Synchronization of Chaotic Systems with Dead Zones via Fuzzy Adaptive Variable-Structure Control

This work is devoted to solving synchronization problem of uncertain chaotic systems with dead zones. Based on the Lyapunov stability theorems, by using fuzzy inference to estimate system uncertainties and by designing eﬀective fuzzy adaptive controllers, the synchronization between two chaotic systems with dead zones is realized and a fuzzy variable-structure control is imple-mented. The stability is proven strictly, and all the states and signals are bounded in the closed-loop system. A simulation example is presented to test the theoretical results ﬁnally.


Introduction
It is widely known that chaos is almost everywhere in the domain of engineering and science. It is a kind of complex dynamic behavior of nonlinear dynamic systems, and chaos has many applications in mechanical, electronics, and biochemistry fields. ere are many common chaotic systems, such as Chen system [1,2], Lorenz system [3], Genesio-Tesi system [4,5], Rössler system [6], and Lur'e system [7]. Chaotic systems, as everyone knows, have deterministic behavior; for example, they are extremely sensitive when the initial conditions alter to a small extent and difficult to predict, but their trajectories are bounded in the phase space [8,9].
People always think that chaotic systems cannot be synchronized due to the characteristics of chaos. It was realized first on electronic circuit by Pecora et al. [10,11]. ey found that if the chaotic system can be decomposed into two subsystems, and, in the response system, all the conditional Lyapunov exponents are less than zero, there will be chaotic synchronization effect in the drive and response system [1,12]. Chaos synchronization means that, for two chaotic systems starting from different initial points, their trajectories gradually tend to be consistent with each other over time, and this synchronization is structurally stable [13][14][15]. After that, synchronization study in chaotic dynamical systems has received much attention. ere are some way to achieve the synchronization of some chaotic systems, for instance, time-delay feedback control [7], active control synchronization [16], impulsive synchronization and synchronization [17,18], sliding mode synchronization [19,20], and projective synchronization [15]. e input nonlinearities, such as dead zones and saturation, may destroy the control performance of the system, and the control problem of uncertain nonlinear systems with nonlinear input is getting more complicated and receiving more considerable attention [21,22]. Control performance of the system may be degraded or even cause the instability of the control system due to input nonlinearities, and their synchronization problem becomes more challenging [23]. Please refer to [15,[24][25][26][27][28][29][30] for some works about synchronization study in chaotic dynamical systems, which are subject to input nonlinearity.
Dead zone is that the value of the output variable does not change with the change of the input variable value. e range of the input variable can be understood as dead zone [31]. Previously, many researchers have used various methods to settle the synchronization problem of the nonlinear systems, that is, the control input with dead zones. For instance, using Laplace transform approach to settle synchronization of nonlinear systems with disturbances subjected to dead-zone and saturation characteristics in control input was studied in [32], projective synchronization of Chua's chaotic systems that control input has dead zones was studied in [33,34], and using adaptive fuzzy sliding mode control to deal with unknown nonlinear chaotic gyros synchronization with unknown dead-zone input was studied in [35].
In this paper, we put forward a fuzzy adaptive variablestructure synchronization scheme to manage the dead-zone nonlinearity and analyze the synchronization properties of chaotic systems. Comparing the related works, for example, [26,32,34], the contribution of this works consists in the following: (1) Input nonlinearity is considered in this paper. However, it is not considered in the above literature. (2) Compared with [26,32], the assumptions of this work are more realistic.
is paper is organized as follows. In Section 2, the notation, problem statement, and preliminaries are raised, including the description of the uncertain chaotic MIMO systems, fuzzy logic system, and input nonlinearity. In Section 3, we present a fuzzy adaptive controller based on the universal approximator property. In Section 4, the effectiveness of the approach is tested and verified by a simulation. In addition, conclusions are contained in Section 5.

