such situation, t→∞

all subsystems of the augmented system evolve on the same attractor in which one of these subsystems evolves (the phase space is reduced to the synchronization manifold). In publications regarding synchronization of dynamical systems, feedback design is often based upon the assumptions that the master system is precisely known, and the slave system can be easily constructed with the well-known parameters. However, in real synchronization, unknown parameters do exist and the external disturbances are always unavoidable. The effects of these uncertainties will destroy the synchronization and even break it. Therefore, adaptive synchronization of dynamical systems in the presence of the system disturbance and unknown parameters is essential [7-11]. This paper studies the possibilities of adaptive synchronization of two coupled oscillators by means of a nonlinear feedback coupling. We consider the synchronization of self-sustained electrical systems described by the Rayleigh-Duffing equation. We show that with a nonlinear 1

Corresponding author: Permanent address B.P. 8210 Yaound´e (Cameroun)

1

feedback coupling, we can currently obtain adaptive synchronization. The proposed nonlinear coupling is a nonlinear resistor which is expressed in terms of the difference of the instantaneous value of the current in one system called the driving system, and the corresponding variable in the other system, called the controlled system. The parameters of both systems are assumed to be identical, but unknown. We use the Lyapunov stability theory to study the asymptotic stability of the solutions of the synchronization error system. Numerical results have good agreement with theoretical ones.

2

The self-sustained electrical system

Figure 1: Schematic of a self-sustained electrical model. The model shown in Fig. 1 is a self-excited electrical system described by the RayleighDuffing oscillator consisting of a nonlinear resistor NLR, a condenser C and an inductor L, all connected in series. Two types of nonlinear components are considered in the model. The voltage of the condenser is a nonlinear function of the instantaneous electrical charge q. It is expressed by VC =

1 q + a3 q 3 , C0

where C0 is the linear value of C and a3 is a nonlinear coefficient depending on the type of the capacitor in use. This is typical of nonlinear reactance components such as varactor diodes widely used in many areas of electrical engineering to design for instance parametric amplifiers, upconverters, mixers, low-power microwave oscillators, etc [12]. The current-voltage characteristic of a resistor is also defined as " µ ¶ µ ¶ # i 3 i + , VR0 = R0 i0 − i0 i0 where Ro and io are respectively the normalization resistance and current. With this nonlinear resistance, the model has the property to exhibit self-sustained oscillations. The nonlinear resistance can be realized using a block consisting of two transistors [13]. The model is described by " µ ¶ # d2 q 1 dq 2 dq L 2 − R0 1 − 2 dτ io dτ dτ +

q + a3 q 3 = 0. C0 2

With the following dimensionless variables q=

i0 x , we q0

t = we τ,

and

R0 a3 i0 1 , we2 = , α= , 2 4 Lwe LC0 Lwe q0 where qo is the reference charge, we find that the above differential equation reduces to the Rayleigh-Duffing equation: µ=

x ¨ − µ(1 − x˙ 2 )x˙ + x + αx3 = 0,

(1)

where the dots over the quantities represent the derivative with respect to time t. µ and α are two positive coefficients. This model has the similar behavior as the Van der Pol-Duffing oscillator [14-16]. It generates the limit cycle, which can (for a low value of the coefficient µ) be approximated by the harmonic function of time defined as x(t) = a cos(w0 t − φ0 ), √ where a = 23 3, φo the phase and the limit cycle frequency corresponding to w02 = 1 + 38 αa2 . This limit cycle is known to be a fairly strong attractor since it attracts all trajectories except the one initiated from the trivial fixed point (x0 , x˙ 0 ) = (0, 0). As the Van der Pol oscillator, a particular characteristic in the Rayleigh-Duffing model is that its phase depends on initial conditions. Consequently, if two Rayleigh-Duffing oscillators are launched with different initial conditions, their trajectory will finally circulate on the same limit cycle, but with different phases φ1 and φ2 . Thus, the objective of the synchronization in this case is to phase-lock so that φ1 − φ2 = 0. This is the aim of section 3.

