FTS and FTB of Conformable Fractional Order Linear Systems

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.


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][5][6][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][13][14][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   (1) = 0 is not satisfied for Riemann Liouville definition, the product of two functions as   () =   () +   () and the chain rule as   ( ∘ )() =   ()(())  (g)() 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 2 Mathematical Problems in Engineering the proposed approach, simulation results are presented in Section 4. Finally, some conclusions are given in Section 5.

Preliminaries
In this section, we recall some definitions, notations, and traditional results.
Definition 5. Let  ∈ (0.1]; the conformable fractional exponential function is defined as with  ∈ R.
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 ( 1 ,  2 , , ) with positive scalars  1 ,  2 , ,  2 >  1 and matrix  > 0, if Definition 8. Consider the integer-order linear time invariant system where () ∈ R  is the disturbance input and satisfies ()  () ≤  , ∀ ∈ [0, ],  ≥ 0,  ∈ R × and  ∈ R × are constant matrices, then, system ( 8) is said to be FTB with respect to ( 1 ,  2 , , , ) with positive scalars  1 ,  2 , , ,  2 >  1 and matrix  > 0, 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.
For the system (10), we have also the following result.
We propose the following theorem.
The following theorem provides a sufficient condition of the FTB for the conformable fractional order LTI system (29).

Mathematical Problems in Engineering
Suppose that ( ] . (42) By using (31) and (42), we get This ends the proof.
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 ( 1 ,  2 , , , ).

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.