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

Authors: Yang Wang; Zhen Wang; Lingyun Kong
Journal: Complexity
 (2021)

---

## 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.


---

## Body

## 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 H∞ 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 Section2, 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: 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.


## 2. Problem Formulation

### 2.1. System Description

In this study, the dynamic of the master chaotic system is described as follows [12]:(1)x˙m1=fm1Xm,t,x˙m2=fm2Xm,t,⋮x˙mn=fmnXm,t,where xmii=1,2,…,n represents the states of the master system, Xm=xm1xm2…xmnT is the state vector, and fmiXm,t,i=1,2,…,n is the nonlinear function and determines the chaotic characteristic.The slave chaotic system is given as follows [12]:(2)x˙s1=fs1Xs,t+Δfs1Xs,t+u1t+d1t,x˙s2=fs2Xs,t+Δfs2Xs,t+u2t+d2t,⋮x˙sn=fsnXs,t+ΔfsnXs,t+unt+dnt,where xsii=1,2,…,n represents the states of the slave system, Xs=xs1xs2…xsnT is the state vector, fsiXs,ti=1,2,…,n is the nonlinear function, uiti=1,2,…,n is the control input, and ΔfsiXs,t and diti=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.The synchronization errors are defined as follows:(3)ei=xsi−xmi,i=1,2,…,n,e¯=xs1−xm1xs2−xm2…xsn−xmnT.Note that if the following condition is satisfied, then the objective of synchronization is realized:(4)e¯⟶0.Considering (1) and (2), for i=1,2,…,n, the error dynamics can be obtained as(5)e˙i=F¯iXm,Xs,t+uit+Dit,where(6)F¯iXm,Xs,t=fsiXs,t−fmiXm,t,Di=ΔfsiXs,t+dit,where Di denotes the lumped uncertainties.The following Assumption is assumed to be valid throughout this study.Assumption 1.
The lumped uncertaintiesDii=1,2,…,n is differentiable and satisfies Di≤Dimax and D˙it≤Dimaxd, where Dimax and Dimaxd 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 as(7)s=ce¯,where(8)s=s1s2…snT,c=diagc1c2…cn,where cii=1,2,…,n is a positive constant.Then, calculating the time derivative ofsii=1,2,…,n along the trajectories of (5) and (7), we have(9)s˙i=ciF¯iXm,Xs,t+Di+uit.To guarantee the sliding mode surfacesi converge to zero, it is necessary to design a robust scheme to suppress the lumped uncertainties Di.Then, the conventional sliding mode controller (SMC) can be designed as(10)uit=ci−1−F¯iXm,Xs,t−kSMCisignsi,where sign⋅ denotes the signum function, and kSMCi is a positive constant.Substituting (10) into (9), we have s˙i=−kSMCi−Disignsi. Then, we can know that sis˙i<0 if kSMCi>Di and si≠0. Thus, the uncertainty Di can be suppressed by the discontinuous switch item kSMCisignsi. However, the constant kSMCi must