#### Abstract

This study addresses an adaptive two-stage sliding mode control (SMC) scheme for the state synchronization between two identical systems, which belong to a kind of *n*-dimensional chaotic system, by considering the appearance of lumped system uncertainties and external disturbances. The controlled system is assumed to be attached to sector nonlinearity for the control input. The proposed adaptive control scheme involves time-variable state-feedback gains, which are updated in accordance with the appropriate adapted rules without foreknown the certain information of nonlinear system dynamics, bounds of lumped system uncertainties and external disturbances, and sector input nonlinearity. The derivation of the control scheme is found based on the introduced sequence of two sliding functions. The stage 1 sliding function is defined by the states of the error dynamical system, where the asymptotical stability is inherent. Then, the stage 2 sliding function is formed by the stage 1 function, where the finite-time stabilization is guaranteed. The proposed adaptive control scheme can cope with the effect of sector nonlinearity for the control to meet the control goal. The sufficient conditions of the stability of the error dynamical system are proven mathematically by means of the Lyapunov theorem. Besides, the capacity of the present scheme is carried out by the numerical studies.

#### 1. Introduction

Since the concept of the drive-driven chaotic synchronization has been first introduced in the commonly cited study [1], the researching topic of synchronization between two chaotic systems received wild attention. Some important applications of synchronization can be listed out such as information processing, energy resource [2], and secure communication [3]. In the literature, many synchronization techniques, including adaptive synchronization [4–9], complete synchronization [10], and sliding mode control synchronization [11–19], have been studied. In the recent, the multisynchronization of memristor-based neural networks [20–22], the cluster synchronization and control on multiple complex networks [23, 24], and the combinational inner and outer synchronization of antistar networks [3, 25] have also been paid much attention.

The object of state synchronization is to achieve that each of state variables of the driven chaotic system is tracking to each of states of the drive one, respectively. Nowadays, many studies have been reported to achieve state synchronization of chaotic systems, such as active control [26], linear feedback control [2, 27], adaptive control [4–9], sliding mode control (SMC) [11–19], dynamic surface approach [28], and finite-time control [29–32].

In [26], the adaptive active control projective synchronization controller was developed to stabilize the error system. In [27], a single-input linear control was proposed to achieve synchronization for the symmetric chaotic systems. The single-input adaptive control with parameter updated rules for the generalized Lorenz chaotic systems and a hyperchaotic system was derived in [4–6, 9]. In [14–16], the robust and adaptive linear and terminal SMCs were performed to realize chaotic synchronization of nonautonomous chaotic systems and neurons. The integral SMC associated with the supertwisting algorithm was applied to the Markovian jump system [18]. In [19], the state synchronization by the fast terminal SMC for unified chaotic systems was accomplished. The problem of chaos suppression in the power system was solved by a fixed-time dynamic surface approach with the high-order SMC [28]. For the control of synchronization within finite time, an adaptive finite-time control technique with robustness was addressed for the synchronization between two chaotic gyros [29]. Synchronizations between two and three complex-variable chaotic systems within finite time by means of the nonsingular terminal SMC control were reported in [31, 32].

The drawbacks of these antecedent studies are not considering the input nonlinearities attached to the control inputs of the driven system. Under the limitation of physical properties, the inputs of control systems are commonly involved in nonlinearity, such as sector nonlinearity, in practical usage. It is revealed that the system performance is caused by a serious degradation by the existence of input nonlinearity for control. Consequently, it would be necessary to design control inputs by taking into account the effects of input nonlinearity [33–37]. In light of the above motivation, by considering the sector nonlinearity for control inputs, the robust and adaptive finite-time controller was reported for synchronization of two different and identical uncertain chaotic systems [33]. For second-order chaotic systems, [35–37] presented that chaotic synchronization between two chaotic oscillators was achieved by the adaptive terminal and PID-type SMCs.

For the jerk chaotic system [38], many previous works have been reported to achieve the chaos control and state synchronization, such as the nonsingular terminal SMC [39], adaptive control [40], and adaptive backstepping control [41]. The main drawback of [39–41] is that the nonlinear dynamics of the error-state dynamical system have to be directly eliminated by the sliding mode or the adaptive controllers. Basically, this active elimination will cause the complexity of the control scheme, and it is unsuitable for practical usage. Furthermore, the input nonlinearity for the control input was not taken into account in these previous works. Motivated by the above discussions, the problem of state synchronization between two identical systems, which belong to a kind of *n*-dimensional () chaotic systems, is studied by taking into account the sector nonlinearity for a control input. In the control problem, the driven chaotic system is considered by attaching to a single input controller with sector nonlinearity for the control input, under the appearance of lumped system uncertainties and external disturbances.

