# Chaos Synchronization of a Finance Chaotic System with an Integral Sliding Mode Controller

Authors: Jingjing Wang; Chunzhi Yang
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, an integral sliding mode controller is proposed to realize the synchroniza-tion of a class of finance chaotic systems. First, an integral sliding surface with tanh function is designed. Then, fuzzy logic systems are used to estimate unknown functions, and then based on the Lyapunov stability theorem, the proposed control method can quickly drive the synchroniza-tion error into a small neighborhood of zero, and the range of this neighborhood can be estimated. The simulation results verify that the proposed method in this paper is better than the traditional method.


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## Body

## 1. Introduction

In the past few decades, chaotic synchronization, due to its unique characteristics, has been widely used in engineering, finance, chemistry, confidential communications, and so on [1–5]. In recent decades, due to the complexity and inherent randomness of economic factors, chaotic nonlinear financial system has attracted extensive attention [6–12]. For example, Chao and Xingyuan [6] studied the existence of Hopf bifurcation and topological horseshoe of a class of finance chaotic system. Yu et al. [7] proposed a feedback control method to stabilize the hyperchaotic finance system to its equilibrium point. Huang and Cao [8] employed active control strategy to realize synchronization and antisynchronization of a class of fractional chaotic financial systems. In [11], a linear feedback control scheme was proposed to realize chaos synchronization of two identical financial chaotic systems. In addition to the feedback control method and active control method, there are other control methods that can realize synchronization of financial chaotic systems, such as adaptive control [13–15], impulsive control [16, 17], sliding mode control [15, 18–24], and so on.Among above methods, more attention is paid to the sliding mode control method because of its simple design and good robustness. Sliding mode control is generally divided into two steps: the first step is to design the sliding mode surface; the second step is to design the controller so that the system state can enter the sliding mode surface. For example, Chiu [21] proposed two integral sliding mode control methods to make the tracking error reach the sliding surface in finite time. Using same two integral sliding modes, Zhao et al. [24] proposed two effective control methods, which inhibited the influence of external disturbances and realized the effective state estimation of the continuous stirred tank reactor. These two types of integral sliding mode are written as (1) s=e+λ∫0tsignedτ; (2) s=e+λ∫0teγsignedτ, γ∈0,1,λ>0. Since these integral sliding mode need to have sign⋅ function, the controllers designed by them will lead to chattering phenomenon. Therefore, to modify the above integral sliding mode has become a research direction.Inspired by the above works, this paper investigates the synchronization of two finance chaotic systems. The main contributions of this paper are summarized as follows.(1)
Compared with the traditional integral sliding mode, the integrated sliding mode proposed in this paper can estimate the ultimate synchronization error bound by its own dynamics.(2)
Compared with the traditional integral sliding mode control method, the proposed method can eliminate the chatter phenomenon in error states and controller.The rest of this work is arranged as follows. Some assumptions and lemmas and the problem statement are presented in Section2. Section 3 gives sliding mode control design and stability analysis. In Section 4, some comparison results are presented to show the validity of the proposed method. At last, the conclusion is included in Section 5.


