# FTS and FTB of Conformable Fractional Order Linear Systems

Authors: Abdellatif Ben Makhlouf; Omar Naifar; Mohamed Ali Hammami; Bao-wei Wu
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 Dα1=0 is not satisfied for Riemann Liouville definition, the product of two functions as Dαfg=fDαg+gDα(f) and the chain rule as Dαf∘g(t)=DαfgtDαg(t)are not satisfied neither for Riemann Liouville derivative nor for Caputo one. In addition, the monotonicity of a function f is not deduced from the sign of Dα(f). 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 Section2, 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 functionf:[0,∞)↦R. Thus, the conformable fractional derivative of f of order α is defined by (1)Tαaft=limε→0⁡ft+εt1-α-ftεfor all t>0, α∈(0.1]. If f is α-differentiable in some (0,a), a>0, and limt→0+⁡Tαft exists, then by definition(2)Tαf0=limt→0+⁡Tαft.Definition 2 ([16, 17]).
Letα∈(0.1]. The conformable fractional integral starting from a point 0 of a function f:[0,∞)↦R of order α is defined as (3)Iαft=∫0tsα-1fsds.Lemma 3 ([16, 17]).
Assume thatf:[0,∞)↦R is continuous and α∈(0.1], so (4)TαIαft=ft,forallt>0.Lemma 4.
Letα∈(0.1]. Consider a continuous function f:[0,∞)↦R such that Tαf(t) exist on 0,∞. If Tαf(t)≥0 (respectively, Tαf(t)≤0), for all t>0, so f is increasing (respectively, decreasing).Definition 5.
Letα∈(0.1]; the conformable fractional exponential function is defined as (5)Eαγ,t=exp⁡γtαα,∀t≥0with γ∈R.Remark 6.
Letx:0,∞↦Rn be an α-differentiable on 0,∞ where xt=x1(t),……,xn(t) and P=PT>0. Then, xTPx is α-differentiable on (0,∞) and TαxTPxt=2xtTPTαxt,∀t>0.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(6)x′t=Axt,x0=x0,t∈0,T,A∈Rn×nis said to be FTS with respect to c1,c2,T,Swith positive scalars c1,c2,T,c2>c1 and matrix S>0, if (7)x0TSx0≤c1⟹xtTSxt<c2,t∈0,T.Definition 8.
Consider the integer-order linear time invariant system(8)x′t=Axt+Dwt,x0=x0,t∈0,Twhere w(t)∈Rm is the disturbance input and satisfies wtTwt≤d,∀t∈[0,T],d≥0, A∈Rn×n and D∈Rn×m are constant matrices, then, system (8) is said to be FTB with respect to c1,c2,T,S,dwith positive scalars c1,c2,T,,c2>c1 and matrix S>0, if (9)x0TSx0≤c1⟹xtTSxt<c2,∀t∈0,T,∀w:wtTwt≤d,∀t∈0,T.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(10)Tαxt=Axt,x0=x0,t∈0,T,where xt=x1(t),…….,xn(t)∈Rn and x0=x01,…….,x0n∈Rn.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 β and a matrix Q>0,Q∈Rn×n, verifying (11)PA+ATP-βP<0,(12)Eαβ,TcondQ<c2c1,where P=S1/2QS1/2. Thus, system (10) is FTS with respect to c1,c2,T,S.Proof.
ConsiderVxt=xtTPx(t); we have (13)TαVxt=xtTPA+ATPxt.From (11), we have (14)TαVxt≤βVxt.Let ht=Eα-β,tVxt. We have (15)Tαht=-βEα-β,tVxt+TαVxtEα-β,t.From (14), we obtain(16)Tαht≤0.Based on Lemma 4, we have (17)ht≤h0⟹Vxt≤Eαβ,tVx0.It yields(18)xtTPxt≤Eαβ,tx0TPx0.Knowing that P=S1/2QS1/2, (18) can be written as (19)xtTS1/2QS1/2xt≤Eαβ,tx0TS1/2QS1/2x0.Then, (20)λminQxtTSxt≤λmaxQEαβ,tx0TSx0.
The two inequalitiesx0TSx(0)≤c1 and (12) leads to xtTSxt<c2,∀t∈0,T. 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 β and a matrix Q>0,Q∈Rn×n, verifying (21)AP+PAT-βP<0,(22)Eαβ,TcondQ<c2c1,where P=S-1/2Q-1S-1/2. Then, system (10) is FTS with respect to c1,c2,T,S.Based on the stability conditions presented in Theorem10, the stabilizing controller design can be formulated. The main objective is to design a state controller ut=Kx(t) for the conformable fractional order LTI system (23)Tαxt=Axt+But,x0=x0,t∈0,T,where xt=x1t,……,xn(t)∈Rn and x0=x01,……,x0n∈Rn such that the feedback system(24)Tαxt=Acxt,t∈0,T,where Ac=A+BK, K∈Rl×n, being FTS with respect to c1,c2,T,S.We propose the following theorem.Theorem 12.
