Abstract
This paper addresses an adaptive control approach for synchronizing two chaotic oscillators with saturated nonlinear function series as nonlinear functions. Mathematical models to characterize the behavior of the transmitter and receiver circuit were derived, including in the latter the adaptive control and taking into account, for both chaotic oscillators, the most influential performance parameters associated with operational amplifiers. Asymptotic stability of the full synchronization system is studied by using Lyapunov direct method. Theoretical derivations and related results are experimentally validated through implementations from commercially available devices. Finally, the full synchronization system can easily be reproducible at a low cost.
1. Introduction
A deep research worldwide on the issue of synchronization and control of chaotic waveforms has received a lot of attention in the last years [1–20]. This is due to the potential applications of chaos in areas such as biomedical engineering, biological systems, data-encryption, MEMs-chaos, robot control based on chaos commonly referred as cooperation, secure communications, and random number generation. Particularly in the second-last area, synchronization of chaotic waveforms plays an important role due to the possibilities of encoding or masking data using as carrier the chaotic signal generated in the transmitter [1–6, 11, 12, 18–20]. The decoding or unmasking process takes place in the receiver, which must be synchronized with the transmitter chaotic behavior and is able to recover the message imbedded into the transmitted chaotic signal. On the one hand, although different synchronization techniques can be found in the literature, each of them has their advantages and disadvantages; see [1–6, 8, 18–20] and the references therein. On the other hand, continuous chaotic waveforms are usually studied and generated from a dimensionless dynamical system and a piecewise linear (PWL) model is often used to model the nonlinear part [6, 8–12, 15, 18–20]. After that, to synchronize the chaotic signal of the receiver with that coming from the transmitter, a synchronization technique is applied. Once it is done, the dynamical systems (receiver and transmitter) along with the synchronization block are commonly built with discrete passive and active electronic components [9, 11, 12, 15, 19]. However, the predicted numerically performance not only of both dynamical systems but also of the synchronization block differs substantially from reality. These differences are more prominent when the operating frequency of the dynamical systems is pushed to operate in higher frequencies, as required into any communication system. The reason of that is the fact that no information related with the real physical active device performance parameters is included in the numerical simulations [18–20]. Only in recent years, the works reported in [17, 21, 22] led the nonlinear behavior of a third-order dynamical system including the most influential performance parameters of Operational Amplifiers (Op Amp) and Current-Feedback Operational Amplifiers (CFOA). The nonlinear behavior of both dynamical systems was validated through experimental tests, confirming a good agreement with numerical tests. In this sense and for realization of communication systems based on chaos with high performance, it is of vital significance to take into account the real physical active device performance parameters, in order to develop communication systems with a practically usable performance [5]. Therefore, as natural evolution from [17, 22], this paper leads to improving the synchronization of multiscroll chaotic attractors by considering those more important performance parameters associated with the active devices used for synthesizing the dynamical systems and the synchronization block, in order to achieve a robust and cheap communication system based on chaos. The outline of the paper is the following: an adaptive control law to synchronize chaotic waveforms of the receiver with that coming from the transmitter is introduced in Section 2, where the adaptive control is embedded in the receiver. Then, the transmitter and receiver dynamical systems along with the synchronization circuit are synthesized with Op Amps in Section 3, and their behavioral models are also introduced. In the next section, experimental results on the synchronization are presented and compared with numerical simulations. Finally, conclusions are listed in Section 5.
2. Synchronization of Multiscroll Chaotic Oscillators with Adaptive Control
A multiscroll chaotic system can be described by the following system of differential equations:where is a parameter that is used to alter the dynamics of (1), the subscript “” is used to denote the transmitter dynamical system, and is a nonlinear function that can be chosen arbitrarily, but in general it depends on the three states of the system. Multiscroll chaotic attractors can be synchronized by some method proposed in the literature. Among them, the control of chaotic systems refers to design stated feedback control laws that stabilizes the chaotic waveforms either on a periodic orbit or around the unstable equilibrium points [2, 7, 10, 13, 14, 18–20]. In this sense, the active control technique is used when system parameters are known whereas adaptive control technique is used when system parameters are unknown, as often occurs in practical situations. The last technique is herein used and hence, by including an adaptive controller in (1), takes the form:where the subscript “” is used to denote the receiver dynamical system and is an external control input which will drag the chaotic waveforms and given aswhere is the estimated feedback gain which is updated according to the following adaption algorithm:and is an adaption gain. Let us now define the state errors between (1) and (2) as follows:Therefore, the controlled resulting error system is described by The synchronization target between (1) and (2) is to achieve the asymptotic stability of the zero solution of (6) in the sense
Preposition 1. The zero solution of (6) is asymptotic stable for .
Proof. We define Lyapunov function asThe time derivative of (8) in the neighborhood of the zero solution of (6) is given asSince and if we take , it is clear that (8) is positive definite whereas (9) is negative definite. From Lyapunov stability theorem one concludes that the equilibrium points given by (7) together with of the system (6) are asymptotically stable [2, 3]. The proof is completed.
