Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

How to impose physically coherent initial conditions to a fractionalsystem?

Jocelyn Sabatier *, Mathieu Merveillaut, Rachid Malti, Alain OustaloupUniversité de Bordeaux – Laboratoire, IMS-UMR 5218 CNRS, 351 cours de la libération, 33405 Talence cedex, France

a r t i c l e i n f o

Article history:Received 25 April 2009Received in revised form 30 May 2009Accepted 30 May 2009Available online 26 June 2009

PACS:02.60.Nm02.30.Jr

Keywords:Fractional order systemDiffusion equationInitialization

1007-5704/$ - see front matter � 2009 Published bdoi:10.1016/j.cnsns.2009.05.070

* Corresponding author.E-mail addresses: Jocelyn.sabatier@u-bordeaux1

a b s t r a c t

In this paper, it is shown that neither Riemann–Liouville nor Caputo definitions for frac-tional differentiation can be used to take into account initial conditions in a convenientway from a physical point of view. This demonstration is done on a counter-example. Thenthe paper proposes a representation for fractional order systems that lead to a physicallycoherent initialization for the considered systems. This representation involves a classicallinear integer system and a system described by a parabolic equation. It is thus also shownthat fractional order systems are halfway between these two classes of systems, and areparticularly suited for diffusion phenomena modelling.

� 2009 Published by Elsevier B.V.

1. Introduction

Fractional calculus has found applications in assorted fields, like engineering [23], physics [21], finance [25], chemistry[24], bioengineering [22]. Fractional order systems are usually represented by fractional differential equations or pseudostate space descriptions. However, other representations of these systems exist among which the ‘‘diffusive representation”,mainly studied by Montseny and Matignon [1,2]. In this paper, this representation is obtained from the impulse response of afractional order system. Then, changes of variables are used to demonstrate that fractional order system output is the com-bination of a linear integer system output and a parabolic differential system output. From a physical point of view, linearfractional order systems are neither quite conventional linear systems nor quite conventional parabolic systems. They arehalfway between these two classes of systems. That explains why fractional order systems are particularly suited for mod-elling of diffusion phenomena.

Using the representation previously introduced, the well known ‘‘time memory effect” translating that the behaviour of afractional system takes into account the system past on an infinite time interval, is converted to ‘‘a spatial memory effect”translating that the behaviour of a fractional system results in the behaviour of an infinite number of systems spatially dis-tributed. As a consequence, a coherent initialization can be proposed using this representation. This is to be opposed to theinitialization proposed by the Riemann–Liouville or Caputo definitions [3,4] that do not permit to take into account initialcondition correctly (compatible with the system physics). This last assertion is demonstrated in the paper on a counter-example.

y Elsevier B.V.

.fr, prenom.nom@laps.ims-bordeaux.fr (J. Sabatier).

J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326 1319

2. How to take into account initial conditions?

In a series of papers, Lorenzo and Hartley [5–8] have demonstrated that initial conditions are not taken into account in thesame way whether Riemann–Liouville or Caputo definitions are considered. The goal of this section is to go further and todemonstrate that neither Riemann–Liouville nor Caputo definitions permit to take into account initial conditions in a coher-ent way with the system physics. The demonstration is done on a counter-example based on the system defined by the dif-ferential equation

dcxðtÞ=dtc ¼ uðtÞ; 0 < c < 1: ð1Þ

As in Lorenzo and Hartley work, the demonstration is based on the shift of the time origin. System (1) is supposed to be atrest when time t < 0. Input uðtÞ is also supposed to be defined by

uðtÞ ¼ HðtÞ � Hðt � t0Þ; ð2Þ

where Hð:Þ denotes the Heaviside function. Time response of system (1) is thus defined by

xðtÞ ¼ tc

Cðcþ 1ÞHðtÞ �ðt � t0Þc

Cðcþ 1ÞHðt � t0Þ: ð3Þ

The change of variable

s ¼ t � 2t0 ð4Þ

is now used so that at time s ¼ 0, the system is not at rest.Using Riemann–Liouville definition, Laplace transform of the fractional derivative of xðsÞ is defined by [9]