be selected as the upper bound of Di, which is difficult to be obtained in advance. And the discontinuous term kSMCisignsi brings undesirable chattering problem.Recently, to avoid using the upper bound ofDi 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 Dii=1,2,…,n, the DO can be designed by using the method in [28–30]:(11)Z˙DOi=ciF¯iXm,Xs,t+D^DOi+uit,D^DOi=−kDOiZDOi−si,where ZDOi is the estimate state of DO, kDOi is the positive observer gain, and D^DOi is the estimation of Di. Then, the DO-based SMC can be designed as(12)uit=−F¯iXm,Xs,t−εDOisi−σDOisiγDOisignsi−D^DOi,where εDOi, σDOi, and γDOi are the positive constants, 0<γDOi<1. The DO (11) can guarantee the estimate error converge to the following region:(13)D^DOi−Di⟶DimaxdkDOici,ift⟶∞,where the constant Ddmax is defined in Assumption 1. From (13), to achieve the high observation precision, the observer gain kDOi should be large enough. However, from (11), it is clear that the initial value D^DOi0 may be very large if kDOi is a high gain and the initial estimate state error ZDOi0−si0≠0. Actually, the initial values of true state si0 cannot be known in advance for most of the cases. Thus, the initial estimate state error ZDOi0−si0 maybe not equal to 0 and even a large value. The large value D^DOi0 may lead to a large overshoot control input uit (see (12)). Then, the large overshoot uit 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 Dii=1,2,…,n also can be estimated by using the method in [31]:(14)Z˙ESOi=ciF¯iXm,Xs,t+D^ESOi+uit−2kESOiZESOi−si,D^˙ESOi=−kESOi2ciZESOi−si,where ZESOi is the estimated state of ESO, D^ESOi is an estimation of Di, and kESOi is the observer gain of ESO. Then, the ESO-based SMC can be designed as(15)uit=−F¯iXm,Xs,t−εESOisi−σESOisiγESOisignsi−D^ESOi,where εESOi, σESOi, and γESOi are the positive constants, 0<γESOi<1. The ESO can guarantee the estimate error converge to following region:(16)D^ESOi−Di⟶DimaxdkESOici,ift⟶∞.Define the estimate error vectorh=ZESOi−siD^ESOi−DiT. From ESO (14), we have(17)h.=Ahh+Δ,where(18)Ah=ci−2kESOi1−kESOi20,Δ=0D˙i.The solution to the differential equation (17) can be easily obtained as(19)h=eAhth0+∫0teAht−τD˙iτdτ,where e is the e constant. Let ∗20ch¯1h¯2T=∫0teAht−τD˙iτdτ. Expanding eAhth0, the estimate error can be rewritten as(20)D^ESOi−Di=−kESOici2te−kESOicitZESOi0−si0+kESOicit+1e−kESOicitD^ESOi0−Di0+h¯2.For the small timet=1/kESOici, we have(21)D^ESOi=−kESOiciZESOi0−si0e−1+2e−1D^ESOi0−Di0+Di+h¯2,ift=1kESOici.It can be known thatkESOiciZESOi0−si0 is a very large value if kESOi is a high gain and ZESOi0−si0≠0. 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.


