# Predefined Time Synchronization Control for Uncertain Chaotic Systems

Authors: Yun Liu; Fang Zhu
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

In this study, the predefined time synchronization problem of a class of uncertain chaotic systems with unknown control gain function is considered. Based on the fuzzy logic system and varying-time terminal sliding mode control technology, the predefined time synchronization between the master system and the slave system can be realized by the proposed control method in this study. The simulation results confirm the theoretical analysis.


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## Body

## 1. Introduction

In recent decades, chaotic synchronization has been a research hotspot. The main reason is its wide application, such as in the fields of secure communication, biological systems, and so on [1–6]. Up to now, there are many synchronization methods between two chaotic systems, such as adaptive control [7–11], active control [12–14], impulsive control [15–17], and sliding mode control [18–23]. Among them, sliding mode control is deeply concerned by scholars because of its simple control principle and good robustness. Under the influence of unknown parameters and disturbances, two kinds of sliding mode synchronization methods were studied in [19]. Subsequently, to realize the state transient performance of the controlled system, many terminal sliding mode control methods were proposed. For example, a terminal sliding mode control method was employed in [22] and the synchronization of coronary artery system was realized. For fractional-order chaotic systems, [23] proposed a fractional-order terminal sliding mode control method, which synchronized two uncertain fractional-order systems. It should be pointed out that the initial value of the system should not be too far from the sliding mode; otherwise, the control performance will be affected. It should also be considered that the gain of the discontinuous controller should not be large, which will increase the serious chattering problem. In order to solve the above problems, a varying-time terminal sliding mode control method will be used to realize the predefined time synchronization of two uncertain chaotic systems.In this study, the predefined time synchronization of the main system and the slave system is considered. The main highlights are as follows: the synchronization of two uncertain chaotic systems is realized by the varying-time sliding mode control method, and the case where the controller gain is unknown is considered. The rest of this study is organized as follows. Some preliminaries are given in Section2. A preset time terminal sliding mode is proposed and main results are investigated in Section 3. A synchronization example is shown in Section 4. Finally, Section 5 gives a brief conclusion.


## 2. Preliminaries

The master system is described as(1)ξ˙1=ξ2,ξ˙2=f1t,ξ1,ξ2,where ξ1,ξ2∈ℝ are the states of system (1), and f1t,ξ1,ξ2∈ℝ is a nonlinear function.The slave system is described as(2)η˙1=η2,η˙2=f2t,η1,η2+gt,η1,η2u,where η1,η2∈ℝ are the states of system (2), f2t,η1,η2∈ℝ is a nonlinear function, u∈ℝ is the control input, and gt,η1,η2 is a control gain function.Define synchronization errorse1=η1−ξ1, e2=η2−ξ2. The aim of this study is to design a varying-time terminal sliding mode control method, so that the synchronization error e1 reaches a small neighborhood of zero in the predefined time. According to (1) and (2), one gets the synchronization error system as(3)e˙1=e2,e˙2=f2t,η1,η2−f1t,ξ1,ξ2+gt,η1,η2u.In order to design the controller in this study, the following assumptions need to be made.Assumption 1.
Statesξ1,ξ2,η1,η2 are measurable, and initial values ξ20=η20.Assumption 2.
f1t,ξ1,ξ2 and f2t,η1,η2 are unknown but bounded.Assumption 3.
gt,η1,η2 is unknown strictly positive and there exists a positive constant χ, such that gt,η1,η2>χ.Remark 1.
ξ20=η20 in Assumption 1 is to ensure that the initial value of error system (3) belongs to the sliding mode, which will be designed later. Assumption 2 ensures that the fuzzy logic system can be used to estimate the unknown function.
In order to achieve the aim of this study, the time-varying terminal sliding mode is considered:(4)z=e2+βe1+2λ1t+λ2+λ3e1+λ1t2+λ2t+αq/p,t≤T,e2+βe1+λ3e1q/p,t>T,where T is a preset time, 0<q/p<1, q and p are the odds, α,β are the design positive constant, and λ1, λ2, and λ3 satisfy the following conditions:(1)
In order to ensure that the initial value of system (3) belongs to the sliding mode (4), i.e.,(5)βe10+λ2+λ3e10+αq/p=0.(2)
The sliding mode (4) is continuous at t=T, i.e.,(6)2λ1T+λ2=0,λ1T2+λ2T+α=0.(3)
In order to ensure that sliding mode (4) can quickly approach the origin, i.e.,(7)λ3>0.
Let(8)Δ=e1+λ1t2+λ2t+α,t≤T,e1,t>T.Remark 2.
The derivation ofΔq/p with respect to time t may appear singular problem, and we modify Δq/p−1Δ˙ as(9)Δq/p−1Δ˙=Δq/p−1e2+2λ1t+λ2,fort≤TandΔ≠0,0,fort≤TandΔ=0,Δq/p−1e2,fort>TandΔ≠0,0,fort>TandΔ=0.


