Model Reference Control of Hyperchaotic Systems

Abstract

We have applied a famous engineering method, called model reference control, to control hyperchaos. We have proposed a general description of the hyperchaotic system and its reference system. By using the Lyapunov stability theorem, we have obtained the expression of the controller. Four examples for the both certain case and the uncertain case show that our method is very effective for controlling hyperchaotic systems with both certain parameters and uncertain parameters.

1. Introduction

Chaos has received increasing attentions in the last thirty years. Compared with the ordinary chaotic systems, the hyperchaotic systems hold at least two positive Lyapunov exponents and then possess more complicated attractors. Hyperchaotic systems have the characteristics of high capacity, high security, and high efficiency and have been studied in many fields, such as secure communication [1, 2], cellular neural network [3, 4], chemical processing [5], nonlinear circuit [6–8], and other fields [9, 10].

Controlling chaos (or hyperchaos) is very meaningful. The research has been started since the pioneering work of OGY method [11] was published. As the development of computational technique and controlling theory, many methods have been proposed for controlling chaos such as, LMI-based approach [12, 13], sliding control [14], active control [15], optimal control [16, 17], and passivity-based control [18, 19]. However, the hyperchaotic systems are more complex than the ordinary chaotic ones. To obtain the satisfying effect of control, we should focus on more advantage algorithms and ideas of controlling techniques. There have been some results on controlling hyperchaotic systems, such as feedback control [20, 21], adaptive control [22], backstepping control [23], and impulsive control [24].

In this paper, we will show that although a hyperchaotic system is complex, its dynamics can still be controlled along an expected trajectory. The controller is generated by an advantage control method, called model reference control (MRC). Nowadays, MRC is widely used in engineering, such as control of robots [25], mechanical oscillators [26], economic cycle [27], and disease spread [28]. Our aim is to control a hyperchaotic system to track with an expected trajectory with the aid of MRC. In the following section, the formulation of the problem will be presented. In Section 3, we will propose the framework of MRC on hyperchaotic systems with certain parameters and the systems with uncertain parameters. In Section 4, we will give two numerical examples to show the effectiveness of MRC on hyperchaotic systems with certain and uncertain parameters and also two examples for the hyperchaotic systems with uncertain parameters. Finally, the conclusion will be given in Section 5.

2. Problem Formulation

The first example of the hyperchaotic systems was presented by Rössler in 1979 [29]. Since then, other hyperchaotic systems have been reported [30], and many researchers are focusing on the discovery of new hyperchaotic systems and their control. In this section, we will describe the basic formulation of generalized hyperchaotic system and the reference system. Recently, nonlinear scientists are focusing on the control problem of chaotic and hyperchaotic systems with uncertain parameters [31–37]. Hence, we will give the formulation of both the hyperchaotic systems with certain parameters and the systems with uncertain parameters.

Since a hyperchaotic system with certain parameters should have quadratic terms at least, we may formulate the system as follows: 𝑋̇=𝑋^𝑇𝐴𝑋+𝐵𝑋+𝑢, (2.1) where 𝑋 is an 𝑛 dimensional column vector, 𝑋^𝑇 is its transposed matrix, 𝐴 is an 𝑛²×𝑛 matrix including 𝑛 matrices 𝐴₁,𝐴₂,…,𝐴_𝑛 in the form of 𝑛×𝑛, 𝐵 is an 𝑛×𝑛 matrix, and 𝑢 is an 𝑛 dimensional input control column vector. It is worth noting that 𝑋^𝑇𝐴𝑋 is only the nominal expression of matrix multiplication, and it describes the quadratic terms in blocks as follows: 𝑋^𝑇𝐴₁𝑋,𝑋^𝑇𝐴₂𝑋,…,𝑋^𝑇𝐴_𝑛𝑋^𝑇. (2.2)

Equation (2.1) can cover a great many hyperchaotic systems, though it may not fit for the systems with higher order terms or even a fractional order term. The uncertain form of system (2.1) is as follows: 𝑋̇=𝑋^𝑇𝐴𝑋+𝐵𝑋+𝐶(𝑋)𝛼+𝑢, (2.3) where 𝐶(𝑋) is an 𝑛×𝑚 matrix, uncertain parameter 𝛼 is an 𝑚 dimensional column vector, and other variables as above. Here, 𝑚 might be equal to 𝑛, also smaller or bigger than 𝑛.

