MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/2572986 2572986 Research Article FTS and FTB of Conformable Fractional Order Linear Systems http://orcid.org/0000-0001-7142-7026 Ben Makhlouf Abdellatif 1 2 Naifar Omar 3 Hammami Mohamed Ali 2 http://orcid.org/0000-0003-3274-8999 Wu Bao-wei 4 Cacace Filippo 1 Department of Mathematics College of Science Jouf University Aljouf Saudi Arabia ju.edu.sa 2 University of Sfax Faculty of Sciences Department of Mathematics BP 1171 3000 Sfax Tunisia univ-sfax.tn 3 National School of Engineering Department of Electrical Engineering CEM lab BP W 3038 Sfax Tunisia enit.rnu.tn 4 College of Mathematics and Information Science Shaanxi Normal University Xi’an 710119 China snnu.edu.cn 2018 572018 2018 17 02 2018 13 06 2018 572018 2018 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.

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 , fault estimation , and observer-based control . Several research works were done to solve FTS and FTB problem for integer-order linear and nonlinear systems . 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  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 Dα1=0 is not satisfied for Riemann Liouville definition, the product of two functions as Dαfg=fDαg+gDα(f) and the chain rule as Dαfg(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 , 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.

2. Preliminaries

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

Definition 1 ([<xref ref-type="bibr" rid="B13">16</xref>, <xref ref-type="bibr" rid="B15">17</xref>] (conformable fractional derivative)).

Consider a function f  :[0,)R. Thus, the conformable fractional derivative of f of order α is defined by (1)Tαaft=limε0ft+εt1-α-ftεfor all t>0, α(0.1]. If f is α-differentiable in some (0,a), a>0, and limt0+Tαft exists, then by definition(2)Tαf0=limt0+Tαft.

Definition 2 ([<xref ref-type="bibr" rid="B13">16</xref>, <xref ref-type="bibr" rid="B15">17</xref>]).

Let α(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 ([<xref ref-type="bibr" rid="B13">16</xref>, <xref ref-type="bibr" rid="B15">17</xref>]).

Assume that f  :  [0,)R is continuous and α(0.1], so (4)TαIαft=ft,forallt>0.

Lemma 4.

Let  α(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.

Let α(0.1]; the conformable fractional exponential function is defined as (5)Eαγ,t=expγtαα,t0with γR.

Remark 6.

Let x  :  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)xt=Axt,x0=x0,t0,T,ARn×nis said to be FTS with respect to c1,c2,T,S  with positive scalars c1,c2,T,c2>c1 and matrix S>0, if (7)x0TSx0c1xtTSxt<c2,t0,T.

Definition 8.

Consider the integer-order linear time invariant system(8)xt=Axt+Dwt,x0=x0,t0,Twhere w(t)Rm is the disturbance input and satisfies wtTwtd  ,t[0,T],  d0, ARn×n and DRn×m are constant matrices, then, system (8) is said to be FTB with respect to c1,c2,T,S,d  with positive scalars c1,c2,T,,c2>c1 and matrix S>0, if (9)x0TSx0c1xtTSxt<c2,t0,T,w:wtTwtd,t0,T.

Remark 9.

The FTS of integer-order systems was studied by F. Amato et al. in , after that Y. Ma et al. in  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,t0,T,where xt=x1(t),.,xn(t)Rn and x0=x01,.,x0nRn.

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,  QRn×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.

Consider Vxt=xtTPx(t); we have (13)TαVxt=xtTPA+ATPxt.From (11), we have (14)TαVxtβVxt.Let ht=Eα-β,tVxt. We have (15)Tαht=-βEα-β,tVxt+TαVxtEα-β,t.From (14), we obtain(16)Tαht0.Based on Lemma 4, we have (17)hth0VxtEαβ,tVx0.It yields(18)xtTPxtEαβ,tx0TPx0.Knowing that P=S1/2QS1/2, (18) can be written as (19)xtTS1/2QS1/2xtEαβ,tx0TS1/2QS1/2x0.Then, (20)λminQxtTSxtλmaxQEαβ,tx0TSx0.

The two inequalities x0TSx(0)c1 and (12) leads to xtTSxt<c2,t0,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,  QRn×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 Theorem 10, 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,t0,T,where xt=x1t,,xn(t)Rn and x0=x01,,x0nRn such that the feedback system(24)Tαxt=Acxt,t0,T,where Ac=A+BK, KRl×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,  QRn×n, and a matrix LRl×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 controller ut=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,t0,T,where xt=x1t,,xn(t)Rn and x0=x01,,x0nRn.

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 βR and two matrices P1>0,  P1Rn×n, and P2>0,P2Rm×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 that Vxt=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-βP2-10.Combining (32) and (33), we have (34)TαVxtxtTwtTβP-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)TαVxtβVxt+r,t0,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)TαgtrEα-β,t-r0,t0,T.Thus, based on Lemma 4 we have (39)gtg0=Vx0,t0,T.Then, (40)VxtEαβ,tVx0+rtαα,t0,TEαβ,tVx0+rTαα.Knowing that P=S-1/2P1S-1/2, the following inequality holds:(41)VxtxtTSxtλmaxP1,Vx0c1λminP1.From (40), we have (42)xtTSxtλmaxP1Eαβ,TrTαα+c1λminP1.By using (31) and (42), we get(43)xtTSxt<c2,t0,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 controllers ut=Kx(t) to stabilize the FTB problem for the conformable fractional order LTI system (44)Tαxt=Axt+But+Dwt,x0=x0,t0,T,where xt=x1t,,xn(t)Rn and x0=x01,,x0nRn.

