Abstract
A solution to the output-feedback generalized synchronization problem for two chaotic systems, namely, the master and the slave, is presented. The solution assumes that the slave is controlled by a single input, and the states of each system are partially known. To this end, both systems are expressed in their corresponding observable generalized canonical form, through their differential primitive element. The nonavailable state variables of both systems are recovered using a suitable Luenberger observer. The convergence analysis was carried out using the linear control approach in conjunction with the Lyapunov method. Convincing numerical simulations are presented to assess the effectiveness of the obtained solution.
1. Introduction
The synchronization of strictly different chaotic systems is a current challenge in Control Theory. In general, accomplishing and understanding this kind of synchronization, referred to as generalized synchronization (GS), are a difficult problem and are very important because its solution can be useful to actual applications. For instance, chaotic signals have been used in secure information transmission. In this regard, interesting secure communication methods based on GS can be found in the literature. In [1], a channel-independent chaotic secure communication scheme based on GS is proposed. A similar scheme, also based on GS, with a remarkable stability to noise and numerically validated using the Rössler system is developed in [2]. Two novel generalized chaos synchronization based secure communication schemes are introduced in [3, 4]. The first scheme is complemented with a transposition function and is tested using the Chen chaotic circuit; the second scheme is oriented to encrypt images and uses a theorem introduced by the authors, which is a generalization of GS to an array of differential equations. From a theoretical point of view, synchronization of strictly different chaotic systems helps to explain how some oscillatory systems, like the intrinsic neuron model, act cooperatively to develop tasks or solve problems [5]. This behavior also occurs in nature and can be responsible for the transition to low-dimensional behavior in systems with many degrees of freedom. A full review of GS is beyond the scope of this study; however, we mention two seminal works. In [6] the authors developed a method for decomposing chaotic systems such that linear GS can be achieved. Finally, we mention work [5], where the authors proposed a method used to detect and study generalized synchronization in drive-response systems. The method uses an identical response system to monitor the synchronized motions. We suggest to the interested reader the following list of remarkable works [7–16].
Loosely speaking the GS phenomenon occurs when the trajectories of one system, through a functional mapping, are equal to trajectories of another. In other words, let us suppose that we have two systems, and , named, respectively, master and slave, with their corresponding trajectories, and [17]. That is, GS is accomplished [18] if there exists a differential primitive element that generates a mapping from the trajectories of in the algebraic manifold to the trajectories of in its space , in which case it holds that . It is noteworthy that for identical systems the functional mapping corresponds to the identity [19]. In our case, we assume that the behavior of the slave system can be either chaotic or bounded and oscillatory. Evidently, to accomplish GS there must be a control action acting over the slave (for simplicity, symbols and refer to the uncontrolled master system and the controlled slave system, resp. In the same way, and , are, resp., the states of systems and .)
In this work, we solve the output-feedback GS problem for systems and , when only a single output of each system is available or measurable, and only a single controller is used in the slave system. This restriction makes the GS problem more challenging and interesting than the works mentioned above. To solve it, the nonavailable state variables of systems and must be reconstructed from the measurable outputs. To overcome this issue, we exploit some ad hoc differential algebraic properties found in some chaotic systems, by choosing the differentiable primitive element [20]. This primitive element allows us to express both systems in a generalized canonical form. Thence, we solve the GS problem as if it were an instance of the trajectory-tracking problem. To this end, it is necessary to split the slave controller into two parts: one devoted to feedback the output tracking error between the master and the slave systems and the other compensating for the underlying and nonavailable state variables of each system. We underscore that our control design basically consists of finding a suitable differentiable primitive element to express the master and slave systems in their corresponding canonical form, in conjunction with a kind of Luenberger observer to reconstruct the nonavailable variable states. Fortunately, there are in nature several chaotic systems that can be expressed in this form.
The remainder of this paper is organized as follows. In Section 2, we introduce some concepts necessary to establish the problem statement properly. In Section 3, we solve the output-feedback GS control problem. Numerical experiments to assess the effectiveness of our solution are presented in Section 4. The final remarks are given in Section 5.
