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.
In regard to the control theory area, finite-time stability investigation is largely used in different areas, for example, stabilization , fault estimation , and observer-based control . 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  fractional order calculus has been used in economy , dielectric polarization , and image processing . 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 , 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 , 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  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 .
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.
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
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 .
Definition 7. The integer-order linear time invariant systemis said to be FTS with respect to with positive scalars and matrix , if
Definition 8. Consider the integer-order linear time invariant systemwhere 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
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.
Proof. Consider ; we have From (11), we have Let . We have From (14), we obtainBased on Lemma 4, we have It yieldsKnowing 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.
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 systemwhere , , 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 isIt is clear that (25) can be rewritten asCombining (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 bywhere and
The following theorem provides a sufficient condition of the FTB for the conformable fractional order LTI system (29).
Proof. Suppose that . We haveTaking into account assumption (30), the pre- and post-multiplying of (30) by giveCombining (32) and (33), we have This together with giveswhere .
Suppose thatThen, From (35), we haveThus, based on Lemma 4 we have Then, Knowing that , the following inequality holds:From (40), we have By using (31) and (42), we getThis 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 asIt is easy to see that (45) can be rewritten aswhich, 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 .
Therefore, system (48) is FTB with respect to
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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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