In this section, we will also introduce a reference system with only linear terms and constants. Our aim is to control the hyperchaotic system track along with the reference model system that exhibits asymptotic stability as follows 𝑋̇_𝑟=𝐵𝑋_𝑟+𝑢, (2.4) where the matrix 𝐵 is a known constant matrix with appropriate dimensions, the eigenvalues of the matrix 𝐵 have negative real part such that the system is asymptotically stable. By letting the system (2.4) be the reference system, we will control the hyperchaotic system (2.1) track along with the system (2.4). Here, 𝑋_𝑟 is called state output vector of the reference model, and 𝑢 the reference input vector.

Particularly, if the reference system has a two-order term, we may write it as 𝑋̇_𝑟=𝑋^𝑇𝐴𝑋+𝐵𝑋_𝑟+𝑢, (2.5) where 𝐴 is a similar matrix as 𝐴.

3. Model Reference Control of Hyperchaotic Systems

In this section, MRC is applied to control a hyperchaotic system with both certain parameters and uncertain parameters. Our objective is to obtain the exact control law 𝑢 such that the original system follows the dynamical behavior of the reference model. We will review the Lyapunov stability theorem for autonomous systems and then give the explicit expression of the controller.

Theorem 3.1 (Lyapunov stability theorem for autonomous systems). Let 𝑥=0 be an equilibrium point for a dynamical system described by 𝑥̇=𝑓(𝑥), (3.1) where 𝑓∶𝐷→ℝ^𝑛 is a locally Lipschitz and 𝐷⊂ℝ^𝑛 a domain that contains the origin. Let 𝑉∶𝐷→ℝ be a continuously differentiable, positive definite function in 𝐷. Then 𝑥=0 is an asymptotically stable equilibrium point, if 𝑉̇(𝑥) is negative definite. The scalar function 𝑉 is a Lyapunov function if 𝑉̇(𝑥) is negative semidefinite in the region 𝐷: 𝑉̇(𝑥)≤0.

Assume that the error system (3.1) satisfies the conditions of Theorem 3.1, and its Lyapunov function has the form as 𝑉(𝑒)=𝑒^𝑇𝑃𝑒, (3.2) where 𝑃 is a symmetric positive definite matrix. The derivative of 𝑉(𝑒) with respect to time is 𝑉̇(𝑒)=𝑒^𝑇̇𝑃𝑒+𝑒^𝑇𝑃𝑒.̇ (3.3)

In the whole process of MRC, the output of reference model and that of the controlled system are compared, and the error vector 𝑒 is 𝑒=𝑋_𝑟−𝑋. (3.4)

If we want to control the system (2.1) to track along with system (2.4), we may have the following results. By using (2.1) and (2.4), we obtain the following error system: 𝑒̇=𝑋̇_𝑟−𝑋̇=𝐵𝑒−𝑋^𝑇𝐴𝑋+𝐵−𝐵𝑋+𝑢−𝑢. (3.5)

We will try to design a control vector 𝑢 such that the objective equation lim_𝑡→∞‖𝑒‖=0. (3.6)

From (3.1) and (3.2), the MRC problem is converted to the asymptotic stability of zero vector of the error system (3.1). Here, we will use the Lyapunov stability theory to determine the proper control law 𝑢. The theorem is as follows.

Theorem 3.2. Let 𝑃 be any symmetric positive definite matrix, 𝜆_max is the largest eigenvalue of the matrix (𝐵^𝑇𝑃+𝑃𝐵), the controller 𝑢=𝐵𝑋+𝑢−𝑋^𝑇𝐴𝑋−𝐵𝑋−𝜆𝐼(𝑋_𝑟−𝑋). The hyperchaotic system (2.1) will asymptotically follow the desired dynamical system (2.4), if 𝜆<−𝜆_max/2 and 𝜆≠0.