Preliminaries
roughout this paper, R represents the real numbers, R n represents the real n-vectors, and R m×n represents the real m × n matrices. ‖ · ‖ represents any suitable vector norm.
Two uncertain chaotic MIMO systems are given as follows. e driving system is expressed by and the response system is given as where x � [x 1 , . . . , x n ] T ∈ R n and y � [y 1 , . . . , y n ] T ∈ R n , respectively, are the overall state vector of the driving system and the response system, which are measurable. u � [u 1 , . . . , u n ] T ∈ R n means the controller, f i (x), f i (y), i � 1, . . . , n are unknown nonlinear functions, g ij , i, j � 1, . . . , n is unknown gain matrix, and . . , f n (y)] T , and G � g 11 . . . g 1n ⋮ ⋱ ⋮ g n1 . . . g nn ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦; then, driving system (1) is rewritten as and (2) can be written as where F(·) and Φ(·) ∈ R n and G(·) ∈ R n×n . Some simple assumptions are set forth.
Assumption 1. G is an unknown positive-definite matrix, and one can find an unknown positive constant σ 0 such that G ≥ σ 0 I n , with I n being the identity matrix.
Remark 1. In fact, the above assumption is not restrictive, because many physical systems satisfy it. is assumption that devoted to adaptive control of MIMO systems is ubiquitous in the literature, and without exaggeration, we can say that the controllability of the system is assured by it.
e response system is driving by controller; however, it does not affect the behavior of the driving system. e control purpose of this paper is to put forward a control input u in response system to synchronize the driving systems; that is, all signals of the driving and response systems must be bounded under the constraint.
First, we define the synchronization errors between driving and response systems as e filtered synchronization errors is given by us, we have Afterwards, (7) will be used to develop the controller and conduct the stability analysis.

Description of the Fuzzy Logic
System. By and large, a fuzzy system contains four aspects, that is, fuzzifier, fuzzy rules, fuzzy inference, and defuzzifier, which is depicted in Figure 1. By using proper fuzzy rules, a fuzzy inference drives the input y T � [y 1 , y 2 , . . . , y n ] ∈ R n becoming an output signal f ∈ R. e i-th fuzzy rule has the following form: . ., and A i n being fuzzy sets. e output of a fuzzy system is given by where μ A I J (y j ) means the membership function of y j to A i j ; suppose that there are m fuzzy rules involved in the fuzzy Complexity that being the fuzzy basis function (FBF) with m i�1 ( n j�1 μ A i j (y j )) > 0. It can be noted that the fuzzy system (8) is ubiquitous in control applications. In light of the universal approximation results, fuzzy system (8), on a compact operating space, is capable of approximating any nonlinear smooth function f to an arbitrary degree of accuracy. It is extremely important to assume that the membership function parameters need the designer to be prespecified. at being said, the structure of the fuzzy system needs the designer decision for determination, namely, the pertinent inputs. However, the parameters θ have to be defined by learning algorithms.

Input Nonlinearity. We conduct the input nonlinearity
where ϕ i+ (u i ) > 0 and ϕ i− (u i ) > 0 are nonlinear functions with respect to u i and u i+ , u i− are positive constants.
Here, Φ i (u i ) has some significant properties as follows: where m * i+ and m * i− are constants known as "gain reduction tolerances." We can make η i � min m * i+ , m * i− . en, we give some reasonable assumptions to study the properties of input nonlinearity in control problems.

Assumption 2
(a) m * i+ , m * i− , namely, the gain reduction tolerances, are unknown, so η i is unknown.
unknown, but we know the properties (11) and assume that u i+ and u i− are known constants.

Remark 2
(1) It can be seen from (10) and (11) that the input nonlinearity Φ i (u i ) can be reduced to the special sector nonlinear function if u i+ � u i− � 0. Consequently, the MIMO system with the input nonlinearities (10), which we considered, is more universal. (2) It can be noted that the model (10) has been widely used in the past, but it has some limitations, and we made some improvements. e limitations are as follows: (i) e chaotic system, which they considered, is a simple SISO system, which is input with sector nonlinearities and/or dead zones (ii) ey assumed that m *

Design of the Fuzzy Adaptive Controller
In this part, for the class of unknown chaotic MIMO systems, we will develop a fuzzy adaptive variable-structure control plan (3).
Substituting (4) into the (7), we get Now posing G 1 � G − 1 , we have Further, for the stability analysis and controller design, (13) can be arranged as where