## 2.1. System Description

In this study, the dynamic of the master chaotic system is described as follows [12]:(1)x˙m1=fm1Xm,t,x˙m2=fm2Xm,t,⋮x˙mn=fmnXm,t,where xmii=1,2,…,n represents the states of the master system, Xm=xm1xm2…xmnT is the state vector, and fmiXm,t,i=1,2,…,n is the nonlinear function and determines the chaotic characteristic.The slave chaotic system is given as follows [12]:(2)x˙s1=fs1Xs,t+Δfs1Xs,t+u1t+d1t,x˙s2=fs2Xs,t+Δfs2Xs,t+u2t+d2t,⋮x˙sn=fsnXs,t+ΔfsnXs,t+unt+dnt,where xsii=1,2,…,n represents the states of the slave system, Xs=xs1xs2…xsnT is the state vector, fsiXs,ti=1,2,…,n is the nonlinear function, uiti=1,2,…,n is the control input, and ΔfsiXs,t and diti=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.The synchronization errors are defined as follows:(3)ei=xsi−xmi,i=1,2,…,n,e¯=xs1−xm1xs2−xm2…xsn−xmnT.Note that if the following condition is satisfied, then the objective of synchronization is realized:(4)e¯⟶0.Considering (1) and (2), for i=1,2,…,n, the error dynamics can be obtained as(5)e˙i=F¯iXm,Xs,t+uit+Dit,where(6)F¯iXm,Xs,t=fsiXs,t−fmiXm,t,Di=ΔfsiXs,t+dit,where Di denotes the lumped uncertainties.The following Assumption is assumed to be valid throughout this study.Assumption 1.
The lumped uncertaintiesDii=1,2,…,n is differentiable and satisfies Di≤Dimax and D˙it≤Dimaxd, where Dimax and Dimaxd 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 as(7)s=ce¯,where(8)s=s1s2…snT,c=diagc1c2…cn,where cii=1,2,…,n is a positive constant.Then, calculating the time derivative ofsii=1,2,…,n along the trajectories of (5) and (7), we have(9)s˙i=ciF¯iXm,Xs,t+Di+uit.To guarantee the sliding mode surfacesi converge to zero, it is necessary to design a robust scheme to suppress the lumped uncertainties Di.Then, the conventional sliding mode controller (SMC) can be designed as(10)uit=ci−1−F¯iXm,Xs,t−kSMCisignsi,where sign⋅ denotes the signum function, and kSMCi is a positive constant.Substituting (10) into (9), we have s˙i=−kSMCi−Disignsi. Then, we can know that sis˙i<0 if kSMCi>Di and si≠0. Thus, the uncertainty Di can be suppressed by the discontinuous switch item kSMCisignsi. However, the constant kSMCi must be selected as the upper bound of Di, which is difficult to be obtained in advance. And the discontinuous term kSMCisignsi brings undesirable chattering problem.Recently, to avoid using the upper bound ofDi 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 Dii=1,2,…,n, the DO can be designed by using the method in [28–30]:(11)Z˙DOi=ciF¯iXm,Xs,t+D^DOi+uit,D^DOi=−kDOiZDOi−si,where ZDOi is the estimate state of DO, kDOi is the positive observer gain, and D^DOi is the estimation of Di. Then, the DO-based SMC can be designed as(12)uit=−F¯iXm,Xs,t−εDOisi−σDOisiγDOisignsi−D^DOi,where εDOi, σDOi, and γDOi are the positive constants, 0<γDOi<1. The DO (11) can guarantee the estimate error converge to the following region:(13)D^DOi−Di⟶DimaxdkDOici,ift⟶∞,where the constant Ddmax is defined in Assumption 1. From (13), to achieve the high observation precision, the observer gain kDOi should be large enough. However, from (11), it is clear that the initial value D^DOi0 may be very large if kDOi is a high gain and the initial estimate state error ZDOi0−si0≠0. Actually, the initial values of true state si0 cannot be known in advance for most of the cases. Thus, the initial estimate state error ZDOi0−si0 maybe not equal to 0 and even a large value. The large value D^DOi0 may lead to a large overshoot control input uit (see (12)). Then, the large overshoot uit 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 Dii=1,2,…,n also can be estimated by using the method in [31]:(14)Z˙ESOi=ciF¯iXm,Xs,t+D^ESOi+uit−2kESOiZESOi−si,D^˙ESOi=−kESOi2ciZESOi−si,where ZESOi is the estimated state of ESO, D^ESOi is an estimation of Di, and kESOi is the observer gain of ESO. Then, the ESO-based SMC can be designed as(15)uit=−F¯iXm,Xs,t−εESOisi−σESOisiγESOisignsi−D^ESOi,where εESOi, σESOi, and γESOi are the positive constants, 0<γESOi<1. The ESO can guarantee the estimate error converge to following region:(16)D^ESOi−Di⟶DimaxdkESOici,ift⟶∞.Define the estimate error vectorh=ZESOi−siD^ESOi−DiT. From ESO (14), we have(17)h.=Ahh+Δ,where(18)Ah=ci−2kESOi1−kESOi20,Δ=0D˙i.The solution to the differential equation (17) can be easily obtained as(19)h=eAhth0+∫0teAht−τD˙iτdτ,where e is the e constant. Let ∗20ch¯1h¯2T=∫0teAht−τD˙iτdτ. Expanding eAhth0, the estimate error can be rewritten as(20)D^ESOi−Di=−kESOici2te−kESOicitZESOi0−si0+kESOicit+1e−kESOicitD^ESOi0−Di0+h¯2.For the small timet=1/kESOici, we have(21)D^ESOi=−kESOiciZESOi0−si0e−1+2e−1D^ESOi0−Di0+Di+h¯2,ift=1kESOici.It can be known thatkESOiciZESOi0−si0 is a very large value if kESOi is a high gain and ZESOi0−si0≠0. 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.