Proof. The fact that the hyperchaotic system (2.1) asymptotically follows the system (2.4) is equivalent to asymptotical stability of the error system. Hence, we will try to prove the inequality of 𝑉̇(𝑒)≤0.
For the symmetric property, the scalar equation (3.3) may have the following description: 𝑉̇(𝑒)=𝑒^𝑇̇𝑃𝑒+𝑒^𝑇𝑃𝑒̇=𝑒^𝑇𝐵^𝑇−𝑋^𝑇𝐴^𝑇𝑋−𝑋^𝑇𝐵^𝑇−𝐵^𝑇−𝑢^𝑇+𝑢^𝑇𝑃𝑒+𝑒^𝑇𝑃𝐵𝑒−𝑋^𝑇𝐴𝑋−𝐵−𝐵𝑋−(𝑢−𝑢)=𝑒^𝑇𝐵^𝑇𝑃+𝑃𝐵𝑒+2𝑒^𝑇𝑃−𝑋^𝑇𝐴𝑋−𝐵−𝐵𝑋−(𝑢−𝑢)=𝑒^𝑇𝐵^𝑇𝑃+𝑃𝐵𝑒+2𝑒^𝑇(𝜆𝐼)𝑒=𝑒^𝑇𝐵^𝑇𝑃+𝑃𝐵+2𝜆𝐼𝑒. (3.7) For 𝜆<−𝜆_max/2, the matrix (𝐵^𝑇𝑃+𝑃𝐵+2𝜆𝐼) is negative definite. Also, the scalar 𝑉̇<0 is obtained.
Using the result of Theorem 3.1, it can be concluded that the error 𝑒 will converge to 0 asymptotically. Equivalently, the controller 𝑢 will make the hyperchaotic system (2.1) asymptotically follow the desired dynamical system (2.4).

If we want to control the system (2.1) to track along with system (2.5), we may have the following results. By using (2.1) and (2.4), we obtain the following error system: 𝑒̇=𝑋̇_𝑟−𝑋̇=𝑋^𝑇_𝑟𝐴𝑋_𝑟+𝐵𝑒−𝑋^𝑇𝐴𝑋+𝐵−𝐵𝑋+𝑢−𝑢. (3.8)

It is easy to obtain the control law in the following theorem with similar process of Theorem 3.2.

Theorem 3.3. Let 𝑃 be any symmetric positive definite matrix, 𝜆_max is the largest eigenvalue of the matrix (𝐵^𝑇𝑃+𝑃𝐵), and the controller 𝑢=𝑋^𝑇_𝑟𝐴𝑋_𝑟+𝐵𝑋+𝑢−𝑋^𝑇𝐴𝑋−𝐵𝑋−𝜆𝐼(𝑋_𝑟−𝑋). The hyperchaotic system (2.1) will asymptotically follow the desired dynamical system (2.4), if 𝜆<−𝜆_max/2 and 𝜆≠0.

We can omit the proof of Theorem 3.3, for it is similar to that of Theorem 3.2.

Theorems 3.2 and 3.3 are fit for the hyperchaotic systems with certain parameters. In the following, we will give two similar theorems for the systems with uncertain parameters. To simplify the control process, we will make 𝑃=𝐼.

Theorem 3.4. Let 𝜆_max is the largest eigenvalue of the matrix 𝐵^𝑇, the controller 𝑢=𝐵𝑋+𝑢−𝑋^𝑇𝐴𝑋−𝐵𝑋−𝜆𝐼(𝑋_𝑟−𝑋)+𝐶(𝑋)𝛼, where 𝛼 is the estimation of uncertain parameter 𝛼, and satisfies the differential equations 𝛼̇=𝐶^𝑇(𝑋)𝑒. The hyperchaotic system (2.3) will asymptotically follow the desired dynamical system (2.4), if 𝜆<−𝜆_max and 𝜆≠0.

Proof. For the uncertain property of system (2.3), we should redefine the Lyapunov function as follows 𝑉₁(𝑒)=𝑒^𝑇𝑒+(𝛼−𝛼)^𝑇(𝛼−𝛼). (3.9) Similar to the proof of Theorem 3.2, we have to prove the inequality of 𝑉̇₁(𝑒)≤0: 𝑉̇₁(𝑒)=2𝑒^𝑇̇𝑒−2𝛼̇^𝑇(𝛼−𝛼)=2𝐵𝑋_𝑟+𝑢−𝑋^𝑇𝐴𝑋−𝐵𝑋−𝑢−𝐶(𝑋)𝛼^𝑇𝑒−2𝑒^𝑇𝐶(𝑋)(𝛼−𝛼)=2𝑒^𝑇𝐵^𝑇+𝜆𝐼𝑒. (3.10) For 𝜆<−𝜆_max, the matrix (𝐵^𝑇+𝜆𝐼) is negative definite. Also, the scalar 𝑉̇<0 is obtained.
Hence, the controller 𝑢 will make the hyperchaotic system (2.3) asymptotically follow the desired dynamical system (2.4).