3

Adaptive synchronization algorithm By using x1 = x and x2 = x, ˙ we may rewrite (1) as follows ½ x˙ 1 = x2 , x˙ 2 = µ(1 − x22 )x2 − x1 − αx31 .

(2)

In order to observe the synchronization behavior in Rayleigh-Duffing equation, the response system is chosen to be ½ y˙ 1 = y2 , (3) y˙ 2 = µ(1 − y22 )y2 − y1 − αy13 + u, where we have introduced the control input u. The control input u is to be determined for the purpose of synchronizing two identical Rayleigh-Duffing oscillators with the same but unknown parameters µ and α in spite of the difference in initial conditions. Let us define the state errors between the response that is controlled and the controlling drive system as e1 = y1 − x1 and e2 = y 2 − x 2 . (4) Subtracting Eq. (3) from Eq. (2) and using the notation (4) yields  e˙ 1 = e2 ,      e˙ 2 = −e1 + µe2 − µe2 (e22 + 3e2 x2 + x22 )      −αe1 (e21 + 3e1 x1 + x21 ) + u. 3

(5)

Or in a matrix equation form as e˙ = Ae + Bf (e, x1 , x2 ) + Bu, where e = (e1 , e2 )T ,

(6)

f (e, x1 , x2 ) = −µe2 (e22 + 3e2 x2 + x22 ) −αe1 (e21 + 3e1 x1 + x21 ), · ¸ 0 1 A= and −1 µ

· B=

0 1

¸ .

Hence, the synchronization problem is now replaced by an equivalent problem of stabilizing the system (6) at the origin, by using a suitable choice of the feedback coupling u. We assume that the following assumptions are pertaining to the error system (6). A1: The states of the Rayleigh-Duffing equation (1) are bounded, and the nonlinear function f (e, x1 , x2 ) with f (0, x1 , x2 ) = 0 satisfies the Lipschitz condition in e with respect to x1 and x2 , i.e., there always exists a positive constant kf such that kf (e, x1 , x2 ) − f (0, x1 , x2 )k ≤ kf kek.

(7)

A2: We can choose a constant matrix C, a feedback gain matrix K and two positive definite matrices P and Q with appropriate dimension satisfying (A − KC)T P + P (A − KC) = −Q,

(8)

B T P = C.

(9)

and Note that the equality (9) implies that the span of rows of B T P belongs to the span of rows of C. Note also that the Lipschitz constant kf is often required to be known for the synchronization design purpose. In fact, it is often difficult to obtain a precise value of kf in some practical systems, hence the Lipschitz constant is selected to be larger, which will induce the feedback gain to be higher, and the obtained results would be conservative. In this paper, we will assume that the Lipschitz constant is unknown, and an adaptive method will be proposed to avoid this difficulty. By adding and subtracting the term KCe, we obtain the following synchronization error equation: e˙ = (A − KC)e (10) +KCe + Bf (e, x1 , x2 ) + Bu. The feedback control law is described as u = −k1 (y2 − x2 ) − k2 (y2 − x2 )3 , (11) = −k1 e2 − k2 e32 , where k1 and k2 are estimated feedback gains updated according to the following adaptive algorithm: k˙ 1 = γ1 e22 and k˙ 2 = γ2 e42 . (12) In this case, C = (0, 1) and the controlling force (11) can be considered as a nonlinear resistance. The schematic circuit of two coupled identical self-sustained electrical models with a unidirectionally homogenous coupling element is shown in Fig. 2. In this circuit, the two selfsustained models, namely, the master and slave systems are coupled by a nonlinear resistor Rc 4

Figure 2: Schematic of unidirectional coupled self-sustained electrical models. and a buffer. The buffer acts a signal-driving element that isolates the master system variable from the slave system variable, thereby providing a one-way coupling or unidirectional coupling. In the absence of the buffer, the system represents two identical self-sustained models coupled by a common resistor Rc , when both the master and slave systems will mutually affect each other. Thus, the resulting error dynamical system can be expressed as follows: e˙ = (A − KC)e + KCe +Bf (e, x1 , x2 )

(13)

−k1 BB T P e − k2 B(B T P e)3 . We define the Lyapunov function candidate as follows: V (e, k1 , k2 ) = eT P e + +

1 (k2 − k¯2 )2 , γ2

1 (k1 − k¯1 )2 γ1

(14)

where k¯1 and k¯2 are positive constants defined in Eq. (19). Taking the time derivative of Eq.

