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

Abstract

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.

1. 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.

Stochastic processes such as electrocardiography, stock market, and Brownian motion have been extensively investigated. Brownian motion is usually described by the Wiener process, which is a continuous-time stochastic process, and taken as the nature uncertainties and perturbations. The stability of stochastic systems has been extensively studied because the uncertainties and perturbations are similar to those in real-world systems [12–15]. Stochastic stability can be guaranteed by the Ito formula, which is derived by using Taylor series expansion. Recently, studies on stochastic chaos synchronization have received a lot of attention. As a special problem of chaotic systems, some kinds of control methods and convergence judgments have been conferred in this decade [16–19].

Sliding-mode control (SMC) is a special case of variable structure systems. By designing a switching surface and using a discontinuous control law, the trajectories of dynamic systems can be forced to slide along the fixed sliding manifold. Then, the system can be compelled to satisfy the desired performance. Generally speaking, the two main advantages of SMC are the uncomplicated dynamic behaviors of the system with the designed switching functions and strong robustness to system uncertainties. Many studies have been conducted on SMC [20–24]. The asymptotic stability theorem with the fundamental theory of Lyapunov has been used to prove the stability of systems. However, the stability over the finite horizon of time cannot be guaranteed. In many applications, it is desirable for the trajectories of the system to converge to a stable equilibrium state in finite time rather than asymptotically. Recently, a particular property of asymptotic stability, finite-time stability, has attracted research interest due to its feasibility and advantages. Based on this property, stability can be achieved within the settling time for many control methods [25–28]. However, few studies have focused on the finite-time stability of stochastic processes. Therefore, the present study designs a sliding-mode control scheme to force the error states of two switched stochastic Chua’s circuits to converge to the sliding surface and for their mean value to converge to zero. In this paper, the partial finite-time stability means that there must be at least one state that can achieve finite-time stability, and other states can obtain asymptotical stability [29]. In order to ensure partial finite-time stability, the Ito formula and Lyapunov stability theory are used to guarantee the synchronization and the mean of the error states reaching zero for the switched stochastic Chua’s circuits.

The rest of this paper is organized as follows. In Section 2, an appropriate switching surface and a sliding-mode controller are designed to drive the system trajectory to reach the sliding surface and stochastically synchronize the master-slave switched stochastic Chua’s circuits on the sliding manifold. A numerical example is given in Section 3. Finally, conclusion is presented in Section 4. Note that throughout the remainder of this paper, the notation 𝐸[𝑥(𝑡)] denotes the mean of 𝑥(𝑡)⋅|𝑥(𝑡)| denotes the modulus of 𝑥(𝑡)⋅sgn(𝑥(𝑡)) is defined as if 𝑥(𝑡)>0, sgn(𝑥(𝑡))=1; if 𝑥(𝑡)=0, sgn(𝑥(𝑡))=0; if 𝑥(𝑡)<0, sgn(𝑥(𝑡))=−1.