For solving the aforementioned control problem, the main contribution of the present study is to develop an adaptive two-stage SMC scheme for achieving the state synchronization. In comparison with the past studies [39–41], the advantage of the proposed adaptive controller is that the control scheme contains only the time-varying state-feedback gains to cope with the nonlinear dynamics without direct elimination. The introduced controller type is similar to the state-feedback control type and more suitable for the applications in practical usage. Meanwhile, the state-feedback gains are updated in line with the appropriate adapted rules without knowing certain information of the nonlinear error system dynamics, bounds of the lumped system uncertainties and the external disturbances, and the sector nonlinearity for control beforehand.

The developed adaptive two-stage SMC scheme is derived based on the introduced sequence of two sliding functions. The first is named the stage 1 sliding function , which is defined by the error states of the dynamical system where the desired stabilization of error states is inherent in the conditions of and . The stage 2 sliding function is formed by the stage 1 sliding function . The novel integral type of the stage 2 sliding function is proposed to guarantee the finite-time stabilization of under the conditions of and which are held. The main purpose of the controller design is that the adaptive control scheme is derived such that and are satisfied. And then, and are guaranteed by means of the definition of . When the conditions of and are met, the inherently asymptotical stabilization of the error states of the dynamical system is accomplished. It means that the trajectory in the phase space for the error states of the dynamical system is converging to the origin. Thus, it is induced that the state synchronization between two identical chaotic systems is completed.

The rest of this paper is organized as follows. The control problem of the state synchronization between two *n*-dimensional identical chaotic systems is described and formulated in Section 2. In Section 3, the adaptive two-stage SMC scheme is derived, and the sufficient conditions are also given in the view-point of the Lyapunov stable theorem. In Section 4, the capacity of the present scheme is carried out by numerical simulations, and the discussion is provided. Finally, some conclusions are made in Section 5.

#### 2. Description of the Control Problem

In this study, the following class of *n*-dimensional () nonlinear chaotic system is considered and described bywhere is the state vector and is a continuous nonlinear scale function. There are many reported chaotic systems belonging to the class shown in equation (1), such as the Genesio–Tesi system [42], the jerk chaotic systems [40,41], and the hyperchaotic hyperjerk systems [38].

Considering the nonlinear chaotic system defined in equation (1), the control problem of state synchronization between two identical systems is discussed in this study. The first chaotic system, named the drive system, is carried out without the control. In the second system, called the driven system, a single control input is applied in the appearance of lumped system uncertainties and disturbances. For conveniently discussing the control problem, the drive system is given by equation (1) to guide the following driven system:where is the state vector and is the bounded lumped system uncertainty satisfying . , satisfying , is the external disturbance, and is the control input. It is assumed that the single input control is additionally attached to sector nonlinearity for the control input and has the form of in system equation (2). is a directed control input without affected by the sector nonlinearity of the control. is the sector nonlinearity for the control input, and it is a continuous function with and satisfies the following inequality:where and are positive constants. An example of the sector nonlinear function is given by . It insides the sector defined in equation (3) with and is shown in Figure 1. The goal of the control problem is to design the single input controller in system equation (2) for achieving the state synchronization between system equations (1) and (2) without foreknown the certain information of nonlinear system dynamics, bounds of lumped system uncertainties and external disturbances, and sector nonlinearity of the control.

The error states between system equations (1) and (2) are defined in the following:

Taking the time derivatives of equation (4) after subtracting equation (1) from (2), the error-state dynamical system is expressed as

Owing to that the chaotic system always exhibits the globally bounded state trajectories, it is fairly assumed that

It is clear to show that the control problem of state synchronization between system equations (1) and (2) mathematically becomes the equivalent problem for stabilizing the error states of dynamical system equation (5) by applying an appropriated control scheme . Thus, the goal of the current problem is to derive the suitable control scheme such that, for any initial states of system equation (5), all error states of the dynamical system converge to zeros, that is, .

