Optimization of the synchronization of the mod- ified ...

Vol. 00 No. 0, pp. 1-24 January 2014

Journal of Advanced Research in Name

Optimization of the synchronization of the modified Duffing system Michaux Kountchou Noube1 , Patrick Louodop1 , Samuel Bowong2,∗ , Hilaire Fotsin1 1

Laboratory of Electronics, Automation and Signal Processing, University of Dschang, PO Box 67 Dschang, Cameroon. 2 Department of Mathematics and Computer Science, University of Douala, PO Box 24157 Douala, Cameroon.

Abstract. This paper addresses the problem of optimization of the synchronization of a modified Duffing system. We first propose a new four-dimensional autonomous system obtained by the modification of the classical two-dimensional Duffing system. The dynamical behaviors of the modified system are investigated. Furthermore, we propose a robust feedback coupling which accomplishes the synchronization of two modified chaotic systems using an optimal tuning scheme based on the Riccatti equation. The approach developed considers incomplete state measurements and no detailed model of the system to guarantee the robust stability. This approach includes an uncertainty estimator and leads to a robust feedback coupling. A finite horizon can be arbitrarily established to ensure that the chaos synchronization is achieved at established time. An advantage is that the proposed scheme accounts the energy wasted by the controller and the closed-loop performance on synchronization. Both stability analysis and numerical simulations are presented to show the effectiveness of the proposed optimization strategy. Pspice analog circuit implementation of the complete master-slave-controller system is presented to show the practical applicability of the proposed scheme. Keywords: Modified Duffing system, Synchronization, Optimization, Ricatti Equation, Pspice simulation.

Mathematics Subject Classification 2010: 20N15, 20C99.

∗

Correspondence to: Samuel Bowong, Department of Mathematics and Computer Science, University of Douala, PO Box 24157 Douala, Cameroon. Email: [email protected] ∗ Received: 3 october 2009 http://www.i-asr.com/Journals/

1

c ⃝2010 Institute of Advanced Scientific Research

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Optimization of the synchronization of the modified Duffing system

Introduction

In 1963, Lorenz found the first chaotic attractor in a 3-dimensional autonomous system when he studied atmospheric convection. In 1979, R¨ossler reported the first hyper chaotic system with two positive Lyapunov exponents [31]. In 1999, Chen produced a 3dimensional autonomous chaotic system based on Lorenz system. Chen et al. introduced a new 4-dimensional hyper chaotic system which had larger Lyapunov exponents in comparison with the previous one [29]. In recent years, the implementation and study of chaotic systems has grown up in many fields and attracts many scientists. Since the idea of synchronizing chaotic systems was introduced by Pecora and Carroll in 1990 [28], chaos synchronization has received increasing attention due to its theoretical challenge and its great potential applications in secure communication [22], chemical reaction and biological systems [6] and so on. Due to the wide potential applications of chaos synchronization, various synchronization schemes have been proposed in the last two decades both in theoretical analysis and experimental implementations, such as generalized synchronization [34,35], phase synchronization [27,32], lag synchronization [22, 33], anti-synchronization [16, 22–24] and many others [3–5, 13, 14, 21, 25, 30, 36, 37]. But despite the amount of theoretical and experimental results already obtained, two current questions are still open from the chaos synchronization problem of nonlinear systems. The former is about how to set an arbitrary time in which the synchronization of nonlinear systems be successfully achieved. The latter concerns to the estimation (and possibly reduction) of the feedback coupling effort wasted during the execution of feedback coupling. There is a few works in the literature about the finite time synchronization of chaotic systems [3, 36]. However the control effort is not accounted. That is, in seeking the optimization of the duration time in the chaos synchronization, the energy demanded to control by a (robust or optimized) feedback can be larger than the available energy by the physical actuator. One example is related to mechanical systems where chaos synchronization and suppression can be desired, the torque demanded by a control scheme can be larger than the maxima torque provided by the motors, which are the actuator for controlling the system motion [10]. Then, as an actuator is saturated as the closed-loop is broken and no feedback control is acting into the chaotic system. Among other situations induced by large control signals, the saturation condition is undesirable, since it implies that in some time interval a constant enters to the system and, as a consequence, the synchronization objective could not be ensured. This paper addresses the problem of optimization of the synchronization of two uncertain modified Duffing systems by accounting the control effort. The main idea behind our proposal is, departing from the discrepancy synchronization error system, to construct an extended nonlinear system which should be dynamically equivalent to the original system. In this way, the discrepancy is lumped into a nonlinear function, which is rewritten into the extended nonlinear system as a state variable. We then propose a robust linear feedback coupling which accomplishes the synchronization using an optimal tuning scheme based on the Riccatti equation. A suboptimal robust feedback coupling is used to get the estimates of the uncertainties, and then the estimated values

Kountchou Noube M et. al.