2. System Description and Main Results

Consider the Chua’s circuit described as follows:𝑥̇(𝑡)𝑦=̇𝑧(𝑡̇()𝑝𝑡=)(𝑦=𝑥(𝑡−()𝑡𝑞)−𝑦−𝑥(𝑡𝑦()𝑡(−)𝑡)−𝑟+𝑧𝑓𝑧(𝑡(()𝑥𝑡,)(𝑡,))),(2.1) 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 ℓ>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:𝑑𝑥_𝑚𝑑(𝑡𝑦)_𝑚=𝜅_𝛿(𝑡)𝑝𝑦_𝑚(𝑡)−𝑥_𝑚(𝑡)−𝑓𝑥_𝑚(𝑡)𝑑𝑡,𝑑𝑧(_𝑚𝑡)=𝜅_𝛿(𝑡)𝑥_𝑚(𝑡)−𝑦_𝑚(𝑡)+𝑧_𝑚(𝑡)𝑑𝑡,𝑑𝑥_𝑠(𝑡)=𝜅_𝛿(𝑡)−𝑞𝑦_𝑚(𝑡)−𝑟𝑧_𝑚(𝑡)𝑑𝑡,𝑑𝑦_𝑠(𝑡)=𝜅_𝛿(𝑡)𝑝𝑦_𝑠(𝑡)−𝑥_𝑠(𝑡)−𝑓𝑥_𝑠(𝑡)𝑑𝑡+𝜎₁(𝑡)𝑒_𝑥(𝑡)𝑑𝑤₁(𝑡),(𝑡𝑑)𝑧=_𝑠𝜅_𝛿(𝑡)𝑥_𝑠(𝑡)−𝑦_𝑠(𝑡)+𝑧_𝑠(𝑡)−𝑢(𝑡)𝑑𝑡+𝜎₂(𝑡)𝑒_𝑦(𝑡)𝑑𝑤₂(𝑡),(𝑡)=𝜅_𝛿(𝑡)−𝑞𝑦_𝑠(𝑡)−𝑟𝑧_𝑠(𝑡)𝑑𝑡+𝜎₃(𝑡)𝑒_𝑧(𝑡)𝑑𝑤₃(𝑡),(2.2) where 𝑓(𝑥_𝑚(𝑡))=𝑏𝑥_𝑚(𝑡)+(1/2)(𝑎−𝑏)[|𝑥_𝑚(𝑡)+1|−|𝑥_𝑚(𝑡)−1|] and 𝑓(𝑥_𝑠(𝑡))=𝑏𝑥_𝑠(𝑡)+(1/2)(𝑎−𝑏)[|𝑥_𝑠(𝑡)+1|−|𝑥_𝑠(𝑡)−1|], 𝑢(𝑡) is the sliding-mode control input, 𝜅_𝛿(𝑡)≥1 is a time-driven switching gain, and 𝛿(⋅)∶[0,∞)→{1,2,…,𝑁} is a piecewise switching signal. Moreover, 𝛿(𝑡)=𝑖 implies that the 𝑖th switching gain is activated. 𝜎_𝑗(𝑡) (𝑗∈1∼3) is the adjustable weight, and 𝑤_ℎ(𝑡) (ℎ∈1∼3) is the Wiener process motion satisfying 𝐸[𝑑𝑤_ℎ(𝑡)]=0 and 𝐸⌊(𝑑𝑤_ℎ(𝑡))²⌋=𝑑𝑡. The synchronization error is defined by 𝑒_𝑥(𝑡)=𝑥_𝑚(𝑡)−𝑥_𝑠(𝑡), 𝑒_𝑦(𝑡)=𝑦_𝑚(𝑡)−𝑦_𝑠(𝑡), and 𝑒_𝑧(𝑡)=𝑧_𝑚(𝑡)−𝑧_𝑠(𝑡).

Define an indicator function that 𝜉(𝑡)=(𝜉₁(𝑡),𝜉₂(𝑡),…,𝜉_𝑁(𝑡))^𝑇 with(i)if 0≤𝑡<𝑇, then 𝜉_𝑖(𝑡)=10,,owthheenrwthisee,𝑖thmodeactivesattime𝑁𝑇(𝑖−1)≤𝑡<𝑁𝑇𝑖,(2.3)(ii)if 𝑇≤𝑡, then 𝜉_𝑖(𝑡+𝑣𝑇)=𝜉_𝑖(𝑡),𝑣=1,2,3,…,(2.4) where 𝑖=1,2,3,…,𝑁 and 𝑇 is a period of time.