Notation. To simplify the development of this work, we adopt the following notation:
2. Preliminaries
Before proceeding, we introduce some needful concepts [20–23]. From the theorem of differential primitive element [20] we posit a single element , which is a differential primitive element, such that ; that is, is differentially generated by and ( and are differential fields).
Consider the following nonlinear system: where is the system state and is the system input. Also, suppose that there exists an auxiliary variable (differential primitive element), defined as which allows us to write system (2) in the following form:where with , for , is a scalar nonlinear function of their arguments and may not be defined everywhere, and is the gain constant of the single input of . The above system representation is well known as the generalized controller canonical form, and variable is the measurable output of the system [21, 24].
2.1. Problem Statement
In this section we introduce formally the main problem of this work, which consists of solving the output-feedback GS problem for two chaotic systems, in a master-slave configuration, where the states of both systems are partially known. To this end, we propose a suitable definition for the GS problem, based on previous works [21]. It should be noted that our definition uses the notion of differential primitive element, already mentioned.
Let us consider the following two nonlinear systems in a master-slave configuration, where the master system, , is given by and the controlled slave system, , is defined bywhere , , , , , , and . , , , and are assumed to be polynomial in their arguments. Before introducing the GS definition, we need to introduce the following two important assumptions, related to some structural properties of and and based on the result found in [21].
Assumption A1. Suppose that the master system, , given in (5), can be written aswhere is the vector of function, with for , is a differential primitive element for the master system, and is a scalar nonlinear function. Additionally, the trajectories of free system (7) must be forward complete in some compact set in .
Assumption A2. Suppose that the slave system , given in (6), can be written aswhere with , for , is a differential primitive element for the slave system, is a gain constant of the single input , and is a scalar nonlinear function.
Notice that the functions vectors, , are generated through the corresponding differentiable primitive element. This construction allows us to solve straightforwardly the GS problem. Finally, we mention that constructions (7) and (8) are in agreement with the canonical form previously introduced in [23, 25]. Now we can introduce the main definition of this study, as follows.
Definition 1 (generalized synchronization). Let us consider systems, and , under Assumptions A1 and A2, with being invertible. The slave and master systems are said to be in a state of GS if there exists a differential primitive element that generates a transformation with and there exist an algebraic manifold and a compact set with such that their trajectories with initial conditions in approach as . This definition leads to the following criterion:
Remark 2. As far as we know, the GS problem has been solved assuming that states are available for measurement. In this work, we moved a step forward, by considering the case when the states of both systems are partially known. To this end, the proposed controller is based on output-feedback stabilization, instead of using full-state control. It is important, because in several actual applications the whole state is not available or is unpractical or impossible to measure. For instance, in the classical Duffing system only the position is available. On the other hand, finding a general solution of this problem is rather difficult.
As it can be seen original systems (5) and (6) are expressed in their corresponding observable generalized canonical form, through their corresponding differential primitive element. It is worthy to mention that, fortunately, several chaotic systems can be expressed in such observable generalized canonical form. Now, if Assumptions A1 and A2 hold, there exists a full-state feedback controller, , such that [21]. We underscore that this result assumes that the whole state of both systems, the master and the slave, is available for measurements.
Now, we can introduce the main problem of this work.
Problem Statement. Consider the uncontrolled master chaotic system (7) and the controlled slave chaotic system (8), satisfying, respectively, Assumptions A1 and A2, and the vector function existing at least locally. The main goal consists of proposing the slave system controller:which solves the GS problem, where are estimations of the unknown .
As this shows, we are dealing with a more difficult problem configuration. Contrary to several previous works [5, 6, 26], we do not use any control action over the master chaotic system, while the slave controller only uses one measurable output from the slave and one from the master. This problem instance can be considered more appealing, because it resembles several actual implementations. On the other hand, it is less expensive to control a system using state estimations than using actual measurements.
To solve this problem, it is necessary to reconstruct the nonavailable dynamics of both systems, the master and the slave. Therefore, we introduce a suitable version of the assumption introduced in [21].
