Abstract
In this letter stability analysis of fractional order nonlinear systems is studied. Some new sufficient conditions on the local (globally) asymptotic stability for a class of fractional order nonlinear systems with order are proposed by using properties of Mittag-Leffler function and the Gronwall inequality. And the corresponding stabilization criteria are also given. The numerical simulations of two systems with order and two systems with order illustrate the effectiveness and universality of the proposed approach.
1. Introduction
During the last decade the fractional calculus has gained importance in both theoretical and applied aspects of several branches of science and engineering. There are two essential differences between integer order derivation and fractional order derivation. Firstly, the integer order derivative indicates a variation or certain attribute at particular time for a mechanical or physical process, while the fractional order derivative is concerned with the whole time domain. Secondly, the integer order derivative describes the local properties of a certain position, while the fractional order derivative is related to the whole space for a physical process. Then many physical systems are well characterized by the fractional order state equations [1–4], such as fractional order Lotka-Volterra equation [1] in biological systems, fractional order Schödinger equation [2] in quantum mechanics, fractional order Langevin equation [3] in anomalous diffusion, and fractional order oscillator equation [4] in damping vibration.
However there are several open problems in this area. Stability of fractional order systems is one of the most fundamental and important issues. On the other hand, because fractional differential operators are nonlocal and have weakly singular kernels, some methods in dealing with interorder systems cannot be simply extended to fractional-order methods. To the best of knowledge, the stability of fractional-order nonlinear systems is still relatively few. Reference [5–7] investigated the necessary and sufficient stability conditions for linear fractional order differential equations and linear time-delayed fractional differential equations. The stability of -dimensional linear fractional order differential systems with order has already been studied in [8]. However, only under some special circumstances or in certain cases, the practical problems may be regarded as linear systems. Therefore, stability of nonlinear system is of great significance, and it also has important value in application. In [9], the stability of fractional nonlinear time-delay systems for Caputo’s derivative are investigated, and two theorems for Mittag-Leffler stability of the fractional order nonlinear time-delay systems are proved. In [10], the authors proposed the finite-time stabilization of a class of multistate time delay of fractional nonlinear systems. In [11, 12], the authors studied the stability of fractional nonlinear dynamic systems using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. In [13], the authors studied fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks. In [14], some new sufficient conditions ensuring asymptotical stability of fractional-order nonlinear system with delay are proposed firstly.
In this paper the stability of nonlinear fractional order nonlinear system is studied. And by using the Gronwall inequality and the properties of Mittag-Leffler function, we proposed some new sufficient conditions on the local (globally) asymptotic stability for a class of fractional order nonlinear systems with order . And the corresponding stabilization criteria are also given. Finally, four numerical simulation examples have illustrated the effectiveness and universality of the proposed methods.
2. Fractional Order Derivative and Mittag-Leffler Function
2.1. Definitions of Fractional Derivative and Mittag-Leffler Function
Fractional calculus plays an important role in modern science [15–17]. Some definitions for fractional derivatives are usually used, such as Grünwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo definition. In this paper, we mainly use the Caputo definitions [15].
Definition 1 (see [15]). The fractional integral of function is defined as follows: where fractional order , and is the gamma function.
Definition 2 (see [15]). The Caputo derivative with order of function is given as
where , .
The formulas for Laplace transform of the Caputo fractional derivative have the following form [16]:
where , and .
As a generalization of the exponential function which is frequently used in the solutions of integer-order systems, the Mittag-Leffler function is frequently used in the solutions of fractional systems. The definition and properties are given in the following.
Definition 3 (see [17]). The Mittag-Leffler function is given as where and .
The generalization of Mittag-Leffler function with two parameters is wildly used and defined as follows: where , , and .
Remark 4. If , we have , especially, .
2.2. Properties of Mittag-Leffler Functions and the Gronwall Inequality
In this section, we give the Gronwall inequality and some important properties of the Mittag-Leffler functions which are used in the following.
Lemma 5 (see [15]). Considering the Laplace transform of Mittag-Leffler function with two parameters, we have where and are, respectively, the variables in the time domain and Laplace domain, stands for the real part of , , and denotes the Laplace transform.
Proof. The proof of this Lemma can be found in [15].
Lemma 6 (see [18, 19]). If , , and satisfies , there exist and such that where , .
Lemma 7 (see [18, 19]). For the Mittage-Leffler function , there exist finite real constants , , and such that where .
Proof. The proof of this Lemma can be found in [18].
