Synchronization of Unified Chaotic Systems Using Sliding Mode Controller

Abstract

This paper presents a method for synchronizing the unified chaotic systems via a sliding mode controller (SMC). The unified chaotic system and problem formulation are described. Two identical unified chaotic systems can be synchronized using the SMC technique. The switching surface and its controller design are developed in detail. Simulation results show the feasibility of a chaotic secure communication system based on the synchronization of the Lorenz circuits via the proposed SMC.

1. Introduction

Chaos theory is an extensively studied branch of the theory of nonlinear systems. Lorenz presented the first well-known chaotic system, which was a third-order autonomous system with only two multiplication-type quadratic terms but displayed very complex dynamical behaviors [1]. A chaotic system is a very complex, dynamical nonlinear system whose response has intrinsic characteristics such as broadband noise-like waveforms, difficult predictability, and sensitivity to initial condition variations [2, 3]. These properties are advantageous in secure communication systems. Therefore, the synchronization of chaotic circuits for secure communication has received a lot of research attention [4–7]. Studies have shown that it is possible to set up a chaotic communication system to obtain secure communication [8].

The synchronization between master (transmitter) and slave (receiver) chaotic systems has potential applications for secure communication [9–13]. Several control schemes have been developed for the synchronization of chaotic systems. Sliding mode control is a popular nonlinear control strategy [14–19]. For sliding mode controller (SMC) design, the Lyaponov stability method is applied to keep the nonlinear system under control. The sliding mode approach transforms a higher-order system into a lower-order system, allowing a simple control algorithm to be applied, making the system very straightforward and robust. Sliding mode control has been applied to the synchronization of chaotic systems [5, 6, 11].

The present study designs an SMC-based chaotic secure communication system. To achieve this goal, a proportional-integral (PI) switching surface is first designed for the considered error dynamics system in sliding motion, and then, based on it, a sliding mode controller is derived. This controller is effective and guarantees both the occurrence of sliding motion and synchronization of the master-slave unified chaotic systems. Finally, an example is given to illustrate the usefulness of the proposed SMC.

2. Problem Formulation and Main Results

A sliding mode control system consisting of master and slave chaotic systems and information𝑝(𝑡) is shown in Figure 1.

Figure 1 

Block diagram of SMC-based synchronized chaotic systems.

Consider the following unified chaotic system: 𝑥̇₁=(25𝛼+10)𝑥₂−𝑥₁𝑥̇₂=(28−35𝛼)𝑥₁+(29𝛼−1)𝑥₂−𝑥₁𝑥₃𝑥̇₃=𝑥₁𝑥₂−(8+𝛼)3𝑥₃, (2.1) where 𝑥₁, 𝑥₂, and 𝑥₃ are states of system (2.1) and 𝛼∈[0,1]. Obviously, the system (2.1) becomes the original Lorenz system for 𝛼=0, while the system (2.1) becomes the original Chen system for 𝛼=1. When 𝛼=4/5, the system (2.1) becomes the critical system. In particular, the system bridges the gap between Lorenz system and Chen system. Moreover, the system is always chaotic in the whole interval 𝛼∈[0,1]. Before the secure communication system can be constructed, the synchronization problem of the system based on the sliding mode control must first be solved. For the unified chaotic system, the master and slave systems are defined as 𝑥̇_𝑚1(𝑡)=(25𝛼+10)𝑥_𝑚2(𝑡)−𝑥_𝑚1(𝑡)𝑥̇_𝑚2(𝑡)=(28−35𝛼)𝑥_𝑚1(𝑡)+(29𝛼−1)𝑥_𝑚2(𝑡)−𝑥_𝑚1(𝑡)𝑥_𝑚3(𝑡)(𝑡)𝑥̇+𝑝_𝑚3(𝑡)=𝑥_𝑚1(𝑡)𝑥_𝑚2(𝑡)−(8+𝛼)3𝑥_𝑚3(𝑡),𝑥̇_𝑠1(𝑡)=(25𝛼+10)𝑥_𝑠2(𝑡)−𝑥_𝑠1(𝑡)𝑥̇_𝑠2(𝑡)=(28−35𝛼)𝑥_𝑠1(𝑡)+(29𝛼−1)𝑥_𝑠2(𝑡)−𝑥_𝑚1(𝑡)𝑥_𝑠3(𝑡)(𝑡)𝑥̇+𝑢_𝑠3(𝑡)=𝑥_𝑚1(𝑡)𝑥_𝑠2(𝑡)−(8+𝛼)3𝑥_𝑠3(𝑡), (2.2) where 𝑢(𝑡)  is the controller output used to synchronize the master and slave systems (2.2), and 𝑝(𝑡)  is the embedded message bounded by 𝑝(𝑡)≤𝜓,𝜓>0. (2.3) The control goal is for the two unified chaotic systems (2.2) to be synchronized such that the resulting error vector satisfies lim_𝑡→∞‖𝑒(𝑡)‖=lim_𝑡→∞𝑥_𝑚𝑖(𝑡)−𝑥_𝑠𝑖(𝑡)=0,𝑖=1,2,3. (2.4) The error vectors and error dynamics are defined as 𝑒₁(𝑡)=𝑥_𝑚1(𝑡)−𝑥_𝑠1(𝑡)𝑒₂(𝑡)=𝑥_𝑚2(𝑡)−𝑥_𝑠2(𝑡)𝑒₃(𝑡)=𝑥_𝑚3(𝑡)−𝑥_𝑠3(𝑡),𝑒₁̇(𝑡)=𝑥̇_𝑚1(𝑡)−𝑥̇_𝑠1(𝑡)𝑒₂̇(𝑡)=𝑥̇_𝑚2(𝑡)−𝑥̇_𝑠2(𝑡)𝑒₃̇(𝑡)=𝑥̇_𝑚3(𝑡)−𝑥̇_𝑠3(𝑡). (2.5) Then, the following error dynamics are obtained: 𝑒₁̇(𝑡)=(25𝛼+10)𝑒₂(𝑡)−𝑒₁(𝑡)𝑒₂̇(𝑡)=(28−35𝛼)𝑒₁(𝑡)+(29𝛼−1)𝑒₂(𝑡)−𝑥_𝑚1𝑒₃(𝑡)(𝑡)(𝑡)𝑒̇+𝑝−𝑢₃(𝑡)=𝑥_𝑚1𝑒₂(𝑡)−(8+𝛼)3𝑒₃(𝑡). (2.6) Therefore, an SMC must be designed such that the resulting error vector satisfies lim_𝑡→∞‖𝐸(𝑡)‖=lim_𝑡→∞𝑒₁(𝑡)𝑒₂(𝑡)𝑒₃(𝑡)→0. (2.7)

