Predefined Time Synchronization Control for Uncertain Chaotic Systems

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.


Introduction
In recent decades, chaotic synchronization has been a research hotspot. e main reason is its wide application, such as in the fields of secure communication, biological systems, and so on [1][2][3][4][5][6]. Up to now, there are many synchronization methods between two chaotic systems, such as adaptive control [7][8][9][10][11], active control [12][13][14], impulsive control [15][16][17], and sliding mode control [18][19][20][21][22][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 fractionalorder 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. e 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. e rest of this study is organized as follows. Some preliminaries are given in Section 2. 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.

Preliminaries
e master system is described as where ξ 1 , ξ 2 ∈ R are the states of system (1), and f 1 (t, ξ 1 , ξ 2 ) ∈ R is a nonlinear function. e slave system is described as where η 1 , η 2 ∈ R are the states of system (2), f 2 (t, η 1 , η 2 ) ∈ R is a nonlinear function, u ∈ R is the control input, and g(t, η 1 , η 2 ) is a control gain function. Define synchronization errors e 1 � η 1 − ξ 1 , e 2 � η 2 − ξ 2 . e aim of this study is to design a varying-time terminal sliding mode control method, so that the synchronization error e 1 reaches a small neighborhood of zero in the predefined time. According to (1) and (2), one gets the synchronization error system as In order to design the controller in this study, the following assumptions need to be made. Assumption 1. States ξ 1 , ξ 2 , η 1 , η 2 are measurable, and initial values ξ 2 (0) � η 2 (0).
Assumption 2. f 1 (t, ξ 1 , ξ 2 ) and f 2 (t, η 1 , η 2 ) are unknown but bounded. Remark 1. ξ 2 (0) � η 2 (0) 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 timevarying terminal sliding mode is considered: where T is a preset time, 0 < q/p < 1, q and p are the odds, α, β are the design positive constant, and λ 1 , λ 2 , and λ 3 satisfy the following conditions: (1) In order to ensure that the initial value of system (3) belongs to the sliding mode (4), i.e., (2) e sliding mode (4) is continuous at t � T, i.e., (3) In order to ensure that sliding mode (4) can quickly approach the origin, i.e., e derivation of Δ q/p with respect to time t may appear singular problem, and we modify Δ q/p−1 _ Δ as
e simulation results are shown in Figures 4-6. Figures 4 and 5 show that states ξ 1 and η 1 are synchronized after T � 5s ( Figure 6). e controller u also has a small fluctuation at t � 5s, and the chattering phenomenon is very small.
Obviously, the proposed control method (12) in this study can ensure the synchronization between master   Computational Intelligence and Neuroscience   Computational Intelligence and Neuroscience system (1) and slave system (2) at a predefined time and can also overcome the influence of unknown gain g(t, η 1 , η 2 ).

Conclusion
In this study, the predefined time synchronization problem of uncertain chaotic systems was investigated. e fuzzy logic system was used to estimate the unknown function. A time-varying sliding mode was constructed. e 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.

Data Availability
e datasets generated for this study are included within the article.