Recibido:17/10/22; Aceptado: 12/12/2022; Publicado 25/01/2023

Introduction

In the last two decades, the synchronization problem of chaotic systems has been widely studied, see, for instance, [1], [2], and [3]. The reasons for the interest in this problem are the potential applications and the synchronization phenomena. The main application is secure communication, where it is proved that relevant information can be transmitted and recovered using chaotic synchronization, and it has been demonstrated that the method is feasible, see for example, [4] and [5]. The other main interest is the dynamical phenomena of synchronization, where some relevant efforts have been made [6], [7], from where some open problems persist. For instance, one problem is the synchronization of strictly different systems, and the other is the synchronization of dynamical systems with different order [8]. The former deal with the problem of synchronize two systems whose models come from two different phenomena. The latter concerns the synchronization of dynamical systems that evolves in different spaces, in other words, a system with nonequal dimension. Synchronization of strictly different systems is still a problem under investigation.

Secure communication is understood as the transmission of signals or messages masked or hidden into the chaotic attractor of the transmitter system. Then the Receiver provides the transmitted signal or message via a particular unmasking algorithm. In this manner a simplex communication scheme is constructed; however, until the authors' knowledge, the secure transmission schemes consist in transmitting a single message or signal provided that the signal does not destroy the chaotic behavior of the transmitter. Several chaos-based communication schemes have been published. Among these, two basic configurations can be identified. (i) An approach consists of adding the signal to the chaotic carrier, which is transmitted to the receiver. That is, the master system comprises the full-state model, whereas the slave system is composed of a reduced model, and the transmitted state completes the receiver system. (ii) Another transmitter/receiver design is based on the full state model of the driving and response systems [9]. That is, both drive and response systems are represented by dynamical systems of the same order. The homogeneous-synchronization configuration has been recently addressed via parameter modulation. Kocarev and Parlitz have proposed a generalization of these approaches, which extends the capabilities for constructing synchronized systems. Their approach enables the message signal to be integrated as a driving signal. However, the message signal can be recovered only under ideal conditions [10].

In this report, we illustrate that multiple signals can be transmitted using a single transmitter system and a single receiver using nonlinear Multiple-Input and Multiple-Output control techniques. In this way, a stereo audio signal can be transmitted, or different conversations can be embedded to be transmitted at the same time. Moreover, chaotic behavior should be maintained.

The control strategy is based on a diffeomorphic transformation obtained from the derivatives of the output functions along the system vector fields for a single signal see [5]. Then as the transmitter system poses many inputs as transmitted signals, it is required the same number of control inputs in the receiver in order to recover each signal.

The writing is organized as follows: Section 2, presents the secure communication based on robust synchronization of similar chaotic systems, in Section 3 an application for secure communication is illustrated and finally Section 4 presents the conclusions of the work.

Materials and Methods

The key procedure consists in synchronize chaotic systems with same order and model. To this end, we consider chaotic systems in the following form

                                 Ec. 1

where vector fields  are sufficiently smooth and similar, with  being the attraction region, ,  are the input vectors with  being unknown and ,   are the real-valued output functions,  are the signals to be transmitted and the signals  stand for the control signals to achieve synchronization, and once the systems are synchronous, the recovered messages are reproduced by the control command signals. These considerations define a Multiple-Input and Multiple-output system.

Therefore, we use the nonlinear control theory for MIMO systems. To begin with, from Master and Slave systems, the synchronization error system is given as

        Ec. 2

where  is the state vector of the synchronization error system, , . The underlying idea of the nonlinear feedback control is to find functions  such that the desired dynamical behavior is induced for any initial condition  in an attraction basin . Then, from the Lie-based geometric approach, it is possible to find a transformation , for , such that the affine form of the synchronization error system takes a linearizable canonical form. We depart from the relative degree definition for a MIMO system [11].

Definition 1. The Multiple Input and Multiple Output affine system (2) has the relative degree vector  at the point  if:

1.  , for all ,  , , and for all  in the neighborhood of
2.  the  matrix

        Ec. 3

is nonsingular at .

Remark 1. The previous definition of the relative degree is for a system with the same number of inputs signals as output signals. Therefore, the relative degree matrix  is square.