To stabilize the error dynamics (2.6) and achieve synchronization, two basic steps are used: first, an appropriate switching surface is selected such that the sliding motion on the sliding manifold is stable and ensures lim_(𝑡→∞)‖𝐸(𝑡)‖=0 ; second, an SMC law which guarantees the existence of the sliding mode 𝑠(𝑡)=0 is established. To guarantee the asymptotic stability of the sliding mode, the PI switching surface 𝑠(𝑡) is defined as 𝑠(𝑡)=𝑒₂(𝑡)+^𝑡₀(25𝛼+10)𝑒₁(𝜏)+𝑥_𝑚1𝑒₃(𝜏)+𝛽𝑒₂(𝜏)𝑑𝜏, (2.8) where 𝛽>0 is given. It is well known that when the system operates in the sliding mode, the following equation must be satisfied [20, 21]: 𝑠(𝑡)=𝑒₂(𝑡)+^𝑡₀(25𝛼+10)𝑒₁(𝜏)+𝑥_𝑚1𝑒₃(𝜏)+𝛽𝑒₂(𝜏)𝑑𝜏=0. (2.9) Since 𝑠(𝑡)=0, we consequently have 𝑠̇(𝑡)=𝑒₂̇(𝑡)+(25𝛼+10)𝑒₁(𝑡)+𝑥_𝑚1𝑒₃(𝜏)+𝛽𝑒₂(𝑡)=0. (2.10) From (2.10), the following is obtained: 𝑒₂̇(𝑡)=−(25𝛼+10)𝑒₁(𝑡)+𝑥_𝑚1𝑒₃(𝑡)+𝛽𝑒₂(𝑡). (2.11) Then, the equivalent sliding mode dynamics is obtained as 𝑒₁̇(𝑡)=(25𝛼+10)𝑒₂(𝑡)−𝑒₁(𝑡)𝑒₂̇(𝑡)=−(25𝛼+10)𝑒₁(𝑡)+𝑥_𝑚1𝑒₃(𝑡)+𝛽𝑒₂(𝑡)𝑒₃̇(𝑡)=𝑥_𝑚1𝑒₂(𝑡)−(8+𝛼)3𝑒₃(𝑡). (2.12) The stability of the sliding mode dynamics (2.11) is analyzed below based on the Lyapunov stability theory.