If we want to control the system (2.3) to track along with system (2.5), we may have the following results. We have the following error system: 𝑒̇=𝑋^𝑇_𝑟𝐴𝑋_𝑟+𝐵𝑒−𝑋^𝑇𝐴𝑋+𝐵−𝐵𝑋+𝑢−𝑢−𝐶(𝑋)𝛼. (3.11) It is easy to obtain the control law in the following theorem with similar process of Theorem 3.4.

Theorem 3.5. Let 𝜆_max is the largest eigenvalue of the matrix 𝐵^𝑇, the controller 𝑢=𝐵𝑋+𝑢−𝑋^𝑇𝐴𝑋−𝐵𝑋−𝜆𝐼(𝑋_𝑟−𝑋)+𝐶(𝑋)𝛼, where 𝛼 is the estimation of uncertain parameter 𝛼, and satisfies the differential equations 𝛼̇=𝐶^𝑇(𝑋)𝑒. The hyperchaotic system (2.3) will asymptotically follow the desired dynamical system (2.5), if 𝜆<−𝜆_max and 𝜆≠0.

We can also omit the proof of Theorem 3.5.

4. Numerical Examples

This section has two parts. The first part is the numerical examples for controlling hyperchaotic systems, where all the parameters are certain. We will use the result in Theorem 3.2 to control hyperchaotic Rössler system and use that of Theorem 3.3 to control the hyperchaotic Lorenz system. The second part is the numerical examples for controlling the systems with uncertain parameters. Especially, Example III has four uncertain parameters, and Example IV has six uncertain parameters.

The whole numerical results show that MRC is very suitable and efficient for controlling hyperchaotic system with both certain parameters and uncertain parameters.

4.1. Example I

The four-variable hyperchaotic Rössler system is described by 𝑥̇₁=−𝑥₂−𝑥₃,𝑥̇₂=𝑥₁+0.25𝑥₂+𝑥₄,𝑥̇₃=𝑥₁𝑥₃𝑥̇+3,₄=−0.5𝑥₃+0.05𝑥₄. (4.1) According to (2.1), we have 𝐴₃=,𝑢000.5000000.50000000,𝐵=0−1−1010.2501000000−0.50.05𝑢=¹𝑢₂𝑢₃𝑢+3₄,𝐴₁=𝐴₂=𝐴₄=diag[0,0,0,0], (4.2) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are equal to 0 in the initial state and need to be determined by Theorem 3.2.

The reference model with asymptotical stability is as follows: 𝑥̇_𝑟1=−𝑥_𝑟2−𝑥_𝑟3,𝑥̇_𝑟2=𝑥_𝑟1+0.25𝑥_𝑟2+𝑥_𝑟4,𝑥̇_𝑟3=20𝑥_𝑟1+10𝑥_𝑟2−50𝑥_𝑟3+100𝑥_𝑟4𝑥̇+3,_𝑟4=−0.5𝑥_𝑟3+0.05𝑥_𝑟4. (4.3) According to (2.4), the matrix 𝐵 and the control vector 𝑢 of the 4-dimensional reference system (4.3) have the following description: 𝐵=,0−1−1010.25012010−5010000−0.50.050030𝑢=. (4.4) According to Theorem 3.2, we may let matrix 𝑃=diag[1,1,1,1], then the matrix 𝐵^𝑇𝑃+𝑃𝐵=0019000.51011910−10099.50199.50.1, (4.5) and its largest eigenvalue is 𝜆_max=63.6297. By considering the conditions 𝜆<−𝜆_max/2 and 𝜆≠0, we should obtain 𝜆=−40. Then, the controller should be determined by Theorem 3.2, 𝑢₁𝑢₂𝑢₃𝑢₄=40𝑥_𝑟1−40𝑥₁40𝑥_𝑟2−40𝑥₂20−𝑥₃𝑥₁+10𝑥₂−90𝑥₃+100𝑥₄+40𝑥_𝑟3+340𝑥_𝑟4−40𝑥₄. (4.6) Figure 1 shows that the controller can make the hyperchaotic Rössler system (4.1) track along with its reference system (4.3). Here, the initials of these two systems are [−10,6,0,10] and [10,−6,10,−10], respectively.