Then, the master and salve systems can be respectively rewritten as𝑑𝑥_𝑚(𝑡)=^𝑁_𝑖=1𝑑𝑦_𝑚𝜉_𝑖(𝑡)𝜅_𝑖𝑝𝑦_𝑚(𝑡)−𝑥_𝑚(𝑡)−𝑓𝑥_𝑚(𝑡)𝑑𝑡,(𝑡)=^𝑁_𝑖=1𝑑𝑧_𝑚𝜉_𝑖(𝑡)𝜅_𝑖𝑥_𝑚(𝑡)−𝑦_𝑚(𝑡)+𝑧_𝑚(𝑡)𝑑𝑡,(𝑡)=^𝑁_𝑖=1𝑑𝑥_𝑠𝜉_𝑖(𝑡)𝜅_𝑖−𝑞𝑦_𝑚(𝑡)−𝑟𝑧_𝑚(𝑡)𝑑𝑡,(𝑡)=^𝑁_𝑖=1𝑑𝑦_𝑠𝜉_𝑖(𝑡)𝜅_𝑖𝑝𝑦_𝑠(𝑡)−𝑥_𝑠(𝑡)−𝑓𝑥_𝑠(𝑡)𝑑𝑡+𝜎₁(𝑡)𝑒_𝑥(𝑡)𝑑𝑤₁(𝑡),(𝑡)=^𝑁_𝑖=1𝑑𝑧_𝑠𝜉_𝑖(𝑡)𝜅_𝑖𝑥_𝑠(𝑡)−𝑦_𝑠(𝑡)+𝑧_𝑠(𝑡)−𝑢(𝑡)𝑑𝑡+𝜎₂(𝑡)𝑒_𝑦(𝑡)𝑑𝑤₂(𝑡),(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝑞𝑦_𝑠(𝑡)−𝑟𝑧_𝑠(𝑡)𝑑𝑡+𝜎₃(𝑡)𝑒_𝑧(𝑡)𝑑𝑤₃(𝑡).(2.5) It is noted that ∑^𝑁𝑖=1𝜉_𝑖(𝑡)=1 under all switching rules.

Then, the dynamics of synchronization error between the master and slave systems, (2.5) can be described by𝑑𝑒_𝑥(𝑡)=^𝑁_𝑖=1𝑑𝑒_𝑦𝜉_𝑖(𝑡)𝜅_𝑖𝑝𝑒_𝑦(𝑡)−𝑒_𝑥(𝑡)−𝑓𝑒_𝑥(𝑡)𝑑𝑡−𝜎₁(𝑡)𝑒_𝑥(𝑡)𝑑𝑤₁(𝑡),(2.6a)(𝑡)=^𝑁_𝑖=1𝑑𝑒_𝑧𝜉_𝑖(𝑡)𝜅_𝑖𝑒_𝑥(𝑡)−𝑒_𝑦(𝑡)+𝑒_𝑧(𝑡)+𝑢(𝑡)𝑑𝑡−𝜎₂(𝑡)𝑒_𝑦(𝑡)𝑑𝑤₂(𝑡),(2.6b)(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝑞𝑒_𝑦(𝑡)−𝑧𝑒_𝑧(𝑡)𝑑𝑡−𝜎₃(𝑡)𝑒_𝑧(𝑡)𝑑𝑤₃(𝑡),(2.6c) 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.𝜓𝑠̇((𝑡𝑡))==−𝑒_𝑦𝛽(𝑡₂)+𝜓(𝑡),(2.7)𝑒^2𝑦(𝑡)||𝑠(𝑡)||sgn(𝑠(𝑡))−𝜂¹+𝑝𝑒_𝑥+𝜂₂||𝑠(𝑡)||^𝛼sgn(𝑠(𝑡))+𝜂₃+𝜂₄||𝑒_𝑦(𝑡)||^𝛼sgn𝑒_𝑦(𝑡)(𝑡)+𝛽₂𝑒_𝑦(𝑡)+⎛⎜⎜⎝𝛽¹𝑒^2𝑥(𝑡)+𝑝ℓ||𝑒_𝑥(𝑡)||²+𝜂₃||𝑒_𝑥(𝑡)||+𝜂₄||𝑒_𝑥(𝑡)||^𝛼+1||𝑒_𝑦(𝑡)||⎞⎟⎟⎠sgn𝑒^𝑦(𝑡),(2.8) where 𝑠∈𝑅¹, 𝛽₁, and 𝛽₂ are two setting positive constant such that 𝛽₁>(1/2)𝜎²¹(𝑡) and 𝛽₂>(1/2)𝜎²²(𝑡), respectively. 𝜂_𝑖(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[𝑠(̇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 𝑒_𝑦̇(𝑡) is𝑒_𝑦(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖𝑒^𝑦(0)+^𝑡0𝑒_𝑥(𝜏)−𝑒_𝑦(𝜏)+𝑒_𝑧(𝜏)+𝑢(𝜏)𝑑𝜏−𝜎₂(𝜏)𝑒_𝑦(𝜏)𝑑𝑤₂(𝜏).(2.9) Combining (2.7) and (2.9) yields𝑠(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖𝑒^𝑦(0)+𝜓(𝑡)+^𝑡0𝑒_𝑥(𝜏)−𝑒_𝑦(𝜏)+𝑒_𝑧(𝜏)+𝑢₁(𝜏)𝑑𝜏−𝜎₂(𝜏)𝑒_𝑦(𝜏)𝑑𝑤₂(𝜏).(2.10) From (2.10), we can obtain the following:𝑑𝑠(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖𝑒_𝑥(𝑡)−𝑒_𝑦(𝑡)+𝑒_𝑧(𝑡)+𝑢(𝑡)+𝜓̇(𝑡)𝑑𝑡−𝜎₂(𝑡)𝑒_𝑦(𝑡)𝑑𝑤₂(𝑡).(2.11) 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: 𝑉̇(𝑡)≤−Δ𝑉^𝛼(𝑡),∀𝑡≥𝑡₀,𝑉𝑡₀≥0,(2.12) where Δ>0 and 0<𝛼<1 are two constants. Then, for any given 𝑡₀, 𝑉(𝑡) satisfies the following inequality: 𝑉^1−𝛼(𝑡)≤𝑉^1−𝛼𝑡₀−Δ(1−𝛼)𝑡−𝑡₀,𝑡₀≤𝑡≤𝑡_𝑟,(2.13)𝑉(𝑡)≡0, for all 𝑡≥𝑡_𝑟, with 𝑡_𝑟 given by 𝑡_𝑟=𝑡₀+𝑉^1−𝛼(𝑡)Δ(1−𝛼).(2.14)
According to the Lyapunov stability theorem and Lemma 2.1, if there is a sliding-mode controller such that 𝑉_𝑠≤−Δ𝑉^𝛼𝑠(𝑡), where 𝑉_𝑠=(1/2)𝑠²(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 is𝑢(𝑡)=−−⎛⎜⎜⎝(1𝛽+𝑝)𝑒_𝑥(𝑡)+𝑒_𝑦(𝑡)−𝑒_𝑧(𝑡)−𝛽₂𝑒_𝑦(𝑡)−𝜂₃+𝜂₄||𝑒_𝑦(𝑡)||^𝛼sgn𝑒_𝑦(𝑡)₁𝑒^2𝑥(𝑡)+𝑝ℓ||𝑒_𝑥(𝑡)||²+𝜂₃||𝑒_𝑥(𝑡)||+𝜂₄||𝑒_𝑥(𝑡)||^𝛼+1||𝑒_𝑦(𝑡)||⎞⎟⎟⎠sgn𝑒^𝑦(𝑡).(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.