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are used to design a robust adaptive feedback coupling to force the dynamics of the slave system to follow the master system in spite of uncertainties errors, mismatch parameter and external perturbations. An advantage of the proposed optimization scheme is that simple a Riccatti equation is used to solve the optimization problem. The proposed approach is algebraic and only requires to solve a differential equation of dimension one. The results show that the control effort can be accounted under some conditions to set the time of convergence. Pspice simulations have been carry on electronic analog models of the complete master-slave systems under the action of the feedback coupling. The rest of the paper is organized as follows. In the next section, a four-dimensional autonomous system obtained by the modification of classical two-dimensional Duffing system is presented and, his properties and dynamics are investigated. In the next section, a design of robust synchronization scheme is proposed. Numerical and Pspice simulations are given to show the effectiveness and applicability of the proposed method. Finally, some conclusions and remarks are drawn in section 4.

2

The system and its dynamics

2.1

The system

The Duffing equation [7] is given by the following equation: d2 x dx +γ − x + x3 = 0, dt2 dt

(2.1)

where γ is a constant parameter. Following Acho et al. (2004) [2], the Duffing equation (2.1) can be converted to a Jerk system: d4 x d3 x d2 x dx + x − x3 . = − − 2 −γ 4 3 dt dt dt dt

(2.2)

The state space representation of this Jerk system is  x˙ 1 = x2 ,          x˙ 2 = x3 ,   x˙ 3 = x4 ,        x˙ 4 = −x4 − x3 − γx2 + x1 − x31 , 2

3

(2.3)

d x d x where x1 = x, x2 = dx dt , x3 = dt2 and x4 = dt3 . The chaotic behavior can be generated by adding two constant parameters b and c with two innovation terms x3 and x4 . Then,

4

Optimization of the synchronization of the modified Duffing system

the system of four-dimensional autonomous differential equations has the following form:  x˙ 1 = x2 + bx3 + cx4 ,          x˙ 2 = x3 , (2.4)   x ˙ = x ,  3 4       x˙ 4 = −x4 − x3 − γx2 + x1 − x31 , where b, c and γ are positive constants.

2.2

Dynamics of the system

In this section, a basic properties and dynamics of system (2.4) are investigated. 2.2.1

Dissipation and existence of chaotic attractor

The divergence [8] of the modified four-dimensional system (2.4) is ⃗ = ∂ x˙ 1 + ∂ x˙ 2 + ∂ x˙ 3 + ∂ x˙ 4 = −1 < 0. ∇V ∂x1 ∂x2 ∂x3 ∂x4

(2.5)

⃗ < 0, one can conclude that system (2.4) is dissipative, that is, a volume Thus, since ∇V element V0 is contracted by the flow into a volume element V0 e−t at time t. This means that each volume containing the trajectory of the dynamical system (2.4) shrinks to zero as t → ∞ at an exponential rate 1. Consequently, all the trajectories of system (2.4) ultimately arrive to an attractor. 2.2.2

Equilibria and their stability

An equilibrium of system (2.4) satisfies the following equations:  x2 + bx3 + cx4 = 0,          x3 = 0,   x4 = 0,        −x4 − x3 − γx2 + x1 − x31 = 0. System (2.4) has three equilibria: P0 = (0, 0, 0, 0), P1 The Jacobian matrix J of system (2.4) is  0 1 b  0 0 1 J =  0 0 0 1 − 3x21 −γ −1

(2.6)

= (1, 0, 0, 0) and P2 = (−1, 0, 0, 0).  c 0  . 1  −1

Kountchou Noube M et. al.

