Passivity-Based Adaptive Hybrid Synchronization of a New Hyperchaotic System with Uncertain Parameters

Abstract

We investigate the adaptive hybrid synchronization problem for a new hyperchaotic system with uncertain parameters. Based on the passivity theory and the adaptive control theory, corresponding controllers and parameter estimation update laws are proposed to achieve hybrid synchronization between two identical uncertain hyperchaotic systems with different initial values, respectively. Numerical simulation indicates that the presented methods work effectively.

1. Introduction

Hyperchaos, characterized as a chaotic attractor with more than one positive Lyapunov exponent, was first reported by Rössler [1]. Due to its great potential in theoretical and engineering applications, hyperchaos has been investigated extensively over the past three decades. Since the hyperchaotic Rössler system was reported, many more hyperchaotic systems have been proposed, such as hyperchaotic Chua’s system, hyperchaotic Chen system, and hyperchaotic LC oscillator system.

Very recently, the authors [2] constructed a new 4D hyperchaotic system by adding one state variable into the 3D Lü chaotic system. The new hyperchaotic system is shown in the following form: 𝑥̇₁=𝑎𝑥₂−𝑥₁+𝑥₄,𝑥̇₂=𝑐𝑥₂−𝑥₁𝑥₃,𝑥̇₃=−𝑏𝑥₃+𝑥₁𝑥₂,𝑥̇₄=𝑑𝑥₁+𝑘𝑥₂𝑥₃, (1) where 𝑥₁, 𝑥₂, 𝑥₃, and 𝑥₄ are state variables; 𝑎, 𝑏, 𝑐, 𝑑, and 𝑘 are system parameters, respectively. System (1) is dissipative and has only one equilibrium point (0,0,0,0). When 𝑎=35, 𝑏=3, 𝑐=12, 𝑑=1, and 𝑘=0.5, system (1) exhibits a hyperchaotic attractor, which is illustrated in Figure 1.

(a)

(b)

(c)

(d)

Figure 1 

Hyperchaotic attractor for system (1).

In recent years, chaos/hyperchaos synchronization has attracted increasingly attentions due to its potential applications in the fields of secure communication and optical, chemical, physical, and biological systems, and so forth [3–5]. Until now, a wide variety of approaches have been proposed for the synchronization of chaotic/hyperchaotic systems, such as linear or nonlinear feedback control [6], delayed feedback control [7], adaptive control [8], backstepping design [9], and sliding mode control [10], just to name a few. Among all kinds of synchronization schemes, hybrid synchronization, which has been proposed by Li [11], is a noticeable one. In hybrid synchronization scheme, the complete synchronization and antisynchronization coexist in the system. So, to apply hybrid synchronization to communication systems, the security and secrecy of communication can be enhanced greatly [12].

Nowadays, the concept of passivity for nonlinear systems has aroused new interest in nonlinear system control. By applying the passivity theory, Yu [13] designed a linear feedback controller to control the Lorenz system. Wei and Luo [14] proposed an adaptive passivity-based controller to control chaotic oscillations in the power system. In [15, 16], Kemih realized chaos control for chaotic Lü system and for nuclear spin generator system, respectively. In [17], Wang and Liu also applied this theory to achieve synchronization between two identical unified chaotic systems. Passivity-based nonlinear controllers were obtained in [18, 19] to synchronize between two identical chaotic systems and between two different chaotic systems, respectively.

In [20], Huang et al. applied the passivity theory to investigate the hybrid synchronization of a hyperchaotic Lü system, but their method was based on exactly knowing the systems structure and parameters. In practical situations, some or all of the systems parameters cannot be exactly known in priori. Therefore, it is necessary to consider hybrid synchronization of hyperchaotic systems in the presence of uncertain parameters. In this paper, we apply the passivity theory to investigate the adaptive hybrid synchronization problem of a new hyperchaotic system with uncertain parameters.

2. Brief Introduction of the Passivity Theory

Consider a nonlinear system modeled by the following ordinary differential equation: ̇(𝑥)(𝑥)(𝑥),𝑥=𝑓+𝑔𝑢,𝑦=ℎ (2) where 𝑥∈𝑅^𝑛 is the state variable; 𝑢∈𝑅^𝑚 and 𝑦∈𝑅^𝑚 are input and output values, respectively. 𝑓(𝑥) and 𝑔(𝑥) are smooth vector fields and ℎ(𝑥) is a smooth mapping. Suppose that the vector field 𝑓 has at least one equilibrium point. Without loss of generality, one can assume that the equilibrium point is 𝑥=0. If the equilibrium point is not at 𝑥=0, the equilibrium point can be shifted to 𝑥=0 by coordinate transform.

Definition 1 (see [21]). System (2) is a minimum phase system if 𝐿_𝑔ℎ(0) is nonsingular and 𝑥=0 is one of the asymptotically stabilized equilibrium points of 𝑓(𝑥).