5

(14) along the trajectories of the closed-loop system (13), we get V˙ (e, k1 , k2 ) = eT [(A − KC)T P + P (A − KC)]e +2eT P KCe + 2eT P Bf (e, x1 , x2 ) −2k1 eT P BB T P e − 2k2 eT P B(B T P e)3 ≤ −eT Qe + 2kB T P ek kf kek

(15)

+2kB T P ek kK T P ek − 2k1 kB T P ek2 −2k2 kB T P ek4 + +

2 (k1 − k¯1 )k˙ 1 γ1

2 (k2 − k¯2 )k˙ 2 . γ2

Notice that 2kB T P ek kf kek ≤



kf2 ε1

kf2 ε1

kB T P ek2 + ε1 kek2 (16)

kB T P ek2 + ε2 kB T P ek4 + ε1 kek2 ,

and 2kB T P ek kK T P ek ≤

1 kB T P ek2 ε3

+ε3 kK T P ek2 1 ≤ kB T P ek2 + ε4 kB T P ek4 ε3

(17)

+ε3 λmax (P KK T P )kek2 , where ε1 , ε2 , ε3 , ε4 are positive scalar and λmax (P KK T P ) the maximum eigenvalue of P KK T P . Substituting Eqs. (16) and (17) into (15), one obtain V˙ (e, k1 , k2 ) = −eT [Q − (ε1 + ε3 λmax (P KK T P ))I2 ]e à +

kf2

1 + ε1 ε3

! kB T P ek2 (ε2 + ε4 ) kB T P ek4

−2k1 kB T P ek2 − 2k2 kB T P ek4 +

2 2 (k1 − k¯1 )k˙ 1 + (k2 − k¯2 )k˙ 2 , γ1 γ2

6

(18)

where I2 is the identity matrix of dimension 2. Let à ! kf2 1 1 k¯1 = + , 2 ε1 ε3

(19)

1 k¯2 = (ε2 + ε4 ) , 2 then, one has

V˙ (e, k1 , k2 ) = −eT [Q − (ε1 + ε3 λmax (P KK T P ))I2 ]e Ã

k˙ 1 +2(k1 − k¯1 ) −kB T P ek2 + γ1 Ã

k˙ 1 +2(k2 − k¯2 ) −kB T P ek4 + γ2

! (20) ! .

Using the adaptive laws (12), Eq. (20) will become V˙ (e, k1 , k2 ) = (21) −eT [Q − (ε1 + ε3 λmax (P KK T P ))I2 ]e. Note that the parameters ε1 and ε3 can be selected to be small enough to let the matrix S = Q − (ε1 + ε3 λmax (P KK T P ))I2 positive definite so that V˙ (e, k1 , k2 ) is semi negative definite. Thus, e1 , e2 ∈ L∞ . Next as V˙ (e, k1 , k2 ) ≤ −eT Se and S is positive definite, we obtain Rt 0

λmin (S)(e21 + e22 )dt ≤

≤−

Rt 0

Rt 0

eT Sedt,

V˙ (e, k1 , k2 )dt,

≤ V (e(0), k1 (0), k2 (0)) − V (e(t), k1 (t), k2 (t)), ≤ V (e(0), k1 (0), k2 (0)), where λmin (S) is the minimum eigenvalue of the positive definite matrix S. It follows that e1 , e2 ∈ L2 . Then, from Eq. (13), we have e˙ 1 , e˙ 2 ∈ L∞ . By Barbalat’s Lemma [17], we have e1 (t), e2 (t) → 0 as t → ∞. This implies that the response system (3) under the adaptive feedback (11), (12) synchronizes with the drive system (2).

