Nonlinear Observer Design of the Generalized Rössler Hyperchaotic Systems via DIL Methodology

Abstract

The generalized Rössler hyperchaotic systems are presented, and the state observation problem of such systems is investigated. Based on the differential inequality with Lyapunov methodology (DIL methodology), a nonlinear observer design for the generalized Rössler hyperchaotic systems is developed to guarantee the global exponential stability of the resulting error system. Meanwhile, the guaranteed exponential decay rate can be accurately estimated. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of proposed approach.

1. Introduction

In recent decades, several kinds of chaotic systems have been widely explored; see, for instance, [1–11] and the references therein. This is due to theoretical interests as well as to an efficient tool for chaos synchronization and chaos control design. As a rule, chaos in many systems is a source of the generation of oscillation and a source of instability. Chaotic systems frequently exist in various fields of application, such as system identification, master-slave chaotic systems, secure communication, and ecological systems.

Form practical considerations, it is either impossible or inappropriate to measure all the elements of the state vector. The state observer has come to take its pride of place in system identification, filter theory, and control design. As we know, the tasks of observer-based control systems (with or without chaos) can be divided into two categories: tracking (or synchronization) and observer-based stabilization (or regulation). The state observer can be skillfully applied in observer-based stabilization, synchronization of master-slave chaotic systems, and secure communication. For more detailed knowledge, one can refer to [1, 2, 7–9, 11–14]. However, the state observer design of dynamic systems with chaos is in general not as easy as that without chaos. Motivated by the above reasons, the observer design of chaotic systems is actually crucial and meaningful. On the other hand, a variety of methods have been proposed for the observer design of systems, such as Chebyshev neural network (CNN), sliding-mode approach, passivation of error dynamics, separation principle, and frequency domain analysis; see, for instance, [15–20] and the references therein.

In this paper, the nonlinear state reconstructor of the generalized Rössler hyperchaotic systems is investigated. Using the DIL methodology, a nonlinear observer for such systems is provided to guarantee the global exponential stability of the resulting error system. Furthermore, the guaranteed exponential decay rate can be correctly estimated. Finally, numerical simulations are given to verify the effectiveness of proposed approach.

2. Problem Formulation and Main Result

In this paper, we consider the generalized Rössler hyperchaotic systems as follows:𝑥̇₁(𝑡)=𝛼₁𝑥₁+𝛼₂𝑥₂+𝑔₁𝑥₃,𝑥₄,𝑥̇₂(𝑡)=𝛼₃𝑥₁+𝛼₄𝑥₂+𝑔₂𝑥₄,𝑥̇₃(𝑡)=𝑟₁𝑥₁𝑥₃+𝑔₃𝑥₄,𝑥̇₄(𝑡)=𝑟₂𝑥₃+𝑔₄𝑥₄,𝑦(𝑡)=𝑥₄, (2.1) where 𝑥(𝑡)∶=[𝑥₁(𝑡)𝑥₂(𝑡)𝑥₃(𝑡)𝑥₄(𝑡)]^𝑇∈ℜ⁴is the state vector, 𝑦(𝑡)∈ℜis the system output, 𝑟₁, 𝑟₂, and 𝛼_𝑖,forall𝑖∈{1,2,3,4} are the system parameters with 𝑟₁𝑟₂≠0. For the existence and uniqueness of system (2.1), we assume that all the functions 𝑔_𝑖(⋅),forall𝑖∈{1,2,3,4}, are sufficiently smooth.

The following assumption is made on system (2.1) throughout this paper.(A1)There exists a constant ℎ₁ such that ℎ₁>𝛼₄,ℎ₁𝛼₄<−𝛼₂𝛼₃. (2.2)

Remark 2.1. It is noted that the Rössler hyperchaotic system [21] is the special cases of system (2.1) with 𝛼₁=0,𝛼₂=−1,𝛼₃=1,𝛼₄=0.25,𝑟₁=1,𝑟₂𝑔=−0.5,₁𝑥₃,𝑥₄=−𝑥₃,𝑔₂𝑥₄=𝑥₄,𝑔₃𝑥₄=3,𝑔₄𝑥₄=0.05𝑥₄. (2.3)
The objective of this paper is to search a nonlinear observer for system (2.1) such that the global exponential stability of the resulting error systems can be guaranteed. Before presenting the main result, let us introduce a definition which will be used in the main theorem.