Suppose that the system (2) gives , then the matrix  is given as follows

c.4

Due to the invertibility of the matrix , a diffeomorphic transformation can be determined. Such a transformation is given by , since the relative degree  the linearizable system is given by

                        Ec. 5

where  and . Thus, from (5) the linearizing controllers are given by

                                  Ec. 6

Where  and  is the new control that leads the system state to a prescribed reference, in this case, it leads the trajectories of the system (2) to the origin.

Remark 2. The controller (6) is the so-called perfect control since it requires the perfect knowledge of the functions  and , in order to accomplish the synchronization.

At this point, we can assume the following:

Assumption 1. Only the state vector  is available for feedback.

Assumption 2.  is an injective function, bounded away from zero.

Assumption 3. There exists a scalar function  available for feedback such that  at any  of .

Thus, for a realistic case, we consider that these functions are unknown, in seek of clarity  represents the way as the control input is affecting the system and  represents the dynamics that should be compensated (uncertain terms and parameters as well as the transmitted signals). These unknown functions represent a source of uncertainty due to parameters and non-modeled dynamics. Moreover, if there exist external perturbations or systems faults, they appear in function  or . For these reasons, the controllers (6) are not physically realizable. To overcome this problem, we propose a representation that lumps the uncertain terms into a new uncontrollable but observable state variable. Following ideas reported in [12], [13] and [14], the system (5) can be written in an extended form by defining , , note that  lumps the uncertainties. Now, let us define . Thus, the system (5) takes the form

                        Ec. 7

where , the augmented state  provides the dynamics of the uncertain function , therefore, the system (7) is dynamically equivalent to system (5). Thus the th linearizing-like control law is modified in the following form

                        Ec. 8

function  is an estimate of the function  and can be used provided that  [14]. Notice that the sign should be constant since any change of sign in function  implies a zero crossing, and the matrix  losses invertibility. Then under the linearizing control feedback (8), the states  converge to zero. Since the feedback controller yield bounded trajectories of state , thus  is also bounded for all . More precisely, if  converges to an equilibrium point, then  also converges to an equilibrium point. Therefore, under feedback (8), the states of system (7) converge asymptotically to zero.

It is important to stress that the controller (8) is not realizable since the new state  is unknown. Therefore, a state estimator is proposed to estimate the uncertain states, which is given as follows

                        Ec. 9

where  and  are tuning parameters chosen in such way that the estimated values .

Proposition 1.  Let  be an estimation error vector whose components are defined as:  and . For sufficiently large , the dynamics of the estimation error decays globally exponentially to zero if .

Proof: Combining systems (7) and (9) the dynamics of the estimation error can be written as follows

                        Ec. 10

where  and the matrix  is

                                Ec. 11

in which  and  which implies  thus  is obviously Hurwitz. There exists a positive definite and symmetric matrix  such that  where  is the identity matrix of dimension 2. Choosing  as a the Lyapunov like function candidate, one has

        Ec. 12

Let  be the estimation error. The function  satisfies  and  for some  and , respectively. Thus,  is a certain bounded function. Moreover, let  be satisfied for some positive constant . In this way, we have

                        Ec. 13

so that  tends to a set bounded by . The dependence on the estimation error  on  deserves special attention. Note that as  increases,  will decrease, which also decreases the exponential estimation error bound. This argument shows that with the proposed method,  should be made as large as possible and this achieves the proof.

The previous proposition states that there exists a Lyapunov function that depends on parameter  and provides the global exponential convergence of the estimation error dynamics. With the estimates of the states, the linearizing control low can be modified as follows

                        Ec. 14

An important characteristic is that this feedback only uses estimated values of the uncertain terms, which gives the feature of robust feedback, moreover it only requires the measure of the output states.

Thus, the Robust Control Scheme for hyperchaotic synchronization can be achieved by means of the state estimator (9) and the robust feedback (14). In this sense, it is required that the state estimator converges to the corresponding states. This result can be stated in the following proposition.

Proposition 2.  Consider two chaotic/hyperchaotic systems with a synchronization error given by the system (2). The system (2) converges asymptotically to zero under the feedback (14) via the stabilization of the extended system (7).