## 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 Dii=1,2,…,n, the TVGDO (22) is constructed.(22)Z˙i=ciF¯iXm,Xs,t+D^i+uit,D^i=−kitZi−si+∫0tk˙iτZi−sidτ,kit=ηi11−e−ηi2t,k˙iτ=ηi1ηi2e−ηi2τ,where ηi1 and ηi2 are the positive constants, and kit is a nonnegative time-varying gain. Assumption 1 is valid. The estimate error of TVGDO is defined as D¯i=D^i−Di. Then, the estimate error D¯i will converge to the following region:(23)D¯i≤Dimaxdηi1ci,ift⟶∞.Proof.
The estimate error of TVGDO is defined asD¯i=D^i−Dii=1,2,…,n. Differentiating D¯i gives(24)Di¯.=Di^.−D˙i.
Considering (9) and (22), (24) can be rewritten as(25)Di¯.=−k˙itZi−si−kitZ˙i−s˙i+k˙itZi−si−D˙i=−kitZ˙i−s˙i−D˙i=−kitciF¯iXm,Xs,t+D^i+uit−ciF¯iXm,Xs,t+Di+uit−D˙i=−kitciD^i−Di−D˙i=−kitciD¯i−D˙i.
Construct the Lyapunov functionJi as(26)Ji=D¯i2/2
Then, calculating the time derivative ofJi along the trajectory of (25), we get(27)J˙i=D¯iDi¯.=−kitciD¯i2−D˙iD¯i.
Considering Assumption1, we have(28)J˙i≤−kitciD¯i2+DimaxdD¯i=−2kitciJi+2DimaxdJi1/2,ift≥0.
Sincekit≥0, from (28), it can be known that(29)J˙i≤2DimaxdJi1/2,ift≥0.
Then, we have(30)Ji−1/2J˙i≤2Dimaxd,ift≥0.
Integrating (30) gives(31)2Ji1/2≤2Dimaxdt+2Ji1/20,ift≥0.
From the expression ofkit, it can be known that the following condition can be satisfied in an arbitrary finite time tif:(32)kit≥kitif>0,ift≥tif.
Combining (28) and (32), we have(33)J˙i≤−2kitciJi+2DimaxdJi1/2,ift≥tif.
From (31), for an arbitrary finite time tif, it is clear that Ji1/2tif is bounded. If Ji1/2>Dimaxd/2kitci, it also can be known from (33) that J˙i<0. Then, we have(34)Ji1/2≤Dimaxd2kitci,ift⟶∞.
Considerlimt⟶∞kit=ηi1; then, we have(35)Ji1/2≤Dimaxd2ηi1ci,ift⟶∞.
From (35) and 2Ji1/2=D¯i, we have(36)D¯i≤Dimaxdηi1ci,ift⟶∞.
The proof is finished.