Without loss of generality, we choose γ = 0.4, point P0 = (0, 0, 0, 0), the Jacobian matrix is:  0 1  0 0 J0 =   0 0 1 −0.4

5

b = 0.5 and c = 0.9. For the equilibrium  0.5 0.9 1 0  . 0 1  −1 −1

The eigenvalues of J0 are λ1 = −1.310114526, λ2 = 0.8226185688, λ3 = −0.2562520212+ 0.9285557546i and λ4 = −0.2562520212 − 0.9285557546i, where i denotes the unit of imaginary number. Since λ2 is a positive real number, λ1 a negative real number, λ3 and λ4 are a pair of complex conjugate eigenvalues with negative real parts, the equilibrium P0 = (0, 0, 0, 0) is a saddle-focus point. In the same way, the eigenvalues corresponding to the equilibrium point P1 = (1, 0, 0, 0) are λ1 = −0.5138922602 + 1.078758353i, λ2 = −0.5138922602 − 1.078758353i, λ3 = 0.01389226028 + 1.183452012i and λ4 = 0.01389226028 − 1.183452012i. For the equilibrium point P2 , we obtain the same eigenvalues. Thus, for each of equilibria P2 and P1 , λ1 and λ2 are a pair of complex conjugate eigenvalues with negative real parts, while λ3 and λ4 are a pair of complex conjugate eigenvalues with positive real parts. Therefore, the equilibria P2 and P1 are also saddle-focus points. Thus, the three equilibria P0 , P1 and P2 are all unstable. 2.2.3

Bifurcation, Lyapunov exponent and chaotic behavior

According to the chaos theory, the Lyapunov exponents measure the exponential rates of divergence and convergence of nearby trajectories in phase space of system (2.4). System (2.4) is solved numerically to define routes to chaos. Here, the types of motion are identified using two indicators. The first indicator is the bifurcation diagram, the second being the largest 1D numerical Lyapunov exponent denoted by ] [ 1 ln(d(t)) , (2.7) λmax = lim t→∞ t where d(t) =

√

(δx1 )2 + (δx2 )2 + (δx3 )2 + (δx4 )2 ,

(2.8)

and computed from the variational equations obtained by perturbing the solutions of system (2.4) as follows: x1 → x1 + δx1 , x2 → x2 + δx2 , x3 → x3 + δx3 and x4 → x4 + δx4 . d(t) is the distance between neighbouring trajectories. Asymptotically, d(t) = eλmax t . Thus, if λmax > 0, the neighboring trajectories diverge and the states of the oscillator are chaotic. For λmax < 0, these trajectories converge and the states of the oscillator are non-chaotic. Finally λmax = 0 corresponds to torus states of the oscillator. The graph on Fig. 1(a) presents a bifurcation diagram showing a transition to chaos whereas Fig. 1(b) shows a largest numerical Lyapunov exponent for increasing γ with b = 0.5 and c = 0.9, one can see that system (2.4) has a positive Lyapunov exponent for a large beach of variation of the parameter γ.

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Optimization of the synchronization of the modified Duffing system

(a)

(b) Figure 1: (a) Bifurcation diagram and (b) 1D largest numerical Lyapunov exponent of system (2.4).

When b = 0.5, c = 0.9, γ = 0.4 with the initial state (x1 (0), x2 (0), x3 (0), x4 (0)) = (0.9, 0.1, 0.003, 0.001), the phase portraits showing the chaotic behavior of the modified system (2.4) are depicted in Fig. 2.

2.3

Electronic circuit implementation

The simple electronic circuit is proposed to investigate the chaotic behavior of system (2.4). The discrete electronics components such as resistors, capacitors, operational amplifiers (T L084 or T L082) are used to construct the circuit (see Fig. 3). The T L084 and T L082 are junction field effect transistors, JFETs, input opamps. Each operational amplifier incorporates well-matched high-voltage JFET and bipolar transistors in the same integrated circuit. The device features are high slew rates, low input bias and offset currents and low offset voltage temperature coefficient. This is significant to reduce sensitivity to circuit parameter values. The electronic multipliers (MULT) are the analog devices AD633JN versions of the AD633 four-quadrant voltage multiplier chips. They are used to implement the nonlinear term of the system.

Kountchou Noube M et. al.

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Phase portraits showing the chaotic behavior of the modified system (2.4).