The Lyapunov function is selected as 𝑉=0.5(𝑒²₁(𝑡)+𝑒²₂(𝑡)+𝑒²₃(𝑡)), which leads to 𝑉̇(𝑡)=𝑒₁(𝑡)𝑒₁̇(𝑡)+𝑒₂(𝑡)𝑒₂̇(𝑡)+𝑒₃(𝑡)𝑒₃̇(𝑡)=(25𝛼+10)𝑒₂(𝑡)−𝑒₁(𝑡)𝑒₁(𝑡)−(25𝛼+10)𝑒₁(𝑡)+𝑥_𝑚1𝑒₃(𝑡)+𝛽𝑒₂(𝑡)𝑒₂(𝑡)+𝑥_𝑚1𝑒₂(𝑡)−(8+𝛼)3𝑒₃(𝑡)𝑒₃(𝑡)=−(25𝛼+10)𝑒²₁(𝑡)−𝛽𝑒²₂(𝑡)−(8+𝛼)3𝑒²₃(𝑡)≤0. (2.13)

According to Lyapunov stability theory, the sliding motion on the sliding manifold is stable and ensures lim_𝑡→∞‖𝐸(𝑡)‖=lim_𝑡→∞‖[𝑒₁(𝑡)𝑒₂(𝑡)𝑒₃(𝑡)]‖=0.

Having established the appropriate switching surface (2.8), as described above, the next step is to design an SMC scheme to drive the system trajectories onto the sliding mode 𝑠(𝑡)=0. This study proposes the following SMC: 𝑢(𝑡)=𝑢₁(𝑡)+𝜂𝜓sign(𝑠(𝑡)),𝜂>1, (2.14) where 𝑢₁(𝑡)=(38+10𝛼)𝑒₁(𝑡)+(29𝛼−1+𝛽)𝑒₂(𝑡).

Before the scheme of the controller is given, the reaching condition of the sliding mode is derived below.

Main Theorem
Consider the error dynamics (2.6). If this system is controlled by 𝑢(𝑡) in (2.14), then the system trajectory converges to the sliding surface 𝑠(𝑡)=0 and satisfies  lim_𝑡→∞‖𝐸(𝑡)‖=lim_𝑡→∞‖[𝑒₁(𝑡)𝑒₂(𝑡)𝑒₃(𝑡)]‖=0.

Proof. Consider the following Lyapunov function candidate: 𝑉(𝑡)=0.5𝑠²(𝑡). (2.15) Taking the derivative of (2.15) with respect to time and using (2.11), (2.14), and (2.15) yields 𝑉̇(𝑡)(𝑡)𝑠̇(𝑡)=𝑠(𝑡)𝑒̇=𝑠₂(𝑡)+(25𝛼+10)𝑒₁(𝑡)+𝑥_𝑚1𝑒₃(𝑡)+𝛽𝑒₂(𝑡)≤|𝑠(𝑡)|𝜓−𝑠(𝑡)𝑢(𝑡). (2.16) Since 𝜓>0 and 𝜂>1, 𝑉̇(𝑡)=𝑠(𝑡)𝑠(̇𝑡)<0 can be derived when 𝑠≠0. Thus, according to Lyapunov stability theory, 𝑠(𝑡) always converges to the switching surface 𝑠=0. Furthermore, since the error dynamics in the sliding manifold is asymptotically stable according to the discussion above, the error dynamic response of lim_𝑡→∞‖𝐸(𝑡)‖=lim_𝑡→∞‖[𝑒₁(𝑡)𝑒₂(𝑡)𝑒₃(𝑡)]‖=0 is satisfied. Hence, the proof is achieved completely.

After the SMC is designed to ensure lim_𝑡→∞‖𝐸(𝑡)‖=lim_𝑡→∞‖[𝑒₁(𝑡)𝑒₂(𝑡)𝑒₃(𝑡)]‖=0, then 𝑒₁̇(𝑡)=(25𝛼+10)𝑒₂(𝑡)−𝑒₁(𝑡)𝑒̇=0₂(𝑡)=(28−35𝛼)𝑒₁(𝑡)+(29𝛼−1)𝑒₂(𝑡)−𝑥_𝑚1𝑒₃(𝑡)(𝑡)(𝑡)𝑒̇+𝑝−𝑢=0₃(𝑡)=𝑥_𝑚1𝑒₂(𝑡)−(8+𝛼)3𝑒₃(𝑡)⇒𝑒̇=0¹(𝑡)𝑒̇=0₂(𝑡)(𝑡)(𝑡)𝑒̇=0+𝑝−𝑢=0₃(𝑡)=0. (2.17) It can be inferred that lim_𝑡→∞𝑝(𝑡)−𝑢(𝑡)=0 (2.18) which means that the message 𝑝(𝑡) can be approximated by the control 𝑢(𝑡). From previous studies [22, 23], the control input 𝑢(𝑡) can be approximated by the following continuous equivalent control 𝑢_eq(𝑡): 𝑢_eq(𝑡)=𝑢₁(𝑡)𝑠(𝑡)+𝜂𝜓|𝑠(𝑡,)|+𝜎 (2.19) where 𝜎 is an arbitrarily small positive constant. When 𝜎 is sufficiently small, then (2.19) will arbitrarily approach (2.14) and input message 𝑝(𝑡) can be recovered using (2.19).

