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Mathematical Problems in Engineering
Volume 2011, Article ID 162490, 13 pages
http://dx.doi.org/10.1155/2011/162490
Research Article

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

1Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
2Department of Electrical Engineering, Far East University, Tainan 744, Taiwan
3Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan

Received 17 May 2011; Revised 21 September 2011; Accepted 22 September 2011

Academic Editor: Kwok W. Wong

Copyright © 2011 Zhang-Lin Wan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [15]. 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 [711]. 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 [1215]. 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 [1619].

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 [2024]. 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 [2528]. 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:̇𝑥(𝑡)=𝑝(𝑦(𝑡)𝑥(𝑡)𝑓(𝑥(𝑡))),̇𝑦(𝑡)=𝑥(𝑡)𝑦(𝑡)+𝑧(𝑡),̇𝑧(𝑡)=𝑞𝑦(𝑡)𝑟𝑧(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)(𝑎𝑏)[|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:𝑑𝑥𝑚(𝑡)=𝜅𝛿(𝑡)𝑝𝑦𝑚(𝑡)𝑥𝑚𝑥(𝑡)𝑓𝑚,(𝑡)𝑑𝑡𝑑𝑦𝑚(𝑡)=𝜅𝛿(𝑡)𝑥𝑚(𝑡)𝑦𝑚(𝑡)+𝑧𝑚,(𝑡)𝑑𝑡𝑑𝑧𝑚(𝑡)=𝜅𝛿(𝑡)𝑞𝑦𝑚(𝑡)𝑟𝑧𝑚,(𝑡)𝑑𝑡𝑑𝑥𝑠(𝑡)=𝜅𝛿(𝑡)𝑝𝑦𝑠(𝑡)𝑥𝑠𝑥(𝑡)𝑓𝑠(𝑡)𝑑𝑡+𝜎1(𝑡)𝑒𝑥(𝑡)𝑑𝑤1,(𝑡)𝑑𝑦𝑠(𝑡)=𝜅𝛿(𝑡)𝑥𝑠(𝑡)𝑦𝑠(𝑡)+𝑧𝑠(𝑡)𝑢(𝑡)𝑑𝑡+𝜎2(𝑡)𝑒𝑦(𝑡)𝑑𝑤2(,𝑡)𝑑𝑧𝑠(𝑡)=𝜅𝛿(𝑡)𝑞𝑦𝑠(𝑡)𝑟𝑧𝑠(𝑡)𝑑𝑡+𝜎3(𝑡)𝑒𝑧(𝑡)𝑑𝑤3(,𝑡)(2.2) where 𝑓(𝑥𝑚(𝑡))=𝑏𝑥𝑚(𝑡)+(1/2)(𝑎𝑏)[|𝑥𝑚(𝑡)+1||𝑥𝑚(𝑡)1|] and 𝑓(𝑥𝑠(𝑡))=𝑏𝑥𝑠(𝑡)+(1/2)(𝑎𝑏)[|𝑥𝑠(𝑡)+1||𝑥𝑠(𝑡)1|], 𝑢(𝑡) is the sliding-mode control input, 𝜅𝛿(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. 𝜎𝑗(𝑡) (𝑗13) is the adjustable weight, and 𝑤(𝑡) (13) is the Wiener process motion satisfying 𝐸[𝑑𝑤(𝑡)]=0 and 𝐸(𝑑𝑤(𝑡))2=𝑑𝑡. The synchronization error is defined by 𝑒𝑥(𝑡)=𝑥𝑚(𝑡)𝑥𝑠(𝑡), 𝑒𝑦(𝑡)=𝑦𝑚(𝑡)𝑦𝑠(𝑡), and 𝑒𝑧(𝑡)=𝑧𝑚(𝑡)𝑧𝑠(𝑡).