## 2. Preliminaries

In general, the finance chaotic system is described as(1)x˙1=x3+x2−α1x1,x˙2=1−β1x2−x12,x˙3=−x1−σ1x3,where x=x1,x2,x3T∈R3 is the state vector, x1 denotes the interest rate, x2 denotes the investment demand, x3 denotes the price exponent, parameter α1 denotes the saving amount, parameter β1 denotes the investment cost, and parameter σ1 denotes the elasticity of the demands of commercials. We regard system (1) as the master system, and the slave system is described as(2)y˙1=y3+y2−α2y1+d1t+satu1,y˙2=1−β2y2−y12+d2t+satu2,y˙3=−y1−σ2y3+d3t+satu3,where y=y1,y2,y3T∈R3 is the state vector and dt=d1t,d2t,d3tT is the external disturbance vector. Parameters α2,β2, and σ2 are the same as α1,β1, and σ1 in system (1). satu=satu1,satu2,satu3T is control input, which is subject to saturation nonlinearity. And satui is described as(3)satui=uib⋅signui,ifui≥uib,ui,ifui<uib,where uib is the unknown bound of ui, i=1,2,3. Define the synchronization error vector e=y−x=y1−x1,y2−x2,y3−x3T=e1,e2,e3T. According to (1) and (2), we obtain the following error dynamic system:(4)e˙1=g1y−f1x+d1t+satu1,e˙2=g2y−f2x+d2t+satu2,e˙3=g3y−f3x+d3t+satu3,where f1x=x3+x2−α1x1, g1y=y3+y2−α2y1, f2x=1−β1x2−x12,g2y=1−β2y2−x22,f3x=−x1−σ1x3,g3y=−y1−σ2y3.In order to achieve effective state synchronization between the master chaotic system and the slave chaotic system, we make the following assumptions aboutfix, giy, and dit,i=1,2,3.Assumption 1.
Nonlinear functionsfix and giy are unknown but bounded, and states x and y are measurable.Assumption 2.
The external disturbancedit is bounded, i.e., dit≤di∗, where di∗ is unknown.Remark 1.
In [11], the linear feedback control method is used to synchronize two identical finance chaotic systems, but the unknown function and saturation nonlinearity input of the finance chaotic system are not considered. At the same time, in order to make the controller have good robustness, an integral sliding mode control method is used to discuss the synchronization of master system (1) and slave system (2).Remark 2.
Fuzzy logic systems are used in this paper to estimate unknown functionsfix and giy, and the related fuzzy estimation theory can be found in the literature [25]. In addition, di∗ is only used in stability analysis and not involved in the design of ui.
By using FLSs,fix and giy can be approximated as(5)fix=θfi∗Tφfix+εfi,εfi≤εfi∗,(6)giy=θgi∗Tφgiy+εgi,εgi≤εgi∗,where θfi∗,θgi∗∈Rmi are ideal weight vectors, εfi and εgi are approximation errors, and εfi∗ and εgi∗ are upper bounds of εfi and εgi, respectively. φfix and φgiy are basis functions.
Defineθ˜fi=θfi∗−θ^fi, θ˜gi=θgi∗−θ^gi where θ^fi and θ^gi are the estimations of θfi∗ and θgi∗, respectively. So, we obtain(7)fix=θ˜fiTφfix+θ^fiTφfix+εfix,giy=θ˜giTφgiy+θ^giTφgiy+εgiy.
Substituting (7) into (4), error system (4) can be rewritten as(8)e˙1=θ˜g1Tφg1y+θ^g1Tφg1y−θ˜f1Tφf1x−θ^f1Tφf1x+d¯1+u1,e˙2=θ˜g2Tφg2y+θ^g2Tφg2y−θ˜f2Tφf2x−θ^f2Tφf2x+d¯2+u2,e˙3=θ˜g3Tφg3y+θ^g3Tφg3y−θ˜f3Tφf3x−θ^f3Tφf3x+d¯3+u3,where d¯i=εgiy−εfix+dit+Δui, Δui=satui−ui. Obviously, there exists d¯i∗ such that d¯i≤d¯i∗.
The following lemma is introduced for the subsequent discussions.Lemma 1.
(see [14]). For any z∈R, 1>γ≥0, and ε>0, the following inequality holds:(9)zγ+1−zzγtanhzε≤ηγεγ+1,where ηγ=ϱγ2ϱ−γ−1>0 and ϱ is the unique solution of ϱ1+tanhϱ=γ+1.Remark 3.
In this paper, inequality (9) of Lemma 1 is used to discuss the ultimate boundary of tracking error ei.
The aim of this paper is to design a novel integral sliding variable such that the error statee reaches a small neighborhood of zero quickly.