4

Simulation results

The fouth-order Runge-Kutta algorithm is used to solve the system of differential equations. The parameters used are µ = α = 1. The initial conditions of the drive system and of the response system are (x1 (0), x2 (0)) = (1, 0) and (y1 (0), y2 (0)) = (2, 0), respectively. Then, e1 (0) = 1 and e2 (0) = 0. In this case, we assume that the drive system and the response system (two coupled identical self-sustained electrical systems) are starting the motion with different initial 7

conditions. In all simulations, we choose k1 (0) = 0, k2 (0) = 0 and γ1 = γ2 = 1. For these systems, we performed the following four set of simulations. The first set of simulations correspond with the adaptive synchronization of two coupled selfsustained electrical systems with a linear feedback coupling. In this case, k2 = 0 and u = −k1 e2 . Figures 3 and 4 show respectively the synchronization errors and, the feedback gain and the feedback coupling. Figure 3 presents the temporal evolutions of the state synchronization errors e1 (t) = y1 (t) − x1 (t) and e2 (t) = y2 (t) − x2 (t). On can observe that after a short transient, the synchronization error converges exactly to the origin and the synchronization objective is attained. The histories of the estimated feedback gain k1 and the feedback coupling u are plotted when the time evolves in Fig. 4. It is found that the feedback gain k1 has a lower bound and the feedback coupling u converges exactly to zero. 1.2

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Figure 3: Adaptive synchronization of two coupled self-sustained electrical systems by means of the feedback coupling u = −k1 e2 . (a) e1 = y1 − x1 and e2 = y2 − x2 .

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Figure 4: (a) Feedback gain k1 and (b) feedback coupling u. In the second set of simulations, we consider the adaptive synchronization of two coupled selfsustained electrical systems with a cubic feedback coupling. In this case k1 = 0 and u = −k2 e32 . The histories of the state synchronization errors e1 (t) = y1 (t) − x1 (t) and e2 (t) = y2 (t) − x2 (t) shown on the left hand side of Fig. 5 were obtained when γ2 = 1. It clearly appears that the synchronization error converges around the origin. This is due to the effects of the cubic term in the feedback coupling which does not dominate the nonlinear function f (e, x1 , x2 ). The dependence of the feedback coupling on k2 deserves special attention. Note that we have k˙ 2 = γ2 e42 . Indeed, as γ2 increases, k2 will increase. Hence, if this term is large, the resulting error will be small as well. Thus, as k2 increases, −k2 e32 will dominate the nonlinear function 8

f (e, x1 , x2 ), which also decreases the asymptotic error bound. To show the effect of increased gain, we also choose γ2 = 10. The graphes of the time variation of the synchronization errors are depicted on the right hand side of Fig. 5. As expected, the asymptotic error is less than those of figures on the left hand side of Fig. 5. Thus, the synchronization error can be made arbitrarily small by increasing the constant γ2 which increasing the feedback gain k2 . Figure 6 delineates the estimated feedback gain k2 and the feedback coupling u when the time evolves for γ1 = 1 (figures on the left hand side) and for γ2 = 10 (figures on the right hand side). As expected, the feedback gain bound on the right hand side of Fig. 6 is less than that on the left hand side of Fig. 6. Thus, this argument shows that with the proposed method when k1 = 0, k2 should be made as large as possible. 0.25

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Figure 5: Adaptive synchronization of two coupled self-sustained electrical systems by means of the feedback coupling u = −k2 e32 performed with γ2 = 1 (Figures on the left hand side) and γ2 = 10 (Figures on the right hand side). The third set of simulations corresponds with the adaptive synchronization of two coupled self-sustained electrical systems with both linear and cubic feedback coupling, i.e., u = −k1 e2 − k2 e32 . Figure 7 presents the synchronization errors e1 (t) and e2 (t) versus the time t while Fig. 8 shows the evolutions of the feedback gains k1 and k2 , and the feedback coupling signal u versus the time t. These numerical simulations demonstrate that the two coupled self-sustained electrical systems have been synchronized using the adaptive feedback coupling (11), (12). In the four set of simulations, we consider the synchronization of two coupled self-sustained electrical systems with parameter mismatch. We simulated system (3) with µ = 1.5, α = 1.5 and u = −k1 e2 − k2 e32 . In this case, the order of the parameter mismatch is 0.701%. The results of the simulation are depicted in Fig. 9. The projections of the attractors from the four-dimensional phase space onto the planes (x1 , y1 ) and (x2 , y2 ), respectively for u = 0 are shown on the left hand side of Fig. 9 (uncontrolled evolution). These projections clearly indicate that these oscillations are not synchronized. The relation between the states of the drive and 9