#### 3. Design of the Control Scheme

In the following, two steps are introduced to design the adaptive two-stage SMC scheme to accomplish the state synchronization between system equations (1) and (2). At first, the sequence of two sliding functions is defined. The first is named the stage 1 sliding function , which is defined by the error states , where the desired stabilization of error states is embedded under the conditions of and . The stage 2 sliding function is formed by the stage 1 sliding function . The novel integral type of the stage 2 sliding function is defined to guarantee the finite-time stabilization of under the conditions of and which are satisfied. Secondly, the adaptive control scheme with the form of in system equation (5) is designed such that and are held. Then, is converging to and maintained at zero even without foreknown the certain information of nonlinear system dynamics, lumped system uncertainty , external disturbance , and sector input nonlinearity. When the status of and is maintained, it means that the trajectory in the phase space for error-state dynamical system equation (5) is stabilized under the embedded converging motion. Then, the state synchronization between two system equations (1) and (2) is achieved.

The stage 1 sliding function is defined aswhere is the positive design parameter and , are positive parameters to be chosen. , are chosen according to the polynomial which is stable in the sense of Hurwitz. When the conditions and are satisfied, can be expressed as

Equation (8) is further mathematically represented bywhere is the error-state vector and matrix is the form of the companion matrix with the coefficients Owing to that the polynomial is Hurwitz-stable, the asymptotical stability of equation (8) is guaranteed.

The novel integral type of the stage 2 sliding function is defined by :where , , , and ; are positive and odd integers. For and , the finite-time stability of is obtained as follows. Taking the time derivatives of equation (10), it is obtained that

Let ; it yields

From equation (12), the finite time, , exists such that the stage 1 sliding function is moving from to .The finite time is given by

At this point, it is concluded that the objective of control scheme design is to force that and are satisfied and is approaching to and retained at zero in finite time. Then, the conditions and are held such that error states , tend to zeros asymptotically according to equations (8) and (9).

Theorem 1. *If the following control scheme in system equation (5) is applied,where and are defined in equations (7) and (10), respectively. The positive and adaptive feedback gains , are updated according to the following adaptation algorithms, respectively:where and , are the positive constants dominating the adaptation process. The stage 2 sliding function will asymptotically be stabilized, and the conditions and are held. It follows that the stage 1 sliding function is approaching to zero in the finite time , which is evaluated in equation (13). When the status of is maintained, it yields that the trajectory in the phase space for the error states of the dynamical system shown in equation (5) is stabilized in the inherent converging motion defined in equations (8) and (9). Then, the state synchronization between systems shown in equations (1) and (2) can be accomplished.*

*Proof. *The mathematical Lyapunov function is selected to bewhere , are sufficient large positive constants. They also satisfy the following inequalities:Taking the time derivative of equation (17) along with the solutions of the error states of dynamical system equation (5) and the selection of the two sliding functions in equations (7) and (10), it yieldsMoreover, from equation (3), one can conduct the following:where . Since , it follows thatBy substituting equations 14–16 into (19) and with the criteria in (18 and 21), it is obtained thatTherefore, the condition of is satisfied. The stage 2 sliding function can be reached to and asymptotically. Then, the stage 1 sliding function is approaching to zero in the finite time according to equation (13). The conditions and are satisfied such that error states , tend to zeros asymptotically by the definitions in equations (8) and (9). It is induced that the state synchronization is accomplished.

*Remark 1. *In the literature, the related SMC schemes for solving the chaos control and state synchronization of the jerk chaotic systems were addressed in [17,39,43]. The nonlinear dynamic terms of the controlled error-state systems have to be involved in the reported SMC schemes in order to directly eliminate the nonlinear system dynamics. The main drawback of the works is that the active elimination will cause the complexity of the control scheme, and the chattering phenomenon of the control signal is obviously serious [39].

In comparison with the past studies [17,39], the main advantage of the proposed controller provided in equation (14) is that the control scheme contains only the time-varying state-feedback gains to cope with the nonlinear dynamics without direct elimination. Furthermore, the state-feedback gains are updated according to the appropriate adapted rules shown in equations (15) and (16), without foreknown the certain information of the nonlinear error system dynamics, etc. The other advantage is that the presented controller is similar to the type of the state-feedback control. Thus, it is more suitable to be applied in the practical applications.

