Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 2572986, 5 pages
https://doi.org/10.1155/2018/2572986

Research Article

FTS and FTB of Conformable Fractional Order Linear Systems

Abdellatif Ben Makhlouf,1,2 Omar Naifar,³ Mohamed Ali Hammami,² and Bao-wei Wu⁴

¹Department of Mathematics, College of Science, Jouf University, Aljouf, Saudi Arabia
²University of Sfax, Faculty of Sciences, Department of Mathematics, BP 1171, 3000 Sfax, Tunisia
³National School of Engineering, Department of Electrical Engineering, CEM lab, BP W, 3038 Sfax, Tunisia
⁴College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China

Correspondence should be addressed to Abdellatif Ben Makhlouf; benmakhloufabdellatif@gmail.com

Received 17 February 2018; Accepted 13 June 2018; Published 5 July 2018

Academic Editor: Filippo Cacace

Copyright © 2018 Abdellatif Ben Makhlouf et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, an extension of some existing results related to finite-time stability (FTS) and finite-time boundedness (FTB) into the conformable fractional derivative is presented. Illustrative example is presented at the end of the paper to show the effectiveness of the proposed result.

1. Introduction

In regard to the control theory area, finite-time stability investigation is largely used in different areas, for example, stabilization [1], fault estimation [2], and observer-based control [3]. Several research works were done to solve FTS and FTB problem for integer-order linear and nonlinear systems [4–7]. Nevertheless, noninteger derivatives constitute a property of many dynamic systems, so fractional order equations are the best way to describe these systems. For example, in electromagnetic systems [8] fractional order calculus has been used in economy [9], dielectric polarization [10], and image processing [11]. In recent decades, and with the growth of complex engineering systems and the development of science, the use of fractional calculus in many contexts of control theory, such as stability, FTS, has increased significantly.

Until now, FTS and FTB for fractional order systems are not well tackled in the literature. Indeed, there are only few works related to this topic. Readers can refer to [12–15] as an example of finite-time stability and finite-time boundedness of fractional order linear systems using Caputo derivative. Nevertheless, there is no works in the literature which investigate FTS and FTB of the new class of fractional order systems using conformable derivative.

Indeed, in [16], R. Khalil et al. proposed a new effective derivative called “the conformable fractional derivative”. Both Riemann Liouville and Caputo derivative share some disadvantages; for example, the property is not satisfied for Riemann Liouville definition, the product of two functions as and the chain rule as are not satisfied neither for Riemann Liouville derivative nor for Caputo one. In addition, the monotonicity of a function is not deduced from the sign of . After that, in [17], T. Abdeljawad developed more properties for the conformable derivative. A lot of investigations on it are currently conducted. In addition, in regard to the physical meaning of such derivative, the work in [18] gave physical and geometrical interpretations of the fractional conformable derivative which thus indicate potential applications in physics and engineering.

In this paper, FTS and FTB, which are well tackled for integer-order LTI systems, are extended to conformable fractional order LTI systems. In addition, the design of a feedback controller for the same class of conformable fractional order systems is described. Note that the solution of LTI fractional conformable systems is given by T. Abdeljawad in [17].

The rest of the paper is organized as follows. In Section 2, preliminaries and useful results with respect to the conformable fractional order calculus, FTS, and FTB are introduced. In Section 3 the main results of the work are introduced detailing the FTS and FTB of a certain class of conformable LTI systems. To show the efficiency of the proposed approach, simulation results are presented in Section 4. Finally, some conclusions are given in Section 5.

2. Preliminaries

In this section, we recall some definitions, notations, and traditional results.

Definition 1 ([16, 17] (conformable fractional derivative)). Consider a function . Thus, the conformable fractional derivative of of order is defined by

for all , . If is -differentiable in some , , and exists, then by definition

Definition 2 ([16, 17]). Let . The conformable fractional integral starting from a point 0 of a function of order is defined as

Lemma 3 ([16, 17]). Assume that is continuous and , so

Lemma 4. Let. Consider a continuous function such that exist on If (respectively, ), for all , so is increasing (respectively, decreasing).

Definition 5. Let ]; the conformable fractional exponential function is defined as

with .