## 3. Sliding Mode Control Design and Stability Analysis

In order to design a sliding mode robust controller to make the synchronization errore stable, an integral sliding variable needs to be designed. In this paper, we design the sliding mode as follows:(10)s1=e1+∫0tμ1Tanhε1γ1e1+μ2Tanhε10e1dτ,s2=e2+∫0tμ1Tanhε1γ1e2+μ2Tanhε10e2dτ,s3=e3+∫0tμ1Tanhε1γ1e3+μ2Tanhε10e3dτ,where Tanhε1γ1ei=eiγitanhei/ε1, Tanhε10ei=tanhei/ε1, μ1,μ2 are designed positive parameters, γ1∈0,1, and ε1 is a small positive constant, i=1,2,3.Remark 4.
Different from traditional sliding mode control method that the occurrence of chatter phenomenon is mainly due to the hysteresis of control switching, the integral sliding variable (10) used in this paper can avoid chatter phenomenon and estimate the ultimate boundary of the tracking error e.
According to (8) and (10), the time derivative of si is obtained as follows:(11)s˙i=θ˜giTφgiy+θ^giTφgiy−θ˜fiTφfix−θ^fiTφfix+d¯it+ui+μ1Tanhϵ1γ1ei+μ2Tanhϵ10ei.
Based on the above analysis, we give the main result as follows.Theorem 1.
Consider the synchronization error system (8) with the external disturbance; if the integral sliding variable is constructed as (10) and the controller ui is designed as(12)ui=−μ3si−μ4Tanhϵ2γ2si−θ^giTφgiy+θ^fiTφfix−μ1Tanhϵ1γ1ei−μ2Tanhϵ10ei,and parameter adaptive laws are proposed as(13)θ^˙gi=ηgsiφgiy−δ1θ^gi,θ^˙fi=ηf−siφfix−δ2θ^fi,where Tanhϵ2γ2si=siγ2tanhsi/ε2, δ1,δ2,μ3,μ4, and ηg,ηf are design positive parameters, i=1,2,3, and ε2 is a small positive constant, then all the closed-loop system signals in (14) are uniformly ultimately bounded.Proof.
Consider the Lyapunov function as(14)V=V1+V2+V3,where(15)V1=∑i=1312si2,V2=∑i=1312ηgθ˜giTθ˜gi,V3=∑i=1312ηfθ˜fiTθ˜fi.
Substituting (12) into (11), the derivative of V1 can be obtained as(16)V˙1=∑i=13sis˙i=∑i=13siθ˜giTφgiy+θ^giTφgiy−θ˜fiTφfix−θ^fiTφfix+d¯it+ui+μ1Tanhϵ1γ1ei+μ2Tanhϵ10ei=∑i=13siθ˜giTφgiy−θ^fiTφfix−μ3si−μ4Tanhϵ2γ2si+d¯it=−μ3∑i=13si2−∑i=1nμ4sisiγ2tanhsiϵ2+∑i=13sid¯it+∑i=13siθ˜giTφgy−∑i=13siθ˜fiTφfx.
From (13), the time derivative of V2 and V3 yields(17)V˙2=∑i=131ηg3θ˜giTθ˜˙gi=∑i=13−siθ˜giTφgiy+δ1θ˜giTθ^gi,(18)V˙3=∑i=131ηfθ˜fiTθ˜˙fi=∑i=13siθ˜fiTφfix+δ2θ˜fiTθ^fi.
Using (16), (17), and (18), one gets(19)V˙=−μ3∑i=13si2−∑i=13μ4sisiγ2tanhsiε2+∑i=13sid¯it+∑i=13δ1θ˜giTθ^gi+∑i=13δ2θ˜fiTθ^fi.