(a)  𝑥1 and 𝑥𝑟1

(b) 𝑥2 and 𝑥𝑟2

(c) 𝑥3 and 𝑥𝑟3

(d) 𝑥4 and 𝑥𝑟4

Figure 1 

Comparison of the corresponding variables in the controlled hyperchaotic Rössler system and its reference system: (a) 𝑥1 and 𝑥𝑟1; (b) 𝑥2 and 𝑥𝑟2; (c) 𝑥3 and 𝑥𝑟3; (d) 𝑥4 and 𝑥𝑟4.

4.2. Example II

The four-variable hyperchaotic Lorenz system [30] is described by 𝑥̇₁=10𝑥₂−𝑥₁,𝑥̇₂=28𝑥₁−𝑥₂−𝑥₁𝑥₃+𝑥₄,𝑥̇₃=𝑥₁𝑥₂−83𝑥₃,𝑥̇₄=−5𝑥₁. (4.7)(4.8)(4.9)(4.10) According to (2.1), we have 𝐴₂=00−0.500000−0.50000000,𝐴₃=,800.5000.500000000000𝐵=−10100028−10100−30𝑢−5000,𝑢=¹𝑢₂𝑢₃𝑢₄,𝐴₁=𝐴₄=diag[0,0,0,0], (4.11) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are equal to 0 in the initial state and need to be determined by Theorem 3.3.

The reference model with asymptotical stability is as follows: ẋ_𝑟1=10𝑥_𝑟2−𝑥_𝑟1−50𝑥_𝑟2,𝑥̇_𝑟2=28𝑥_𝑟1−𝑥_𝑟2−𝑥_𝑟1𝑥_𝑟3+𝑥_𝑟4,𝑥̇_𝑟3=𝑥_𝑟1𝑥_𝑟2−83𝑥_𝑟3,𝑥̇_𝑟4=−5𝑥_𝑟1. (4.12) According to (2.5), the matrix 𝐴_𝑖(𝑖=1,…,4) and 𝐵 and the control vector 𝑢 of the 4-dimensional reference system (4.12) have the following description: 𝐴₂=,00−0.500000−0.50000000𝐴₃=,00.5000.500000000000𝐵=8−10−400028−10100−30,−50000000,𝐴𝑢=₁=𝐴₄=diag[0,0,0,0]. (4.13) According to Theorem 3.3, we may let matrix 𝑃=diag[1,1,1], then the matrix 𝐵^𝑇𝑃+𝑃𝐵=−20−120−5−12−20100−1630−5100, (4.14) and its largest eigenvalue is 𝜆_max=5.8387. By considering the conditions 𝜆<−𝜆_max/2 and 𝜆≠0, we should obtain 𝜆=−10. Then, the controller should be determined by Theorem 3.3, 𝑢₁𝑢₂𝑢₃𝑢₄=−10𝑥₂−40𝑥_𝑟2𝑥₁𝑥₃−9𝑥₂−𝑥₄+9𝑥_𝑟2−𝑥_𝑟1𝑥_𝑟3+𝑥_𝑟4−𝑥₁𝑥₂−223𝑥₃+𝑥_𝑟1𝑥_𝑟2+223𝑥_𝑟35𝑥₁−10𝑥₄−5𝑥_𝑟4+10𝑥_𝑟4. (4.15)

Figure 2 shows that the controller can make the hyperchaotic Lorenz system (4.7)–(4.12) track along with its reference system (4.12). Here, the initials of these two systems are [10,0,5,8] and [2,−10,1,10], respectively.

(a) 𝑥1 and 𝑥𝑟1

(b) 𝑥2 and 𝑥𝑟2

(c) 𝑥3 and 𝑥𝑟3

(d) 𝑥4 and 𝑥𝑟4

Figure 2 

Comparison of the corresponding variables in the controlled hyperchaotic Lorenz system and its reference system: (a) 𝑥1 and 𝑥𝑟1; (b) 𝑥2 and 𝑥𝑟2; (c) 𝑥3 and 𝑥𝑟3; (d) 𝑥4 and 𝑥𝑟4.

