#### Abstract

For synchronization of a class of chaotic systems in the presence of nonvanishing uncertainties, a novel time-varying gain observer-based sliding mode control is proposed. First, a novel time-varying gain disturbance observer (TVGDO) is developed to estimate the uncertainties. Then, by using the output of TVGDO to modify sliding mode control (SMC), a new TVGDO-based SMC scheme is developed. Although the observation and control precision of conventional fixed gain disturbance observer-based control (FGDOC) for chaotic systems can be guaranteed by a high observer gain, the undesirable spike problem may be caused by the high gain if the initial values of estimate and true states are not equal. The most attractive feature of this work is that the newly proposed TVGDO can eliminate the spike problem by developing a time-varying gain scheme. Finally, the effectiveness of the proposed method is demonstrated by the numerical simulation.

#### 1. Introduction

In the past decades, with the development of theoretical analysis methods of chaos, many chaos systems such as the Lorenz system [1], Rossler system [2], and Chen system [3] have been wildly studied. These theoretical advancements of chaotic systems have been influentially applicated in many fields, such as power electrical systems [4, 5], robotics [6], lasers [7], and secure communications [8]. Among these applications, to achieve the desired chaotic characteristic, the high-precision synchronization problem is the key problem that must be solved. The objective of synchronization control between the master and slave chaotic systems can be achieved when the instantaneous states of the two systems become identical. Note that unmodeling dynamic, environmental disturbance, and uncertainty parameters usually exist in the slave chaotic systems [9]. These uncertainties can greatly affect the synchronization performance. To improve the robustness of chaos synchronization in allusion to uncertainties, many modern robust control theories have been applied to design synchronization controllers, including robust control [10, 11], adaptive control [12, 13], neural network control [14, 15], observer-based control [16, 17], and sliding mode control (SMC) [18–22]. Among these schemes in [10–22], due to its advantage of low sensitivity to uncertainties and fast dynamic response, SMC is a good candidate to achieve high-precision synchronization in the presence of uncertainties. In [18, 19], the authors adopted the linear sliding mode surface to design a synchronization controller for chaotic systems. In [20, 21], the terminal sliding mode method was investigated to guarantee the fast finite-time synchronization of uncertain chaotic systems. In [22], to establish invariance of the system with uncertainties from the initial time instant, the integral sliding mode control scheme was investigated to design the synchronization controller. However, the robustness of these conventional SMC schemes in [18–22] is guaranteed by using the discontinuous control terms. The discontinuous terms can bring an undesirable chattering problem.

It is well known that the chattering problem may affect the synchronization precision and cause the instability of the closed-loop system [23–25]. Thus, research on chattering-free SMC synchronization scheme has the important practical and theoretical significance. Based on observer techniques, the chattering problem of conventional SMC can be eliminated by using the estimation of uncertainties to replace the discontinuous control terms of SMC. In [26, 27], the authors adopted the high-order sliding mode (HOSM) observers to estimate the uncertainties of the chaotic system. However, the HOSM observer used in [26, 27] must know the upper bound of uncertainties in advance. Since the characteristics of uncertainties are complex, it is difficult to know the upper bound. In [28–30], the SMC schemes were proposed by employing the disturbance observer (DO) to estimate the uncertainties in chaotic systems. In [31], the authors adopted the estimation of extended state observer (ESO) to modify the conventional SMC. Unlike HOSM observer in [26, 27], the DO and ESO in [28–31] does not require the upper bound of uncertainties.

For the DO-based and ESO-based SMC schemes in [28–31], to guarantee the observation and control precision, the observer gains of DO and ESO should be chosen as the high gains. However, the undesirable spike problem can be caused by the high observer gains when the initial values of estimate and true states are not equal. The spike problem may cause the saturation of control input and even the instability. Actually, it is difficult to obtain the initial values of state in advance for the most of practical systems. Thus, the DO-based and ESO-based SMC schemes in [28–31] may not work well in many practical situations.

In this study, a new time-varying gain disturbance observer (TVGDO) is proposed to estimate the lumped uncertainties of the chaotic system. And then, a novel TVGDO-based SMC scheme is designed based on the output of TVGDO. The main contributions of this study lie in the following aspects:(1)Compared with conventional DO and ESO used in the uncertain chaotic systems, the most attractive feature of the proposed TVGDO is that the spike problem can be eliminated on the condition of the initial values of estimate and true states are not equal.(2)The proposed TVGDO-based SMC is spike-free and chattering-free. And the TVGDO-based SMC can guarantee the synchronization without using the upper bound information of uncertainties.

