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Guopeng Zhou, Jinhua Huang, Xiaoxin Liao, Shijie Cheng, "Stability Analysis and Control of a New Smooth Chua's System", Abstract and Applied Analysis, vol. 2013, Article ID 620286, 10 pages, 2013. https://doi.org/10.1155/2013/620286
Stability Analysis and Control of a New Smooth Chua's System
This paper is concerned with the stability analysis and control of a new smooth Chua's system. Firstly, the chaotic characteristic of the system is confirmed with the aid of the Lyapunov exponents. Secondly, it is proved that the system has globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to globally exponentially stabilize the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua's systems are globally synchronized, and the estimation errors of the uncertain parameters converge to zero as tends to infinity. Finally, the numerical simulations are also presented.
It is well known that Chua’s system is the first analog circuit to realize chaos in experiments. The original Chua’s system is described by the following ordinary differential equations : where are state variables and , , , and , are constants. Due to the form of a simple circuit, there are a large literature on the dynamical behavior of Chua’s system [2–8]. By changing the parameters or the corresponding functions of Chua’s system, the chaotic phenomenon is very rich, and it is more convenient to study the chaotic mechanism and characteristics [6–8].
For the chaotic systems, Lagrange stability, stability of equilibrium points and, synchronization are three important problems which attracted more and more attention (refer to [9–16] and the reference therein). In [11–13], the authors studied the Lagrange stability by applying the attractive set and positive invariant set of the chaotic systems. Moreover, the researchers examined the stabilization of the unstable equilibrium points and the synchronization control for the chaotic systems with linear controllers [7, 14]. Recently, adaptive controllers are used in synchronous control of chaotic systems when the parameters of the systems are uncertain [15–17].
Motivated by the previous results, the main purpose of this paper is to construct a new smooth Chua’s system and investigate the stability and control problems. More precisely, we will consider the following smooth Chua’s system: where are state variables and , are constants.
We will show that the chaotic characteristics are depended on the parameters and and the initial state values of the system (2). All equilibrium points of system (2) are examined to be unstable when and (see in Section 3). By computing with MATLAB, the maximum Lyapunov exponent of the system (2) is 0.0021, where the embedding dimension is 3 and the delay time is 5. Since the maximum Lyapunov exponent is greater than 0, the Chua’s system is chaotic. It will be of great significance if the solution of (2) is ultimately bounded (Lagrange asymptotically stable). The chaotic phase diagrams of such system is obtained by simulation with MATLAB. Figure 1 shows the phase diagrams of the system (2) with , , and , the phase diagrams of Chua’s system exhibits chaotic.
The remains of this paper are organized as follows. The existence of globally exponential attractive set and positive invariant set for the system (2) will be discussed in Section 2. The asymptotic stability of the equilibrium points will be studied in Section 3, and the synchronization control for two similar types of the Chua’s systems will be discussed in Section 4. In Section 5, we will give the numerical simulations to demonstrate the correctness of our results, and finally we will give the conclusions in Section 6.
2. Existence of Globally Exponential Attractive Set and Positive Invariant Set
Definition 1. If there exists a radially unbounded, positive definite Lyapunov function and positive numbers , such that for all , when , the solution of the system (2), along satisfies , then system (2) has a globally exponential attractive set .
Definition 2. Let , if and for all , , then is called positive invariant set of the system (2).
It is easy to prove that the globally exponential attractive set is positive invariant. A system with global attractive set is always called Lagrange globally asymptotically stable system or ultimately bounded dissipative system. For the system (2), we will prove the following the Lagrange stability results.
Theorem 3. The system (2) has the following globally exponential attractive and positive invariant set: where , are symmetric positive definite matrices, and are symmetric negative definite matrices. , , , and are the maximum eigenvalues of , , , and , respectively. , and in (3) are chosen to guarantee that , are positive definite and , are negative definite. , , and are chosen such that , , , , and .
Proof. The proof is divided into three steps.
Step I. The existence of , such that is positive definite and is negative definite.