## 3. Main Result

Sinceξ1,ξ2,η1,η2 are measurable, unknown functions f1t,ξ1,ξ2, f2t,η1,η2, and gt,η1,η2 can be estimated by fuzzy logic systems [24, 25]. For f1t,ξ1,ξ2, f2t,η1,η2, and gt,η1,η2, there exist θf1∗Tφf1ξ1,ξ2, θf2∗Tφf2η1,η2, and θg∗Tφgη1,η2, such that(10)f1t,ξ1,ξ2=θf1∗Tφf1ξ1,ξ2+εf1ξ1,ξ2,f2t,η1,η2=θf2∗Tφf2η1,η2+εf2η1,η2,gt,η1,η2=θg∗Tφgη1,η2+εgη1,η2,where εf1ξ1,ξ2, εf2εgη1,η2, and εgη1,η2 are the bounded fuzzy estimation errors, θf1∗T, θf2∗T, and θg∗T are the ideal weight vectors, and φf1ξ1,ξ2, φf2η1,η2, and φgη1,η2 are the Gaussian functions.From (3) and (4), the derivative of z with respect to t can be obtained as(11)z˙=f2t,η1,η2−f1t,ξ1,ξ2+βe2+gt,η1,η2u+2λ1+λ3qpΔp/q−1Δ˙,t≤Tλ3qpΔp/q−1Δ˙,t>T=θf2∗Tφf2η1,η2−θf1∗Tφf1ξ1,ξ2+βe2+εf2η1,η2−εf1ξ1,ξ2+θg∗Tφgη1,η2ue+gt,η1,η2u+2λ1+λ3qpΔp/q−1Δ˙,t≤T,λ3qpΔp/q−1Δ˙,t>T.Now, design the controller as(12)u=ue+us,ue=θ^gTφgη1,η2θ^gTφgη1,η22+1−θ^f2Tφf2η1,η2+θ^f1Tφf1ξ1,ξ2−k1z−βe2+−2λ1−λ3qpΔp/q−1Δ˙,t≤T−λ3qpΔp/q−1Δ˙,t>T≜θ^gTφgη1,η2Πθ^gTφgη1,η22+1,us=−|Π|signzθ^gTφgη1,η22+1χ,where k1 is a design positive constant, and θ^f1, θ^f2, and θ^g are the estimations of θf1∗, θf2∗, and θg∗. The parameter adaptation laws of θ^f1 and θ^f2 are given by(13)θ^.f1=γf1−zφf1ξ1,ξ2−δf1θ^f1,θ^.f2=γf2zφf2η1,η2−δf2θ^f2,θ^.g=γgzueφgη1,η2−δgθ^g,where γf1, γf2, γg, δf1, δf2, and δg are the design positive constants. Let εt=εf1ξ1,ξ2+εf2η1,η2+εgη1,η2ue. Obviously, εt is bounded, i.e., there exists a positive constant ε∗, such that εt≤ε∗.Theorem 1.
Under Assumptions1–3, if the time-varying terminal sliding mode (4), controller (12), and parameter adaptive laws (13) are employed, then all signals in (14) are bounded.Proof.
Consider the following Lyapunov function:(14)V1=12z2+1γf1θ˜f1Tθ˜f1+1γf2θ˜f2Tθ˜f2+1γgθ˜gTθ˜g,where θ˜f1=θf1∗−θ^f1, θ˜f2=θf2∗−θ^f2, and θ˜g=θg∗−θ^g. From (11), derivation of V1 with respect to t yields(15)V˙1=zz˙−1γf1θ˜f1Tθ^.f1−1γf2θ˜Tθ^.f2=zθf2∗Tφf2η1,η2−θf1∗Tφf1ξ1,ξ2+βe2+εt+θg∗Tφgη1,η2ue+gt,η1,η2us+2λ1+λ3qpΔp/q−1Δ˙,t≤Tλ3qpΔp/q−1Δ˙,t>T−1γf1θ˜f1Tθ^.f1−1γf2θ˜f2Tθ^.f2−1γgθ˜gTθ^.g=zθ˜f2Tφf2η1,η2−θ˜f1Tφf1ξ1,ξ2+βe2+εtθ^f2Tφf2η1,η2−θ^f1Tφf1ξ1,ξ2+θ˜gTφgη1,η2ue+θ^gTφgη1,η2ue+gt,η1,η2us+2λ1+λ3qpΔp/q−1Δ˙,t≤Tλ3qpΔp/q−1Δ˙,t>T−1γf1θ˜f1Tθ^.f1−1γf2θ˜f2Tθ^.f2−1γgθ˜gTθ^.g.
Substituting (12) and (13) to (15) yields(16)V˙1=−k2z2−zΠθ^gTφgη1,η22+1−gt,η1,η2|z‖|π|θ^gTφgη1,η22+1χ+zεt+δf1θ˜f1Tθ^f1+δf2θ˜f2Tθ^f2+δgθ˜gTθ^g≤−k2z2+zεt+δf1θ˜f1Tθ^f1+δf2θ˜f2Tθ^f2+δgθ˜gTθ^g.
Since the following inequalities hold:(17)zεt≤14z2+ε∗2,δf1θ˜f1Tθ^f1≤δf12−θ˜f1Tθ˜f1+θf1∗Tθf1∗,δf2θ˜f2Tθ^f2≤δf22−θ˜f2Tθ˜f2+θf2∗Tθf2∗,substituting (12) into V˙1 yields(18)V˙1≤−k1−14z2−δf12θ˜f1Tθf1−δf22θ˜f2Tθ˜f2+R∗,where R∗=ε∗2+δf1/2θf1∗Tθf1∗+δf1/2θf2∗Tθf2∗. Selecting k1, δ1, and δ2, such that ı1≜min2k1−1/2,γf1δf1,γf2δf2>0, then(19)V˙1≤−ı1V1+R∗.
According to (19), we can conclude that all signals in (14) are bounded. This completes the proof.Remark 3.
Fort>T, e2=e˙1=z−βe1−λ3e1q/p, with the boundedness of z; there exists unknown constant b∗, such that z≤b∗. Let V2=1/2e12, and one has(20)V˙2=e1e˙1=e1z−βe1−λ3e1q/p≤−β−14e12−λ3e1q/p−1+b∗2.
Letβ>1/4 and define(21)Ωe=e1|λ3e1q/p−1≤b∗21−ν,where ν∈0,1. Obviously, if e1∈¯Ωe, V˙2≤−λ3e1q/p+1+b∗2<−λ3νe1q/p+1<0, V2 will monotonically decrease only to enter Ωe. Therefore, we obtain the convergence range of the tracking error e1.


