Robust Synchronization of Class Chaotic Systems Using Novel Time-Varying Gain Disturbance Observer-Based Sliding Mode Control

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. +en, 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 problemmay be caused by the high gain if the initial values of estimate and true states are not equal. +e 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.


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. ese 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. e 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]. ese 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 H ∞ robust control [10,11], adaptive control [12,13], neural network control [14,15], observerbased control [16,17], and sliding mode control (SMC) [18][19][20][21][22]. Among these schemes in [10][11][12][13][14][15][16][17][18][19][20][21][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][19][20][21][22] is guaranteed by using the discontinuous control terms. e 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][24][25]. us, 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][29][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][29][30][31] does not require the upper bound of uncertainties.
For the DO-based and ESO-based SMC schemes in [28][29][30][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. e 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. us, the DO-based and ESO-based SMC schemes in [28][29][30][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 TVGDObased SMC scheme is designed based on the output of TVGDO. e 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) e 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.
e 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.
e 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. e following notations will be used in this study: t denotes the time and the initial time is 0. Let ‖ · ‖ denote the Euclidean norm of a vector and its induced norm of a matrix.

System Description.
In this study, the dynamic of the master chaotic system is described as follows [12]: where x mi (i � 1, 2, . . . , n) represents the states of the master system, X m � x m1 x m2 . . . x mn T is the state vector, and f mi (X m , t), (i � 1, 2, . . . , n) is the nonlinear function and determines the chaotic characteristic. e slave chaotic system is given as follows [12]: where x si (i � 1, 2, . . . , n) represents the states of the slave system, X s � x s1 x s2 . . . x sn T is the state vector, f si (X s , t)(i � 1, 2, . . . , n) is the nonlinear function, u i (t)(i � 1, 2, . . . , n) is the control input, and Δf si (X s , t) and d i (t)(i � 1, 2, . . . , n) 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. e 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 i � 1, 2, . . . , n, the error dynamics can be obtained as where where D i denotes the lumped uncertainties. e following Assumption is assumed to be valid throughout this study.

Problem Description and Purposes of is Study.
To satisfy condition (4), like [32], a simple sliding mode surface vector can be chosen as where s � s 1 s 2 . . . s n T , where c i (i � 1, 2, . . . , n) is a positive constant. en, calculating the time derivative of s i (i � 1, 2, . . . , n) along the trajectories of (5) and (7), we have To guarantee the sliding mode surface s i converge to zero, it is necessary to design a robust scheme to suppress the lumped uncertainties D i . en, the conventional sliding mode controller (SMC) can be designed as where sign(·) denotes the signum function, and k SMCi is a positive constant. Substituting (10) into (9), we have _ s i � − (k SMCi − D i )sign(s i ). en, we can know that s i _ s i < 0 if k SMCi > D i and s i ≠ 0. us, the uncertainty D i can be suppressed by the discontinuous switch item k SMCi sign(s i ). However, the constant k SMCi must be selected as the upper bound of D i , which is difficult to be obtained in advance. And the discontinuous term k SMCi sign(s i ) brings undesirable chattering problem.
Recently, to avoid using the upper bound of D i and solve the chattering problem, some observer-based SMC schemes have been developed in [28][29][30][31].
In [26][27][28], the disturbance observer (DO) is designed to estimate nonvanishing disturbances and model uncertainties in the chaotic system. For the uncertainties D i (i � 1, 2, . . . , n), the DO can be designed by using the method in [28][29][30]: where Z DOi is the estimate state of DO, k DOi is the positive observer gain, and D DOi is the estimation of D i . en, the DO-based SMC can be designed as where ε DOi , σ DOi , and c DOi are the positive constants, 0 < c DOi < 1. e DO (11) can guarantee the estimate error converge to the following region: where the constant D d max is defined in Assumption 1. From (13), to achieve the high observation precision, the observer gain k DOi should be large enough. However, from (11), it is clear that the initial value D DOi (0) may be very large if k DOi is a high gain and the initial estimate state error Z DOi (0) − s i (0) ≠ 0. Actually, the initial values of true state s i (0) cannot be known in advance for most of the cases. us, the initial estimate state error Z DOi (0) − s i (0) maybe not equal to 0 and even a large value. e large value D DOi (0) may lead to a large overshoot control input u i (t) (see (12)). en, the large overshoot u i (t) 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. e uncertainties D i (i � 1, 2, . . . , n) also can be estimated by using the method in [31]: where Z ESOi is the estimated state of ESO, D ESOi is an estimation of D i , and k ESOi is the observer gain of ESO. en, the ESO-based SMC can be designed as where ε ESOi , σ ESOi , and c ESOi are the positive constants, 0 < c ESOi < 1. e ESO can guarantee the estimate error converge to following region: Define the estimate error vector where e solution to the differential equation (17) can be easily obtained as where e is the e constant. Let * 20ch 1 h 2 Expanding e A h t h(0), the estimate error can be rewritten as For the small time t � 1/(k ESOi c i ), we have It can be known that us, the undesirable spike problem also exists in ESO.

Motivation of is 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. us, if the initial value of true states is unknown, to avoid the spike problem, the ESObased 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. is 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.