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Figure 7: Adaptive synchronization of two coupled self-sustained electrical systems by means of the feedback coupling u = −k1 e2 − k2 e32 . (a) e1 = y1 − x1 and (b) e2 = y2 − x2 .

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Figure 8: (a) Feedback gain k1 ; (b) feedback gain k2 and (c) feedback coupling u.

response systems under feedback actions is depicted on the right hand side of Fig. 9. Note that the phases of the drive and response systems are locked, which is a common measure of the degree of synchronization. Thus, it clearly appears that the manifolds x1 = y1 and x2 = y2 are stable, and one can conclude that the chaotic oscillations of the drive and response systems are synchronized in phase and complete sense.

5

Conclusions

In this paper, we introduce adaptive synchronization problems for the coupled self-sustained electrical systems described by the Rayleigh-Duffing equations, which is taken as an example of nonlinear dynamical systems of great importance in physics. We have studied the possibilities of adaptive synchronization of two coupled Rayleigh-Duffing oscillators by means of a nonlinear feedback coupling. The stability analysis is derived from the Lyapunov direct method. Numerical simulations are given to explain the effectiveness of the proposed synchronization scheme. We hope that the methodology developed for this system will be applicable to other types of dynamics systems such as Duffing oscillator, Van der Pol equation, Chua circuit, R¨ossler system, etc. We think that this technique provides a strong tool for synchronization and is full of promise because it could be applied in a great range of problems: stabilization, tracking, implementation, etc. Although in this paper, the implementation is performed via numerical simulations, it is not hard to see that the physical application of the proposed coupling technique can be performed. In fact, experimental results are under progress and they will be reported elsewhere.

11

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Figure 9: Relation between the drive and response systems without feedback coupling (figures on the left hand side) and under feedback actions (figures on the right hand side).

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[9] Feki M. An adaptive chaos synchronization scheme applied to secure communication. Chaos, Solitons & Fractals, 18 (2003), 141-148. [10] Han X., Lu J. A., Wu X. Adaptive feedback synchronization of L¨ u system, Chaos, Solitons & Fractals, 22 (2004), 221-227. [11] Elabbasy E. M., Agiza H. N., El-Dessoky M. Controlling and Synchronization of Rossler System with Uncertain Parameters, International Journal of Nonlinear Sciences and Numerical Simulation, 5(2), (2004), 171-182 [12] Oksasoglu A., Vavriv D. Interaction of low- and High-frequency oscillations in a nonlinear RLC circuit, IEEE Trans. Circ. Syst-I, 41 (1994) 669-672. [13] Hasler M. J. Electrical circuits with chaotic behavior. Proc. IEEE, 75 (1987), 1009-1021. [14] Szempli´ nska-Stupnicka W., Rudowski J. Neimark bifurcation, almost-periodicity and chaos in the forced Van der Pol-Duffing system in the neighbourhood of the principal resonance, Phys. Lett. A, 192 (1994), 201-206. [15] Venkatesan A., Lakshamanan M. Bifurcation and chaos in the double-well Duffing-Van der Pol oscillator: Numerical and analytical studies, Phys. Rev. E 56 (1997), 6321-6330. [16] Zhang Y., Xu J. Classification and Computation of Non-resonant Double Hopf Bifurcations and Solutions in Delayed van der Pol-Duffing System, International Journal of Nonlinear Sciences and Numerical Simulation, 6(1) (2005), 63-68 [17] Khalil H. K. Nonlinear systems, third edition, United Kingdom: Springer-Verlag, 1995.

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