Proof. Define Lyapunov function 𝑉_𝑠(𝑡)=12𝑠²(𝑡)=12||𝑠(𝑡)||².(2.16) By using the Ito formula, one can obtain that 𝐸𝑉_𝑠(𝑡)=𝐸⎡⎢⎢⎣^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖⎧⎪⎨⎪⎩𝑠⎛⎜⎜⎝𝑒_𝑥+𝑒_𝑦(𝑡)−𝑒_𝑦(𝑡)+𝑒_𝑧(𝑡)−(1+𝑝)𝑒_𝑥(𝑡)−⎛⎜⎜⎝𝛽(𝑡)−𝑒_𝑧(𝑡)−𝛽₂𝑒_𝑦(𝑡)−𝜂₃+𝜂₄||𝑒_𝑦(𝑡)||^𝛼sgn𝑒_𝑦(𝑡)₁𝑒^2𝑥(𝑡)+𝑝ℓ||𝑒_𝑥(𝑡)||²+𝜂₃||𝑒_𝑥(𝑡)||+𝜂₄||𝑒_𝑥(𝑡)||^𝛼+1||𝑒_𝑦(𝑡)||⎞⎟⎟⎠sgn𝑒^𝑦+𝑝𝑒_𝑥(𝑡)(𝑡)+𝛽₂𝑒_𝑦(𝑡)−𝛽²𝑒^2𝑦(𝑡)||𝑠(𝑡)||sgn(𝑠(𝑡))−𝜂¹+⎛⎜⎜⎝𝛽+𝜂₂||𝑠(𝑡)||^𝛼sgn(𝑠(𝑡))₁𝑒^2𝑥(𝑡)+𝑝ℓ||𝑒_𝑥(𝑡)||²+𝜂₃||𝑒_𝑥(𝑡)||+𝜂₄||𝑒_𝑥(𝑡)||^𝛼+1||𝑒_𝑦(𝑡)||⎞⎟⎟⎠sgn𝑒^𝑦+𝜂(𝑡)₃+𝜂₄||𝑒_𝑦(𝑡)||^𝛼sgn𝑒_𝑦(𝑡)⎞⎟⎟⎠+12𝜎²(𝑡)𝑒_𝑦(𝑡)²⎫⎪⎬⎪⎭⎤⎥⎥⎦.(2.17) Since 𝛽₂>(1/2)𝜎²²(𝑡), regarding the above inequality, 𝐸𝑉_𝑠(𝑡)≤𝐸^𝑁_𝑖=1≤𝐸_𝑁𝜉_𝑖(𝑡)𝜅_𝑖𝑠−𝜂₁+𝜂₂||𝑠(𝑡)||^𝛼sgn(𝑠(𝑡))_𝑖=1≤𝐸_𝑁𝜉_𝑖(𝑡)𝜅_𝑖−𝜂¹||𝑠(𝑡)||−𝜂₂||𝑠(𝑡)||^𝛼+1_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−2^(𝛼+1)/2𝜂₂𝑉_𝑠(𝑡)^(𝛼+1)/2.(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.