Definition 2.2. System (2.1) is exponentially state reconstructible if there exist an observer 𝐸𝑥̇(𝑡)=𝑔(𝑥(𝑡),𝑦(𝑡))and positive numbers 𝑘 and 𝛼 such that ‖𝑒(𝑡)‖∶=‖𝑥(𝑡)−𝑥(𝑡)‖≤𝑘exp(−𝛼𝑡),∀𝑡≥0, (2.4) where 𝑥(𝑡) expresses the reconstructed state of system (2.1). In this case, the positive number 𝛼 is called the exponential decay rate.
Now we present the main result for the state observer of system (2.1).

Theorem 2.3. System (2.1) with (A1) is exponentially state reconstructible. Besides, a suitable nonlinear observer is given by 𝑟₁𝑥₃𝑥̇₁=𝛼₁+ℎ₁𝑥̇₃−𝑔₃𝑦−ℎ₁𝑟₁𝑥₁𝑥₃+𝑟₁𝑥₃𝛼₂𝑥₂+𝑔₁𝑥₃,̇,𝑦𝑥₂=𝛼₃𝑥₁+𝛼₄𝑥₂+𝑔₂𝑦,𝑥₃=1𝑟₂𝑦̇−𝑔₄𝑦,𝑥₄=𝑦. (2.5) In this case, the guaranteed exponential decay rate is given by 𝛼∶=1/𝜆_max(𝑃), where 𝑃>0 is the unique solution to the following Lyapunov equation: −ℎ₁𝛼₂𝛼₃𝛼₄^𝑇𝑃+𝑃−ℎ₁𝛼₂𝛼₃𝛼₄=−200−2. (2.6)