## 4. Numerical Simulations

In this section, the chaotic gyroscope system [26] is taken as an example to show the effectiveness of the proposed method (12). For the master system (1), define f1t,ξ1,ξ2=−1021−cosξ12/sin3ξ1+sinξ1−0.5ξ2−0.05ξ23+35.7sin2tsinξ1. For the slave system (2), define f2t,η1,η2=−1021−cosη12/sin3η1+sinξ1−0.5η2−0.05η23+35.5sin2tsinη1, gt,η1,η2=5+sinη2. Obviously, gt,η1,η2>χ≜3. The initial values ξ10=−1, ξ20=1, η10=2, and η20=1. The fuzzy membership functions are selected as(22)φρ=exp−12ρ+7.5−2.5j1.22,where ρ=ξ1,ξ2,η1,η2; j=1,2,3,4,5. First, select a group of parameters as T=2,k1=3,q=3,p=5,β=3,α=−5,λ1=−5/4,λ2=5,λ3=14/23/5, and the simulation results are shown in Figures 1–3. Figures 1 and 2 show that the state ξ1 of master system (1) and the state η1 of slave system (2) are synchronized after T=2s. In order to overcome the influence of unknown gain gt,η1,η2, Figure 3 shows that the controller u fluctuates at T=2s, and then, the controller has a small chattering phenomenon.Figure 1
Time response trajectory ofe1 by using the proposed method (12) with T=2s.Figure 2
Time response trajectories ofξ1 and η1 by using the proposed method (12) with T=2s.Figure 3
Time response trajectory of controlleru by using the proposed method (12) with T=2s.Extend the predefined time toT=5s, and parameters modify as λ1=−1/5,λ2=2,λ3=11/23/5; other parameters remain unchanged. The simulation results are shown in Figures 4–6. Figures 4 and 5 show that states ξ1 and η1 are synchronized after T=5s (Figure 6). The controller u also has a small fluctuation at t=5s, and the chattering phenomenon is very small.Figure 4
Time response trajectory ofe1 by using the proposed method (12) with T=5s.Figure 5
Time response trajectories ofξ1 and η1 by using the proposed method (12) with T=5s.Figure 6
Time response trajectory of controlleru by using the proposed method (12) with T=5s.Obviously, the proposed control method (12) in this study can ensure the synchronization between master system (1) and slave system (2) at a predefined time and can also overcome the influence of unknown gain gt,η1,η2.


## 5. Conclusion

In this study, the predefined time synchronization problem of uncertain chaotic systems was investigated. The fuzzy logic system was used to estimate the unknown function. A time-varying sliding mode was constructed. The proposed varying-time terminal sliding mode control method in this study made all signals bounded and the synchronization error entered a small neighborhood of zero after the predefined time. Simulation results show the effectiveness of the method.



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