4.3. Example III

The four-variable hyperchaotic Rössler system with uncertain parameters is described by 𝑥̇₁=𝑎𝑥₂+𝑥₃,𝑥̇₂=𝑥₁+𝑏𝑥₂+𝑐𝑥₄,𝑥̇₃=𝑥₁𝑥₃𝑥̇+3,₄=−0.5𝑥₃+𝑑𝑥₄. (4.16)

According to (2.3), we have 𝐴₃=,𝑢000.5000000.50000000,𝐵=00001000000000−0.50𝑢=¹𝑢₂𝑢₃𝑢+3₄(𝑋)=𝑥,𝐶²+𝑥₃0000𝑥₂𝑥₄00000000𝑥₄,𝐴₁=𝐴₂=𝐴₄=diag[0,0,0,0], (4.17) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are equal to 0 in the initial state and need to be determined by Theorem 3.4. In Figure 3, 𝛼 is the estimation value of the column vector 𝛼, where 𝛼=(𝑎,𝑏̂,𝑐,̂𝑑)^𝑇.

(a) The estimation of 𝑎

(b) The estimation of 𝑏

(c) The estimation of 𝑐

(d) The estimation of 𝑑

Figure 3 

The estimation of the uncertain parameters: (a) 𝑎 is the estimation of 𝑎; (b) 𝑏̂ is the estimation of 𝑏; (c) 𝑐̂ is the estimation of 𝑐; (d) 𝑑 is the estimation of 𝑑.

The estimation system of uncertain parameters is 𝑎̇=𝑥₂+𝑥₃𝑥_𝑟1−𝑥₁,̇̂𝑏=𝑥₂𝑥_𝑟2−𝑥₂,𝑐̇̂=𝑥₄𝑥_𝑟2−𝑥₂,̇𝑑=𝑥₄𝑥_𝑟4−𝑥₄. (4.18)

The reference model is the same as Example I, 𝑥̇_𝑟1=−𝑥_𝑟2−𝑥_𝑟3,𝑥̇_𝑟2=𝑥_𝑟1+0.25𝑥_𝑟2+𝑥_𝑟4,𝑥̇_𝑟3=20𝑥_𝑟1+10𝑥_𝑟2−50𝑥_𝑟3+100𝑥_𝑟4𝑥̇+3,_𝑟4=−0.5𝑥_𝑟3+0.05𝑥_𝑟4,𝐵=,0−1−1010.25012010−5010000−0.50.050030𝑢=. (4.3) We have the largest eigenvalue 𝜆_max=−0.9675. By considering the conditions 𝜆<−𝜆_max and 𝜆≠0, we should obtain 𝜆=−1.

Figure 4 shows that the controller can make the hyperchaotic Rössler system with four uncertain parameters (4.1) track along with its reference system (4.3). Here, the initials of the hyperchaotic system, reference system, and estimation system of uncertain parameters are [−10,6,0,10], [10,−6,10,−10], and [1,2,1,2], respectively.

(a) 𝑥1 and 𝑥𝑟1

(b) 𝑥2 and 𝑥𝑟2

(c) 𝑥3 and 𝑥𝑟3

(d) 𝑥4 and 𝑥𝑟4.

Figure 4 

Comparison of the corresponding variables in the controlled hyperchaotic Rössler system and its reference system: (a) 𝑥1 and 𝑥𝑟1; (b) 𝑥2 and 𝑥𝑟2; (c) 𝑥3 and 𝑥𝑟3; (d) 𝑥4 and 𝑥𝑟4.

Comparing with Figures 1(c), 1(d), 4(c), and 4(d), we find that our method has even a better performance in the uncertain case than the certain case. In Figure 4(a), we can see some waves in time domain [0,2], and it is not better than the certain case. In Figures 4(b) and 1(b), the controlling performance is similar.

4.4. Examples IV

The four-variable hyperchaotic Lorenz system with uncertain parameters is described by 𝑥̇₁=𝑎𝑥₂−𝑥₁,𝑥̇₂=𝑏𝑥₁−𝑐𝑥₂−𝑥₁𝑥₃+𝑑𝑥₄,𝑥̇₃=𝑥₁𝑥₂−𝑓𝑥₃,𝑥̇₄=𝑔𝑥₁, (4.20) where 𝐴₂=00−0.500000−0.50000000,𝐴₃=,𝑢00.5000.500000000000𝐵=0000000000000000,𝑢=¹𝑢₂𝑢₃𝑢₄𝑎𝑏𝑐𝑑𝑓𝑔,𝐶(𝑋)=𝑥,𝛼=₂−𝑥₁000000𝑥₁−𝑥₂𝑥₄000000−𝑥₃000000𝑥₁,𝐴₁=𝐴₄=diag[0,0,0,0], (4.21) where 𝑢₁, 𝑢₂, 𝑢₃, and 𝑢₄ are equal to 0 in the initial state and need to be determined by Theorem 3.5. Figure 5 shows the estimation of 𝛼, and 𝛼 is the estimation value of the column vector 𝛼, where 𝛼=(𝑎,𝑏̂,𝑐,̂𝑑,𝑓,𝑔)^𝑇.