Remark 6. Let be an -differentiable on where and . Then, is -differentiable on and .

In what follows, some definitions and sufficient conditions of the FTS and the FTB for the integer-order linear time invariant system are introduced. Readers can refer to [7, 12].

Definition 7. The integer-order linear time invariant system

is said to be FTS with respect to with positive scalars and matrix , if

Definition 8. Consider the integer-order linear time invariant system

where is the disturbance input and satisfies , and are constant matrices, then, system (8) is said to be FTB with respect to with positive scalars and matrix , if

Remark 9. The FTS of integer-order systems was studied by F. Amato et al. in [7], after that Y. Ma et al. in [12] have generalized the work of Amato et al., 2001, to the Caputo fractional derivative.

3. Main Results

Consider the conformable fractional order LTI system

where and

Sufficient conditions of the FTS for the conformable fractional order LTI system (10) are stated through the following theorem.

Theorem 10. For the conformable fractional order LTI system (10), assume that there exist a scalar and a matrix , verifying

where . Thus, system (10) is FTS with respect to

Proof. Consider ; we have

From (11), we have

Let . We have

From (14), we obtain

Based on Lemma 4, we have

It yields

Knowing that , (18) can be written as

Then,

The two inequalities and (12) leads to . This ends the proof.

For the system (10), we have also the following result.

Corollary 11. For the conformable fractional order linear time invariant system (10), assume that there exist a scalar and a matrix , verifying

where . Then, system (10) is FTS with respect to

Based on the stability conditions presented in Theorem 10, the stabilizing controller design can be formulated. The main objective is to design a state controller for the conformable fractional order LTI system

where and such that the feedback system

where , , being FTS with respect to

We propose the following theorem.

Theorem 12. For the conformable fractional order linear time invariant system (23), assume that there exist a scalar , a matrix , and a matrix verifying

where . Thus, system (23) is finite-time stable with respect to under the feedback control .

Proof. Apply the state feedback controller to (23) such that the conformable fractional closed-loop system is

It is clear that (25) can be rewritten as

Combining (26) and using Theorem 10, the FTS of the closed-loop system (23) can be given.

Now, consider the LTI conformal fractional order system presented by

where and

The following theorem provides a sufficient condition of the FTB for the conformable fractional order LTI system (29).

Theorem 13. Consider system (29) and assume that and two matrices , and , verifying

where . Thus, system (29) is FTB with respect to

Proof. Suppose that . We have

Taking into account assumption (30), the pre- and post-multiplying of (30) by give

Combining (32) and (33), we have

This together with gives

where .
Suppose that

Then,

From (35), we have

Thus, based on Lemma 4 we have

Then,

Knowing that , the following inequality holds:

From (40), we have

By using (31) and (42), we get

This ends the proof.

Remark 14. The obtained results of the FTS and the FTB for the conformable fractional order LTI systems are the generalization of the integer-order LTI systems.

Now, consider the design of state feedback controllers to stabilize the FTB problem for the conformable fractional order LTI system

where and .

Theorem 15. For the conformable fractional order linear time invariant system (44), assume that there exist a scalar, two matrices , and , and a matrix such that the inequality (31) and

hold, where .
Thus, system (44) is FTB with respect to under the control law .

Proof. Applying the state feedback controller to (44) results in the closed-loop system as

It is easy to see that (45) can be rewritten as

which, by Theorem 13, implies that the closed-loop system (44) is FTB with respect to .

4. Numerical Example

Let us consider the conformable fractional order LTI system with a disturbance defined by

with , , . The parameters are given as , , , , , and the matrix .

Applying Theorem 13, the feasible solution of the condition (30) is given by

Therefore, system (48) 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 (48) is FTB with respect to from Figure 2.

Figure 1: The state of system (48) versus time.

Figure 2: The trajectory of of system (48) versus time.

5. Conclusion

The FTS and FTB problem for integer-order systems have been subjected to several research works. However, much less interest is given to the new generalized mathematical representation, namely, conformable fractional order systems. In this paper, some existing results in the literature, related to FTS for linear integer-order systems, are extended into the new conformable fractional derivative. Finally, some simulation results are given to validate theoretical results.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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