Partial Finite-Time Synchronization of Switched Stochastic Chua ’ s Circuits via Sliding-Mode Control

This paper considers the problem of partial finite-time synchronization between switched stochastic Chua’s circuits accompanied by a time-driven switching law. Based on the Ito formula and Lyapunov stability theory, a sliding-mode controller is developed to guarantee the synchronization of switched stochastic master-slave Chua’s circuits and for the mean of error states to obtain the partial finite-time stability. Numerical simulations demonstrate the effectiveness of the proposed methods.


Introduction
The concept of chaos synchronization in message transmission has been extensively studied.The synchronization of chaos is a key technology in generating identical chaotic waveforms in the transmitter and receiver for signal decoding.Under the assumption that the structure of nonlinearity or matching condition is known, studies on chaos synchronization have been concerned with control methods and applications 1-5 .Many natural physical systems such as chemical processes, mechanical systems, and a variety of power systems can be described by hybrid models comprising continuous and discrete dynamic behaviors.A special case is a hybrid system composed of many subsystems and a rule that governs the switching between these subsystems.By neglecting the details of the discrete behavior and instead of considering all possible switching patterns for a certain class, a switched system may be derived from a hybrid system 6 .Recently, the problems of stability analysis and synchronization of switched systems have attracted a lot of attention 7-11 .In the present study, special chaotic systems whose gain is changed by switching rules are designed to force the speed of system response to be fast or slow, as with frequency modulation.where x t , y t , and z t are system states; f x t is a three-segment piecewise linear function f x t bx t 1/2 a−b |x t 1|−|x t −1| that satisfies the Lipschitz condition with Lipschitz constant > 0. a < −1, −1 < b < 0, p > 0, q > 0, and r > 0 are system parameters.
A set of nonlinear stochastic Chua's circuits are derived with the separate switching rules of the switched system.Master and slave stochastic switched systems are respectively described as follows:

2.5
It is noted that N i 1 ξ i t 1 under all switching rules.Then, the dynamics of synchronization error between the master and slave systems, 2.5 can be described by where f e x t f x m t − f x s t .The main objective of control development in this paper is to select an appropriate switching surface and to design a sliding-mode controller to guarantee partial finite-time synchronization between the master and slave switched stochastic Chua's circuit systems.The first step is to select an appropriate switching surface to ensure the stochastic stability of the sliding motion on the sliding manifold.In order to derive the main results, the following lemma is needed.
Lemma 2.1 see 30 .Assume that a continuous, positive-definite function V t satisfies the following differential inequality: where Δ > 0 and 0 < α < 1 are two constants.Then, for any given t 0 , V t satisfies the following inequality: V t ≡ 0, for all t ≥ t r , with t r given by According to the Lyapunov stability theorem and Lemma 2.1, if there is a sliding-mode controller such that V s ≤ −ΔV α s t , where V s 1/2 s 2 t is the defined Lyapunov function, and Δ > 0 and 0 < α < 1 are two real constants, the error dynamics converging to the sliding surface and E s t 0 reaching in finite time can be achieved.Therefore, the second step is to design the proposed sliding-mode controller u t which is

2.18
From Lemma 2.1, it implies that E s t 0 in finite time with the controller in 2.15 , completing the proof.Theorem 2.3.Based on the design-switching surface in 2.7 and the controller in 2.15 , the partial finite-time synchronization of the sliding motion on the sliding manifold is guaranteed.Then, the mean of E e t on the sliding manifold can achieve the partial finite-time stability.

2.19
By using the Ito formula, one can obtain that

2.21
From the above equation, we can obtain that

2.22
Then, they can be rewritten as 1 2 e y t 2 .

2.23
It implies that

2.24
Therefore, we can get that

2.25
From Lemma 2.1, E e x t and E e y t can converge to zero in finite time t r along the sliding surface.Then, from the error dynamic 2.6c , E e z t can tend to zero as E e y t converging to zero in finite-time t r .It implies that the asymptotical stability of E e z t can be achieved after the time t r .Based on the above proof, the partial finite-time synchronization of the sliding motion on the sliding manifold is guaranteed, completing the proof.