Proof:  Substituting the robust feedback (14) into (7) and considering the estimation error system (10), the closed-loop dynamics is given by

        Ec. 15

Since  and  it follows that  (which can be computed from the first integral of , i.e., . Then, according to the Contraction Mapping Theorem, the state  can be expressed globally and uniquely as a function of the coordinates . Now, note that since the matrix  is Hurwitz by construction, and the nonlinear function  is bounded, the estimation error system (15) is asymptotically stable. In this sense, given a compact set of initial conditions  containing the origin, there exists an upper bound  such that  and a high gain estimator parameter  such that  is contained into the attraction basin. Hence, the closed-loop system is semi-globally practically stable, i.e., , and this achieves the proof.

Therefore, with these results, one can design controllers that achieve synchronization in spite of uncertainties. Thus, we apply these results to construct a secure communication system based on synchronization.        

Results and Discussion

To illustrate our proposal, we consider two chaotic Rössler systems given by

                        Ec. 16

one for the Transmitter and the other for the Receiver. For this case, the parameters were considered equal in both systems. Then we desire to send three signals without excessive deformation or destroy the chaotic Transmitter attractor. In this example we transmit a square periodic signal and two sinusoids, all with different amplitudes and frequencies. Once the signals have been injected into the Transmitter system, the attractor of the Transmitter is illustrated in Figure 1, where there is a slight deformation of the attractor, and it still provides a chaotic behavior.

Figure 1. Chaotic attractor for the Rössler Transmitter system when the transmitted signals are injected.

Therefore, the synchronization error system obtained from the Transmitter and Receiver system is given by

Ec. 17

where the function  stands for the terms given by the nonlinear operation between systems. As was stated in the previous section, system (17) is a nonlinear Multiple-Input and Multiple-Output one, moreover the relative degree matrix  which is invertible and satisfies the propositions of the previous section. Following the corresponding procedure, one has that the diffeomorphic transformation is given by . Once we have the transformation, the system for the synchronization error can be written as

                Ec. 18

where the function  comprises the uncertain terms since it represents the derivatives of the th output along the vector field of the synchronization error system and the function  stands for the function of the input control and is considered uncertain, but by Assumption 3 there exists an estimated value given by . Therefore, we can calculate the controllers as follows

                Ec. 19

where the new dynamic is given by , however as was discussed above, the function  is uncertain and the previous controller is not realizable. The idea is to lump the uncertain terms into a new state. Thus, the transformed system can be extended as follows

                Ec. 20

This new system does not have the uncertainties of the synchronization error system; however, it has new states that provide the dynamic of the uncertainties. Again, the new states are not available from the system but can be estimated to have an estimated value, and the controllers can counteract them.

The state estimators for system (20) are given by

        Ec. 21

where this estimator provides the corresponding estimated values of the uncertain states . With these estimates, we have that the controllers are given by

                Ec. 22

Therefore, it is possible to determine the estimators and controller gains such that the closed-loop systems be stable, and applying these controllers to the corresponding systems, we obtain that the synchronization is achieved as illustrated in Figure 2, where the controllers were activated at  sec., after that the signals were applied or transmitted for  sec. Thus, it is observed that neither the chaotic attractor nor the synchronization is affected considerably when the signals are injected.

Figure 2.  Synchronization of the Transmitter and Receiver systems, at  sec. the synchronization is initiated, whereas for  sec. the signals are injected into the Transmitter system.

Once the chaotic synchronization and the signals have been injected, the recovered transmitted signals are presented in Figure 3.

Note that the recovered signals are very close to the transmitter. Therefore, digital and analogous signals can be transmitted simultaneously in a secure way in spite of uncertainties in the Transmitter and Receiver systems.

Figure 3. Transmitted and recovered signals.

Conclusions

In this contribution, we presented that using chaos synchronization, multiple signals can be transmitted in a secure way. The proposal is such that analog and digital signals can be transmitted without deforming or destroying the chaotic behavior of the Transmitter and Receiver. Also, we considered the case when the controllers do not pose full information about the controlled system, in this way, the communication scheme is robust against unknown parameter values or external (signal transmitted) perturbations. The scheme is general for  order systems, therefore, in principle, it can be used to transmit  signals. The key assumption was that the systems are similar and the parameter variations were not considered, but it is being an understudy.

Conflicts of Interest

The authors solemnly declare that we are not and shall not be in any situation which could give rise to a conflict of interest.

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