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

Huangfu, Yongbing; Xue, Kaijuan

doi:https://doi.org/10.1155/2021/3224416

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Dynamic Analysis, Learning, and Robust Control of Complex Systems

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Research Article | Open Access

Volume 2021 | Article ID 3224416 | https://doi.org/10.1155/2021/3224416

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

Yongbing Huangfu¹and Kaijuan Xue¹

Academic Editor: Heng Liu

Received12 Apr 2021

Accepted09 Jul 2021

Published19 Jul 2021

Abstract

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 implemented. 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.

1. 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. There 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]. They 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 [13–15]. After that, synchronization study in chaotic dynamical systems has received much attention. There 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].

The 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, 24–30] 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. The 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 variable-structure 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.

This 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.

2. Preliminaries

Throughout this paper, represents the real numbers, represents the real -vectors, and represents the real matrices. represents any suitable vector norm.

Two uncertain chaotic MIMO systems are given as follows. The driving system is expressed byand the response system is given aswhere and , respectively, are the overall state vector of the driving system and the response system, which are measurable. means the controller, , are unknown nonlinear functions, is unknown gain matrix, and is input nonlinearity.

Let us denote , , and ; then, driving system (1) is rewritten asand (2) can be written aswhere and .

Some simple assumptions are set forth.

Assumption 1. is an unknown positive-definite matrix, and one can find an unknown positive constant such that , with being the identity matrix.

Remark 1. In fact, the above assumption is not restrictive, because many physical systems satisfy it. This 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.

The response system is driving by controller; however, it does not affect the behavior of the driving system. The control purpose of this paper is to put forward a control input 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

The filtered synchronization errors is given by

Thus, we have

Afterwards, (7) will be used to develop the controller and conduct the stability analysis.

2.1. 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 becoming an output signal . The -th fuzzy rule has the following form:

: if is and and is then is , with , and being fuzzy sets. The output of a fuzzy system is given bywhere means the membership function of to ; suppose that there are fuzzy rules involved in the fuzzy system. can be adjusted online, andthat being the fuzzy basis function (FBF) with .

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 to an arbitrary degree of accuracy. It is extremely important to assume that the membership function parameters need the designer to be prespecified. That being said, the structure of the fuzzy system needs the designer decision for determination, namely, the pertinent inputs. However, the parameters have to be defined by learning algorithms.

2.2. Input Nonlinearity

We conduct the input nonlinearity aswhere and are nonlinear functions with respect to and are positive constants.

Here, has some significant properties as follows:where and are constants known as “gain reduction tolerances.” We can make .

Then, we give some reasonable assumptions to study the properties of input nonlinearity in control problems.

Assumption 2. (a), namely, the gain reduction tolerances, are unknown, so is unknown.(b)The explicit mathematical equation of is unknown, but we know the properties (11) and assume that and are known constants.

Remark 2. (1)It can be seen from (10) and (11) that the input nonlinearity can be reduced to the special sector nonlinear function if . 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. The limitations are as follows:(i)The chaotic system, which they considered, is a simple SISO system, which is input with sector nonlinearities and/or dead zones(ii)They assumed that and or are known in their control scheme

3. 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 , we have

Further, for the stability analysis and controller design, (13) can be arranged aswhere and .

Assumption 3. There is an unknown continuous positive function satisfying , where .

Remark 3. The reasons why the above Assumption 3 is not restrictive are as follows:(i)We assume that the upper bound is unknown(ii)As is a function about and is continuous, is always there

The unknown continuous positive function over compact set can be approximated by the fuzzy system (8) as follows:where is the FBF vector, which is determined in advance by the designer, and is the adjustable parameter vector in the fuzzy system.

Letbe the optimal value of .

It is worth noting that, for the sake of analysis, we put forward artificial constant quantities , and when implementing the controller, their values are not needed.

Fix the parameter estimate error asand the fuzzy approximation error aswhere .

In this work, assume the compact set and the fuzzy systems we used do not infringe the universal approximator property, and is supposed to be large enough, so that it can contain the input vector of the fuzzy system in a closed-loop control system. So, it is rational to suppose that is bounded for all , i.e., , , where is an unknown constant.

Then, we have

In order to achieve the control objective, let us propose a suitable fuzzy adaptive variable-structure controller:with andwhere , , , , are design constants and and are the online estimates of the uncertain terms and , respectively.

Remark 4. From adaptive laws (21) and (22), we can get their solutions satisfied and , for so that and .

Multiplying (9) by and using Assumption 3, we have

From (19) and (23), we getwhere and .

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)The system enclosed is asymptotically stable

Proof. of Theorem 1. Let the Lyapunov function beThe time derivative of iswith .
It can be noticed from (24) that for and for . Thus, from (20) and (24), we can get that for ,and for ,Then, for and , we haveSince and , then from (29), we haveFor all , we haveUsing (21), (22), (24), and (26), (31) becomesObviously, we haveThen, (32) becomesSince , then we haveFrom (34) and (35), we havewhereMultiplying (36) by yieldsIntegrating (38) over , we haveFrom the above analysis, we can get , , , and that are uniformly ultimately bounded. Thus, is bounded. Then, by using (25), can be written asAs is symmetric positive-definite (i.e., there is an unknown positive constant , such that ), from (25) and (39), we haveWe can get that the solution of exponentially converges to a bounded region . This completes the proof of the theorem.;

Remark 5. If (or when , and ), 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)There is a special case that , and (20) can be simplified to the following form: where .(2)It is worth mentioning that the function can be replaced by any equivalent smooth function: , , , and so on. The chattering effect caused by the discontinuous control term in (20) and (42) can be removed.

4. 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 asand the response system iswhere , , and , , are state variables of the driving system and the response system, respectively, and , , 2, 3, are the inevitable input nonlinear models. Let , , , and .

Then, system (43) can be written asand system (44) can be written as

The input nonlinearities , for , 3, are described byand the input nonlinearity is supposed to be

It is worth noting that , , are unknown, except for some structural natures, for example:(i)The symmetry and sign of (ii)The natures (11) of and an understanding of and , with , 2, 3

The initial conditions of the driving system are , and the initial conditions of the response system are .

The adaptive fuzzy systems, , with , have the vector as input, and three triangular membership functions that uniformly distribute on the intervals are given for each entry variable of these fuzzy systems. The design parameters are taken as , , , , , , , , . The initial conditions are given as , , .

It is worth noting that , which is a discontinuous function, has been replaced with , which is a smooth function, with , , 2, 3.

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. The transient behaviors of controller are presented in Figure 4.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

5. Conclusion

This 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. The validity of the approach has been tested and verified by means of example and simulation. The 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

The authors declare that there are no conflicts of interest related to this article.

Acknowledgments

This work was supported by the Key R&D Projects in Shanxi Province (Grant no. 201703D121002).

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Copyright © 2021 Yongbing Huangfu and Kaijuan Xue. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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