### 3.2. The Spike-Free Characteristic Analysis

Let∫0tkiτcidτ=kiSt. Considering (22) and solving the differential equation (25), the solution of estimate error D¯i in time domain can be easily obtained as(37)D¯i=e−kiStD^i0−Di0−∫0tD˙iτekiSτdτ,where e is the e constant. Substituting the detailed expression of D^i0 into (37), we have(38)D¯i=e−kiStD^i0−Di0−e−kiSt∫0tD˙iτekiSτdτ=e−kiSt−ki0Zi0−si0+∫00k˙iτZi−sidτ−Di0−e−kiSt∫0tD˙iτekiSτdτ.From the previous discussion in Section2.2, it can be known that the undesirable spike problem is caused by the spike term kDOiZDOi0−si0 in DO (11) or the spike term kESOiciZESOi0−si0 in ESO (14). Since ki0Zi0−si0=0 and ∫00k˙iτZi−sidτ=0, it is clear that the expression of D¯i 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 ηi1 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 i=1,2,…,n, the TVGDO-based sliding mode controller is constructed as(39)uit=−F¯iXm,Xs,t−εisi−σisiγisignsi−D^i,where the sliding mode surface si is defined in (7). εi, σi, and γi are the positive constants. 0<γi<1. D^i is given in TVGDO (22). Assumption 1 is valid. The synchronization error ei can converge to following small region:(40)ei≤minDimaxd/εiηi1cici,Dimaxd/σiηi1ci1/γici,ift⟶∞.Proof.
Construct the Lyapunov functionPii=1,2,…,n as(41)Pi=si22.
Then, calculating the time derivative ofPi along the trajectory of (9), we get(42)Pi=sis˙i=siciF¯iXm,Xs,t+Di+uit.
Substituting the control input (39) into (42), we have(43)P˙i=cisiDi−εisi−σisiγisignsi−D^i=ci−εisi2−σisiγi+1−D^i−Disi=−εicisi2−σicisiγi+1−ciD¯isi≤−2εiciPi−2γi+1/2σiciPiγi+1/2+ci2D¯iPi1/2,where the estimate error D¯i=D^i−Di.
From (43), we know that ei affected the estimate error D¯i. Thus, the following proof will consist of two steps. In the first step, it will be proved that ei 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 ei will converge to a neighborhood of zero after D¯i converges to a small neighborhood of zero.Step 1.
From (31) in Theorem 1, we have known that the estimate error D¯i is bounded as(44)2Ji1/2≤2Dimaxdt+2Ji1/20,ift≥0.
For the arbitrary finite timet≤ts, it can be known that(45)D¯i≤Dimaxdt+2Ji1/20≤Dimaxdts+2Ji1/20,ift≤ts.
Let the positive constantDPi=Dimaxdts+2Ji1/20. Then, combining (43) with (45), we have(46)P˙i≤−2εiciPi+ci2DPiPi1/2≤Pi1/2−2εiciPi1/2+ci2DPi,ift≤ts,(47)P˙i≤−2γi+1/2σiciPiγi+1/2+ci2DPiPi1/2=Pi1/2−2γi+1/2σiciPiγi/2+ci2DPi,ift≤ts.
From (46) and (47), we can know that P˙i<0 if Pi>DPi/2γi/2σiγi/2 or Pi>DPi/2εi2. Thus, for t≤ts, Pi will not escape to infinity and is bounded as(48)Pi≤maxDPi2γi/2σi2/γi,DPi2εi2,ift≤ts.
Let the constantPimax=maxDPi/2γi/2σi2/γi,DPi/2εi2. Then, we can know that ei will not escape to infinity and is bounded as ei≤2Pimax/ci if t≤ts.Step 2.
From (43), we have(49)P˙i≤−2ciPi1/22εiPi1/2−D¯i,P˙i≤−2ciPi1/22γi/2σiPiγi/2−D¯i.
Then, we can know that(50)P˙i<0,if2εiPi1/2>D¯i≥0,(51)P˙i<0,if2γi/2σiPiγi/2>D¯i≥0.
According to (50) and (51), we have(52)Pi≤D¯i2εi2,ift⟶∞,(53)Pi≤D¯i2γi/2σi2/γi,ift⟶∞.
From Theorem1, it can be known that the estimate error D¯i is bounded as(54)D¯i≤Dimaxdηi1ci,ift⟶∞.
Combining (52)–(54), we have(55)Pi≤minDimaxd2εiηi1ci2,Dimaxd2γi/2σiηi1ci2/γi,ift⟶∞.
From (55), it can be known that si will converge to following region:(56)si≤minDimaxdεiηi1ci,Dimaxdσiηi1ci1/γi,ift⟶∞.
Then, the synchronization error will converge to following small region:(57)ei≤minDimaxd/εiηi1cici,Dimaxd/σiηi1ci1/γici,ift⟶∞.
The proof is finished.Remark 2.
It can be known from (57) that, if large enough observer parameter ηi1 and the control parameters σi, εi, and ci 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.