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Optimization of the synchronization of the modified Duffing system

The circuit equation described by   V˙ Cx1 = VCx2 + 104 R1b Cx VCx3 +   1         V˙ Cx2 = VCx3 ,           

1 V , 104 Rc Cx1 Cx4

(2.9) V˙ Cx3 = VCx4 , V˙ Cx4 = −VCx4 − VCx3 −

1 V 104 Rγ Cx4 Cx2

+ VCx1 − VC3x , 4

where b=

1 , 104 Rb Cx1

VCx1 = x1 ,

c=

VCx2 = x2 ,

1 , 104 Rc Cx1 VCx3 = x3

γ=

1 , 104 Rγ Cx4

and VCx4 = x4 .

Assume that Cx1 = Cx2 = Cx3 = C = 10nF , R = 10kΩ, Rb = 20kΩ, Rc = 11.111kΩ, Rγ = 25kΩ, the voltage sources is set at ±15Vdc. The Pspice simulation results for the proposed chaotic circuit are shown in Fig. 4. From this figure, one can easily observe the good correspondence with the phase portraits obtained using numerical simulations (see Fig. 2). We provide in Fig. 5 the phase portraits obtained experimentally. Note that the graphs of Fig. 5 are also very close to the numerically computed phase portraits of Fig. 2. Electronic circuit realizations, and oscilloscope outputs of modified system is seen in Fig. 6

Figure 3: Analog circuit of system. (2.4).

Kountchou Noube M et. al.

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(a)

(b)

(c)

(d)

Figure 4: Pspice phase portraits obtained by using the analog circuit of Fig. (3).

(a)

(c)

(b)

(d)

Figure 5: Phase portraits of the electronic experimental circuit showing the chaotic behavior of system (2.4). (a) x1 versus x3 ; (b) x4 versus x1 ; (c) x1 versus x2 and (d) x2 versus x3 (0.5 V/div).

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Optimization of the synchronization of the modified Duffing system

Figure 6: Electronic circuit realizations, and oscilloscope outputs of system (2.4).

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11

Robust synchronization of two modified Duffing systems

3.1

Synchronization problem

In this section, we state the synchronization problem. Let us consider the following chaotic system as the drive system:  x˙ 1 = f (x1 , x2 , x3 , x4 ),          x˙ 2 = x3 , (3.1)   x ˙ = x ,  3 4       x˙ 4 = −x4 − x3 − γx2 + x1 − x31 , where f (x) = x2 + bx3 + cx4 with x = (x1 , x2 , x3 , x4 )T is a smooth function. The slave system is constructed as follows:  y˙ 1 = f (y1 , y2 , y3 , y4 ) + u,          y˙ 2 = y3 ,   y˙ 3 = y4 ,        y˙ 4 = −y4 − y3 − γy2 + y1 − y13 ,

(3.2)

where u is the feedback coupling and f (y) = y2 + by3 + cy4 with y = (y1 , y2 , y3 , y4 )T is a smooth function. The synchronization problem can be stated as follows: given the transmitted signal x1 and least prior information about the structure of system (3.1), to design a feedback coupling u such that which synchronizes the orbits of both the drive and response systems at an established finite time T , i.e., lim y1 (t) ≈ x1 (t).

t→T

(3.3)

Now, let us define the synchronization error as follows: ei = yi − xi ,

i = 1, 2, 3, 4.

(3.4)

Then, the synchronization error dynamics is  e˙ 1 = ∆F + u,          e˙ 2 = e3 ,   e˙ 3 = e4 ,        e˙ 4 = −e4 − e3 − γe2 + e1 + x31 − y13 ,

(3.5)

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Optimization of the synchronization of the modified Duffing system

where △F = e2 + be3 + ce4 is a smooth vector field. In this way, the synchronization problem can be seen as the stabilization of Eq. (3.5) at the origin in a finite horizon. In other words, the problem is to find a robust feedback coupling u such that lim ∥e(t)∥ ≈ 0 t→T