The remaining parts of this study are as follows. In Section 2, the synchronization control model, the design objective, and the motivation of this study are expounded. The main results are presented in Section 3. In Section 3.1, a novel observer TVGDO is developed, and the stability proof of TVGDO is presented. In Section 3.2, the spike-free characteristic of TVGDO is analyzed. In Section 3.3, a TVGDO-based SMC scheme is proposed, and the stability of the proposed controller is also obtained. In Section 4, a simulation verifies the effectiveness of both TVGDO and the proposed TVGDO-based SMC. In Section 5, the conclusion of the whole study is presented.

*Notations*. The following notations will be used in this study: denotes the time and the initial time is 0. Let denote the Euclidean norm of a vector and its induced norm of a matrix.

#### 2. Problem Formulation

##### 2.1. System Description

In this study, the dynamic of the master chaotic system is described as follows [12]:where represents the states of the master system, is the state vector, and is the nonlinear function and determines the chaotic characteristic.

The slave chaotic system is given as follows [12]:where represents the states of the slave system, is the state vector, is the nonlinear function, is the control input, and and are the bounded uncertainty and disturbance, respectively. Like the conventional SMC, the uncertainties considered in this study are matched uncertainties, which imply that the uncertainties and control inputs exist in the same channel. It is assumed that all states of systems (1) and (2) are measured and noise-free.

The synchronization errors are defined as follows:

Note that if the following condition is satisfied, then the objective of synchronization is realized:

Considering (1) and (2), for , the error dynamics can be obtained aswherewhere denotes the lumped uncertainties.

The following Assumption is assumed to be valid throughout this study.

*Assumption 1. *The lumped uncertainties is differentiable and satisfies and , where and are the unknown positive constants.

##### 2.2. Problem Description and Purposes of This Study

To satisfy condition (4), like [32], a simple sliding mode surface vector can be chosen aswherewhere is a positive constant.

Then, calculating the time derivative of along the trajectories of (5) and (7), we have

To guarantee the sliding mode surface converge to zero, it is necessary to design a robust scheme to suppress the lumped uncertainties .

Then, the conventional sliding mode controller (SMC) can be designed aswhere denotes the signum function, and is a positive constant.

Substituting (10) into (9), we have . Then, we can know that if and . Thus, the uncertainty can be suppressed by the discontinuous switch item . However, the constant must be selected as the upper bound of , which is difficult to be obtained in advance. And the discontinuous term brings undesirable chattering problem.

Recently, to avoid using the upper bound of and solve the chattering problem, some observer-based SMC schemes have been developed in [28–31].

In [26–28], the disturbance observer (DO) is designed to estimate nonvanishing disturbances and model uncertainties in the chaotic system. For the uncertainties , the DO can be designed by using the method in [28–30]:where is the estimate state of DO, is the positive observer gain, and is the estimation of . Then, the DO-based SMC can be designed aswhere , , and are the positive constants, . The DO (11) can guarantee the estimate error converge to the following region:where the constant is defined in Assumption 1. From (13), to achieve the high observation precision, the observer gain should be large enough. However, from (11), it is clear that the initial value may be very large if is a high gain and the initial estimate state error . Actually, the initial values of true state cannot be known in advance for most of the cases. Thus, the initial estimate state error maybe not equal to 0 and even a large value. The large value may lead to a large overshoot control input (see (12)). Then, the large overshoot may reduce the dynamic performance of synchronization and even lead to the instability, and this is the undesirable spike problem of DO.

In [31], the extended state observer (ESO) is developed to estimate the uncertainties and chaotic nonlinear function. The uncertainties also can be estimated by using the method in [31]:where is the estimated state of ESO, is an estimation of , and is the observer gain of ESO. Then, the ESO-based SMC can be designed aswhere , , and are the positive constants, . The ESO can guarantee the estimate error converge to following region:

Define the estimate error vector . From ESO (14), we havewhere

The solution to the differential equation (17) can be easily obtained aswhere *e* is the *e* constant. Let . Expanding , the estimate error can be rewritten as

For the small time , we have

It can be known that is a very large value if is a high gain and . Thus, the undesirable spike problem also exists in ESO.

###### 2.2.1. Motivation of This Study

From the previous discussion, an undesirable spike problem can be caused by the high observer gain in ESO and DO if the initial values of estimate and true states are not equal. Thus, if the initial value of true states is unknown, to avoid the spike problem, the ESO-based and DO-based controller cannot adopt the high observer gains to guarantee the control precision. Actually, the initial value of true states cannot be known in advance in most of cases. This motivates the research topic of this study, that is, for the chaotic system in the presence of uncertainties, designing a new TVGDO and TVGDO-based SMC schemes not only can guarantee high control precision but also eliminate the undesirable spike problem.

