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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.

In regard to the control theory area, finite-time stability investigation is largely used in different areas, for example, stabilization [

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 [

Indeed, in [

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 [

The rest of the paper is organized as follows. In Section

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

Consider a function

Let

Assume that

Let

Let

Let

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 [

The integer-order linear time invariant system

Consider the integer-order linear time invariant system

The FTS of integer-order systems was studied by F. Amato et al. in [

Consider the conformable fractional order LTI system

Sufficient conditions of the FTS for the conformable fractional order LTI system (

For the conformable fractional order LTI system (

Consider

The two inequalities

For the system (

For the conformable fractional order linear time invariant system (

Based on the stability conditions presented in Theorem

We propose the following theorem.

For the conformable fractional order linear time invariant system (

Apply the state feedback controller

Combining (

Now, consider the LTI conformal fractional order system presented by

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

Consider system (

Suppose that

Suppose that

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

For the conformable fractional order linear time invariant system (

Thus, system (

Applying the state feedback controller

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

Applying Theorem

Therefore, system (

The state trajectory over

The state

The trajectory of

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.

No data were used to support this study.

The authors declare that they have no conflicts of interest.