(which implies that xi (t) ≈ yi (t) for all t ≥ T > 0, i = 1, · · ·, 4). Now, let ζ1 = e2 , ζ2 = e3 , ζ3 = e4 . Then, system (3.5) can be changed into a canonical form [10–12, 17, 20] as follows:   e˙ 1 = ∆F + u,          ζ˙1 = ζ2 , (3.6)   ˙2 = ζ3 ,  ζ        ζ˙ = −ζ − ζ − γζ + e − e3 + G, 3 3 2 1 1 1 where ∆F = ζ1 + bζ2 + cζ3 and G = −3x1 e21 − 3x21 e1 . Let ∆F = η. Then, following [10–12, 17, 20], system (3.6) can be rewritten in the following extended form:  e˙ 1 = η + u,      η˙ = Γ(e1 , η, ζ, u, u), ˙ (3.7)      ˙ ζ = Ψ(e1 , ζ), where ζ = (ζ1 , ζ2 , ζ3 )⊤ ,

Ψ(e1 , ζ) = (ζ2 , ζ3 , −ζ3 − ζ2 − γζ1 + e1 − e31 + G)⊤

Γ(e1 , η, ζ, u, u) ˙ = ζ3 (b − c) + ζ2 (1 − c) − c(γζ1 − e1 + e31 − G). For the error system (3.7), we suppose that the only state available for measurements is e1 . This assumption is realistic. For instance, in the secure communication case, only the transmitted signal and receiver signal are available for feedback from measurements. Another example can be found in neuron synchronization where master neuron transmits a scalar signal. The slave neuron tracks the signal of the master neuron. At this stage, we point out that system (3.7) is minimum phase. This means that the zero dynamics ζ˙ = Ψ(0, ζ) converges to the origin. In other words, the closed-loop system is internally stable [20] From the control viewpoint this is reasonable du to the boundedness of the chaotic attractor in the state space and the interaction of all the trajectories inside the attractor. So, when we have taken actions to achieve lim e1 (t) = 0, t→T

the part Ψ(e1 , ζ) → Ψ(0, ζ) → 0 as t → ∞ asymptotically for the so-called minimumphase character.

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To illustrate that system (3.7) satisfies the minimum phase property, one may prove that ζ˙1 = ζ2 , ζ˙2 = ζ3 and ζ˙3 = −ζ3 − ζ2 − γζ1 + e1 + e31 + G converge asymptotically to zero when e1 = 0. Note that ζ = (ζ1 , ζ2 , ζ3 )⊤ is bounded. Thus, the zero dynamics can be written as ζ˙ = Eζ, where 

 0 1 0 0 1 , E= 0 −γ −1 −1 which is Hurwitz because γ = 0.4 > 0. Thus, the zero dynamics subsystem ζ˙ = Eζ is asymptotically stable. Hence, system (3.7) is minimum phase.

3.2

Design of the feedback coupling

To achieve the finite time synchronization stated in the previous section, a suitable robust feedback coupling u will be designed. In what follows, the problem of designing u is addressed in such a manner that the energy wasted by the feedback coupling is accounted. Towards the optimization, the first step in our approach is to consider the transitive of states. To this end, the following quadratic criterion is defined by quantifying the transient trajectory of the synchronization error: ∫ T 2 Qe21 (t)dt, (3.8) J(e1 , u) = Qf e1 (T ) + t0

where t0 ≥ 0 is the time at which the control starts and T > t0 is the time for which the synchronization error system (3.7) achieves the desired trajectory (e = 0); Q > 0 and Qf ≥ 0 are the suitable positive constants. The feedback coupling is designed as follows: 1 u = −η − P (t)e1 , t0 ≤ t ≤ T, (3.9) 2 where T is given, P ∈ R be a positive function, solution of the differential Riccati equation:   −P˙ (t) = −P 2 (t) + Q, (3.10)  P (T ) = Qf , Equation (3.10) will be useful to prove that the dynamics of the closed-loop error system is globally asymptotically stable at the origin. We have the following result. Proposition 3.1. : Under the feedback coupling (3.9), the synchronization error e1 (t) converges asymptotically to zero at an established finite time T , with suitable positive constants Q and Qf . Moreover, the closed loop performs a value of the functional (3.8) P (0)e21 (0), where e1 (0) is the initial state of the error e1 .

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Optimization of the synchronization of the modified Duffing system

Proof : Substituting the feedback coupling (3.9) into the synchronization error system (3.7) yields  1   e˙1 = − P (t)e1 , 2 (3.11)   η˙ = Γ(e1 , η, ζ, u, u), ˙ and ζ˙ = Ψ(e1 , ζ).