Theorem 15.

For the conformable fractional order linear time invariant system (44), assume that there exist a scalar, two matrices P1>0,  P1Rn×n    , and P2>0,P2Rm×m, and a matrix LRl×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,d  under the control law ut=Kxt=LP-1x(t).

Proof.

Applying the state feedback controller ut=Kxt to (44) results in the closed-loop system as(46)Tαxt=A+BLP-1xt+Dwt,t0,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,t0,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 Theorem 13, 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 over 010s 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.

The state x(t) of system (48) versus time.

The trajectory of xtTRx(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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Hong Y. Finite-time stabilization and stabilizability of a class of controllable systems Systems & Control Letters 2002 46 4 231 236 10.1016/s0167-6911(02)00119-6 MR2010240 Zbl0994.93049 2-s2.0-0037162403 Shen Q. Wang D. Zhu S. Poh E. K. Finite-time fault-tolerant attitude stabilization for spacecraft with actuator saturation IEEE Transactions on Aerospace and Electronic Systems 2015 51 3 2390 2405 2-s2.0-84942874078 10.1109/TAES.2015.130725 Zavala-Río A. Fantoni I. Sanahuja G. Finite-time observer-based output-feedback control for the global stabilisation of the PVTOL aircraft with bounded inputs International Journal of Systems Science 2016 47 7 1543 1562 2-s2.0-85000049427 10.1080/00207721.2014.938906 Zbl1333.93202 Bhat S. P. Bernstein D. S. Finite-Time Stability of Continuous Autonomous Systems SIAM Journal on Control and Optimization 2006 38 3 751 766 Amato F. Cosentino C. Merola A. Sufficient conditions for finite-time stability and stabilization of nonlinear quadratic systems IEEE Transactions on Automatic Control 2010 55 2 430 434 10.1109/tac.2009.2036312 MR2604419 2-s2.0-76949098976 Zbl1368.93524 Kamenkov G. V. On stability of motion over a finite interval of time Journal of Applied Mathematics and Mechanics 1953 17 529 540 MR0061237 Amato F. Ariola M. Dorato P. Finite-time control of linear systems subject to parametric uncertainties and disturbances Automatica 2001 37 9 1459 1463 2-s2.0-0035452456 10.1016/S0005-1098(01)00087-5 Zbl0983.93060 Engheta N. On fractional calculus and fractional multipoles in electromagnetism Institute of Electrical and Electronics Engineers. Transactions on Antennas and Propagation 1996 44 4 554 566 10.1109/8.489308 MR1382017 Laskin N. Fractional market dynamics Physica A: Statistical Mechanics and its Applications 2000 287 3-4 482 492 10.1016/S0378-4371(00)00387-3 MR1802416 Sun H. H. Abdelwahab A. A. Onaral B. Linear approximation of transfer function with a pole of fractional order IEEE Transactions on Automatic Control 1984 29 5 441 444 Zbl0532.93025 2-s2.0-0021424253 10.1109/TAC.1984.1103551 Oustaloup A. La Dérivation Non Entiére 1995 Ma Y. Wu B. Wang Y. Finite-time stability andfinite-time boundedness of fractional order linear systems Neurocomputing 2016 Vol 173 2076 2082 10.1016/j.neucom.2015.09.080 Ma Y. Wu B. Wang Y. Input-output finite time stability of fractional order linear systems with 0 < a < 1 Transactions of the Institute of Measurement and Control 2015 10 1177 10.1177/0142331215617237 Yang Y. Li J. Chen G. Finite-time stability and stabilization of nonlinear stochastic hybrid systems Journal of Mathematical Analysis and Applications 2009 356 1 338 345 Moulay E. Perruquetti W. Finite time stability and stabilization of a class of continuous systems Journal of Mathematical Analysis and Applications 2006 323 2 1430 1443 10.1016/j.jmaa.2005.11.046 MR2260193 Zbl1131.93043 2-s2.0-33751079336 Khalil R. Al Horani M. Yousef A. Sababheh M. A new definition of fractional derivative Journal of Computational and Applied Mathematics 2014 264 65 70 10.1016/j.cam.2014.01.002 MR3164103 Zbl1297.26013 2-s2.0-84893186929 Abdeljawad T. On conformable fractional calculus Journal of Computational and Applied Mathematics 2015 279 57 66 10.1016/j.cam.2014.10.016 MR3293309 Zbl1304.26004 2-s2.0-84911406110 Zhao D. Luo M. General conformable fractional derivative and its physical interpretation Calcolo 2017 54 3 903 917 2-s2.0-85009170962 10.1007/s10092-017-0213-8 Zbl1375.26020