Definition 2 (see [13]). System (2) is passive if there exists a real constant 𝛽 such that for for all 𝑡≥0, the following inequality holds: ^𝑡₀𝑢^𝑇(𝜏)𝑦(𝜏)𝑑𝜏≥𝛽, (3) or there exists a 𝜌≥0 and a real constant 𝛽 such that ^𝑡₀𝑢^𝑇(𝜏)𝑦(𝜏)𝑑𝜏+𝛽≥^𝑡₀𝜌𝑦^𝑇(𝜏)𝑦(𝜏)𝑑𝜏. (4)
If system (2) has relative degree [1,…,1] at 𝑥=0 (i.e., 𝐿_𝑔ℎ(0) is nonsingular) and the distribution spanned by the vector field 𝑔₁(𝑥),…,𝑔_𝑚(𝑥) is innovative, then it can be represented as the following normal form: 𝑧̇=𝑓₀(𝑧)̇+𝑝𝑧,𝑦𝑦,𝑦=𝑏𝑧,𝑦+𝑎𝑧,𝑦𝑢, (5) where 𝑎(𝑧,𝑦) is nonsingular for any (𝑧,𝑦).

Theorem 3 (see [13]). If system (2) is a minimum phase system and has relative degree [1,1,…] at 𝑥=0, then system (5) will be equivalent to a passive system and will be asymptotically stable at any equilibrium points through the following local feedback control: 𝑢=𝑎𝑧,𝑦⁻¹−𝑏^𝑇−(𝑧)𝑧,𝑦𝜕𝑊𝑝𝜕𝑧𝑧,𝑦−𝛼𝑦+𝑣. (6)

3. Hybrid Synchronization of the New Hyperchaotic System

Let system (1) be the drive system, and the response system is given by the following form: 𝑤̇₁=𝑎𝑤₂−𝑤₁+𝑤₄,𝑤̇₂=𝑐𝑤₂−𝑤₁𝑤₃+𝑢₁,𝑤̇₃=−𝑏𝑤₃+𝑤₁𝑤₂,𝑤̇₄=𝑑𝑤₁+𝑘𝑤₂𝑤₃+𝑢₂, (7) where 𝑎, 𝑏, 𝑐, 𝑑, and 𝑘 are unknown parameters; 𝑢₁ and 𝑢₂ are controllers to be determined.

To investigate the hybrid synchronization, we define the state errors between the drive system (1) and the response system (7) as 𝑒₁=𝑤₁+𝑥₁,𝑒₂=𝑤₂+𝑥₂,𝑒₃=𝑤₃−𝑥₃,𝑒₄=𝑤₄+𝑥₄. (8) Then the following error dynamical system can be obtained 𝑒₁̇=𝑎𝑒₂−𝑒₁+𝑒₄,𝑒₂̇=𝑐𝑒₂−𝑒₁𝑒₃+𝑥₁𝑒₃−𝑥₃𝑒₁+𝑢₁,𝑒₃̇=−𝑏𝑒₃+𝑒₁𝑒₂−𝑥₁𝑒₂−𝑥₂𝑒₁,𝑒₄̇=𝑑𝑒₁+𝑘𝑒₂𝑒₃−𝑥₂𝑒₃+𝑥₃𝑒₂+𝑢₂. (9)

Let 𝑧₁=𝑒₁, 𝑧₂=𝑒₃, 𝑦₁=𝑒₂, and 𝑦₂=𝑒₄; the error dynamical system (9) can be rewritten as 𝑧̇₁=𝑎𝑦₁−𝑧₁+𝑦₂,𝑧̇₂=−𝑏𝑧₂+𝑧₁𝑦₁−𝑥₁𝑦₁−𝑥₂𝑧₁,𝑦̇₁=𝑐𝑦₁−𝑧₁𝑧₂+𝑥₁𝑧₂−𝑥₃𝑧₁+𝑢₁,𝑦̇₂=𝑑𝑧₁+𝑘𝑦₁𝑧₂−𝑥₂𝑧₂+𝑥₃𝑦₁+𝑢₂, (10) which is a normal formal 𝑧̇=𝑓₀(𝑧)̇+𝑝𝑧,𝑦𝑦,𝑦=𝑏𝑧,𝑦+𝑎𝑧,𝑦𝑢, (11) where 𝑧=[𝑧₁,𝑧₂]^𝑇,  𝑦=[𝑦₁,𝑦₂]^𝑇 and 𝑓₀(𝑧)=−𝑎𝑧₁−𝑥₂𝑧₁−𝑏𝑧₂,𝑝=𝑧𝑧,𝑦𝑎1₁−𝑥₁0,𝑏=𝑐𝑦₁−𝑧₁𝑧₂+𝑥₁𝑧₂−𝑥₃𝑧₁𝑑𝑧₁+𝑘𝑦₁𝑧₂−𝑥₂𝑧₂+𝑥₃𝑦₁. (12)

Theorem 4. The error dynamical system (9) is a minimum phase system.