Define an indicator function that 𝜉(𝑡)=(𝜉1(𝑡),𝜉2(𝑡),,𝜉𝑁(𝑡))𝑇 with(i)if 0𝑡<𝑇, then 𝜉𝑖𝑇(𝑡)=1,whenthe𝑖thmodeactivesattime𝑁𝑇(𝑖1)𝑡<𝑁𝑖,0,otherwise,(2.3)(ii)if 𝑇𝑡, then 𝜉𝑖(𝑡+𝑣𝑇)=𝜉𝑖(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 as𝑑𝑥𝑚(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑝𝑦𝑚(𝑡)𝑥𝑚𝑥(𝑡)𝑓𝑚,(𝑡)𝑑𝑡𝑑𝑦𝑚(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑥𝑚(𝑡)𝑦𝑚(𝑡)+𝑧𝑚,(𝑡)𝑑𝑡𝑑𝑧𝑚(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑞𝑦𝑚(𝑡)𝑟𝑧𝑚,(𝑡)𝑑𝑡𝑑𝑥𝑠(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑝𝑦𝑠(𝑡)𝑥𝑠𝑥(𝑡)𝑓𝑠(𝑡)𝑑𝑡+𝜎1(𝑡)𝑒𝑥(𝑡)𝑑𝑤1,(𝑡)𝑑𝑦𝑠(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑥𝑠(𝑡)𝑦𝑠(𝑡)+𝑧𝑠(𝑡)𝑢(𝑡)𝑑𝑡+𝜎2(𝑡)𝑒𝑦(𝑡)𝑑𝑤2,(𝑡)𝑑𝑧𝑠(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑞𝑦𝑠(𝑡)𝑟𝑧𝑠(𝑡)𝑑𝑡+𝜎3(𝑡)𝑒𝑧(𝑡)𝑑𝑤3.(𝑡)(2.5) It is noted that 𝑁𝑖=1𝜉𝑖(𝑡)=1 under all switching rules.