Assumption P1. Suppose that there exists a primitive element, , of the master system “” (5) that transforms it into the new form “”, given bywhere , is a scalar nonlinear function that depends on the output , is a vector of constants, and is a scalar Lipschitz function in on the open set . That is,
Assumption P2. Suppose that there exists a primitive element, , of the slave system “” (6) that transforms it into the new form “”, given bywhere , is a scalar nonlinear function that depends on the output , is a constant vector, and is a scalar Lipschitz function in , with its corresponding constant . Finally, is a fixed constant and is the single input of the system.
Comment 1. Assumptions A1 and A2 are needed to solve the GS problem, when the states of both systems, the master and the slave, are known. Assumptions P1 and P2 are needed to solve the GS problem when only a single output of each system can be measurable. As we can see, our goal is to move a step forward with respect to the previous works. On the other hand, there are many chaotic systems that admit the representation, given by (11) and (13).
3. Solving the GS Control Problem for a Class of Chaotic Nonlinear Systems
In this section we present a solution to the main problem of this study. To this end, we design a Lyapunov-based observer to recover the underlying dynamics of the master and slave systems, and . Afterwards, using the recovered dynamics by a static feedback , we solve the GS problem.
Reconstructing the Nonavailable Dynamic of or . Suppose that we have a nonlinear system, defined bywhere is the single measurable output, is the partially known state, is any scalar function, and is a scalar Lipschitz function in , with as the Lipschitz constant, and constant. Now, consider the observer of this nonlinear system, defined aswhere is an estimation of , is a vector of constants, and is the system input. Notice that system (14) has a similar structure to that of the systems defined in Assumptions P1 and P2. The following proposition provides sufficient conditions to ensure that the estimation error converges to zero, as long as .
Proposition 3. Consider systems (14) and (15). If the vector of constants, , is selected, according to where and satisfyingwhere ( is in fact the last row of matrix ), then, the error converges asymptotically and exponentially to zero.
Proof. From (11) and (15), it is easy to show that the dynamic of the error is given byIn order to analyze the convergence of the state , we propose the Lyapunov function , where , whose time derivative, along of (18), is given byNow, it is quite easy to see that the following inequality holds: Therefore, from (19) and (20), we have that can be upper bounded, asEvidently, , for all , in the case that inequality (17) is fulfilled.
The following remark allows us to propose the needed observers for and .
Remark 4. Assumptions P1 and P2, in conjunction with Proposition 3, suggest proposing the observer for the master system (11) aswhere . Similarly, observer for the slave system can be proposed aswith , where and must be selected, such that the following matrices are obtained:being Hurwitz and fulfilling the Lyapunov equation , , with the following restrictions:
In the next section, we use the Lyapunov-based observer to solve the GS control problem.
3.1. Generalized Synchronization
Now we are ready to introduce the main proposition of this paper.
Proposition 5. Assuming that the observers, (22) and (23), satisfy the conditions in Remark 4, the following static feedback,where,ensures that the synchronization error exponentially converges to zero.
Proof. From Assumptions P1 and P2, we have that systems and can be expressed according to (11) and (13), respectively. Computing the dynamic error, , we haveTherefore, introducing , proposed in (26), into (28), we obtainwhere and are defined byRemembering that and , we have the following relations:Similarly, Substituting these two relations into (29), we obtainUsing simple algebra, we have that can be upper bounded, as follows:Definingprovided that is Hurwitz, (33) can be written in the following simple way:where is given byHowever, according to (34), can be upper bounded, as follows:By assumption, and converge exponentially and asymptotically to zero. Then, there are some and being strictly positives, such that . Consequently, the error of system (36) asymptotically and exponentially converges to zero.
4. Numerical Simulations
The goal of this example consists of solving the GS problem for the controlled slave Chua system and the free master Colpitts oscillator, provided that a single output of each system is available for measurements. On the other hand, we underscore that, in the present case, the trajectories of the Colpitts system are bounded in some compact set and exhibit chaotic or oscillatory behavior.