Proof. Choose the following storage function: 𝑉(𝑧)+1𝑧,𝑦=𝑊2𝑦^𝑇1𝑦+2𝑎₁−𝑎²+12𝑏₁−𝑏²+12𝑐₁−𝑐²+12𝑑₁−𝑑²+12𝑘₁−𝑘², (13) where 𝑊(𝑧)=(𝑁²/4𝑎𝑏)𝑧²₁+(1/2)𝑧²₂ is a Lyapunov function of 𝑓₀(0), 𝑁 is a bound of 𝑥₂, namely, |𝑥₂|≤𝑁, and 𝑎₁, 𝑏₁, 𝑐₁, 𝑑₁, and 𝑘₁ are estimated values of the uncertain parameters 𝑎, 𝑏, 𝑐, 𝑑, and 𝑘, respectively.
The zero dynamics of system (11) describes the internal dynamics, which is consistent with the external constraint 𝑦=0, that is, 𝑧̇=𝑓₀(𝑧), then we have 𝑑𝑊(𝑧)=(𝑧)𝑑𝑡𝜕𝑊𝑓𝜕𝑧⁰(𝑧)𝑁=−²𝑧2𝑏²¹−𝑏𝑧²₂−𝑥₂𝑧₁𝑧₂=−𝑏𝑧₂+𝑥₂𝑧2𝑏¹²+𝑥²₂𝑧4𝑏²¹−𝑁²𝑧2𝑏²¹≤−𝑏𝑧₂+𝑥₂𝑧2𝑏¹²−𝑁²𝑧4𝑏²¹≤0. (14) Then, 𝑓₀(𝑧) is globally asymptotically stable. Meanwhile, 𝐿_𝑔ℎ(0)=¹⁰₀₁ is nonsingular. In the light of Definition 1, system (9) is a minimum phase system.

Theorem 5. If we choose the controllers as 𝑢₁=−𝑐₁+𝛼𝑦₁+𝑥₃−𝑁²𝑧2𝑏¹+𝑣₁,𝑢₂=−𝑑₁+𝑁²𝑧2𝑎𝑏¹−𝑘₁𝑦₁𝑧₂+𝑥₃𝑦₁−𝑥₂𝑧₂−𝛼𝑦₂+𝑣₂, (15) and the parameter estimation update laws as 𝑎̇₁𝑏̇=0,₁𝑐=0,₁=𝑦²₁,𝑑₁=𝑧₁𝑦₂,𝑘₁=𝑤₂𝑤₃+𝑥₂𝑥₃𝑦₂, (16) where 𝑣=[𝑣₁,𝑣₂]^𝑇 is an external signal vector which is connected with the reference input, the error dynamical system (9) will be asymptotically stable at any desired equilibrium points with different values of 𝑣, and the hybrid synchronization between the two hyperchaotic systems (1) and (7) with different initial values will be achieved.