Then, the dynamics of synchronization error between the master and slave systems, (2.5) can be described by𝑑𝑒𝑥(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑝𝑒𝑦(𝑡)𝑒𝑥𝑒(𝑡)𝑓𝑥(𝑡)𝑑𝑡𝜎1(𝑡)𝑒𝑥(𝑡)𝑑𝑤1,(𝑡)(2.6a)𝑑𝑒𝑦(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑒𝑥(𝑡)𝑒𝑦(𝑡)+𝑒𝑧(𝑡)+𝑢(𝑡)𝑑𝑡𝜎2(𝑡)𝑒𝑦(𝑡)𝑑𝑤2(𝑡),(2.6b)𝑑𝑒𝑧(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑞𝑒𝑦(𝑡)𝑧𝑒𝑧(𝑡)𝑑𝑡𝜎3(𝑡)𝑒𝑧(𝑡)𝑑𝑤3,(𝑡)(2.6c) where 𝑓(𝑒𝑥(𝑡))=𝑓(𝑥𝑚(𝑡))𝑓(𝑥𝑠(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.𝑠(𝑡)=𝑒𝑦𝛽(𝑡)+𝜓(𝑡),(2.7)̇𝜓(𝑡)=2𝑒2𝑦(𝑡)||||𝜂𝑠(𝑡)sgn(𝑠(𝑡))1+𝜂2||||𝑠(𝑡)𝛼𝜂sgn(𝑠(𝑡))+3+𝜂4||𝑒𝑦||(𝑡)𝛼𝑒sgn𝑦(𝑡)+𝑝𝑒𝑥(𝑡)+𝛽2𝑒𝑦𝛽(𝑡)+1𝑒2𝑥||𝑒(𝑡)+𝑝𝑥||(𝑡)2+𝜂3||𝑒𝑥||(𝑡)+𝜂4||𝑒𝑥||(𝑡)𝛼+1||𝑒𝑦||𝑒(𝑡)sgn𝑦,(𝑡)(2.8) where 𝑠𝑅1, 𝛽1, and 𝛽2 are two setting positive constant such that 𝛽1>(1/2)𝜎21(𝑡) and 𝛽2>(1/2)𝜎22(𝑡), respectively. 𝜂𝑖(𝑖14) 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 ̇𝑒𝑦(𝑡) is𝑒𝑦(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑒𝑦(0)+𝑡0𝑒𝑥(𝜏)𝑒𝑦(𝜏)+𝑒𝑧(𝜏)+𝑢(𝜏)𝑑𝜏𝜎2(𝜏)𝑒𝑦(𝜏)𝑑𝑤2.(𝜏)(2.9) Combining (2.7) and (2.9) yields𝑠(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑒𝑦(0)+𝜓(𝑡)+𝑡0𝑒𝑥(𝜏)𝑒𝑦(𝜏)+𝑒𝑧(𝜏)+𝑢1(𝜏)𝑑𝜏𝜎2(𝜏)𝑒𝑦(𝜏)𝑑𝑤2.(𝜏)(2.10) From (2.10), we can obtain the following:𝑑𝑠(𝑡)=𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑒𝑥(𝑡)𝑒𝑦(𝑡)+𝑒𝑧(𝑡)+𝑢(𝑡)+̇𝜓(𝑡)𝑑𝑡𝜎2(𝑡)𝑒𝑦(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: ̇𝑉(𝑡)Δ𝑉𝛼(𝑡),𝑡𝑡0𝑡,𝑉00,(2.12) where Δ>0 and 0<𝛼<1 are two constants. Then, for any given 𝑡0, 𝑉(𝑡) satisfies the following inequality: 𝑉1𝛼(𝑡)𝑉1𝛼𝑡0Δ(1𝛼)𝑡𝑡0,𝑡0𝑡𝑡𝑟,(2.13)𝑉(𝑡)0, for all 𝑡𝑡𝑟, with 𝑡𝑟 given by 𝑡𝑟=𝑡0+𝑉1𝛼(𝑡).Δ(1𝛼)(2.14)
According to the Lyapunov stability theorem and Lemma 2.1, if there is a sliding-mode controller such that 𝑉𝑠Δ𝑉𝛼𝑠(𝑡), where 𝑉𝑠=(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)=(1+𝑝)𝑒𝑥(𝑡)+𝑒𝑦(𝑡)𝑒𝑧(𝑡)𝛽2𝑒𝑦𝜂(𝑡)3+𝜂4||𝑒𝑦||(𝑡)𝛼𝑒sgn𝑦𝛽(𝑡)1𝑒2𝑥||𝑒(𝑡)+𝑝𝑥||(𝑡)2+𝜂3||𝑒𝑥||(𝑡)+𝜂4||𝑒𝑥||(𝑡)𝛼+1||𝑒𝑦||𝑒(𝑡)sgn𝑦(.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 𝑉𝑠1(𝑡)=2𝑠21(𝑡)=2||||𝑠(𝑡)2.(2.16) By using the Ito formula, one can obtain that 𝐸𝑉𝑠(𝑡)=𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑠𝑒𝑥(𝑡)𝑒𝑦(𝑡)+𝑒𝑧(𝑡)(1+𝑝)𝑒𝑥(𝑡)+𝑒𝑦(𝑡)𝑒𝑧(𝑡)𝛽2𝑒𝑦𝜂(𝑡)3+𝜂4||𝑒𝑦||(𝑡)𝛼𝑒sgn𝑦𝛽(𝑡)1𝑒2𝑥||𝑒(𝑡)+𝑝𝑥||(𝑡)2+𝜂3||𝑒𝑥||(𝑡)+𝜂4||𝑒𝑥||(𝑡)𝛼+1||𝑒𝑦||𝑒(𝑡)sgn𝑦(𝑡)+𝑝𝑒𝑥(𝑡)+𝛽2𝑒𝑦𝛽(𝑡)2𝑒2𝑦(𝑡)||||𝜂𝑠(𝑡)sgn(𝑠(𝑡))1+𝜂2||𝑠||(𝑡)𝛼+𝛽sgn(𝑠(𝑡))1𝑒2𝑥||𝑒(𝑡)+𝑝𝑥||(𝑡)2+𝜂3||𝑒𝑥||(𝑡)+𝜂4||𝑒𝑥||(𝑡)𝛼+1||𝑒𝑦||𝑒(𝑡)sgn𝑦+𝜂(𝑡)3+𝜂4||𝑒𝑦||(𝑡)𝛼𝑒sgn𝑦+1(𝑡)2𝜎2(𝑡)𝑒𝑦(𝑡)2.(2.17) Since 𝛽2>(1/2)𝜎22(𝑡), regarding the above inequality, 𝐸𝑉𝑠(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑠𝜂1+𝜂2||||𝑠(𝑡)𝛼sgn(𝑠(𝑡))𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂1||||𝑠(𝑡)𝜂2||||𝑠(𝑡)𝛼+1𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖2(𝛼+1)/2𝜂2𝑉𝑠(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 𝑉𝑒1(𝑡)=2𝑒2𝑥1(𝑡)+2𝑒2𝑦(𝑡).