#### 3. Main Result

##### 3.1. Observer Design and Stability Analysis

In this section, a novel time-varying gain disturbance observer (TVGDO) will be proposed. The TVGDO can guarantee the high precision and avoid the undesirable spike problem even if the initial values of estimate and true states are not equal.

The expression of TVGDO and the stability analysis are given in the following Theorem.

Theorem 1. *Taking the master and slave chaotic systems (1) and (2) into consideration, for the uncertainties , the TVGDO (22) is constructed.where and are the positive constants, and is a nonnegative time-varying gain. Assumption 1 is valid. The estimate error of TVGDO is defined as . Then, the estimate error will converge to the following region:*

*Proof. *The estimate error of TVGDO is defined as . Differentiating givesConsidering (9) and (22), (24) can be rewritten asConstruct the Lyapunov function asThen, calculating the time derivative of along the trajectory of (25), we getConsidering Assumption 1, we haveSince , from (28), it can be known thatThen, we haveIntegrating (30) givesFrom the expression of , it can be known that the following condition can be satisfied in an arbitrary finite time :Combining (28) and (32), we haveFrom (31), for an arbitrary finite time , it is clear that is bounded. If , it also can be known from (33) that . Then, we haveConsider ; then, we haveFrom (35) and , we haveThe proof is finished.

##### 3.2. The Spike-Free Characteristic Analysis

Let . Considering (22) and solving the differential equation (25), the solution of estimate error in time domain can be easily obtained aswhere is the *e* constant. Substituting the detailed expression of into (37), we have

From the previous discussion in Section 2.2, it can be known that the undesirable spike problem is caused by the spike term in DO (11) or the spike term in ESO (14). Since and , it is clear that the expression of in (38) does not contain any spike term. Thus, the spike problem is avoided in the TVGDO.

*Remark 1. *It can be known from (26) that a small enough estimate error can be guaranteed by choosing reasonably. Thus, the proposed TVGDO not only can eliminate the undesirable spike problem but also can guarantee high observation precision.

##### 3.3. Observer-Based Controller Design and Stability Analysis

Then, a novel TVGDO-based sliding mode controller will be developed in this section. The expression of the proposed controller and the stability analysis are given in the following Theorem.

Theorem 2. *Taking the master and slave chaotic systems (1) and (2) into consideration, for , the TVGDO-based sliding mode controller is constructed aswhere the sliding mode surface is defined in (7). , , and are the positive constants. . is given in TVGDO (22). Assumption 1 is valid. The synchronization error can converge to following small region:*

*Proof. *Construct the Lyapunov function asThen, calculating the time derivative of along the trajectory of (9), we getSubstituting the control input (39) into (42), we havewhere the estimate error .

From (43), we know that affected the estimate error . Thus, the following proof will consist of two steps. In the first step, it will be proved that will not escape to infinity in arbitrary finite time (before converges to a neighborhood of zero). In the second step, it will be proved that will converge to a neighborhood of zero after converges to a small neighborhood of zero.

*Step 1. *From (31) in Theorem 1, we have known that the estimate error is bounded asFor the arbitrary finite time , it can be known thatLet the positive constant . Then, combining (43) with (45), we haveFrom (46) and (47), we can know that if or . Thus, for , will not escape to infinity and is bounded asLet the constant . Then, we can know that will not escape to infinity and is bounded as if .

*Step 2. *From (43), we haveThen, we can know thatAccording to (50) and (51), we haveFrom Theorem 1, it can be known that the estimate error is bounded asCombining (52)–(54), we haveFrom (55), it can be known that will converge to following region:Then, the synchronization error will converge to following small region:The proof is finished.

*Remark 2. *It can be known from (57) that, if large enough observer parameter and the control parameters , , and are chosen, then the convergence region of synchronization error will be small enough. It means that the synchronization error can be made arbitrarily small through adjusting parameters properly.

#### 4. Simulation Results

In this section, to illustrate the effectiveness of the proposed methods, the mathematical simulation is presented. The master and slave chaotic systems are selected as the three-dimensional chaotic systems given in [32]. Thus, for systems (1) and (2), we select . The chaotic nonlinear function and uncertainties are chosen as

The initial system states are set as , , , and . The simulation method is chosen as the fixed step Dormand–Prince method. The step size of simulation is set as 0.001s. Let , and the chaotic behavior of the master system (1) is shown in Figure 1.

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For the comparison, the proposed TVGDO-based SMC (46), DO-based SMC (12), and ESO-based SMC (15) are considered in this section.