Proof. Define Lyapunov function 𝑉_𝑒(𝑡)=12𝑒^2𝑥(𝑡)+12𝑒^2𝑦(𝑡).(2.19) By using the Ito formula, one can obtain that 𝐸𝑉_𝑒̇(𝑡)=𝐸⎡⎢⎢⎣^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖⎧⎪⎨⎪⎩𝑒_𝑥(𝑡)−𝑝𝑒_𝑥(𝑡)+𝑝𝑒_𝑦(𝑡)−𝑝𝑓𝑒_𝑥(𝑡)+12𝜎²¹+𝑒_𝑦𝑒^2𝑥(𝑡)−⎛⎜⎜⎝𝛽(𝑡)𝑒_𝑥(𝑡)−𝑒_𝑦(𝑡)+𝑒_𝑧(𝑡)−(1+𝑝)𝑒_𝑥(𝑡)+𝑒_𝑦(𝑡)−𝑒_𝑧(𝑡)−𝛽₂𝑒_𝑦(𝑡)₁𝑒^2𝑥(𝑡)+𝑝ℓ||𝑒_𝑥(𝑡)||²+𝜂₃||𝑒_𝑥(𝑡)||+𝜂₄||𝑒_𝑥(𝑡)||^𝛼+1||𝑒_𝑦(𝑡)||⎞⎟⎟⎠sgn𝑒^𝑦−𝜂(𝑡)₃+𝜂₄||𝑒_𝑦(𝑡)||^𝛼sgn𝑒_𝑦(𝑡)+12𝜎²²(𝑡)𝑒^2𝑦(𝑡)⎫⎪⎬⎪⎭⎤⎥⎥⎦.(2.20) Based on the Lipschitz condition and setting 𝛽₁>(1/2)𝜎²¹(𝑡) and 𝛽₂>(1/2)𝜎²²(𝑡), regarding the above inequality, 𝐸𝑉_𝑒̇(𝑡)≤𝐸^𝑁_𝑖=1≤𝐸_𝑁𝜉_𝑖(𝑡)𝜅_𝑖−𝜂³||𝑒_𝑥(𝑡)||−𝜂₄||𝑒_𝑥(𝑡)||^𝛼+1−𝜂₃||𝑒_𝑦(𝑡)||−𝜂₄||𝑒_𝑦(𝑡)||^𝛼+1_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝜂⁴||𝑒_𝑥(𝑡)||^𝛼+1−𝜂₄||𝑒_𝑦(𝑡)||^𝛼+1.(2.21) From the above equation, we can obtain that 𝐸𝑉_𝑒̇(𝑡)≤𝐸^𝑁_𝑖=1𝐸𝑉_𝑒̇𝜉_𝑖(𝑡)𝜅_𝑖−𝜂⁴||𝑒_𝑥(𝑡)||^𝛼+1,(𝑡)≤𝐸^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝜂⁴||𝑒_𝑥(𝑡)||^𝛼+1.(2.22) Then, they can be rewritten as 𝐸𝑉̇^{𝑒2/(𝛼+1)}(𝑡)≤𝐸^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−2𝜂^42/(𝛼+1)12||𝑒^𝑥𝐸𝑉̇(𝑡)||²_{𝑒2/(𝛼+1)},(𝑡)≤𝐸^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−2𝜂^42/(𝛼+1)12||𝑒^𝑦(𝑡)||².(2.23) It implies that 𝐸2𝑉̇^{𝑒2/(𝛼+1)}(𝑡)≤𝐸^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−2𝜂^42/(𝛼+1)12||𝑒^𝑥(𝑡)||²−2𝜂^42/(𝛼+1)12||𝑒^𝑦𝐸𝑉̇(𝑡)||²_{𝑒2/(𝛼+1)},(𝑡)≤𝐸^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝜂^42/(𝛼+1)12||𝑒^𝑥(𝑡)||²+||𝑒_𝑦(𝑡)||².(2.24) Therefore, we can get that 𝐸𝑉_𝑒̇(𝑡)≤𝐸^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝜂⁴21||𝑒^𝑥(𝑡)||²+||𝑒_𝑦(𝑡)||²^(𝛼+1)/2≤𝐸_𝑁_𝑖=1𝜉_𝑖(𝑡)𝜅_𝑖−𝜂⁴𝑉_𝑒^(𝛼+1)/2.(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.