(3.12)

Since the subsystem (3.12) is stable as (e1 , η) → (0, 0) for any (e1 (0), η(0), ζ(0)) in a subset U 0 ∈ R5 containing the (regular) point e01 ∈ R. Therefore, the proof is focused on the first equation of (3.11). Consider the following Lyapunov candidate function: V = P (t)e21 .

(3.13)

Its time derivative satisfies V˙

= 2e˙ 1 P (t)e1 + P˙ (t)e21 , (

) 1 = 2 − P (t)e1 P e1 + (P 2 (t) − Q)e21 , 2

(3.14)

= −Qe21 ≤ 0. Thus, the state e1 (t) converges to zero for all t ≥ 0. This implies that e1 = 0 is a ˙ stable point. Since e1 (t) and ζ(t) belong to some attractor, function Γ(e1 , η, ζ, u, u) in the error system (3.7) is bounded and smooth. Moreover, the state ζ(t) under the state feedback coupling is bounded and converges to zero since the system is in cascade form. In addition, the positive constant Q determine the convergence rate. Then, by integrating Eq. (3.14) from t0 to T and using Eq. (3.13), one obtains ∫ P (t)e21 (T ) − P (0)e21 (0) = −

T

Qe21 (t)dt,

(3.15)

t0

from where J(e1 , u) = P (0)e21 (0). This completes the proof.

 The feedback coupling (3.9) is defined in the interval t0 ≤ t ≤ T and the stabilization of the error system at the origin is achieved for some positive constants Q and Qf if solutions of Eq. (3.10) exist in such interval. The proposed feedback coupling requires availability of the complete state. This can be seen as drawback; hence an approach of the state feedback is required to avoid dependence on the full information of the system. Additionally, other approach to derive result in Proposition 1 can be stated by introducing the following definition uu = −η. The control uu can be named the “unavoidable” part of the feedback coupling because it

Kountchou Noube M et. al.

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represents the force necessary to compensate the non-linear function η. In this sense, as a second step is derived to design the feedback coupling such that the control effort can be accounted. That is, the criterion (3.8) can be re-defined to include the ‘avoidable’ control effort as follows: ∫ T 2 ˜ J(e1 , u ˜) = Qf e1 (t) + [Qe21 (t) + R˜ u2 ]dt, (3.16) t0

with a given constant R > 0 and u ˜ = u − uu . Hence, the feedback coupling becomes uu = −η + u ˜. Therefore, the closed loop system (3.11)-(3.12) takes the form:  e˙ 1 = u ˜,      η˙ = Γ(e1 , η, ζ, u, u), ˙      ˙ ζ = Ψ(e1 , ζ),

(3.17)

(3.18)

which allows to set the standard LQ problem as in [10]: ˜ 1, u min J(e ˜),

(3.19)

e˙ 1 = u ˜,

(3.20)

u

such that whose solution for the feedback coupling is given by u ˜ = −R−1 P (t)e1 ,

t0 ≤ t ≤ T,

(3.21)

where P (t) is now the solution of the following Riccati equation:   −P˙ (t) = −P (t)R−1 + Q, 

(3.22) P (T ) = Qf ,

and a value of functional (3.16) given by ˜ 1, u J(e ˜) = P (0)e21 (0).

(3.23)

We stress that, at this point, the full knowledge of states is required by this approach. Nevertheless, the linearizing feedback coupling (3.9) is not physically realizable because it requires measurements of uncertain state η which is a very stringent demand for the literature of chaotic secure communication. In order to increase the security of communication, the least possible information about the transmitter should be contained in the communication channel. Some strategies have been developed to achieve this, such