Proof. From (2.1), (2.5) with 𝑒_𝑖(𝑡)∶=𝑥_𝑖(𝑡)−𝑥_𝑖(𝑡),∀𝑖∈{1,2,3,4}, (2.7) it can be readily obtained that 𝑒₄(𝑡)=𝑥₄(𝑡)−𝑥₄(𝑡)𝑒=0,∀𝑡≥0,₃(𝑡)=𝑥₃(𝑡)−𝑥₃(𝑡)=𝑥₃(𝑡)−1𝑟₂𝑦̇−𝑔₄𝑦=𝑥₃(𝑡)−1𝑟₂𝑥̇₄−𝑔₄𝑥₄=𝑥₃(𝑡)−1𝑟₂𝑟₂𝑥₃(𝑡)+𝑔₄𝑥₄−𝑔₄𝑥₄=0,∀𝑡≥0,𝑒₂̇(𝑡)=𝑥̇₂(𝑡)−𝑥̇₂(𝑡)=𝛼₃𝑥₁+𝛼₄𝑥₂+𝑔₂𝑥₄−𝛼₃𝑥₁−𝛼₄𝑥₂−𝑔₂𝑥₄=𝛼₃𝑥₁−𝑥₁+𝛼₄𝑥₂−𝑥₂+𝑔₂𝑥₄−𝑔₂𝑥₄=𝛼₃𝑒₁+𝛼₄𝑒₂+𝑔₂𝑦−𝑔₂𝑦=𝛼₃𝑒₁(𝑡)+𝛼₄𝑒₂(𝑡),∀𝑡≥0,𝑒₁̇(𝑡)=𝑥̇₁(𝑡)−𝑥̇₁(𝑡)=𝛼₁𝑥₁+𝛼₂𝑥₂+𝑔₁𝑥₃,𝑥₄−𝛼₁+ℎ₁𝑥̇₃−𝑔₃𝑦𝑟₁𝑥₃+ℎ₁𝑥₁−𝛼₂𝑥₂−𝑔₁𝑥₃,𝑦=𝛼₁𝑥₁+𝛼₂𝑒₂+𝑔₁𝑥₃,𝑥₄−𝛼₁+ℎ₁𝑥̇₃−𝑔₃𝑥₄𝑟₁𝑥₃+ℎ₁𝑥₁−𝑔₁𝑥₃,𝑥₄=𝛼₁𝑥₁+𝛼₂𝑒₂−𝛼₁+ℎ₁𝑔₃𝑥₄+𝑟₁𝑥₁𝑥₃−𝑔₃𝑥₄𝑟₁𝑥₃+ℎ₁𝑥₁=𝛼₁𝑥₁+𝛼₂𝑒₂−𝛼₁+ℎ₁𝑥₁+ℎ₁𝑥₁=ℎ₁𝑒₁(𝑡)+𝛼₂𝑒₂(𝑡),∀𝑡≥0. (2.8) This implies that 𝑒₁̇(𝑡)𝑒₂̇(𝑡)=−ℎ₁𝛼₂𝛼₃𝛼₄𝑒₁(𝑡)𝑒₂(𝑡),𝑒₃(𝑡)𝑒₄(𝑡)=0,∀𝑡≥0, (2.9) with 𝜎(^−ℎ1𝛼2_𝛼3𝛼4)⊆𝐶⁻, in view of (A1):
Let 𝑊(𝑡)∶=𝑒₁(𝑡)𝑒₂(𝑡)𝑃𝑒₁(𝑡)𝑒₂(𝑡). (2.10) The time derivative of 𝑊(𝑡) along the trajectories of dynamical error system, with (2.9), (2.10), and (2.6), is given by 𝑊̇(𝑡)=𝑒₁̇𝑒₂̇𝑃𝑒₁𝑒₂+𝑒₁𝑒₂𝑃𝑒₁̇𝑒₂̇=𝑒₁𝑒₂−ℎ₁𝛼₂𝛼₃𝛼₄^𝑇𝑃𝑒₁𝑒₂+𝑒₁𝑒₂𝑃−ℎ₁𝛼₂𝛼₃𝛼₄𝑒₁𝑒₂=𝑒₁𝑒₂−ℎ₁𝛼₂𝛼₃𝛼₄^𝑇𝑃+𝑃−ℎ₁𝛼₂𝛼₃𝛼₄𝑒₁𝑒₂=−2𝑒₁𝑒₂𝑒₁𝑒₂2≤−𝜆_max(𝑃)𝑊(𝑡)=−2𝛼𝑊(𝑡),∀𝑡≥0. (2.11) Thus, one has 𝑒^2𝛼𝑡⋅𝑊̇+𝑒^2𝛼𝑡𝑑⋅2𝛼𝑊=𝑒𝑑𝑡^2𝛼𝑡⋅𝑊≤0,∀𝑡≥0. (2.12) It follows that ^𝑡₀𝑑𝑒𝑑𝜏^2𝛼𝜏⋅𝑊(𝜏)𝑑𝜏=𝑒^2𝛼𝑡⋅𝑊(𝑡)−𝑊(0)≤^𝑡₀0𝑑𝜏=0. (2.13) Consequently, we conclude that ‖𝑒(𝑡)‖=𝑒²₁(𝑡)+𝑒²₂(𝑡)+𝑒²₃(𝑡)+𝑒²₄(𝑡)=𝑒²₁(𝑡)+𝑒²₂(𝑡)≤𝑊(𝑡)𝜆_min(𝑃)≤𝑒^−2𝛼𝑡𝑊(0)𝜆_min(𝑃)=𝑊(0)𝜆_min(𝑃)⋅𝑒^−𝛼𝑡,∀𝑡≥0, (2.14) in view of (2.8), (2.10), and (2.13). This completes the proof.

