Output-Feedback Controller Based Projective Lag-Synchronization of Uncertain Chaotic Systems in the Presence of Input Nonlinearities

1LAJ, Automatic Control Department, University of Jijel, BP 98, Ouled-Aissa, 18000 Jijel, Algeria 2LARA, École Nationale d’Ingénieurs de Tunis, Université de Tunis El Manar, BP 37, Le Belvédère, 1002 Tunis, Tunisia 3Department of Electronics, University of Jijel, BP 98, Ouled-Aissa, 18000 Jijel, Algeria 4Department of Telecommunications and Systems Engineering, Universitat Autònomade Barcelona, Bellaterra, 08193 Barcelona, Spain 5Departamento de Ingenieŕıa, Facultad de Ciencias Naturales e Ingenieŕıa, Universidad de Bogotá Jorge Tadeo Lozano, 22 Street, No. 4-96, Mod. 7A, Bogotá 110311, Colombia


Introduction
The chaos synchronization has attracted great attention and has been extensively studied [1][2][3][4][5][6][7][8][9][10][11][12][13][14], since it was suggested originally by Pecora and Carroll in [15].The basic configuration of chaos synchronization consists of two chaotic systems: a drive (master) system and a response (slave) system.These systems can be identical but with different initial conditions (IC) or quite different.The response system is driven via some transmitted (drive) signals so that the trajectories of the response system synchronize with that of the drive system.
In real applications of chaos synchronization, the state vectors of drive-response systems are not available for measurement, except the outputs of drive-response systems.Thus, designing a synchronization scheme based on an outputfeedback controller (i.e., an observer-based controller) is required.Based on state observer, some adaptive control systems were designed in [29][30][31][32].These systems involve strictly positive real (SPR) concept on the observation-error dynamics.The dynamics of the observation errors, which are originally not SPR, are augmented by an appropriate low-pass filter designed to meet the SPR concept.
On the other hand, most of the above works are only valid for chaotic systems without dynamical disturbances and input nonlinearity.However, in practice, the chaotic systems are inevitably affected by uncertain dynamical disturbances.The existence of these disturbances can generally lead to the synchronization failure and cause undesirable results.How to enhance the disturbance compensation or attenuation is of great significance [33,34].Besides, owing to the physical limitations, the practical implementations of the control systems are frequently exposed to input nonlinearities (backlash, dead-zone, and saturation).It has been shown that these input nonlinearities can cause a serious degradation of the system performances and in a worst-case system failure.So, the design of a controller for chaos synchronization by considering of the external disturbances and input nonlinearities is of significant importance [31][32][33][34][35][36][37][38][39].To effectively deal with these problems, the control schemes have been generally designed in a variable-structure control framework.
Motivated by the above discussions, in this paper, we aim at addressing the problem of projective lag-synchronization for a class of uncertain chaotic systems subject to uncertain external dynamical disturbances and input nonlinearities (sector nonlinearities with dead-zone).This synchronization can be realized through an appropriate fuzzy adaptive variable-structure controller based on a state observer.Compared with the previous works on the chaos synchronization and control [8-14, 16-20, 31-39], the main contributions of this paper are the following: (i) A novel projective lag-synchronization system based on fuzzy adaptive variable-structure output-feedback control is designed for unknown perturbed chaotic systems containing dead-zone nonlinearity.
(ii) The model of the chaotic drive-response system is assumed to be completely different, unknown (except its relative degree), subject to dynamical disturbances, with input dead-zone and sector nonlinearities, and immeasurable states.Besides, its dynamics should not satisfy the SPR property.To authors' best knowledge, such a class of chaotic (drive-response) systems with all these properties has not been previously considered in the open synchronization literature.
(iv) By designing a linear observer to estimate the lag-synchronization errors, only the outputs of the response-drive system are assumed to be measurable in this synchronization scheme.
(v) The designed fuzzy adaptive control is very simple and has only two adaptive parameters.So, this controller is of practical significant importance.