Remark 3.
e reasons why the above Assumption 3 is not restrictive are as follows: (i) We assume that the upper bound ηα i (y) is unknown (ii) As ] is a function about (y, y d ), y d ∈ L ∞ and α i (y, ]) is continuous, α i (y) is always there e unknown continuous positive function α i (y) over compact set Ω y can be approximated by the fuzzy system (8) as follows: where ψ i (y) is the FBF vector, which is determined in advance by the designer, and θ i is the adjustable parameter vector in the fuzzy system. Let be the optimal value of θ i . It is worth noting that, for the sake of analysis, we put forward artificial constant quantities θ * i , and when implementing the controller, their values are not needed.
Fix the parameter estimate error as and the fuzzy approximation error as where α i (y, θ * i ) � θ * T i ψ i (y). In this work, assume the compact set Ω y and the fuzzy systems we used do not infringe the universal approximator property, and Ω y is supposed to be large enough, so that it can contain the input vector of the fuzzy system in a closedloop control system. So, it is rational to suppose that ε i (y) is bounded for all y ∈ Ω y , i.e., |ε i (y)| ≤ ε i , ∀y ∈ Ω y , where ε i is an unknown constant. en, we have In order to achieve the control objective, let us propose a suitable fuzzy adaptive variable-structure controller: where c 0i , c 1i , σ 0i , σ 1i , k 1i > 0 are design constants and k 0i and θ i are the online estimates of the uncertain terms k * 0i � ε i and θ * i , respectively. (21) and (22), we can get their solutions satisfied k 0i (t) ≥ 0 and θ i (t) ≥ 0, for t > 0 so that k 0i (0) ≥ 0 and θ i (0) ≥ 0.

Theorem 1. For system (3), if Assumptions 1-3 are satisfied, the control law (20)-(22) can guarantee the following properties: (i) It is no exaggeration to say that, in the closed-loop system, all signals are uniformly ultimately bounded (ii) e system enclosed is asymptotically stable
Proof of eorem 1. Let the Lyapunov function be e time derivative of V is with _ G 1 � 0. It can be noticed from (24) that u i < − u i− for S i > 0 and u i > u i+ for S i < 0. us, from (20) and (24), we can get that for S i > 0, and for S i < 0, en, for S i > 0 and S i < 0, we have Since S 2 i > 0 and S i sign(S i ) � |S i |, then from (29), we have For all S i , we have 4 Complexity Using (21), (22), (24), and (26), (31) becomes Obviously, we have en, (32) becomes Since G ≥ σ g0 I n , then we have From (34) and (35), we have where From the above analysis, we can get k 0i , θ i , S i , E and y that are uniformly ultimately bounded. us, u i is bounded. en, by using (25), V(0) can be written as As G 1 is symmetric positive-definite (i.e., there is an unknown positive constant σ g1 , such that G 1 ≥ σ g1 I n ), from (25) and (39), we have We can get that the solution of S i exponentially converges to a bounded region Ω S i � S i ||S i | ≤ ((2η/ σ g1 )(π/μ)) 1/2 }. is completes the proof of the theorem.; □ , namely, there are neither dead zones, nor sector nonlinearities in the input, we can prove that the controller is still applicable for these MIMO chaotic systems.

Remark 6
(1) ere is a special case that u i+ � u i− � u i0 , and (20) can be simplified to the following form: where (2) It is worth mentioning that the function sign(·) can be replaced by any equivalent smooth function: tanh(·), arctan(·), Sat(·), and so on. e chattering effect caused by the discontinuous control term in (20) and (42) can be removed.

Simulation Results
In order to demonstrate the effectiveness of the proposed adaptive fuzzy controller for uncertain chaotic MIMO systems, we consider the Lotka-Volterra system: the driving system is given as and the response system is where x 1 , x 2 , x 3 and y 1 , y 2 , y 3 are state variables of the driving system and the response system, respectively, and Φ i (u i ), i � 1, 2, 3, are the inevitable input nonlinear models.
Finally, the simulation results are shown in Figures 2-4. Figure 2 shows the chaotic phenomenon of the driving system. Figure 3 shows that the driving system and the response system are basically synchronized after 0.2 seconds, and the results indicate the effectiveness of our method. e transient behaviors of controller are presented in Figure 4.

Conclusion
is paper proposes a fuzzy adaptive variable-structure controller for the synchronization of the MIMO unknown chaotic system that has sector nonlinearities and dead zones. Based on the Lyapunov stability theory, the whole system can achieve asymptotic stability; namely, all the closed-loop signals and states are bounded, and the synchronization performance of two systems can be achieved. To be specific, a smooth function can reduce the chattering phenomena in the process of control. e validity of the approach has been tested and verified by means of example and simulation. e approach can be applied to settle the synchronization of a large class of chaotic system with dead zones. How to obtain accurate control performance is one of our future research directions.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest related to this article.