3. Numerical Simulation and Analysis

In this section, the continuous equivalent control 𝑢_eq(𝑡) (2.19) is utilized for the synchronization of unified chaotic circuits (2.1). When 𝛼=0, the system becomes the original Lorenz system as 𝑥̇₁=10𝑥₂−𝑥₁𝑥̇₂=28𝑥₁−𝑥₂−𝑥₁𝑥₃𝑥̇₃=𝑥₁𝑥₂−83𝑥₃. (3.1)

As mentioned in [24], when directly implementing nonlinear Lorenz systems with electronic circuits, the major difficulty is that the state variables of the system (3.1) occupy a wide dynamic range with values that exceed reasonable power supply limits. However, this difficulty can be eliminated by a simple transformation of variables. Let the Lorenz equations be transformed into 𝑥̇_𝑚1=10𝑥_𝑚2−𝑥_𝑚1𝑥̇_𝑚2=28𝑥_𝑚1−𝑥_𝑚2−𝑘₁𝑥_𝑚1𝑥_𝑚3𝑥̇_𝑚3=𝑘₁𝑥_𝑚1𝑥_𝑚2−83𝑥_𝑚3. (3.2)

The system (3.2) with 𝑘₁=10, which is referred to as the transmitter, can be more easily implemented because the state variables never exceed the range of typical power supply limits.

Now consider the following Lorenz circuits:

Master Lorenz circuit as 𝑥̇_𝑚1(𝑡)=10𝑥_𝑚2(𝑡)−𝑥_𝑚1(𝑡)𝑥̇_𝑚2(𝑡)=28𝑥_𝑚1(𝑡)−𝑥_𝑚2(𝑡)−10𝑥_𝑚1(𝑡)𝑥_𝑚3(𝑡)(𝑡)𝑥̇+𝑝_𝑚3(𝑡)=10𝑥_𝑚1(𝑡)𝑥_𝑚2(𝑡)−83𝑥_𝑚3(𝑡), (3.3) Slave Lorenz circuit as 𝑥̇_𝑠1(𝑡)=10𝑥_𝑠2(𝑡)−𝑥_𝑠1(𝑡)𝑥̇_𝑠2(𝑡)=28𝑥_𝑠1(𝑡)−𝑥_𝑠2(𝑡)−10𝑥_𝑚1(𝑡)𝑥_𝑠3(𝑡)(𝑡)𝑥̇+𝑢_𝑠3(𝑡)=10𝑥_𝑚1(𝑡)𝑥_𝑠1(𝑡)𝑥_𝑠2(𝑡)−83𝑥_𝑠3(𝑡), (3.4) where 𝑥̇_𝑚 and 𝑥̇_𝑠 denote the derivatives of 𝑥_𝑚 and 𝑥_𝑠 with respect to time 𝑡, respectively. The input message 𝑝(𝑡) is a sine wave (0.5 V, 5 Hz) embedded into the chaotic transmitter and the equivalent SMC synchronization scheme (2.9) is given in the receiver. The initial value conditions [𝑥_𝑚1(0),𝑥_𝑚2(0),𝑥_𝑚3(0)]=[0.40.10.2] and [𝑥_𝑠1(0),𝑥_𝑠2(0),𝑥_𝑠3(0)]=[0.10.10.1] are used in this example. The control toolbox of MATLAB was used to simulate the proposed secure commutation system. Figure 2 shows the states of the master and slave systems. Figure 3 shows the simulation results of the switch surface 𝑠(𝑡). Figure 4 shows the simulation results of errors between state 𝑥_𝑚 and state𝑥_𝑠. These figures show that the switching surface 𝑠(𝑡) converges to almost zero within 8 s and that the synchronization errors converge to almost zero after 0.5 s. The master and slave are then synchronous. Figure 5 shows the simulation results of the continuous equivalent control 𝑢_eq(𝑡) and input message 𝑝(𝑡). The input message  𝑝(𝑡) can be recovered.

Figure 2 

States of master and slave systems.

Figure 3 

Simulation results of switch surface 𝑠(𝑡).

Figure 4 

Simulation results of errors between state 𝑥𝑚 and state 𝑥𝑠.

Figure 5 

Simulation results of continuous equivalent control 𝑢eq(𝑡) and input message𝑝(𝑡).

4. Conclusion

This study presented a method for synchronizing the unified chaotic systems via an SMC. The simulation results show the feasibility of the chaotic secure communication system based on the synchronization of the Lorenz circuits via an SMC.

Acknowledgments

The authors thank the National Science Council of Taiwan for supporting this work under Grants NSC 101-2622-E-269-012-CC3 and NSC 101-2221-E-241-002. The authors also wish to thank the anonymous reviewers for providing constructive suggestions.