System Description and Problem Formulation
Consider the following class of drive-response chaotic systems: or equivalently of the form with  = [ [ ], where  = [ 1 , . . .,   ]  ∈   and  = [ 1 , . . .,   ]  ∈   are the state vectors of the drive and response systems, respectively.  () and   () are unknown nonlinear smooth functions and  = (V) is the input nonlinearity, with V being the control input which will be designed later.  (, ) and   (, ) are the external disturbances of the drive and response systems, respectively.The input nonlinearity  = (V) under consideration is given by [48][49][50] with  + (V) > 0 and  − (V) > 0 being nonlinear smooth functions of V, V + > 0 and V − > 0. Note that this model contains both sector nonlinearity and dead-zone.The nonlinearity (V) also has the following features: with  * + and  * − being so-called "the gain reduction tolerances" [48][49][50].
Design Objective.Determine an output-feedback control law V to achieve a projective lag-synchronization between the drive system and the response one, while ensuring that all involved signals in the closed-loop system remain bounded.
To facilitate the control system design, the following usual assumptions are considered and will be used in the subsequent developments.
where   and   are some unknown positive constants.
Definition 4. The drive and response systems (2) are projective lag-synchronized if there exists a scaling factor  such that  =  − ( − ) → 0 as  → ∞, where  > 0 is a constant propagation delay or transmission delay.This means that the transmitted signal is received  time late after it was sent.The value of  depends on the channel or the distance between drive and response system.
Remark 5. From Definition 4, it is easy to see that, for  = 0, the complete synchronization, antisynchronization, and projective synchronization are the special cases when the scaling factor takes the values  = +1,  = −1, and  ̸ = 1 and −1, respectively.And when  = 1, one obtains the lagsynchronization.
From (2) and Definition 4, one can write the dynamics of the lag-synchronization error as where   = ( − ) and Note that one can easily show the existence of a constant  1 > 0 such as | 1 | ≤  1 , for the following reasons:  evolves in a compact set (an intrinsic property of the (noncontrolled) chaotic systems), also the delayed state   is bounded and the external disturbances,   (, ) and   (, ), are already assumed to be bounded, and finally the function   (  ) is smooth and with a bounded argument.
Since   () is unknown and the vector  is immeasurable, in this paper, one will use (1) a fuzzy adaptive system to approximate the uncertain functions, (2) an observer to estimate the projective lagsynchronization error .

Controller Design for Projective Lag-Synchronization
This section proposes a fuzzy adaptive output-feedback controller for lag-projective synchronization of the driveresponse system (2) using Lyapunov stability theory.The proposed synchronization scheme is shown in Figure 1.One can rewrite the dynamics of the lag-synchronization errors as follows: where  = [1 0 ⋅ ⋅ ⋅ 0].Note that the pair (, ) is observable.
(1) The controllability property of the pair (, ) guarantees the existence of a feedback gain vector,   , so that the characteristic polynomial of  −    is strictly Hurwitz.
(2) The observability property of the pair (, ) ensures the existence of an observer gain vector,   , so that the characteristic polynomial of  −    is strictly Hurwitz.
According to fuzzy approximation theorem [28], the unknown function   ( + ( − )) can be optimally approximated by a linearly parameterized fuzzy system, as follows [47]: with () being the vector of FBFs (which are assumed to be designed a priori), (, (−)) being the fuzzy approximation error, and  * being the optimal value of the adjustable parameter vector of the fuzzy system (9) which is defined as According to [28], the fuzzy approximation error (, (−)) is bounded.

Mathematical Problems in Engineering
Since the lag-synchronization-error vector  is not available for measurement, one designs the following linear observer to estimate it: where ê is the estimate of ,   = [ 1 , . . .,   ]  ∈   is the gains vector of observer,   =  −    , and   = [ 1 , . . .,   ]  ∈   is the feedback gain vector.Now, one defines the observation-error vector as ẽ = [ẽ 1 , . . ., ẽ ]  =  − ê.From ( 12) and ( 11), the dynamics of this observation error can be obtained as follows: with   =  −    and Then, we can rewrite (13) using the time-frequency (mixed) notation as follows [51,52]: where  is the Laplace variable and () = ( −   ) −1  is the stable transfer function of (13).It is worth noting that this mixed notation is very valuable in the adaptive control literature [51][52][53][54][55][56].It also refers to the convolution between the inverse Laplace transform () and the term  *  () + (V) +  3 .Since () is not SPR, one introduces a low-pass filter () such that () = () −1 () becomes SPR: with  ] being an unknown positive vector, and Let us define a novel error  1 , called the modified error, as follows: with  1 being the auxiliary error.Its dynamics are given by where   is the estimate of the unknown vector  *  and  > 0 is a small design constant.tanh(⋅) designates the usual hyperbolic tangent function.
To achieve our objective, the control input can be determined as with  > 1/, and  = min{ * − ,  * + }, where where  1 is a design positive constant and  2 is an adaptive parameter estimating the upper bound of ‖ * ‖; that is,  * 2 ≥ ‖ * ‖.
The adaptive laws for the control law (23) where   ,   ,   , and   are strictly positive design parameters.
can be simply rewritten as In (40), the sign function, that is, sign( 1 ), can cause the undesirable chattering phenomenon.In practice, the latter is generally replaced by an equivalent and smooth function (e.g., tanh( 1  1 )): with  1 > 0 being a high constant value.
Remark 11.More importantly, the design of a lag-synchronization system based on output-feedback controller for a class of uncertain drive-response systems with input nonlinearities has a major interest in both theory and practice.
(a) Theoretical Interests.Compared to previous works [8][9][10][11][12][13][14][16][17][18], our theoretical contributions are the following: (1) Design of a projective lag-synchronization system by considering the ubiquitous input nonlinearities (i.e., sector nonlinearities and dead-zone), the uncertain dynamics of both models, and the immeasurability of the states of drive-response system is theoretically challenge.To the best of authors' knowledge, the projective lag-synchronization for this class of driveresponse systems with all these features has rarely been studied in the literature.
(2) The proposed fuzzy adaptive output-feedback control requires the so-called SPR condition on the lagsynchronization errors.It should be noted that the design of an output-feedback controller dealing with input nonlinearities (particularly, sector nonlinearity and dead-zone) and by using a SPR approach is not theoretically simple.This is why in the literature there are few fundamental results dealing with this control problem.
(b) Practical Interests.The proposed synchronization approach has the following practical interests: (1) The proposed projective lag-synchronization approach is characterized by one scalar transmitted signal.This feature is of practical significant importance.
(2) The effect of ubiquitous input nonlinearities (sector nonlinearities and dead-zone) has been taken into account in the stability analysis and the design of the control system.In practice, it is well known that the nonconsideration of the latter may lead to a serious degradation of the system's performances and even cause system instability.
(3) In particular, this projective lag-synchronization approach has also a prospective application in secure communication due to its safety against attack and unmasking.