## 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 Dii=1,2,…,n, the TVGDO (22) is constructed.(22)Z˙i=ciF¯iXm,Xs,t+D^i+uit,D^i=−kitZi−si+∫0tk˙iτZi−sidτ,kit=ηi11−e−ηi2t,k˙iτ=ηi1ηi2e−ηi2τ,where ηi1 and ηi2 are the positive constants, and kit is a nonnegative time-varying gain. Assumption 1 is valid. The estimate error of TVGDO is defined as D¯i=D^i−Di. Then, the estimate error D¯i will converge to the following region:(23)D¯i≤Dimaxdηi1ci,ift⟶∞.Proof.
The estimate error of TVGDO is defined asD¯i=D^i−Dii=1,2,…,n. Differentiating D¯i gives(24)Di¯.=Di^.−D˙i.
Considering (9) and (22), (24) can be rewritten as(25)Di¯.=−k˙itZi−si−kitZ˙i−s˙i+k˙itZi−si−D˙i=−kitZ˙i−s˙i−D˙i=−kitciF¯iXm,Xs,t+D^i+uit−ciF¯iXm,Xs,t+Di+uit−D˙i=−kitciD^i−Di−D˙i=−kitciD¯i−D˙i.
Construct the Lyapunov functionJi as(26)Ji=D¯i2/2
Then, calculating the time derivative ofJi along the trajectory of (25), we get(27)J˙i=D¯iDi¯.=−kitciD¯i2−D˙iD¯i.
Considering Assumption1, we have(28)J˙i≤−kitciD¯i2+DimaxdD¯i=−2kitciJi+2DimaxdJi1/2,ift≥0.
Sincekit≥0, from (28), it can be known that(29)J˙i≤2DimaxdJi1/2,ift≥0.
Then, we have(30)Ji−1/2J˙i≤2Dimaxd,ift≥0.
Integrating (30) gives(31)2Ji1/2≤2Dimaxdt+2Ji1/20,ift≥0.
From the expression ofkit, it can be known that the following condition can be satisfied in an arbitrary finite time tif:(32)kit≥kitif>0,ift≥tif.
Combining (28) and (32), we have(33)J˙i≤−2kitciJi+2DimaxdJi1/2,ift≥tif.
From (31), for an arbitrary finite time tif, it is clear that Ji1/2tif is bounded. If Ji1/2>Dimaxd/2kitci, it also can be known from (33) that J˙i<0. Then, we have(34)Ji1/2≤Dimaxd2kitci,ift⟶∞.
Considerlimt⟶∞kit=ηi1; then, we have(35)Ji1/2≤Dimaxd2ηi1ci,ift⟶∞.
From (35) and 2Ji1/2=D¯i, we have(36)D¯i≤Dimaxdηi1ci,ift⟶∞.
The proof is finished.