16

Optimization of the synchronization of the modified Duffing system

as the linearizing feedback-based estimator [1, 9, 12, 18, 19, 22], the nonlinear geometricbased estimator, and so on. In the next section, the full knowledge situation is relaxed by using a state estimator. The cost to pay is a higher control effort because of estimation than those due to the use of all states in feeding back. As it has been established in [1, 17, 22, 26], the problem of estimating η can be addressed using a high-gain observer. Thus, the dynamics of the state η can be reconstructed from measurements of the output e1 in the following way [10, 15]:   eˆ˙ 1 = ηˆ + u + 2L(e1 − eˆ1 ), (3.24)  ˙ 2 ηˆ = L (e1 − eˆ1 ). where (ˆ e1 , ηˆ) are estimated values of (e1 , η), and L > 0 is the so-called high-gain parameter. It has been proved that there exists a sufficiently large value of the high-gain parameter L > L∗ , the dynamics of the estimation error converges exponentially to zero (see for instance [15]. In addition, the closed-loop is stable [17]. By using the estimated values (ˆ e1 , ηˆ), the feedback coupling (3.9) can be written as 1 u(ˆ e1 ) = −ˆ η − P (t)ˆ e1 , 2

t0 ≤ t ≤ T,

(3.25)

while, after transient, the feedback coupling can be expressed by 1 u(ˆ e1 ) = −ˆ η − P¯ eˆ1 , 2

t > T,

(3.26)

where P¯ is a positive constant such that the error system (3.7) remains at e = 0 for t > T , thus P¯ is also a tuning parameter. Note that the control effort on the suboptimal robust feedback coupling is higher than the ideal one. As a matter of fact, the waste of energy increases due to the estimation since ∫ T ˜ 1, u J(e ˜) = P (0)e21 (0) + Q(e1 − eˆ1 )2 dt. (3.27) t0

As a consequence, the choice of parameters becomes important and present a trade-off between optimization and estimation.

3.3

Numerical and implementation results

In this section, we present the results of numerical simulations and implementation to illustrate and validate the analytical results obtained in the previous section. The parameter values of the drive and response systems are as in Fig. 2. The initial conditions of the master and slave systems were chosen to be (x1 (0), x2 (0), x3 (0), x4 (0)) = (0.9, 0.1, 0.003, 0.001) and (y1 (0), y2 (0), y3 (0), y4 (0)) = (2, 0.15, 0.0035, 0.0015). We choose (ˆ e1 (0), ηˆ(0)) = (0, 0). The high gain parameter value was chosen as L = 20, the feedback coupling parameters were Q = 100, Qf = 9 and P¯ = 10. The finite horizon is established at T = 1 sec (i.e., the convergence should be attained at time t ≡ T ).

Kountchou Noube M et. al.

17

Figure 7 presents the time evolution of the synchronization error. From this figure, one can observe that the synchronization error is stabilized at the origin by the outputfeedback coupling (3.24)-(3.26). From Fig. 7(a), it clearly appears that a fairly good convergence of e1 is obtained in about 1 sec which corresponds to the finite horizon. Note that although the feedback coupling is acting only on the state e1 , the synchronization errors e2 , e3 and e4 are also stabilized at the origin. The performance of the proposed robust adaptive feedback is depicted in Fig. 8. Figure 8(a) presents the time evolution of the control signal. It is evident that the control signal converge to zero. Figure 8(b) presents the current term η and its estimates ηˆ. It illustrates that after a short transient, η evolves very closely to ηˆ .

(a)

(b)

(c)

(d)

Figure 7: Time evolution of the synchronization errors using numerical simulations. (a) e1 = y1 − x1 ; (b) e2 = y2 − x2 ; (c) e3 = y3 − x3 and (d) e4 = y4 − x4

(a)

(b)

Figure 8: Performance of the proposed feedback control. Time evolution of (a) the feedback coupling u and (b) the current value of η (red line) and its estimated value ηˆ (blue line).

Now, we investigate the robustness of the proposed scheme with respect to parameter

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Optimization of the synchronization of the modified Duffing system

mismatching. The parameter values of the master system (3.1) are chosen as b = 0.5, c = 0.9, γ = 0.4 while the parameter values of the slave system (3.2) were chosen to be b = 0.4, c = 0.72 and γ = 0.32 corresponding to 20% of parameter mismatches. Figure 9 presents the time evolution of the synchronization error e1 . From this figure, it is evident that a fairly good convergence is obtained in about 1sec which corresponds to the finite horizon in spite of the fact that both master and slave systems have different parameters values and different initial states.

(a) Figure 9: Time evolution of the synchronization error e1 (t) with 20% of parameter mismatch between the master and slave systems.