Observer Design and Stability Analysis.
In this section, a novel time-varying gain disturbance observer (TVGDO) will be proposed. e 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. e expression of TVGDO and the stability analysis are given in the following eorem. Theorem 1. Taking the master and slave chaotic systems (1) and (2) into consideration, for the uncertainties where η i1 and η i2 are the positive constants, and k i (t) is a nonnegative time-varying gain. Assumption 1 is valid. e estimate error of TVGDO is defined as D i � D i − D i . en, the estimate error D i will converge to the following region: Proof. e estimate error of TVGDO is defined as Considering (9) and (22), (24) can be rewritten as Construct the Lyapunov function J i as en, calculating the time derivative of J i along the trajectory of (25), we get Considering Assumption 1, we have Since k i (t) ≥ 0, from (28), it can be known that en, we have 4 Complexity Integrating (30) gives From the expression of k i (t), it can be known that the following condition can be satisfied in an arbitrary finite time t if : Combining (28) and (32), we have From (31), for an arbitrary finite time t if , it is clear that , it also can be known from (33) that _ J i < 0. en, we have Consider lim t⟶∞ k i (t) � η i1 ; then, we have From (35) and e proof is finished.

e Spike-Free Characteristic Analysis. Let
Considering (22) and solving the differential equation (25), the solution of estimate error D i in time domain can be easily obtained as where e is the e constant. Substituting the detailed expression of D i (0) 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 k DOi (Z DOi (0) − s i (0)) in DO (11) or the spike term (k ESOi c i )(Z ESOi (0) − s i (0)) in ESO (14). Since k i (0)(Z i (0) − s i (0)) � 0 and 0 0 _ k i (τ)(Z i − s i )dτ � 0, it is clear that the expression of D i in (38) does not contain any spike term. us, 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 η i1 reasonably.
us, the proposed TVGDO not only can eliminate the undesirable spike problem but also can guarantee high observation precision.

Observer-Based Controller Design and Stability Analysis.
en, a novel TVGDO-based sliding mode controller will be developed in this section. e expression of the proposed controller and the stability analysis are given in the following eorem.

Theorem 2.
Taking the master and slave chaotic systems (1) and (2) into consideration, for i � 1, 2, . . . , n, the TVGDObased sliding mode controller is constructed as where the sliding mode surface s i is defined in (7). ε i , σ i , and c i are the positive constants. 0 < c i < 1. D i is given in TVGDO (22). Assumption 1 is valid. e synchronization error e i can converge to following small region: Proof. Construct the Lyapunov function P i (i � 1, 2, . . . , n) as en, calculating the time derivative of P i along the trajectory of (9), we get Substituting the control input (39) into (42), we have where the estimate error D i � D i − D i .

Complexity 5
From (43), we know that e i affected the estimate error D i . us, the following proof will consist of two steps. In the first step, it will be proved that e i will not escape to infinity in arbitrary finite time (before D i converges to a neighborhood of zero). In the second step, it will be proved that e i will converge to a neighborhood of zero after D i converges to a small neighborhood of zero.
Step 1. From (31) in eorem 1, we have known that the estimate error D i is bounded as For the arbitrary finite time t ≤ t s , it can be known that . en, combining (43) with (45), we have From (46) and (47), we can know that _ us, for t ≤ t s , P i will not escape to infinity and is bounded as en, we can know that e i will not escape to infinity and is bounded as Step 2. From (43), we have en, we can know that According to (50) and (51), we have From eorem 1, it can be known that the estimate error D i is bounded as Combining (52)-(54), we have From (55), it can be known that s i will converge to following region: en, the synchronization error will converge to following small region: (57) e proof is finished.
Remark 2. It can be known from (57) that, if large enough observer parameter η i1 and the control parameters σ i , ε i , and c i 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.

Simulation Results
In this section, to illustrate the effectiveness of the proposed methods, the mathematical simulation is presented. e master and slave chaotic systems are selected as the threedimensional chaotic systems given in [32]. us, for systems (1) and (2), we select n � 3. e chaotic nonlinear function and uncertainties are chosen as 6 Complexity e initial system states are set as x m1 (0) � 4, x m2 (0) � 3.5, x m3 (0) � 2.5, and x s1 (0) � x s2 (0) � x s3 (0) � 0. e simulation method is chosen as the fixed step Dormand-Prince method. e step size of simulation is set as 0.001s. Let u i (t) � 0(i � 1, 2, 3), and the chaotic behavior of the master system (1) is shown in Figure 1.
us, the observer gain of TVGDO k i (t)(i � 1, 2, 3) is close to 50. And, in this case, we consider that the initial value of sliding mode surface s i (0)(i � 1, 2, 3) is known. en, the initial values of estimate states in DO, ESO, and TVGDO can be chosen as us, 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.
us, 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. us, 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.
us, for DO, ESO, and TVGDO, the initial values of estimate and true states are not equal.     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. en, the spike output values of observer are transmitted into the control inputs to lead the large overshoot of synchronization errors. e 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. us, the spike problem is avoided by the proposed scheme of this study.
According to the simulation results, the following can be concluded: (2) Since the uncertainties have been estimated by proposed TVGDO, the TVGDO-SMC has no      us, 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.

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. e 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 ESObased SMC is solved by the proposed controller. Second, the chattering problem in the conventional SMC also is avoided in the proposed method. ird, unlike the conventional SMC, the proposed method requires no information on the uncertainties. Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.