*Case 1. *(Initial values of estimate and true states are equal). In this case, the SMC parameters of the three methods are selected as , , , , , , , and . Then, it can be known that the initial values of sliding mode surface are , , and . The observer gains of DO and ESO are chosen as and , respectively. The time-varying parameters are chosen as and . Thus, the observer gain of TVGDO is close to 50. And, in this case, we consider that the initial value of sliding mode surface is known. Then, the initial values of estimate states in DO, ESO, and TVGDO can be chosen asThus, for DO, ESO, and TVGDO, the initial values of estimate and true states are equal. Figures 2–5 show the simulation results for Case1. From Figure 2, it is clear that the DO-based SMC, ESO-based SMC, and proposed TVGDO-based SMC all can guarantee the synchronization errors that converge to a small neighborhood of zero. Figure 3 shows that DO, ESO, and TVGDO can ensure the estimation errors converge to a small neighborhood of zero. From Figure 4, it is clear that the control inputs of the three observer-based sling mode controllers are chattering-free. Thus, the undesirable chattering problem in conventional SMC can be solved by employing these observer-based schemes. Figure 5 shows the time-varying observer gains of proposed TVGDO. Thus, the results in Case 1 proved that the control parameters used in Case 1 for the three methods can achieve a similar good performance when the initial values of estimate and true states are equal.

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*Case 2. *(The initial values of estimate and true states are not equal). From the simulation result of Case 1, we have known that the three methods can achieve a similar control performance under the control parameters chosen in Case 1. To ensure the fairness of comparison, the control parameters are selected, the same parameters as in Case 1. And, in this case, we consider that the initial value is unknown. The initial values of estimate states of DO, ESO, and TVGDO are chosen asThus, for DO, ESO, and TVGDO, the initial values of estimate and true states are not equal.

Figures 6–8 show the simulation results for Case 2. From Figure 6, for DO-based SMC and ESO-based SMC, the undesirable large overshoot of synchronization errors can be observed. And the proposed TVGDO-based SMC can achieve the faster convergence rate than the DO-based and ESO-based schemes. As mentioned before in Section 2.2, the reason is that the spike problem of DO and ESO can be caused by choosing a large observer gain if the initial values of estimate and true states are not equal. Then, the spike output values of observer are transmitted into the control inputs to lead the large overshoot of synchronization errors. The undesirable spike phenomenon of DO and ESO can be observed from Figure 7. Figure 8 shows the undesirable large spike control inputs of DO-based SMC and ESO-based SMC. From Figures 6–8, it is clear that the undesirable spike phenomenon is eliminated in the proposed TVGDO and TVGDO-based SMC. An excellent control performance which is similar to Case 1 still can be guaranteed by the proposed controller and observer. Thus, the spike problem is avoided by the proposed scheme of this study.

According to the simulation results, the following can be concluded:(1)For conventional ESO and DO, an undesirable spike problem can be caused if the initial values of estimate and true states are not equal. The proposed TVGDO can eliminate the undesirable spike problem (Figures 2–4 and 6–8).(2)Since the uncertainties have been estimated by proposed TVGDO, the TVGDO-SMC has no discontinuous control term. Thus, the chattering problem in conventional SMC is solved (Figures 4 and 8). And, unlike the conventional SMC, the proposed controller does not need the upper bound of uncertainties.

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#### 5. Conclusion

(1)In this study, a novel TVGDO was proposed to estimate the lumped uncertainties and disturbances in the slave chaotic system, which can solve the spike problem in the conventional DO and ESO on the condition of the initial values of estimate and true states are not equal. Moreover, the proposed TVGDO does not need to know the upper bound of uncertainties in advance.(2)Subsequently, a novel TVGDO-based SMC was proposed to synchronize the chaotic systems. The newly proposed SMC scheme has several advantages over existing SMC. First, the spike problem in the observer-based SMC such as the DO-based and ESO-based SMC is solved by the proposed controller. Second, the chattering problem in the conventional SMC also is avoided in the proposed method. Third, unlike the conventional SMC, the proposed method requires no information on the uncertainties.(3)Finally, mathematical simulation result illustrated the effectiveness of the TVGDO and the proposed TVGDO-based SMC.#### Data Availability

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

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

This work was supported by the Natural Science Basic Research Program of Shaanxi (2021JQ-880, 2021JM-533, and 2020JM-646), the Innovation Capability Support Program of Shaanxi (2018GHJD-21), the Science and Technology Program of Xi’an (2019218414GXRC020CG021-GXYD20.3), and the Fund of Excellent Doctoral Innovation of Xi’an University of Technology and the Scientific Research Foundation of Xijing University (XJ200202).