## 3.2. The Spike-Free Characteristic Analysis

Let∫0tkiτcidτ=kiSt. Considering (22) and solving the differential equation (25), the solution of estimate error D¯i in time domain can be easily obtained as(37)D¯i=e−kiStD^i0−Di0−∫0tD˙iτekiSτdτ,where e is the e constant. Substituting the detailed expression of D^i0 into (37), we have(38)D¯i=e−kiStD^i0−Di0−e−kiSt∫0tD˙iτekiSτdτ=e−kiSt−ki0Zi0−si0+∫00k˙iτZi−sidτ−Di0−e−kiSt∫0tD˙iτekiSτdτ.From the previous discussion in Section2.2, it can be known that the undesirable spike problem is caused by the spike term kDOiZDOi0−si0 in DO (11) or the spike term kESOiciZESOi0−si0 in ESO (14). Since ki0Zi0−si0=0 and ∫00k˙iτZi−sidτ=0, it is clear that the expression of D¯i 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 ηi1 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 i=1,2,…,n, the TVGDO-based sliding mode controller is constructed as(39)uit=−F¯iXm,Xs,t−εisi−σisiγisignsi−D^i,where the sliding mode surface si is defined in (7). εi, σi, and γi are the positive constants. 0<γi<1. D^i is given in TVGDO (22). Assumption 1 is valid. The synchronization error ei can converge to following small region:(40)ei≤minDimaxd/εiηi1cici,Dimaxd/σiηi1ci1/γici,ift⟶∞.Proof.
Construct the Lyapunov functionPii=1,2,…,n as(41)Pi=si22.
Then, calculating the time derivative ofPi along the trajectory of (9), we get(42)Pi=sis˙i=siciF¯iXm,Xs,t+Di+uit.
Substituting the control input (39) into (42), we have(43)P˙i=cisiDi−εisi−σisiγisignsi−D^i=ci−εisi2−σisiγi+1−D^i−Disi=−εicisi2−σicisiγi+1−ciD¯isi≤−2εiciPi−2γi+1/2σiciPiγi+1/2+ci2D¯iPi1/2,where the estimate error D¯i=D^i−Di.
From (43), we know that ei affected the estimate error D¯i. Thus, the following proof will consist of two steps. In the first step, it will be proved that ei 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 ei will converge to a neighborhood of zero after D¯i converges to a small neighborhood of zero.Step 1.
From (31) in Theorem 1, we have known that the estimate error D¯i is bounded as(44)2Ji1/2≤2Dimaxdt+2Ji1/20,ift≥0.
For the arbitrary finite timet≤ts, it can be known that(45)D¯i≤Dimaxdt+2Ji1/20≤Dimaxdts+2Ji1/20,ift≤ts.
Let the positive constantDPi=Dimaxdts+2Ji1/20. Then, combining (43) with (45), we have(46)P˙i≤−2εiciPi+ci2DPiPi1/2≤Pi1/2−2εiciPi1/2+ci2DPi,ift≤ts,(47)P˙i≤−2γi+1/2σiciPiγi+1/2+ci2DPiPi1/2=Pi1/2−2γi+1/2σiciPiγi/2+ci2DPi,ift≤ts.
From (46) and (47), we can know that P˙i<0 if Pi>DPi/2γi/2σiγi/2 or Pi>DPi/2εi2. Thus, for t≤ts, Pi will not escape to infinity and is bounded as(48)Pi≤maxDPi2γi/2σi2/γi,DPi2εi2,ift≤ts.
Let the constantPimax=maxDPi/2γi/2σi2/γi,DPi/2εi2. Then, we can know that ei will not escape to infinity and is bounded as ei≤2Pimax/ci if t≤ts.Step 2.
From (43), we have(49)P˙i≤−2ciPi1/22εiPi1/2−D¯i,P˙i≤−2ciPi1/22γi/2σiPiγi/2−D¯i.
Then, we can know that(50)P˙i<0,if2εiPi1/2>D¯i≥0,(51)P˙i<0,if2γi/2σiPiγi/2>D¯i≥0.
According to (50) and (51), we have(52)Pi≤D¯i2εi2,ift⟶∞,(53)Pi≤D¯i2γi/2σi2/γi,ift⟶∞.
From Theorem1, it can be known that the estimate error D¯i is bounded as(54)D¯i≤Dimaxdηi1ci,ift⟶∞.
Combining (52)–(54), we have(55)Pi≤minDimaxd2εiηi1ci2,Dimaxd2γi/2σiηi1ci2/γi,ift⟶∞.