2.26
Therefore, we can have

2.27
From the definition of the Chua's circuit 31 , |a| > |b| can be obtained.It implies that the following inequality is achieved:

2.28
From the above reasoning, it can be sure that f x t satisfies the Lipschitz condition with Lipschitz constant ≥ |a|.
Remark 2.5.In order to avoid chattering, sgn s t is replaced with s t / |s t | ℘ in the simulation, where ℘ is an appropriate minimal value.

An Illustrative Example
Consider the proposed synchronization of switched stochastic Chua's circuits with the parameters given by a −1.28, b −0.69, p 10, q 15, r 0.0385, and T 50 sec .The  parameters for the sliding surface and sliding mode controller are given by α 0.9, β 1 β 2 1, 1.3, and the sliding-mode controller gains are given by η 1 η 2 η 3 η 4 0.3.w h t is the Wiener process motion with time T , shown in Figure 1, and σ 1 t σ 2 t σ 3 t 0.1.The real constant ℘ 10 −3 is given.Time response of the piecewise switching signal κ δ t is shown in Figure 2.With the modulation of the time-driven switching rule, the state responses behave like frequency modulation.The state responses of the stochastic switched Chua's circuits are shown in Figure 3, and the speed of response is different with different system gains.Based on the proposed controller, the partial finite-time stability of the sliding motion on the sliding manifold is shown in Figure 4 which displays the synchronization errors of the stochastic switched Chua's circuits.The mean values of synchronization errors on the sliding manifold reaching the partial finite-time stability are shown in Figure 5.In Figure 6, time responses and mean value of the sliding surface function s t are shown, and it also reveals that E s t 0 is reached in finite time.According to the above simulation, partial finite time synchronization between switched stochastic Chua's circuits and the mean value of the error states reaching zero in finite time on the sliding manifold are guaranteed by the proposed controller.

Conclusion
This study investigated the partial finite-time synchronization problem of stochastic Chua's circuits with switched gains which depend on a time-driven switching law.Based on the Ito formula and Lyapunov stability theory, a sliding-mode controller was proposed to synchronize the switching master and slave stochastic Chua's circuits.The mean value of the error states reaching zero in finite time was demonstrated.Numerical simulations show the effectiveness of the proposed method.

Figure 1 :
Figure 1: Time responses of the Wiener process motion w h t .
|e x t | 2 η 3 |e x t | η 4 |e x t | α 1 R 1 , β 1 , and β 2 are two setting positive constant such that β 1 > 1/2 σ 2 1 t and β 2 > 1/2 σ 22 t , respectively.ηi i ∈ 1 ∼ 4 are the sliding-mode controller gains which are positive constants.When system 2.6a -2.6c is in the sliding mode, the condition E s t E ṡ t 0 has to be satisfied.Then, the stochastic process of the sliding surface s t is considered as follows.The time integration of the error dynamic equations ėy t is t 0 e x τ − e y τ e z τ u τ dτ − σ 2 τ e y τ dw 2 τ .i t κ i e x t − e y t e z t u t ψ t dt − σ 2 t e y t dw 2 t .2.11 e y t − e z t − β 2 e y t − η 3 η 4 e y t α sgn e y t |e x t | 2 η 3 |e x t | η 4 |e x t | α 1 2.15 Theorem 2.2.By setting the sliding-mode controller in 2.15 , the error dynamics in 2.6a -2.6c will converge to the sliding surface, and E s t 0 is reached in finite time.x t − e y t e z t − 1 p e x t e y t − e z t − β 2 e y t − η 3 η 4 e y t α sgn e y t − ⎛ ⎜ ⎝ β 1 e 2 x t p |e x t | 2 η 3 |e x t | η 4 |e x t | α 1