Master System. Consider the state equations for the Colpitts oscillator , given bywhere , , , and (to avoid confusion, the master parameters use the subscript , and the slave parameters use the subscript ). It is well known that this oscillator exhibits, for certain values of its parameters, a chaotic behavior. We will show that, selecting as the differential primitive element, we can write it according to expression (11). We use to refer to the vector state of the Colpitts system. From , we obtain the following representation:where is invertible. Furthermore, the time derivative of can be written aswhereNotice that the term can be expressed in the coordinates of , asNotice that (41) and (43) are in agreement with the condition in Assumption P1. According to Proposition 3 and Remark 4, the observer can be proposed aswhere has to be selected, such that is Hurwitz. Using the following actual parameters reported in [27],it is easy to show that selecting allows us to obtain the following matrix:which is Hurwitz. Therefore, proposed observer (44) ensures that the estimation error converges exponentially and asymptotically to zero.
For the numerical simulation, the experimental setup assumed the above defined parameter values, the master initial conditions as , and the master observer as . The obtained results are shown in Figure 1, where we can see that the estimation error of each state variable settles to almost zero after nearly 4 seconds.
Slave System. Consider the Chua chaotic system as the controlled slave system, . This system is formed by three linear energy-storage elements (an inductor and two capacitors), a linear resistor, and a single nonlinear resistor. It is well known that this circuit is described by the following differential equations:withwhere parameters , , , and are chosen, such that system (47) exhibits a chaotic behavior when . Notice that is the input of the slave system that is necessary to achieve the GS problem. Particularly, in the case where and the parameters values are in the neighbourhood of [28],it is known that we have the so-called double scroll chaotic attractors. As before, we use symbol to indicate the states of slave system (47). In a similar fashion, we choose as the primitive element . Therefore, to obtain the corresponding canonical form, we derive two times, obtainingwhere is an invertible map. From the above equation, the time derivative of is given bywhereNotice that can be rewritten in the coordinates of aswhere andOn the other hand, it is clear that is Lipchitz, with (see Appendix). Notice that (51) and (53) are in agreement with Assumption P2. Now, according to Proposition 3 and Remark 4, the observer can be proposed asUsing and the list of data (49), we have the following matrix:which is stable, and . Solving the Lyapunov equation, it is easy to see that , ensuring that . That is, the corresponding observer proposed in (55) ensures that the estimation error converges exponentially and asymptotically to zero.
Similarly to the master system, the slave experimental setup uses the above defined parameter values, and the initial conditions of the slave and its observer were fixed as and , respectively. The slave input was fixed as . In Figure 2, it is easy to see that the errors are very close to zero after 3.5-second elapse.
Colpitts Oscillator and Chua Circuit Generalized Synchronization. For this numerical simulation, we used the same experimental setup and the same observer as in the previous experiments. The setup for the slave controller was fixed according to Proposition 5. The used static feedback waswhere
The synchronization error of this experiment is shown in Figure 3, where we can see that, after 9-second elapse, the difference between each pair of transformed coordinates is almost zero, for the case .
5. Conclusions
In this paper, we proposed a solution to the output-feedback GS problem for a class of chaotic systems, namely, the free master and the controlled slave, assuming that the master system trajectories are bounded in a compact set. The problem was solved given that the slave was controlled using a single input, and a single output of each system was available or measurable. The solution consisted of expressing both systems in their corresponding observable generalized canonical form, through the differentiable primitive element. To this end, a suitable diffeomorphism transformation was applied, between the original state space and the corresponding canonical observability form of each system. Due to the fact that the controlled slave system uses its own whole state and the whole state of the master, a kind of a Luenberger observer for each system was proposed. Finally, the information obtained through the observers is used by the controlled slave system, solving the GS problem. The corresponding convergence analysis was carried out using the linear control approach and the Lyapunov method. The effectiveness of the obtained method was assessed by convincing numerical simulations, using the chaotic Colpitts oscillator as the master and the chaotic Chua circuit as the slave.
Appendix
Computing the Lipchitz Constant for Function
From the definition given in (48), we have thatAfter using the well-known inequality, (A.1) can be rewritten as where
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the Secretaría de Investigación y Posgrado of the Instituto Politécnico Nacional (SIP-IPN) under Research Grants 20150929 and 20150351 and by the Departamento de Control Automático of the CINVESTAV-IPN.