Lemma 8 (Gronwall inequality [19, 20]). Let , is a nonnegative function locally integrable on and is a nonnegative, nondecreasing continuous function defined on , (constant), and suppose is nonnegative and locally integrable on with on this interval. Then Moreover, if is a nondecreasing function on , we have
3. Stability and Stabilization of Fractional Order Nonlinear System
3.1. Stability and Stabilization of Fractional Order Nonlinear System with Order
Firstly, we consider the Caputo fractional nonlinear systems [16, 21] with the initial condition , where denotes the state vector of the system, is the order of the fractional-order derivative, defines a nonlinear vector field in the -dimensional vector space, and and denote the linear and nonlinear parts of , respectively. If , the constant is called the equilibrium point of Caputo fractional nonlinear system (12). Without loss of generality, we suppose the equilibrium point is .
Theorem 9. The fractional order nonlinear system (12) is local asymptotically stable, if it satisfies the following conditions: and , where is the gamma function; satisfies as .
Proof. Applying the Laplace transform on (12), we have that is, where is the Laplace transform of , is an identity matrix, and denotes the Laplace transform. By using the Laplace inverse transform, we obtain the solution of (16), It follows from Lemma 7 that there exist constants and such that Since matrix is stable, there is a constant such that . Substituting it into (16), one has Based on the condition (2), that is, , there is a constant , such that where . And when and , then Multiplying the inequality by , we will get Applying Lemma 8 (Gronwall inequality) to (20), we have Then Therefore, when , for , which implies that the system (12) is asymptotically stable.
Theorem 10. The fractional order nonlinear system (12) is globally asymptotically stable, if it satisfies the following conditions: satisfies and the Lipschitz condition with respect to , that is, ; and , where and satisfy and .
Proof. Applying the Laplace transform and Laplace inverse transform on (12), we obtain the solution of (12), It follows from Lemma 7 that there exist constants , , and such that Multiplying the inequality by , we will get Applying Lemma 8 (Gronwall inequality) to (25), we have Then Therefore, when , for , which implies that the system (12) is globally asymptotically stable.
The controlled fractional order nonlinear system with linear feedback control input is given as where is the linear feedback control input, , and the feedback gain matrix needs to be determined.
Therefore, our aim is to design a suitable feedback gain matrix such that the controlled system is local (globally) asymptotically stable.
Theorem 11. The controlled fractional order nonlinear system (28) is local asymptotically stable, if it satisfies the following conditions: and ; satisfies as .
Proof. The proof is similar to that of Theorem 9.
Theorem 12. The controlled fractional order nonlinear system (28) is globally asymptotically stable, if it satisfies the following conditions: satisfies and the Lipschitz condition with respect to , that is, ; and , where and satisfy and .
Proof. The proof is similar to that of Theorem 10.
3.2. Stability and Stabilization of Fractional Order Nonlinear System with Order
Firstly, we consider the Caputo fractional nonlinear systems [16, 21] with the initial conditions and , where denotes the state vector of the system, is the order of the fractional order derivative, defines a nonlinear vector field in the -dimensional vector space, and and denote the linear and nonlinear parts of , respectively.
Theorem 13. The fractional order nonlinear system (29) is local asymptotically stable, if it satisfies the following conditions: and ; satisfies as .
Proof. Applying the Laplace transform on (29), we have that is, where is the Laplace transform of , and is an identity matrix. By using the Laplace inverse transform, we obtain the solution of (29), It follows from Lemma 7 that there exist constants and such that Since matrix is stable, there is a constant such that . Substituting it into (33), one has Based on the condition (2), that is, , there is a constant , such that where . And when , and then Multiplying the inequality by , we will get Applying Lemma 8 (Gronwall inequality) and Lemma 6 to (37), we have Then Therefore, when , for , which implies that the system (29) is asymptotically stable.
Theorem 14. The fractional order nonlinear system (29) is globally asymptotically stable, if it satisfies the following conditions: satisfies and the Lipschitz condition with respect to , that is, ; and , where and satisfy and .
Proof. Applying the Laplace transform and Laplace inverse transform on (29), we obtain the solution of (29), It follows from Lemma 7 that there exist constants , , and such that Multiplying the inequality by , we will get Applying Lemma 8 (Gronwall inequality) and Lemma 6 to (42), we have Then Therefore, when , for , which implies that the system (29) is globally asymptotically stable.
The controlled fractional order nonlinear system with linear feedback control input is given as where is the linear feedback control input, , and the feedback gain needs to be determined.
Therefore, our aim is to design a suitable feedback gain matrix such that the controlled system is local (globally) asymptotically stable.
Theorem 15. The controlled fractional order nonlinear system (45) is local asymptotically stable, if it satisfies the following conditions: and ; satisfies as .