3. Synthesizing the Transmitter, Receiver, and Adaptive Control with Op Amps
Several nonlinear functions can be used for modeling in (1) and (2), such as functions, staircase functions, hysteresis functions, saw-tooth functions, and, herein, a saturated nonlinear function series (SNFS), which is traditionally modeled by using a PWL approach [15]. Moreover, the dimensionless dynamical systems (1) and (2) can be built with either Op Amp [17] or CFOA [22]. In this paper, the former case is considered and the equivalent circuit, taken from [17], is shown in Figure 1. According to [17] and Figure 1, the system of equations is given bywhere are the breakpoints, is the DC gain, is the gain-bandwidth product, is the slew rate, and each voltage signal is limited by , where are the positive and negative saturation voltages and the difference of them is the dynamic range (DR) for each amplifier. Additional information about Figure 1 and the deduction of (10) can be found in [17]. Notice that , , , and are the main performance parameters of an Op Amp [17, 22]. It is worth mentioning that (10) involves the nonlinear model for SNFS including the most influential performance parameters associated with Op Amps, whereas the rest of amplifiers, those placed horizontally in Figure 1, are ideally considered. This is due to the fact that not only do the amplifiers placed horizontally have negative feedback loops, improving their stability into a wide bandwidth, but also the nonlinear behavior of the SNFS has a high influence in the generation of chaotic waveforms, as has been demonstrated in [21]. Nevertheless, whether performance parameters for all amplifiers are taken into account during the numerical analysis, the chaotic waveforms in the time domain are not drastically modified. We want to point out that Figure 1 becomes the transmitter block in a communication system based on chaos and its chaotic behavior must be synchronized with the receiver, which is a similar circuit. Moreover, the control law given by (3) and (4) can be rewritten asFigure 2 shows the design of (11) by using Op Amps and analog multipliers. According to the AD633JN data sheet [23], the output voltage on the node of Figure 2 is given byThe behavior of a continuous integrator (omitting ) is obtained on the node and from (12) takes the form Note that is a resistor with high value and in practice it is used to prevent the integration from a zero voltage drop. Otherwise, the voltage on the node is given asCombining (13) and (14) along with the second multiplier, we obtainThe transmitter and receiver circuits shown in Figure 1 are connected by using Figure 2 and hence, the full synchronization system is got as shown in Figure 3. According to (2), (10), (15), and Figure 3, the behavioral model for the receiver circuit including the adaptive control is given bywhere . Now that we have the behavioral models for the transmitter and receiver along with the synchronization scheme given by (10) and (16), respectively, we can use them to do numerical simulations by using the fourth-order Runge-Kutta algorithm with time step . It is worth mentioning that unlike the synchronization methods previously reported, PWL functions are used to design the nonlinear part of the chaotic oscillator and as a consequence a level of inaccuracy is introduced into a numerical analysis, which is more evident when numerical and experimental results are compared [17, 21, 22]. Meanwhile, the behavioral models previously deduced take into account the real physical active device parameters and hence, the synchronization of chaotic waveforms between the transmitter and receiver is better forecasted. It is important to stress that finite performance parameters of Op Amps are the main limitation of that, in practice, the real behavior of the SNFS (see Figure 7(b)) and the generation of chaotic attractors are both degraded when the operating frequency of the chaotic system increases. As one consequence, the performance of the synchronization scheme is also worsened. Therefore, to gain insight and improve the synchronization schemes used in communication systems based on chaos, performance parameters of the active devices must be taken into account during the numerical simulations.
4. Experimental Results
The chaotic oscillators were designed to oscillate in a center frequency of 3.45 kHz. Table 1 gives the numerical values of Figure 3; meanwhile Table 2 shows the performance parameters of the UA741 Op Amp. In a first step and with the synchronization circuit deactivated, chaotic waveforms for and variables are generated and shown in Figure 4. For these graphics, the following initial conditions were used: , , and , , . Note that the dynamical systems are extremely sensitive with respect to differences in initial conditions and although here it is not shown, the chaotic waveforms for , , , and variables are also different. In a second step, the synchronization circuit was activated and the results are depicted in Figure 5. From this figure one can observe that the synchronization occurs after 2.5 ms. Figure 6 shows the evolutions of state adaptive synchronization errors between the transmitter and receiver oscillators, respectively.
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To validate the results previously derived and demonstrate the real behavior of the synchronization scheme, the proposed circuit illustrated in Figure 3 was built and experimentally tested. The numerical values of the discrete components used during the numerical simulations were also used in the experimental tests, but with a slight modification of in order that the discrete integrator is operating to 3.45 kHz. Figure 7(a) shows the experimental results corresponding to the generation of 4-scrolls in the -versus- plane and Figure 7(b) shows the real behavior of the SNFS. A similar behavior is obtained for the SNFS and scrolls on the receiver circuit. Moreover, whereas Figure 8(a) shows time-domain chaotic waveforms for and when the synchronization circuit is deactivated, Figure 8(b) shows the chaotic waveforms when the synchronization circuit is activated. Finally, Figure 9 shows the synchronized signals in the -versus- plane and a similar behavior also occurs for the other state variables.
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5. Conclusions
Unlike the synchronization techniques reported in the literature where a PWL function is used to model the nonlinear part of a dynamical system, herein the behavioral model of the dynamical system deduced in [17] was used as transmitter and receiver circuit, in order to a posteriori be used into a communication system based on chaos. The most influential performance parameters associated to Op Amps were included to model, principally, the real behavior of the SNFS. In this context, it is clear that the use of PWL models introduces a high level of inaccuracy, as was already demonstrated in [17, 21, 22] and hence, the use of PWL models are only feasible at low frequency. Furthermore, an adaptive control law was proposed and synthesized with Op Amps and analog multipliers, showing that the synchronization between two nonlinear dynamical systems is more accurate and realistic than PWL models. Lyapunov direct method was used to study the asymptotic stability of the full synchronization system. It is important to stress that the full synchronization system is of low cost, approximately $74.00 USD. Experimental data using commercial available Op Amps for synchronizing four-scrolls at 3.45 kHz were gathered showing good agreement with numerical approximations.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Council for Science and Technology (CONACyT), Mexico, under Grant 222843 and in part by the Universidad Autónoma de Tlaxcala (UATx), Tlaxcala de Xicohtencatl, TL, Mexico, under Grants UAT-121AD-R and CACyPI-UATx-2014.