From (55), it can be known that si will converge to following region:(56)si≤minDimaxdεiηi1ci,Dimaxdσiηi1ci1/γi,ift⟶∞.
Then, the synchronization error will converge to following small region:(57)ei≤minDimaxd/εiηi1cici,Dimaxd/σiηi1ci1/γici,ift⟶∞.
The proof is finished.Remark 2.
It can be known from (57) that, if large enough observer parameter ηi1 and the control parameters σi, εi, and ci 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 n=3. The chaotic nonlinear function and uncertainties are chosen as(58)fm1Xm,t=4014xm1−xm2xm3+10,fm2Xm,t=−10xm2+xm1xm3,fm3Xm,t=−4xm3+xm1xm2,fs1Xs,t=4014xs1−xs2xs3+10,fs2Xs,t=−10xs2+xs1xs3,fs3Xs,t=−4xs3+xs1xs2,Δfs1Xs,t=3%xs1+5%xs2,Δfs2Xs,t=2%xs1+7%xs2+3%xs3,Δfs3Xs,t=−2%xs1+3%xs2+6%xs3,d1t=60cost5+20,d2t=50cost5+20,d3t=40cosu5+10.The initial system states are set asxm10=4, xm20=3.5, xm30=2.5, and xs10=xs20=xs30=0. The simulation method is chosen as the fixed step Dormand–Prince method. The step size of simulation is set as 0.001s. Let uit=0i=1,2,3, and the chaotic behavior of the master system (1) is shown in Figure 1.Figure 1
Chaotic behavior of the system without the control input. (a) Phase portrait of the system. (b) System statesx1 and x2.
(a)(b)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 asci=1, εDOi=2, εESOi=0.5, εi=0.5, σDOi=12, σESOi=7, σi=2, and γDOi=γESOi=γi=0.7i=1,2,3. Then, it can be known that the initial values of sliding mode surface are s10=−4, s20=−3.5, and s30=−2.5. The observer gains of DO and ESO are chosen as kDOi=50 and kESOi=50i=1,2,3, respectively. The time-varying parameters are chosen as ηi1=50 and ηi1=25i=1,2,3. Thus, the observer gain of TVGDO kiti=1,2,3 is close to 50. And, in this case, we consider that the initial value of sliding mode surface si0i=1,2,3 is known. Then, the initial values of estimate states in DO, ESO, and TVGDO can be chosen as(59)ZDO1=s10=−4,ZDO2=s20=−3.5,ZDO3=s30=−2.5,ZESO1=s10=−4,ZESO2=s20=−3.5,ZESO3=s30=−2.5,Z1=s10=−4,Z2=s20=−3.5,Z3=s30=−2.5.
Thus, for DO, ESO, and TVGDO, the initial values of estimate and true states are equal. Figures2–5 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.Figure 2
Synchronization error (Case1).
(a)(b)(c)Figure 3
Estimate error (Case1).
(a)(b)(c)Figure 4
Control input (Case1).
(a)(b)(c)Figure 5
Observer gain of TVGDO (Case1).
(a)(b)(c)Case 2.
(The initial values of estimate and true states are not equal). From the simulation result of Case1, 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 si0i=1,2,3 is unknown. The initial values of estimate states of DO, ESO, and TVGDO are chosen as(60)ZDO1=0≠s10,ZDO2=0≠s20,ZDO3=0≠s30,ZESO1=0≠s10,ZESO2=0≠s20,ZESO3=0≠s30Z1=0≠s10,Z2=0≠s20,Z3=0≠s30.
Thus, for DO, ESO, and TVGDO, the initial values of estimate and true states are not equal.
Figures6–8 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 (Figures2–4 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 (Figures4 and 8). And, unlike the conventional SMC, the proposed controller does not need the upper bound of uncertainties.Figure 6
Synchronization error (Case2).
(a)(b)(c)Figure 7
Estimate error (Case2).
(a)(b)(c)Figure 8
Control input (Case2).
(a)(b)(c)


## 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.



---

*Source: 8845553-2021-04-21.xml*