3.4

Pspice implementation of the synchronization scheme

The aim of this section is to implement a practical set-up for the synchronization strategy presented above, and to perform Pspice simulation to verify the practical feasibility of the proposed strategy. Using the values of the parameters obtained above, we determine the corresponding electronic coefficients to design and implement the electronic circuit of the synchronization scheme. The values of the parameters of the feedback coupling can be obtained from the circuit component values as follows: R 1 = , 2 R1

2L =

1 104 RL1 Ce1

and

L2 =

1 . 104 RL2 Cη

The values of the parameters Q and P¯ of the the Riccati equation can be obtained from the circuit component values as follows: Q = V1

R RQ

and P¯ = V2 .

The circuit diagrams of the drive and response systems, and the feedback coupling depicted in Figs. 10, 11 and 12, respectively. On the feedback coupling circuit diagram of Fig. 12, the temporized switches ‘tOpen’ and ‘tClose’ open themselves and close up respectively at the synchronization established

Kountchou Noube M et. al.

19

time, these switches are used to select the tuning parameter P¯ when the synchronization is established. According to the selected synchronization time of the previous section TM = 1sec for numerical simulations, the corresponding synchronization time for the Pspice simulation can be obtained as follows TS = RCTM = 10−4 sec,

(3.28)

where TS is the established synchronization time through Pspice simulations, TM the established synchronization time using numerical simulations, R = 10kΩ and C = 10nF . Then, the temporized switches are set as tOpen = tClose = 10−4 sec. Assume that the initial conditions of the master and slave systems and the feedback coupling are (VCx1 (0), VCx2 (0), VCx3 (0), VCx4 (0)) = (0.9, 0.15, 0.003, 0.001), (VCy1 (0), VCy2 (0), VCy3 (0), VCy4 (0)) = (2, 0.1, 0.0035, 0.0015) and (VCP (0), Ve1 (0), VCη (0)) = (10V, 0V, 0V ). The circuit component values of the feedback coupling were chosen to be Ce1 = Cη = C = 10nF , CP = 20nF , R = 10kΩ, R1 = 20kΩ, RQ = 100Ω, RL1 = 250Ω, RL2 = 25Ω, V1 = 1V and V2 = 10V . The voltage sources is set at ±15Vdc. it is noted that, the component values of the master are the same with the component values of the slave. The Pspice simulation results of the proposed synchronization scheme are shown in Figs. 13 and 14. These graphs confirm the practical applicability of the proposed method. One can also observe the good correspondence with numerical simulations of the previous section. Note that a fairly good convergence of error e1 is obtained in about 10−4 sec which corresponds to the finite horizon (see Fig. 13(a)). Fig. 14 presents the time evolution of the feedback coupling.

Figure 10: Circuit diagram of the drive system.

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Optimization of the synchronization of the modified Duffing system

Figure 11: Circuit diagram of the response system.

Figure 12: Circuit diagram of the feedback coupling.

Kountchou Noube M et. al.

21

(a)

(b)

(c)

(d)

Figure 13: Time evolution of the synchronization errors using Pspice simulations. (a) e1 = y1 − x1 ; (b) e2 = y2 − x2 ; (c) e3 = y3 − x3 and (d) e4 = y4 − x4 .

Figure 14: Time evolution of the feedback coupling.

22

4

Optimization of the synchronization of the modified Duffing system

Conclusion

In this paper, a four-dimensional autonomous system obtained by the modification of the classical two-dimensional Duffing system is presented. The dynamics of the proposed system have been investigated. We found that the proposed system exhibits a chaotic behavior. Furthermore, a novel robust control scheme using the Ricatti equation for the synchronization of two modified Duffing systems is presented. The main idea is to construct an augmented dynamical system from the synchronization error system, which is itself uncertain. Then, we have proposed a robust feedback coupling that takes into account the behavior of transient response and the feedback coupling effort (i.e., the energy wasted by the feedback coupling action). Thus, the proposed strategy allows to set specifically the time horizon for the synchronization of two modified Duffing systems. Both stability analysis and numerical simulations are presented to show the effectiveness of the optimization strategy. Also, Pspice simulations are presented to show the feasibility of the proposed scheme. More practical implementation of the proposed scheme will be provided in the near future. It is believed that the system will have broad applications in various chaos-based information systems.

Acknowledgments Samuel Bowong acknowledges the financial support of the ICTP in Trieste-Italy under the Associate Federation Scheme.

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