Abstract
This paper investigates a fractional-order linear system in the frame of Atangana–Baleanu fractional derivative. First, we prove that some properties for the Caputo fractional derivative also hold in the sense of AB fractional derivative. Subsequently, several sufficient criteria to guarantee the finite-time stability and the finite-time boundedness for the system are derived. Finally, an example is presented to illustrate the validity of our main results.
1. Introduction
The topic of fractional calculus is more than three hundred years old. The number of applications of fractional calculus rapidly grows in recent decades. Fractional differential equations (FDEs) [1–7] have been successfully confirmed to be useful tools in various fields such as electrical circuits, diffusion, economy, and control problem.
Recently, many authors try to find new fractional operators with different kernels in order to better describe these phenomena. In 2015, Caputo and Fabrizi [8] present a new definition of fractional derivative with exponential kernel which is called CF fractional derivatives. In 2016, based on CF fractional derivatives, Atangana and Baleanu [9] introduced another new definition of fractional derivatives called AB fractional derivatives with a nonlocal and nonsingular kernel which are built by the generalized Mittag–Leffler function. In addition, the study in [9] derived the fractional integral associate to AB derivatives by taking the inverse Laplace transform and using the convolution theorem. The detail definitions of ABR fractional derivative, ABC fractional derivative, and AB fractional integral will be introduced in Section 2.
On the other hand, many basic properties of AB fractional differintegral have already been studied in recent years [10–12], such as integration by parts, mean value theorem and Taylor’s theorem, semigroup property, and product rule and chain rule. And some problems about existence of solution, stability, controllability, and optimal control are also studied by several researchers [13–18]. Having the advantage of nonlocal and nonsingular kernel, AB fractional derivatives have been widely applied in many fields such as diffusion equation [19], electromagnetic waves in dielectric media [20], chaos [21], and circuit model [22].
Finite-time stability (FTS) and finite-time boundedness (FTB) analysis is an important problem in control theory [23–29]. Since it always considers the behavior of systems in finite time, it is very useful in many practical application systems. Moreover, it should be noted that FTS and Lyapunov asymptotic stability are independent concepts. In [23], the authors have introduced the definition of FTS and FTB and given the sufficient condition for the FTB of linear time-invariant (LTI) systems with integer order. The FTS and FTB for the fractional-order systems with have been given in [24]. Motivated by the above works, this paper studies the FTS and FTB for the class of fractional-order LTI systems in the sense of this new type of AB fractional derivatives. First, we establish one new property of this new type of AB fractional derivatives. Second, we introduce the concept of FTS and FTB for the fractional-order LTI system in the sense of AB fractional derivative corresponding with integer-order system and provide the sufficient conditions which guarantee FTS and FTB for a class of fractional-order LTI systems in the sense of AB fractional derivative. Third, the problem of designing feedback controllers to keep FTS and FTB for fractional closed-loop systems in the sense of AB is studied.
Throughout this paper, denotes the n-dimensional Euclidean space, is real matrix, is the transpose of matrix , is the inverse of matrix , indicates that the matrix is positive (negative) definite, and and are the maximum and minimum eigenvalues of matrix .
2. Preliminaries
In this section, we present the definition of AB fractional derivatives and integral.
In addition, some properties of AB fractional calculus and Mittag–Leffler function are introduced.
Definition 1. (see [9]). Let , , and . Then, the AB fractional integral operator is defined bywhere is the classical Riemann–Liouville fractional integral, denotes a real-valued normalization function satisfying , and .
Definition 2. (see [9]). Let , , and . Then, the ABR fractional derivative of order is defined bywhere is one-parameter Mittag–Leffler function denoted by .
Definition 3. (see [9]). Let , , and . Then, the ABC fractional derivative of order is defined byIn [9, 12], the authors have introduced some basic properties of AB fractional differintegral which will be used in this paper. For , the relation between AB integral operators and AB differential operators is given asThe Laplace transform of AB fractional derivative is defined as follows:where .
In addition, the Mittag–Leffler function which plays an important role in AB fractional derivative will appear frequently in this paper. It is indispensable to introduce some more properties of the Mittag–Leffler function. The generalized Mittag–Leffler functions (two parameters) are denoted by , , and . For , . The Laplace transform of the Mittag–Leffler function is given asIn [30], Wang et al. provide some properties which are useful to prove Lemma 2. It is easy to note that . So, we haveAnd for , the functions and are nonnegative, and for , we have .
3. Lemmas
This section will introduce two important lemmas. In [29], Ye et al. have obtained the generalized Gronwall inequality in the sense of Caputo derivative which has wide applications in fractional differential equations. On this basis, Jarad et al. [15] proposed the Gronwall inequality in the frame of AB fractional derivative.
Lemma 1. Suppose that , is a nonnegative, nondecreasing, and locally integrable function on , is nonnegative and bounded on , and is nonnegative and locally integrable on with
Then,
In [31], Norelys et al. present a new property for the Caputo fractional derivative that is , holds for a continuous and derivable function , and . According to this, we can prove that this property also holds in the sense of AB fractional derivative.