3. Numerical Simulations

Consider the generalized hyperchaotic system: 𝑥̇₁(𝑡)=−𝑥₂−𝑥₃,𝑥̇₂(𝑡)=𝑎𝑥₁+𝑏𝑥₂+𝑥₄,𝑥̇₃(𝑡)=𝑥₁𝑥₃+3,𝑥̇₄(𝑡)=−0.5𝑥₃+0.05𝑥₄,𝑦(𝑡)=𝑥₄. (3.1)

Case 1 ((𝑎=1,𝑏=0.25) or, equivalently, the Rössler hyperchaotic system). It can be verified that condition (A1) is satisfied with ℎ₁=1.2. By Theorem 2.3, we conclude that system (3.1) with 𝑎=1 and 𝑏=0.25 is exponentially state reconstructible by the nonlinear observer: 𝑥₃𝑥̇₁=1.2𝑥̇₃−3−1.2𝑥₁𝑥₃−𝑥₂𝑥₃−𝑥²₃,𝑥̇₂=𝑥₁+0.25𝑥₂+𝑦,𝑥₃=0.1𝑦−2𝑦,̇𝑥₄=𝑦, (3.2) with the guaranteed exponential decay rate 𝛼=0.164.

Case 2 (𝑎=−20,𝑏=−50). It can be verified that condition (A1) is satisfied with ℎ₁=10. By Theorem 2.3, we conclude that system (3.1) with 𝑎=−20 and 𝑏=−50 is exponentially state reconstructible by the nonlinear observer: 𝑥₃𝑥̇₁=10𝑥̇₃−3−10𝑥₁𝑥₃−𝑥₂𝑥₃−𝑥²₃,𝑥̇₂=−20𝑥₁−50𝑥₂+𝑦,𝑥₃=0.1𝑦−2𝑦,̇𝑥₄=𝑦, (3.3) with the guaranteed exponential decay rate 𝛼=8.47.

Case 3 (𝑎=30,𝑏=−40). It can be verified that condition (A1) is satisfied with ℎ₁=5. By Theorem 2.3, we conclude that system (3.1) with 𝑎=30 and 𝑏=−40 is exponentially state reconstructible by the nonlinear observer: 𝑥₃𝑥̇₁=5𝑥̇₃−3−5𝑥₁𝑥₃−𝑥₂𝑥₃−𝑥²₃,𝑥̇₂=30𝑥₁−40𝑥₂+𝑦,𝑥₃=0.1𝑦−2𝑦,̇𝑥₄=𝑦, (3.4) with the guaranteed exponential decay rate 𝛼=3.79.

The time response of error states for system (3.1) with Case 1–Case 3 is depicted in Figures 1, 2, and 3, respectively. From the foregoing simulations results, it is seen that system (3.1) with Case 1–Case 3, regardless of chaotic system or nonchaotic system, is exponentially state reconstructible by the nonlinear observers (3.2)–(3.4), respectively.

Figure 1 

The time response of error states, with 𝑎=1 and 𝑏=0.25.

Figure 2 

The time response of error states, with 𝑎=−20 and 𝑏=−50.

Figure 3 

The time response of error states, with 𝑎=30 and 𝑏=−40.

4. Conclusion

In this paper, the generalized Rössler hyperchaotic systems have been presented, and the state observation problem of such systems has been investigated. Based on the DIL methodology, a nonlinear state reconstructor of the generalized Rössler hyperchaotic systems has been developed to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be accurately estimated. However, the state observation design for more general uncertain hyperchaotic system still remains unanswered. This constitutes an interesting future research problem.

Nomenclature

ℜ𝑛:	The 𝑛-dimensional real space
𝐶−:	The set of {𝑎+𝑏𝑗∣𝑎<0,𝑏∈ℜ}
|𝑎|:	The modulus of a real number 𝑎
‖𝑥‖:	The Euclidean norm of the vector 𝑥∈ℜ𝑛
‖𝐴‖:	The induced Euclidean norm of the matrix 𝐴
𝐴𝑇:	The transpose of the matrix 𝐴
𝜎(𝐴):	The set of all eigenvalues of the matrix 𝐴
𝑃>0:	The symmetric matrix 𝑃 is positive definite
𝜆max(𝑃):	The maximum eigenvalue of the symmetric matrix 𝑃 with real eigenvalues
𝜆min(𝑃):	The minimum eigenvalue of the symmetric matrix 𝑃 with real eigenvalues.

Acknowledgments

The author thanks the National Science Council of Republic of China for supporting this work under Grant NSC-100-2221-E-214-015. The author also wishes to thank the anonymous reviewers for providing constructive suggestions.