(a) The estimation of 𝑎

(b) The estimation of 𝑏

(c) The estimation of 𝑐

(d) The estimation of 𝑑

(e) The estimation of 𝑓

(f) The estimation of 𝑔

Figure 5 

The estimation of the uncertain parameters: (a) 𝑎 is the estimation of 𝑎; (b) 𝑏̂ is the estimation of 𝑏; (c) 𝑐̂ is the estimation of 𝑐; (d) 𝑑 is the estimation of 𝑑; (e) 𝑓 is the estimation of 𝑓; (f) 𝑔 is the estimation of 𝑔.

The estimation system of uncertain parameters is 𝑎̇=𝑥₂−𝑥₁𝑥_𝑟1−𝑥₁,̇̂𝑏=𝑥₁𝑥_𝑟2−𝑥₂,𝑐̇̂=−𝑥₂𝑥_𝑟2−𝑥₂,̇𝑑=𝑥₄𝑥_𝑟2−𝑥₂,̇𝑓=−𝑥₃𝑥_𝑟3−𝑥_𝑟3,𝑔̇=𝑥₁𝑥_𝑟4−𝑥₄. (4.22)

The reference model is the same as Example II with asymptotical stability is as follows: 𝑥̇_𝑟1=10𝑥_𝑟2−𝑥_𝑟1−50𝑥_𝑟2,𝑥̇_𝑟2=28𝑥_𝑟1−𝑥_𝑟2−𝑥_𝑟1𝑥_𝑟3+𝑥_𝑟4,𝑥̇_𝑟3=𝑥_𝑟1𝑥_𝑟2−83𝑥_𝑟3,𝑥̇_𝑟4=−5𝑥_𝑟1, (4.23) and the largest eigenvalue of 𝐵^𝑇 is 𝜆_max=0.1767. By considering the conditions 𝜆<−𝜆_max and 𝜆≠0, we should obtain 𝜆=−1.

Figure 6 shows that the controller can make the hyperchaotic Lorenz system with uncertain parameters (4.20) track along with its reference system (4.23). Here, the initials of the hyperchaotic system, the reference system, and the estimation system of uncertain parameters are [10,0,5,8], [2,−10,1,10], and [1,2,1,3,1,0], respectively.

(a) 𝑥1 and 𝑥𝑟1

(b) 𝑥2 and 𝑥𝑟2

(c) 𝑥3 and 𝑥𝑟3

(d) 𝑥4 and 𝑥𝑟4

Figure 6 

Comparison of the corresponding variables in the controlled hyperchaotic Lorenz system and its reference system: (a) 𝑥1 and 𝑥𝑟1; (b) 𝑥2 and 𝑥𝑟2; (c) 𝑥3 and 𝑥𝑟3; (d) 𝑥4 and 𝑥𝑟4.

The performance in Figures 6(a)-6(b) is much better than the one in Figures 2(a)-2(b). Other parts of the two figures have almost the same performance.

5. Conclusion

In this paper, we have used an MRC technique to control the hyperchaotic system to track with an expected trajectory. The expression of the controller has been given. Four numerical examples show that the MRC method is very effective for controlling both the hyperchaotic system with all certain parameters and the systems with uncertain parameters. By comparing the results in the corresponding figures, we find that our method does not only fit for controlling hyperchaotic systems with certain parameters, but also is a robust for the systems with uncertain parameters.

Acknowledgments

This paper is supported by China Scholarship Council. P. F. Zhao expresses sincere thanks to Professor Y. Li for his guidance, Professor D. Liu and Professor B. Yang for their instructions. Also, many thanks to the NSFC (grant number 11071026, 11001100, 11171131, 61133011, 61170092, 60973088, 61202308 and 11026043) and the Basic Research Program of Jilin University (450060481098). This work is supported in part by the National Basic Research Program of China (973) under Grant 2009CB219301, the National Public Benefit Scientific Research Foundation of China under Grant 201011078, the China Scholarship Council and the National Innovation Research Project for Exploration and Development of Oil Shale under Grant OSP-02 and OSR-02. The authors gratefully acknowledge the anonymous reviewers for their hard work and good patience.