Lemma 2. Let and be a continuous and differentiable function. Then, for any , the following inequality holds:
Proof. By the definition of AB fractional derivative, we haveEquation (10) is equivalent toDefine . Then, equation (12) can be reformulated asLet and . By the integration by parts, equation (13) is equivalent toIndeed, equation (14) holds obviously since and deduce that the first term equals zero and the nonnegativity of , , , and leads to the nonnegativity of the latter two terms.
From Lemma 2, we can easily obtain the following two results.
Corollary 1. Let and be a continuous and differentiable function on . Then, for any , the following inequality holds:
Proof. Assume that and , . By Lemma 2, we have
Corollary 2. Let be a symmetric positive definite matrix, , and be a continuous and differentiable function on . Then, for any , the following inequality holds:
Proof. From matrix theory, we know that there exists a nonsingular matrix such that . Define ; then,
4. Main Results
In this section, the concept of FTS and FTB for fractional-order LTI systems in the sense of AB fractional derivative corresponding with integer-order systems is introduced. Several sufficient conditions of FTS and FTB for such systems are given.
Consider the following fractional order LTI system in the sense of AB fractional derivative:where , , and is a constant matrix.
Definition 4. System (19) is said to be of finite-time stability with respect to , with positive scalars , , and a matrix , if
Theorem 1. Assume that there exists a scalar and a matrix , , satisfyingwhere . Then, system (19) is FTS with respect to .
Proof. Define the function . From Corollary 2 and condition equation (21), we haveThen, there exists a nonnegative function satisfyingTaking the Laplace transform on both sides of equality (24), we obtainIt can be reformulated thatApplying the inverse Laplace transform, one hasDue to the nonnegativity of the second term in equation (27), it is easy to obtainTaking and into equation (28) yieldsThis implies thatAccording to condition equation (22), we can conclude thatFor system (19), we also have the following results similar to Theorem 1.
Corollary 3. Assume that there exists a scalar and a matrix , , satisfyingwhere . Then, system (19) is FTS with respect to .
Based on system (19), we will consider the finite-time stabilization of fractional-order systems in the AB fractional derivative sense with control function:where is a control function and is a constant matrix. We assume that all the state variables are available for state feedback. The problem is to design a state feedback controller , where is the control gain matrix to be designed, such that the closed-loop system iswhere , being FTS with respect to .
Theorem 2. Assume that there exists a scalar , a matrix , , and a matrix satisfyingwhere . Then, under the feedback controller , system (33) is FTS with respect to .
Proof. Apply the state feedback controller to system (33) such that the corresponding closed-loop system isBy Corollary 3, it is clear that system (34) is FTS under the condition equation (35) and equation (36).
Next, we will study the problem of FTB for fractional-order LTI systems in the frame of AB fractional derivative. Consider the following systems:where is the disturbance input and satisfies , . is a constant matrix.
Definition 5. System (38) is said to be of finite-time boundedness with respect to , with positive scalars , , and a matrix , if
Theorem 3. Assume that there exists a scalar and two matrices , , and , , satisfyingwhere and . Then, system (38) is FTB with respect to .
Proof. Let , and taking AB fractional derivative of , by Corollary 2, we haveBy premultiplying and postmultiplying equation (40) by the symmetric positive definite matrix , we haveThen, combining equations (43) and (44) leads toIntegrating with order from to in the frame of AB fractional integral on both sides of equality (45), thenwhere . By equation (42) and applying Lemma 1, we haveIt follows from thatCombining equations (47), (48), and (49), it follows thatBy the condition equations (41) and (50), it implies thatNow, we consider the problem of designing state feedback controllers to stabilize the FTB problem for fractional-order LTI systems in the frame of AB fractional derivative:
Theorem 4. Assume that there exist a scalar , two matrices , , and , , and a matrix satisfying equations (41) and (42) andwhere . Then, under the feedback controller , system (53) is FTS with respect to .
Proof. Apply the state feedback controller to system (53) such that the corresponding closed-loop system isBy Theorem 3, it is clear that system (52) is FTB under the condition equations (41), (42), and (53).
5. Example
An example is presented to illustrate the main results.
Example 1. Consider the following fractional-order LTI system in the sense of AB fractional derivative with a disturbance defined bywhere the initial value and the disturbance . The parameters are given as , , , , , and . Applying the condition equations (40), (41), and (42) of Theorem 3 and setting , we can obtain the feasible matrices and as follows:Therefore, system (55) is FTB with respect to . The state trajectory over with the initial state is shown in Figure 1. It is easy to see that system (55) is FTB with respect to from Figure 2.
6. Conclusion
The contribution of this paper is to present one new property of AB fractional derivatives and to provide the sufficient conditions which guarantee FTS and FTB for a class of fractional-order LTI systems in the sense of AB fractional derivative, as well as to study the problem of designing feedback controllers. Because of the interesting nonsingular kernel of this new fractional derivative, there is much work on this type of fractional calculus that is worth thinking about and discussing in the future.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors contributed equally to the manuscript. All authors have read and approved the final manuscript..
Acknowledgments
This work was supported by the Natural Science Foundation of Anhui Province (2008085QA19) and National Natural Science Foundation of China (110131013 and 11471015).