Since the following inequalities hold:(20)sid¯it≤si22+d¯i2t2≤si22+d¯i∗2t2,δ1θ˜giTθ^gi≤−δ12θ˜gi2+δ12θgi∗2,δ2θ˜fiTθ^fi≤−δ22θ˜fi2+δ22θfi∗2,by using Lemma 1, one has(21)−∑i=13μ4sisiγ2tanhsiε2≤−∑i=13μ4siγ2+1+3μ4ηγ2ε2γ2+1.
Substituting (20) and (21) into (19) results in(22)V˙≤−∑i=13μ3−12si2−∑i=13μ4siγ2+1−δ12∑i=13θ˜gi2−δ22∑i=13θ˜fi2+r0,where r0=∑i=13d¯i∗2/2+3μ4ηγ2ε2γ2+1+δ2/2∑i=13θgi∗2+δ2/2∑i=13θfi∗2. And let rmin=min2μ3−1,ηgδ1,ηfδ2. It can be obtained as(23)V˙≤−rminV+r0.
From (23), we know that(24)V≤r0rmin+V0−r0rmine−rmint≤2V0+r0rmin.
Obviously, it can be concluded that all signals in (14) are ultimately uniformly bounded. The proof is completed.
Theorem1 shows that the designed sliding mode controller (12) guarantees that all signals in (14) are bounded. Now, we further study the convergence of the synchronization error ei. According to (10), (11), and (12), we have(25)e˙i=s˙i−μ1Tanhϵ1γ1ei−μ2Tanhϵ10ei=ψit−μ1Tanhϵ1γ1ei−μ2Tanhϵ10ei,where ψit=θ˜giTφgiy−θ˜fiTφfix+d¯it−μ3si−μ4Tanhϵ2γ2si. Through the conclusion of Theorem 1, there exists a positive constant ϖi such that(26)ψit≤ϖi.
Consider the following Lyapunov function:(27)V¯=12∑i=13ei2.
The derivative ofV¯ is calculated as(28)V¯˙=∑i=13eiψit−μ1eiTanhϵ1γ1ei−μ2eiTanhϵ10ei≤e12+e22+e32ϖ12+ϖ22+ϖ32−∑i=13μ1eieiγ1tanheiϵ1−∑i=13μ2eitanheiϵ1.
By using Lemma1, the following inequalities hold:(29)−∑i=13μ1eieiγ1tanheiε1≤−∑i=13μ1eiγ1+1+3μ1ηγ1ε1γ1+1≤−μ1e12+e22+e32γ1+1/2+3μ1ηγ1ε1γ1+1,(30)−∑i=1nμ2eitanheiε1≤−∑i=1nμ2ei+3μ2ηγ0ε1≤−μ2e12+e22+e32+3μ2ηγ0ε1.
Substituting (29) and (30) into (28), we have(31)V¯˙≤−μ2−ϖ¯e12+e22+e32−μ1e12+e22+e32γ1+1/2+ε0,where ϖ¯=ϖ12+ϖ22+ϖ32,ε0=3μ1ηγ1ε1γ1+1+3μ2ηγ0ε1. Selecting μ2 satisfies μ2>ϖ¯, and define the compact set as(32)Ωe=e∈R3|μ1e12+e22+e32γ1+1/2≤ε01−ς,where ς∈0,1 is any number. If e∉Ωe, then(33)V¯˙≤−μ1e12+e22+e32γ+1/2+ε0<−μ1ςe12+e22+e32γ1+1/2=−2γ1+1/2μ1ςV¯γ1+1/2.
Obviously,V¯ decreases monotonically outside the set Ωe until it enters the minimal level set of V¯ containing Ωe. Therefore, we obtain the following theorem.Theorem 2.
Based on the conditions and results of Theorem1 and selecting the appropriate parameters μ1 and μ2 in (10), the tracking error vector e is ultimately bounded and the ultimate bound is e≤ε0/μ11−ς1/γ1+1.Remark 5.
The synchronization method of chaotic systems in this paper can be extended to generalized projective synchronization, lag synchronization, and so on.