It is well known that is positive definite if and only if all the order principal minors of are positive . That is, Let ; it is easy to see that is positive definite since , .
Next, let Then,
From Schur theorem , is negative definite if and only if , . Let , satisfy It is easy to verify that is negative definite.
Step II. Existence of globally exponential attractive set of .
Let and be the minimum eigenvalues of and , respectively. By constructing a radially unbounded Lyapunov function as follows:
then, one can obtain that
The time derivative of along the system (2) is given by
If , one has Then, the following inequality holds Thus, the trajectory of the system (2) will exponentially decay into the area if .
When the trajectory of the system (2) is in the area , it holds that
Let . If and , it holds that
Let , . Then, exponentially decays when . Thus, is the globally exponential attractive set of .
Step III. Existence of globally exponential attractive set of and .
Let then, is a positive matrix if , and is a negative matrix if .
At the same time, it is easy to examine that is positive and is negative if let , since and .
Let , be, respectively, minimum and maximum eigenvalues of , and let and be the minimum and maximum eigenvalues of , respectively. Since , a radially unbounded and positive definite Lyapunov function about and is constructed as follows: Then, the time derivative of along the system (2) yields where , and are chosen such that , , , , and .
Let ; if and , it holds that
Similarly, if , the state trajectory will stay in the area such that holds. Hence, and are exponentially decreased and ultimately enter into the attractive region , that is,
Combining Steps I, II, and III, Theorem 3 is obtained and the proof is completed.
Remark 4. In this section, a constructive method is proposed to prove the main results of existence of globally exponential attractive set and positive invariant set for the Chua’s systems. By constructing the matrices , , , , and Lyapunov function candidate and , the problem is solved ingeniously.
3. Global Linear Stabilization of the Equilibrium Points
In this section, the stability of the equilibrium points for the system (2) will be discussed with the aid of a linear controller.
Firstly, it is easy to examine that the system (2) has three equilibrium points: Moreover, the equilibrium points of the system is independent of parameters and . It should be mentioned that, however, the stability of the equilibrium points is depended on and . In the following, we will design a linear controller to stabilize the unstable equilibrium points.
Let , , and be any equilibrium point of the system (2), the corresponding Jacobian matrix is given by Then the characteristic equation of the corresponding local linearization system of system (2) is as follows: where , , , , and .(i)If , one has , . According to Hurwitz stability criterion , the necessary condition of stable equilibrium point is the same sign of the coefficients of the characteristic equation. Consequently, the equilibrium is unstable.(ii)If , one has , , and . Similarly, one can obtain that and are the unstable equilibrium points.
Now, we will discuss how to design a linear feedback controller such that the unstable equilibrium points are exponentially stable. For this purpose, we add the control terms to the system (2):
Let be any of the three unstable equilibrium points, let be the solution of the system (25), and , then the error system is given by
Definition 5. , if is appropriately selected such that holds (). Then, the control input can globally exponentially stabilize the equilibrium point .
Theorem 6. If the following linear controller is added to the error system (26), where is any parameter given beforehand such that ; then the equilibrium point is globally exponentially stable.
Proof. The proof is divided into two steps.(1) We will find the existence of such that is positive definite and is negative definite, where
It is easy to obtain that is positive definite if . Now we focus on choosing such that is negative definite. By the Schur theorem , is negative definite if and only if
Obviously, is negative definite if , .(2) We construct a positive definite and radially unbounded Lyapunov function to prove the stability of closed-loop systems (26) with controller (28) which is written as
Suppose and are minimum and maximum eigenvalues of the positive definite matrix , respectively. Then, we have
Let . Obviously, is a monotonically increasing odd function. Then, (i)if , then and (ii)if , then and Thus
Differentiating with respect to time yields where is the maximum eigenvalues of the negative definite matrix . Then,
Hence, , , and converge to zero exponentially. According to Definition 5, the equilibrium point is globally exponentially stable. The proof is complete.
Remark 7. A constructive method to stabilize the unstable equilibrium points is proposed in this section, matrices , , and Lyapunov function candidate are given. Then, a linear controller is obtained to solve the problem. Comparing with nonlinear controller, linear controller is easy to implement in reality.