For the conformable fractional order linear time invariant system (23), assume that there exist a scalar β, a matrix Q>0,Q∈Rn×n, and a matrix L∈Rl×n verifying (25)AP+PAT+BL+LTBT-βP<0,(26)Eαβ,TcondQ<c2c1,where P=S-1/2Q-1S-1/2. Thus, system (23) is finite-time stable with respect to c1,c2,T,S under the feedback control ut=Kxt=LP-1x(t).Proof.
Apply the state feedback controllerut=LP-1x(t) to (23) such that the conformable fractional closed-loop system is(27)Tαxt=A+BLP-1xt.It is clear that (25) can be rewritten as(28)A+BLP-1P+PA+BLP-1T-βP<0.
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(29)Tαxt=Axt+Dwt,x0=x0,t∈0,T,where xt=x1t,……,xn(t)∈Rn and x0=x01,……,x0n∈Rn.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 ∃β∈R and two matrices P1>0,P1∈Rn×n, and P2>0,P2∈Rm×m, verifying (30)AP+PAT-βPDP2P2DT-βP2<0,(31)λmaxP1Eαβ,TβdTαλminP2α+c1λminP1<c2,where P=S-1/2P1S-1/2. Thus, system (29) is FTB with respect to c1,c2,T,S,d.Proof.
Suppose thatVxt=xtTP-1x(t). We have(32)TαVxt=xtTwtTP-1A+ATP-1P-1DDTP-10xtwt.Taking into account assumption (30), the pre- and post-multiplying of (30) by P-100P2-1 give(33)P-1A+ATP-1-βP-1P-1DDTP-1-βP2-1≤0.Combining (32) and (33), we have (34)TαVxt≤xtTwtTβP-100βP2-1xtwt=βVxt+βwtTP2-1wt.This together with w(t)TP2-1w(t)≤λmax(P2-1)w(t)Tw(t)≤d/λmin(P2) gives(35)TαVxt≤βVxt+r,t∈0,T,where r=βd/λmin(P2).
Suppose that(36)gt=Eα-β,tVxt-rtαα.Then, (37)Tαgt=-βEα-β,tVxt+Eα-β,tTαVxt-r.From (35), we have(38)Tαgt≤rEα-β,t-r≤0,t∈0,T.Thus, based on Lemma 4 we have (39)gt≤g0=Vx0,t∈0,T.Then, (40)Vxt≤Eαβ,tVx0+rtαα,t∈0,T≤Eαβ,tVx0+rTαα.Knowing that P=S-1/2P1S-1/2, the following inequality holds:(41)Vxt≥xtTSxtλmaxP1,Vx0≤c1λminP1.From (40), we have (42)xtTSxt≤λmaxP1Eαβ,TrTαα+c1λminP1.By using (31) and (42), we get(43)xtTSxt<c2,∀t∈0,T.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 controllersut=Kx(t) to stabilize the FTB problem for the conformable fractional order LTI system (44)Tαxt=Axt+But+Dwt,x0=x0,∀t∈0,T,where xt=x1t,……,xn(t)∈Rn and x0=x01,……,x0n∈Rn.Theorem 15.
For the conformable fractional order linear time invariant system (44), assume that there exist a scalar, two matrices P1>0,P1∈Rn×n, and P2>0,P2∈Rm×m, and a matrix L∈Rl×n such that the inequality (31) and (45)AP+PAT+BL+LTBT-βPDP2P2DT-βP2<0hold, where P=S-1/2P1S-1/2.
Thus, system (44) is FTB with respect to c1,c2,T,S,dunder the control law ut=Kxt=LP-1x(t).Proof.
Applying the state feedback controllerut=Kxt to (44) results in the closed-loop system as(46)Tαxt=A+BLP-1xt+Dwt,∀t∈0,T.It is easy to see that (45) can be rewritten as(47)A+BLP-1P+PA+BLP-1T-βPDP2P2DT-βP2<0,which, by Theorem 13, implies that the closed-loop system (44) is FTB with respect to c1,c2,T,S,d.


## 4. Numerical Example

Let us consider the conformable fractional order LTI system with a disturbance defined by(48)Tαxt=Axt+Dwt,t∈0,T,with A=-11-1.50.5, D=1001, wt=0.2sin(t)0.2cos⁡(t)T. The parameters are given as c1=1, c2=56, d=0.04, T=10, β=0.04, and the matrix R=I.Applying Theorem13, the feasible solution of the condition (30) is given by (49)P=7.5243-3.8465-3.84655.7412,P2=0.1090-0.0244-0.02440.0726.Therefore, system (48) is FTB with respect to 1,56,20,R,0.04.The state trajectory over0–10s with the initial state x0=01 is shown in Figure 1. It is easy to see that system (48) is FTB with respect to 1,56,10,R,0.04 from Figure 2.Figure 1
The statex(t) of system (48) versus time.Figure 2
The trajectory ofxtTRx(t) 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.



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