## 4. Synchronization of Finance Chaotic System

In this section, the chaotic finance system [11] is introduced to demonstrate the effectiveness of the proposed control method (12). Parameters in the master system (1) and the slave system (2) are selected as α1=0.8,α2=0.9,β1=0.2,β2=0.25,σ1=1.9,σ2=1.8, and external disturbances d1t=2.0sint, d2t=2.0cos2t, d3t=2.0sin3t. Here, initial values of x and y are chosen as x0=1,1.5,1.2T and y0=2,−1,−2.5T, and other parameters are chosen as ηg=ηf=50, μ1=μ2=μ3=μ4=5, ς=1/7, ε1=ε2=0.005, γ1=1/5,γ2=1/5,u1b=u2b=u3b=8,δ1=δ2=0.01. The fuzzy membership functions are selected as(34)θf1ζ=exp−12ζ+4.5−1.5k10.72,θg1xi=exp−12ξ+4.5−1.5k10.72,θf2ζ=exp−12ζ+5.5−1.5k10.72,θg2ξ=exp−12ξ+5.5−1.5k10.72,θf3ζ=exp−12ζ+3.0−1.5k20.72,θg3ξ=exp−12ξ+3.0−1.5k20.72,where ζ=x1,x2,x3;ξ=y1,y2,y3,k1=1,2,3,4,5;k2=1,2,3.According to Theorem2, the ultimate bound of the tracking error e can be estimated in the interval −2.4×10−3,2.4×10−3 when γ1=1/5. Now, we give the traditional integral sliding mode control method as follows:(35)si=ei+∫0tμ1Signeidτ,ui=−μ3si−μ4Signγ2si−θ^giTφgiy+θ^fiTφfix−μ1Signei,θ^˙gi=ηgsiφgiy−δ1θ^gi,θ^˙fi=ηf−siφfix−δ2θ^fi,where Signei=signei, Signγ2si=siγ2signsi,i=1,2,3. Obviously, the approximation range of e cannot be obtained directly from (35).Firstly, by using method (35), the corresponding simulation results are shown in Figures 1 and 2. From Figure 1, it can be found that all three synchronization errors e1,e2,e3 can be controlled into an estimated range, however, $e_1, e_2$, and $e_3$ appear the chattering phenomenon. Meanwhile, Figure 2 shows that satuiandui also appear a large chattering phenomenon, which is mainly caused by the sign⋅ function in method (35). In order to overcome the influence of the sign⋅ function, this paper uses the tanh⋅ function to design the controller. The simulation results are shown in Figures 3 and 4. From Figure 3, we can find that e1, e2, and e3 are limited within a small range, which is smaller than the estimated range, and there is no chattering phenomenon for e1, e2, and e3. Finally, comparing Figures 2 and 4, it can be seen that method (12) is significantly better than method (35) in terms of controller stability. Based on the above discussion, it can be seen that the proposed method (12) in this paper can effectively and quickly synchronize the master system and the slave system, can overcome the chattering phenomenon, and can estimate the predetermined range of synchronization error.Figure 1
The trajectories of tracking errorse1,e2,e3 by method (35) in time period (a) [0, 10 s] and (b) [5 s, 10 s].
(a)(b)Figure 2
The trajectories of (a)satu1,satu2,satu3 by method (35) in time period [0, 10 s] and (b) u1,u2,u3 by method (35) in time period [5 s, 10 s].
(a)(b)Figure 3
The trajectories of tracking errorse1,e2,e3 by the proposed method (12) in time period (a) [0, 10 s] and (b) [5 s, 10 s].
(a)(b)Figure 4
The trajectories of (a)satu1,satu2,satu3 by method (12) in time period [0, 10 s] and (b) u1,u2,u3 by method (12) in time period [5 s, 10 s].
(a)(b)


## 5. Conclusion

This paper considered the synchronization problem of a class of finance chaotic systems by using an integrated sliding mode control method. Firstly, the unknown functionsfix and giy are estimated by using fuzzy logic systems. Then, a novel integral sliding variable si is proposed. The proposed control strategy ui and parameter adaptive rules θ^f¯i can ensure that all signals of the closed-loop system are semiglobally uniformly stable, and the ultimate bound of the tracking error can be estimated. Comparison results of simulation show that the proposed control method in this paper can ensure that the error fluctuation range is completely within the estimated range and the trajectories of errors and controllers have no chattering phenomenon. Therefore, compared with the traditional integrated sliding mode control method, the proposed control method in this paper has better control performance.



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