Proof. Taking the time derivative of 𝑉(𝑧,𝑦) along the trajectory of the error dynamical system (9) yields 𝑑𝑉=(𝑧)𝑑𝑡𝑧,𝑦𝜕𝑊̇𝜕𝑧𝑧+𝑦^𝑇𝑦̇+𝑎₁−𝑎𝑎̇₁+𝑏₁−𝑏𝑏̇₁+𝑐₁−𝑐𝑐₁̇+𝑑₁−𝑑𝑑̇₁+𝑘₁−𝑘𝑘̇₁=𝜕𝑊(𝑧)𝑓𝜕𝑧⁰(𝑧)+𝜕𝑊(𝑧)𝑝𝜕𝑧𝑧,𝑦𝑦+𝑦^𝑇𝑏𝑧,𝑦+𝑦^𝑇𝑎𝑧,𝑦𝑢+𝑎₁−𝑎𝑎̇₁+𝑏₁−𝑏𝑏̇₁+𝑐₁−𝑐𝑐₁̇+𝑑₁−𝑑𝑑̇₁+𝑘₁−𝑘𝑘̇₁. (17)
According to Theorem 4, the error dynamical system (9) is a minimum phase system, that is, (𝜕𝑊(𝑧)/𝜕𝑧)𝑓₀(𝑧)≤0, then (17) becomes 𝑑𝑉≤(𝑧)𝑑𝑡𝑧,𝑦𝜕𝑊𝑝𝜕𝑧𝑧,𝑦𝑦+𝑦^𝑇𝑏𝑧,𝑦+𝑦^𝑇𝑎𝑢+𝑎𝑧,𝑦₁−𝑎𝑎̇₁+𝑏₁−𝑏𝑏̇₁+𝑐₁−𝑐𝑐₁̇+𝑑₁−𝑑𝑑̇₁+𝑘₁−𝑘𝑘̇₁. (18)
Substituting (15) and (16) into (18) yields 𝑑𝑉𝑑𝑡𝑧,𝑦≤−𝛼𝑦^𝑇𝑦+𝑣^𝑇𝑦. (19)
Then, taking integration on both sides of (19), we get 𝑉𝑧,𝑦−𝑉𝑧₀,𝑦₀≤−^𝑡₀𝛼𝑦^𝑇(𝜏)𝑦(𝜏)+𝑑𝜏_𝑡0𝑣^𝑇(𝜏)𝑦(𝜏)𝑑𝜏. (20)
For 𝑉(𝑧,𝑦)≥0, let 𝑉(𝑧₀,𝑦₀)=𝜇; the above inequality can be rewritten as ^𝑡₀𝑣^𝑇(𝜏)𝑦(𝜏)𝑑𝜏+𝜇≥^𝑡₀𝛼𝑦^𝑇(𝜏)𝑦(𝜏)≥𝑑𝜏+𝑉𝑧,𝑦_𝑡0𝛼𝑦^𝑇(𝜏)𝑦(𝜏)𝑑𝜏. (21)
According to Definition 2, system (9) is a passive system. Because 𝑊(𝑧) is radially unbounded, it follows from (13) that 𝑉(𝑧,𝑦) is also radially unbounded, so that the closed-loop system is bounded state stable for [𝑧^𝑇,𝑦^𝑇]^𝑇. This means that we can use the controllers (15) and parameter estimation update laws (16) to regulate the error dynamical system (9) to the equilibrium points, and the two hyperchaotic systems (1) and (7) with different initial values will be synchronized.

4. A Numerical Simulation

In this section, a numerical simulations is carried out to verify the theoretical results obtained in Section 3. In the following numerical simulation, the fourth order Runge-Kutta method is applied to solve the equations with time step size 0.001. The system parameters are selected as 𝑎=35, 𝑏=3, 𝑐=12, 𝑑=1, and 𝑘=0.5, so that system (1) can exhibit a hyperchaotic attractor.

For the hybrid synchronization of the new hyperchaotic system, we consider the drive system (1) and the response system (7). The initial values for them are given as 𝑥₁(0)=1, 𝑥₂(0)=1, 𝑥₃(0)=1, 𝑥₄(0)=1, and 𝑤₁(0)=2, 𝑤₂(0)=2, 𝑤₃(0)=2, 𝑤₄(0)=2, respectively. Thus, the initial errors are 𝑒₁(0)=3, 𝑒₂(0)=3, 𝑒₃(0)=1, 𝑒₄(0)=3. And the initial values of the parameter estimation update laws are 𝑎₁(0)=𝑏₁(0)=𝑐₁(0)=𝑑₁(0)=𝑘₁(0)=0.1. We choose 𝛼=1 and 𝑣₁=𝑣₂=0. Figure 2 shows the time response of states determined by the drive system (1) and the response system (7) with the controllers (15) and the parameter estimation update laws (16). Figures 2(a), 2(b), and 2(d) illustrate antisynchronization of 𝑥₁ versus 𝑤₁, 𝑥₂ versus 𝑤₂, and 𝑥₄ versus 𝑤₄, and Figure 2(c) illustrates complete synchronization of 𝑥₃ versus 𝑤₃. As expected, one can observe that the trajectories of the error dynamical system (9) are asymptotically stabilized at the equilibrium point 𝑂(0,0,0,0), as illustrated in Figure 3. From Figures 2 and 3, we can conclude that the hybrid synchronization between the drive system (1) and the response system (7) starting from different initial values is achieved. And the estimations of the parameters are shown in Figure 4, which converge to constants as time goes.

(a)

(b)

(c)

(d)

Figure 2 

The time response of states for the drive system (1) and the response system (7).

Figure 3 

The time response of error states for the error dynamical system (9).

Figure 4 

The estimations of the parameters 𝑐1, 𝑑1, 𝑘1.

5. Conclusions

In this paper, we have investigated the adaptive hybrid synchronization of a new hyperchaotic system with unknown parameters, which includes complete synchronization and antisynchronization. Based on the passivity theory and the adaptive control theory, hybrid synchronization between two identical hyperchaotic systems with uncertain parameters starting from different initial values is achieved. A numerical simulation is presented to illustrate and verify the theoretical analysis. The simulation result and the theoretical analysis agree quite well.

Acknowledgments

This work was supported by the Natural Science Foundation of Yunnan Province under Grant no. 2009CD019 and the Natural Science Foundation of China under Grants no. 61065008, no. 61005087, and no. 61263042.