(2.19) By using the Ito formula, one can obtain that 𝐸̇𝑉𝑒(𝑡)=𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝑒𝑥(𝑡)𝑝𝑒𝑥(𝑡)+𝑝𝑒𝑦𝑒(𝑡)𝑝𝑓𝑥+1(𝑡)2𝜎21𝑒2𝑥(𝑡)+𝑒𝑦𝑒(𝑡)𝑥(𝑡)𝑒𝑦(𝑡)+𝑒𝑧(𝑡)(1+𝑝)𝑒𝑥(𝑡)+𝑒𝑦(𝑡)𝑒𝑧(𝑡)𝛽2𝑒𝑦𝛽(𝑡)1𝑒2𝑥||𝑒(𝑡)+𝑝𝑥||(𝑡)2+𝜂3||𝑒𝑥||(𝑡)+𝜂4||𝑒𝑥||(𝑡)𝛼+1||𝑒𝑦||𝑒(𝑡)sgn𝑦𝜂(𝑡)3+𝜂4||𝑒𝑦||(𝑡)𝛼𝑒sgn𝑦+1(𝑡)2𝜎22(𝑡)𝑒2𝑦.(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, 𝐸̇𝑉𝑒(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂3||𝑒𝑥||(𝑡)𝜂4||𝑒𝑥||(𝑡)𝛼+1𝜂3||𝑒𝑦||(𝑡)𝜂4||𝑒𝑦||(𝑡)𝛼+1𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂4||𝑒𝑥||(𝑡)𝛼+1𝜂4||𝑒𝑦||(𝑡)𝛼+1.(2.21) From the above equation, we can obtain that 𝐸̇𝑉𝑒(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂4||𝑒𝑥||(𝑡)𝛼+1,𝐸̇𝑉𝑒(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂4||𝑒𝑥||(𝑡)𝛼+1.(2.22) Then, they can be rewritten as 𝐸̇𝑉𝑒2/(𝛼+1)(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖2𝜂42/(𝛼+1)12||𝑒𝑥||(𝑡)2,𝐸̇𝑉𝑒2/(𝛼+1)(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖2𝜂42/(𝛼+1)12||𝑒𝑦||(𝑡)2.(2.23) It implies that 𝐸2̇𝑉𝑒2/(𝛼+1)(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖2𝜂42/(𝛼+1)12||𝑒𝑥||(𝑡)22𝜂42/(𝛼+1)12||𝑒𝑦||(𝑡)2,𝐸̇𝑉𝑒2/(𝛼+1)(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂42/(𝛼+1)12||𝑒𝑥||(𝑡)2+||𝑒𝑦||(𝑡)2.(2.24) Therefore, we can get that 𝐸̇𝑉𝑒(𝑡)𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂412||𝑒𝑥||(𝑡)2+||𝑒𝑦||(𝑡)2(𝛼+1)/2𝐸𝑁𝑖=1𝜉𝑖(𝑡)𝜅𝑖𝜂4𝑉𝑒(𝛼+1)/2.(2.25) From Lemma 2.1, 𝐸[𝑒𝑥(𝑡)] and 𝐸[𝑒𝑦(𝑡)] can converge to zero in finite time 𝑡𝑟 along the sliding surface. Then, from the error dynamic (2.6c), 𝐸[𝑒𝑧(𝑡)] can tend to zero as 𝐸[𝑒𝑦(𝑡)] converging to zero in finite-time 𝑡𝑟. It implies that the asymptotical stability of 𝐸[𝑒𝑧(𝑡)] can be achieved after the time 𝑡𝑟. 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)(𝑎𝑏)[|𝑥(𝑡)+1||𝑥(𝑡)1|] can be rewritten as ||𝑥||𝑓(𝑥(𝑡))=𝑏𝑥(𝑡)+𝑎𝑏,if𝑥(𝑡)>1,𝑎𝑥(𝑡),if(𝑡)1,𝑏𝑥(𝑡)𝑎+𝑏,if𝑥(𝑡)<1.(2.26) Therefore, we can have ||𝑓𝑥𝑚(𝑥𝑡)𝑓𝑠(||=||𝑡)𝑏𝑥𝑚(𝑡)𝑏𝑥𝑠||||(𝑡),if𝑥(𝑡)>1,𝑎𝑥𝑚(𝑡)𝑎𝑥𝑠||||𝑥||||(𝑡),if(𝑡)1,𝑏𝑥𝑚(𝑡)𝑏𝑥𝑠||(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: ||𝑓𝑥𝑚𝑥(𝑡)𝑓𝑠||||𝑥(𝑡)|𝑎|𝑚(𝑡)𝑥𝑠||(𝑡).(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.

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, =1.3, and the sliding-mode controller gains are given by 𝜂1=𝜂2=𝜂3=𝜂4=0.3. 𝑤(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 =103 is given. Time response of the piecewise switching signal 𝜅𝛿(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.

162490.fig.001
Figure 1: Time responses of the Wiener process motion 𝑤(𝑡).
162490.fig.002
Figure 2: Time responses of piecewise switching signal 𝜅𝛿(𝑡).
162490.fig.003
Figure 3: State responses of the switched stochastic Chua’s system.
162490.fig.004
Figure 4: Time responses of the synchronization error.
162490.fig.005
Figure 5: Time responses of the mean values of the synchronization error.
162490.fig.006
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.

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