Proof. The proof is similar to that of Theorem 13.
Theorem 16. The controlled fractional-order nonlinear system (45) is globally asymptotically stable, if it satisfies the following conditions: satisfies and the Lipschitz condition with respect to , that is, ; and , where and satisfy and .
Proof. The proof is similar to that of Theorem 14.
Remark 17. There are many fractional order chaotic (hyperchaotic) systems which satisfy as or the Lipschitz condition, such as fractional order Lorenz system, fractional order Chen system, fractional order Lü system, fractional order Liu system, and so forth [22]. Therefore, Theorems 9–16 can be used as the criteria to control chaos in a class of fractional-order systems. Compared with nonlinear control methods, the advantage of linear control lies in reducing control cost and is easy to implement.
Remark 18. The obtained sufficient conditions could be applied to a class of fractional order hyperchaotic systems [23–25]. On the one hand, complex multiscroll chaotic systems have garnered much attention in recent years. J. H. Lü has done a large amount of remarkable work. In fact, the sufficient conditions could be applied to a class of complex multiscroll chaotic systems, which could also generate a complex four-scroll chaotic attractor.
4. Four Illustrative Examples
In this section, we apply the proposed method in stabilizing a fractional order Chen system, Chua system, Lü system, and Liu system to verify its effectiveness and universality.
4.1. Stabilization of Fractional Order Chaotic Chen System
The fractional order Chen system [21, 22] with order can be de described by When the parameters are chosen as , , , and , system (46) exhibits the chaotic behavior, as shown in Figure 1. We consider system (46) as form (12) where
(a)
(b)
(c)
(d)
Adding control input to system (47), the controlled system can be rewritten as . It is easy to demonstrate that satisfies that is, . The feedback gain matrix is selected as which satisfies the conditions and in Theorem 11. The simulation result is shown in Figure 2, which shows that the zero solution of the controlled system is asymptotically stable.
(a)
(b)
(c)
(d)
4.2. Stabilization of Fractional Order Chaotic Chua System
The fractional order Chua system [26] with order can be described by where . When the parameters are chosen as , , , , , and , system (55) exhibits the chaotic behavior, as shown in Figure 3.
(a)
(b)
(c)
(d)
We consider system (51) as form (12) where
Adding control input to system (52), the controlled system can be rewritten as . It is easy to demonstrate that satisfies that is, . The feedback gain matrix is selected as which satisfies the conditions and in Theorem 11. The simulation result is shown in Figure 4, which shows that the zero solution of the controlled system is asymptotically stable.
(a)
(b)
(c)
(d)
4.3. Stabilization of Fractional Order Chaotic Lü System
The fractional order Lü [27] system with order can be de described by When the parameters are chosen as , , , and , system (56) exhibits the chaotic behavior, as shown in Figure 5.
(a)
(b)
(c)
(d)
We consider system (56) as form (33) where
Adding control input to system (57), the controlled system can be rewritten as . It is easy to demonstrate that satisfies that is, . The feedback gain matrix is selected as which satisfies the conditions and in Theorem 15. The simulation result is shown in Figure 6, which shows that the zero solution of the controlled system is asymptotically stable.
(a)
(b)
(c)
(d)
4.4. Stabilization of Fractional Order Chaotic Liu System
The fractional order Liu system [28, 29] with order can be de described by When the parameters are chosen as , , , , , and , system (65) exhibits the chaotic behavior, as shown in Figure 7.
(a)
(b)
(c)
(d)
We consider system (61) as form (29) where
Adding control input to system (62), the controlled system can be rewritten as . It is easy to demonstrate that satisfies that is, . The feedback gain matrix is selected as which satisfies the conditions and in Theorem 15. The simulation result is shown in Figure 8, which shows that the zero solution of the controlled system is asymptotically stable.
(a)
(b)
(c)
(d)
5. Conclusion
Stability of the nonlinear dynamical systems is important for scientists and engineers. Fractional dynamic systems were used intensively during the last decade in order to describe the behavior of complex systems in physical and engineering. In this paper the stabilization of nonlinear fractional order dynamic system is studied. And by using the Gronwall inequality and the properties of Mittag-Leffler function, we proposed some new sufficient conditions on the local (globally) asymptotic stability for a class of fractional order nonlinear systems. Finally the corresponding stabilization criteria are also given. Four numerical simulation examples have illustrated the effectiveness and universality of the proposed methods.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the Artificial Intelligence Key Laboratory of Sichuan Province 2014RYJ05 and Sichuan University of Science and Engineering Grants 2012PY17, 2014PY06, 2014RC03, and 2013KY02.