Illustrative Simulation Examples
Three academic examples are provided in this section to validate the effectiveness of this proposed synchronization approach.
Example 1.Consider the projective lag-synchronization between chaotic Gyros system and Duffing oscillator.
Then, this chaotic drive-response system can be rewritten as follows: where  = [ 0 1 0 0 ],  = [ 0 1 ], and   = [ 1 0 ]. = (V) is the input nonlinearity which is defined below, and V is the control input to be designed.
(a) Simulation by Using the Discontinuous Controller (40). Figure 2 shows that the proposed controller performs well.In fact, one can see from Figures 2(a) and 2(b) that the states of response system ( 1 ,  2 ) effectively track that of the drive system ( 1 ( − ),  2 ( − )), despite the presence of the immeasurable states, uncertain dynamics, dead-zone at the input, and external disturbances.From Figure 2(c), it is clear also that the estimates of the synchronization errors are bounded and asymptotically converge towards small values.The corresponding control signal is bounded and not smooth in Figure 2(d).
(b) Simulation by Using the Smooth Controller (41). Figure 3 provides the simulation results.From Figures 3(a) and 3(b), one can observe that the states of the response system ( 1 ,  2 ) effectively follow the corresponding desired trajectories ( 1 ( − ),  2 ( − )).From Figure 3(c), one can see that the estimates of the synchronization errors are wellbounded and converge to a small value.In Figure 3(d), the control signal is smooth, bounded, and admissible.
Example 2. Now, we will consider the projective lagsynchronization between two uncertain similar chaotic systems of the third order.
The Response System (Genesio Chaotic System) [61] where   (, ) = sin(6) and  = [ 1 ,  2 ,  3 ]  is the state vector of the response system.Then, this chaotic drive-response system can be rewritten as follows:    where  = [ ], and  = [1 0 0]. = (V) is the input nonlinearity which is defined below, and V is the control input to be designed.
From the expression of (), one can find that  = ] and solving (22), one gets The initial conditions are chosen as , and   (0) = [0, 0, 0, 0]  .The projective lag-synchronization response of system (51) is presented in Figure 4.It is obvious from the latter that the trajectories of response system ( 1 ,  2 ,  3 ) effectively track that of the drive system ( 1 ( − ),  2 ( − ),  3 ( − )), despite the presence of the immeasurable states of chaotic systems, uncertain dynamics, dead-zone at the input, and external disturbances.The corresponding control signal is bounded.
Example 3. We consider now the projective lag-synchronization between two different chaotic systems of the third order.
Note that this synchronization scheme is designed as that of the second example, except that the response system is selected as an Arneodo chaotic system and  = −0.25.
Therefore, the fuzzy system and the synchronization-error observer are designed as in Example 2, and the initial conditions and the design parameters are also selected as in Example 2.
Figure 5 provides the simulation results of this example, from which it can be clearly seen that the trajectories of response system ( 1 ,  2 ,  3 ) effectively track the trajectories of the drive system ( 1 (−),  2 (−),  3 (−)), despite the presence of immeasurable states of chaotic systems, uncertain dynamics, input nonlinearities, and external disturbances.

Conclusion
The problem of adaptive fuzzy output-feedback controlbased projective lag-synchronization for unknown driveresponse (or master-slave) chaotic systems has been investigated in this paper.In the design process, the input nonlinearities (dead-zone together with sector nonlinearities) have been considered.To effectively handle the unknown functions in the drive-response system, fuzzy adaptive systems have been incorporated in the control system.To deal with the input nonlinearities, the proposed controller has been designed in a variable-structure framework.And to estimate the synchronization-error states, a simple linear observer has been constructed.Finally, three academic examples have been given to demonstrate the effectiveness of the proposed lagsynchronization approach.In our future work, the investigation for chaotic fractional-order drive-response systems ė = ż −  ẋ ( − ) =  +  [−  (  ) −   (,   ) +   () +  +   (, )] =  +  [  () +  +  1 ] ,

( 17 )
Remark 7. () is SPR, with  =  +  if the following conditions are satisfied [57]: (a) When  is real, () is real.(b) The poles of () are not in the right half-plane.(c) For any real , the real part of () is positive; that is, Re[()] ≥ 0.