4. Globally Exponential Synchronization of Two Chua’s Systems
In this section, the globally exponential synchronization of two Chua’s systems will be discussed. The drive system is given by and the response system is described as follows: where the subscripts and denote the drive and response systems and is feedback control input which satisfies .
Let , , and ; one obtains that
Theorem 8. If the following controller is added to the error system (41), where is any parameter given beforehand with ; then the zero solution of (41) is globally exponentially stable and the systems (39) and (40) are globally exponentially synchronized.
Proof. Since the proof of this theorem is parallel to that of Theorem 6, we omit it here.
If the parameters and in the drive system are uncertain, one can construct the following controlled response system: where and are the estimates of the uncertain parameters and , respectively. Let , , , , and , then, the error system of (39) and (43) is given by
Theorem 9. If the following adaptive controller is added to the error system (44), then, we have the following.(1)The equilibrium points (, , , , ) of the system (44) with adaptive control law (45) are globally stable; in addition, , , and . And thus, the two systems (39) and (44) are globally synchronized.(2)The parameter estimates and will, respectively, converge to and as tends to infinity.
Proof. The proof contains two steps.
Firstly, a Lyapunov function candidate is constructed as follows:
Then, one has
By LaSalle-Yoshizawa theorem , all the equilibrium points of the closed-loop systems are globally stable. Additionally, , , and . And thus, the two systems (39) and (43) are globally synchronized.
By referring to Lemma 4.1 in , , if there exist two functions , which satisfy persistency of excitation condition, such that
Since , , it is easy to obtain from (45) that Add and from (45) to (44), we have Let and ; it is easy to see that and satisfy persistency of excitation condition ; thus, we have The proof is complete.
Remark 10. Synchronization control methods for two Chua’s systems are proposed in this section. A linear controller is given when parameters and are known. In the case of uncertain parameters and , an adaptive controller is proposed to solve the problem. Comparing with the results in reference [22, 23], the output tracking error only converges to a small neighborhood of the origin, yet the synchronization errors of Chua’s systems can exponentially approach zero. On the other hand, compared with , a novel exponentially convergent method is used to solve the convergence of the estimation error of the uncertain parameters.
5. Numerical Simulations
In this section, several examples of numerical simulations are proposed to illustrate the theoretical results obtained in the previous sections. A fourth-order Runge-Kutta method is used to obtain the simulation results with MATLAB.
Chua’s system (2) and the error systems (26), (41), and (44) are considered in this section for the numerical simulations. Let , , the initial state , , and . Figure 2 shows the state trajectories of the closed-loop system (26) with linear control input (28); it is easy to see that the equilibrium point is asymptotically stable. Similarly, Figure 3 shows that, with the corresponding control input (28), the equilibrium point is asymptotically stable. Figure 4 shows the synchronous errors of the system (39) and (40) with linear controller (42); it is easy to see that the two systems are globally asymptotically synchronized. When parameters and are uncertain, by using adaptive controller (45), the synchronous errors are asymptotically convergent to , , (see Figure 5), and the estimate values of the uncertain parameters and asymptotically converge to the real values , (see Figure 6).
A new smooth Chua’s system is constructed, and the chaotic characteristics is confirmed by computing the Lyapunov exponents of the system. A Constructive method is used to prove the existence of globally exponential attractive set and positive invariant set. For the three unstable equilibrium points of the system, a linear controller is designed to achieve globally exponential stability of the equilibrium points. Then, a linear controller and an adaptive controller are, respectively, proposed so that two similar types of smooth Chua’s systems are globally synchronized, and the estimate errors of the uncertain parameters converge to zero as tends to infinity.
The authors want to express their sincere thanks to the editor and the referee for their invaluable comments and suggestions which helped improve the paper greatly. This work was supported by the National Natural Science Foundation of China (50937002 and 51207063) and the Project of the Education Department of Hubei Province (T200910, T201009, and D20132801).
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