#### 4. Numerical Studies and Discussion

In the following section, the proposed adaptive two-stage SMC scheme is carried out by the numerical simulations for two identical 3D jerk chaotic systems reported in [40]. The fourth-order Runge–Kutta method is applied to implement the whole system with a time step size of _{.} The 3D jerk chaotic system reported in [40] is described as follows:

The system in equation (23) with the system parameters can perform the chaotic phenomena. The chaotic system with the initial conditions is shown in Figure 2. It exhibits that the free control 3D jerk chaotic system has bounded trajectories.

For the drive and driven 3D jerk chaotic systems (), the error-state dynamical system can be expressed aswhere the nonlinear functions of and are upper-bounded because the chaotic system always depicts the globally bounded state trajectories. In the following numerical simulation, the initial conditions of the drive system are set with the same values in Figure 2, and the driven systems are chosen: . It is presumed that the external disturbance, the lumped system uncertainty, and the sector nonlinearity for the control input are , and respectively.

The guideline for choosing the positive design parameters of the proposed control scheme involves three basic steps. Firstly, according to equation (7), the positive parameters, , are chosen such that the polynomial is stable in Hurwitz. Then, the positive parameter is properly tuned to avoid the huge initial value of . Then, the positive parameter and the odd integers are selected for a prescribed finite time defined in equation (13) under the requirements of , , and . Finally, suitably tuning , numerically to prevent the huge magnitude of the control input and chattering effect.

According to the aforementioned guideline, for the adaptive two-stage SMC scheme in equation (14) associated with equations (7), (10), (15), and (16), the positive design parameters are chosen as and . The control scheme for the illustrated example is computed in the following:where , .

Time responses for error states of system equation (24) are demonstrated in Figure 3. It is shown that the state synchronization is achieved.

Figure 4 depicts the time responses of two-stage sliding functions, and , and the state trajectory in the () plane, respectively. It is shown that asymptotically converges to zero. After and , is rapidly stabilized within finite time. In the () plane, it is exhibited the state trajectory is firstly holded on the status of and , and then the status of , is satisfied.

Figure 5 shows the time response of the adaptive feedback gains , and the applied control input . The time-varying gains , become steady-state when the state synchronization is completed. In the appearance of lumped system uncertainties, external disturbances, and sector input nonlinearity, it is shown that the control input is continuous and chattering-free. Figure 6 demonstrates the time responses for states of systems (1) and (2), respectively. It is obviously exhibited that state synchronization is accomplished.

For the control scheme given by equations (25)–(30), it is clearly found that the certain information of nonlinear system dynamics, bounds of lumped system uncertainties and external disturbances, and sector input nonlinearity is not included. By changing the related conditions of the system uncertainties and external disturbance, the robustness of the proposed control scheme is easy to be verified in the numerical simulations, and the detailed results are omitted here.

To demonstrate the effectiveness of the proposed method, the existing control method introduced in [39] is added. For error dynamical system (24) without attached to an input nonlinearity, the control scheme according to [39] is listed in the following:where is the saturation function defined by

Time responses for error states , and the applied control input of system equation (24) utilized by the control scheme in equations (31)–(34) are shown in Figure 7. It is shown that the error states are stable. However, the chattering effect of the control signal developed in [39] is obviously serious in comparison with the proposed control input depicted in Figure 5.

#### 5. Conclusions

In this study, the adaptive two-stage SMC scheme for achieving state synchronization between two systems, which are pertaining to a class of the *n*-dimensional () chaotic system defined in (1), has been addressed by taking into account lumped system uncertainties, external disturbances, and the sector nonlinearity for the control input. In the procress of computing the proposed adaptive control terms, it is not required to know beforehand the certain information of nonlinear system dynamics, bounds of lumped system uncertainties and external disturbances, and sector input nonlinearity.

The proposed adaptive control scheme including time-variable feedback gains updated by the suitably adaptive rules can cope with the effect of sector input nonlinearity for achieving the goal of control. The mathematically sufficient conditions are given to guarantee the stability of synchronization by means of the Lyapunov stable theorem. Besides, numerical studies for two 3D jerk chaotic systems reported in [40] are carried out to verify the effectiveness of the proffered schemes.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Disclosure

This study was carried out as part of the Intelligent Manufacturing Program coordinated by the beautiful China Research Institute of Sanming University.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was financially supported by the Operational Funding of the Advanced Talents for Scientific Research (19YG04) of Sanming University and the Science and Technology Department of Fujian Province (Grant no. 2017H6019). The authors also acknowledge the support from the College of Mechanical and Electrical Engineering in Sanming University.