Remark 2.4. From the system (2.1), 𝑓(𝑥(𝑡))=𝑏𝑥(𝑡)+(1/2)(𝑎−𝑏)[|𝑥(𝑡)+1|−|𝑥(𝑡)−1|] can be rewritten as 𝑓(𝑥(𝑡))=⎧⎪⎨⎪⎩𝑏𝑎𝑏𝑥𝑥𝑥(((𝑡𝑡𝑡))),+−𝑎𝑎−+𝑏𝑏,,iiifff𝑥𝑥||𝑥((𝑡𝑡())𝑡)><||≤1−,11,.(2.26) Therefore, we can have ||𝑓𝑥_𝑚(𝑡)−𝑓𝑥_𝑠(𝑡)||=⎧⎪⎨⎪⎩||𝑏𝑥^𝑚||𝑎𝑥_𝑚(𝑡)−𝑏𝑥_𝑠(𝑡)||,if𝑥(𝑡)>1,||𝑏𝑥_𝑚(𝑡)−𝑎𝑥_𝑠(𝑡)||,if||𝑥(𝑡)||≤1,(𝑡)−𝑏𝑥_𝑠(𝑡)||,if𝑥(𝑡)<−1.(2.27) From the definition of the Chua’s circuit [31], |𝑎|>|𝑏| can be obtained. It implies that the following inequality is achieved: ||𝑓𝑥_𝑚(𝑡)−𝑓𝑥_𝑠(𝑡)||≤|𝑎|||𝑥_𝑚(𝑡)−𝑥_𝑠(𝑡)||.(2.28) From the above reasoning, it can be sure that 𝑓(𝑥(𝑡)) satisfies the Lipschitz condition with Lipschitz constant ℓ≥|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.

3. 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, 𝛽₁=𝛽₂=1, ℓ=1.3, and the sliding-mode controller gains are given by 𝜂₁=𝜂₂=𝜂₃=𝜂₄=0.3. 𝑤_ℎ(𝑡) is the Wiener process motion with time 𝑇, shown in Figure 1, and 𝜎₁(𝑡)=𝜎₂(𝑡)=𝜎₃(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.

Figure 1 

Time responses of the Wiener process motion 𝑤ℎ(𝑡).

Figure 2 

Time responses of piecewise switching signal 𝜅𝛿(𝑡).

Figure 3 

State responses of the switched stochastic Chua’s system.

Figure 4 

Time responses of the synchronization error.

Figure 5 

Time responses of the mean values of the synchronization error.

Figure 6 

Time responses and mean values of the sliding surface function 𝑠(𝑡).

4. 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.

Acknowledgment

This work was supported by the National Science Council of Republic of China under contract 100-2221-E-366-001 and 100-2632-E-366-001-MY3.