LfDcxðsÞg ¼ sc�xðsÞ �Xn�1

k¼0

sk½Dc�k�1xðsÞ�s¼0 ð5Þ

with n� 1 < c < n and where �xðsÞ denotes the Laplace transform of xðsÞ with zero initial conditions. Laplace transform ap-plied to Eq. (1), leads to

sc�xðsÞ �Xn�1

k¼0

sk½Dc�k�1xðsÞ�s¼0 ¼ 0 ð6Þ

(here n ¼ 1) and the corresponding time response is defined by

xðsÞ ¼L�1 1sc

� �½I1�cxðsÞ�s¼0; ð7Þ

where

½I1�cxðsÞ�s¼0 ¼L�1 1s1�c

e2t0s

scþ1 �et0s

scþ1

� �� �s¼0¼L�1 e2t0s

s2 �et0s

s2

� �� �s¼0¼ 2t0 � t0 ¼ t0: ð8Þ

The time response of system (1) is thus given by

xðsÞ ¼ t0sc�1

CðcÞ ; s P 0: ð9Þ

Using Caputo definition, Laplace transform of the fractional derivative of xðsÞ is defined by [9]

LfDcxðtÞg ¼ sc�xðsÞ �Xn�1

k¼0

sc�k�1½DkxðsÞ�s¼0: ð10Þ

Laplace transform applied to Eq. (1), leads to

sc�xðsÞ �Xn�1

k¼0

sc�k�1½DkxðsÞ�s¼0 ¼ 0 ð11Þ

(here n ¼ 1) and the corresponding time response is given by

xðsÞ ¼L�1 sc�1

sc

� �½xðsÞ�s¼0; ð12Þ

where, using (3),

½xðsÞ�s¼0 ¼ð2t0Þc

Cðcþ 1Þ �ð2t0 � t0Þc

Cðcþ 1Þ : ð13Þ

1320 J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326

Time response of system (1) is thus given by

Fig

xðsÞ ¼ ð2t0Þc � tc0

Cðcþ 1Þ HðsÞ; s P 0: ð14Þ

In Fig. 1, system (1) response to the input given by relation (2) is compared to the response obtained using Riemann–Liou-ville and Caputo definitions, initial conditions being taken into account. This figure highlights that the three time responsesare completely different.

Using a counter-example, it has been demonstrated in this section that neither the Riemann–Liouville nor the Caputo def-initions are compatible with the real behaviour of a fractional system. Using these definitions, system response uðtÞ does notmach system response to non-zero initial conditions. To solve this problem another representation is introduced in the fol-lowing section.

3. On representation of fractional systems

Fractional systems are usually described by fractional differential equations or state space like representations. In thissection another representation of fractional systems is used. It is based on the impulse response of a fractional system thatis first detailed for a fractional system of the first kind. Then a generalization to a larger class of fractional system is discussedand finally a physical interpretation linked to the obtained representation is proposed.

3.1. Impulse response of a fractional system of the first kind

Consider the following fractional system of the first kind:

GðsÞ ¼ 1sc � a

ð15Þ

with 0 < c < 2 and a > 0 (stability condition).Using the poles pk definition given in Section 3.2, the impulse response of such a system is given by [10]

gðtÞ ¼ 1ac

Xk

pketpk þ sinðcpÞp

Z 1

0

xce�tx

a2 � 2axc cosðcpÞ þ x2c dx: ð16Þ

Response of system (1) to an input uðtÞ is defined as the convolution product of the impulse response gðtÞ and the inputuðtÞ:

yðtÞ ¼Z t

0gðt � sÞuðsÞds; ð17Þ

and thus using relation (16):

yðtÞ ¼Z t

0

1ac

Xk

pkeðt�sÞpk uðsÞdsþZ t

0

sinðcpÞp

Z 1

0

xce�ðt�sÞx

a2 � 2axc cosðcpÞ þ x2c dx� �

uðsÞds ð18Þ

-20 -15 -10 -5 0 5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

10

Time (s)

Res

pons

e co

mpa

rison

s

Input

Exact solutionRieman-Liouville

Caputo

. 1. Comparison of the exact response of system (1) with the responses obtained with Riemann–Liouville and Caputo definitions (t0 ¼ 10 s).

J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326 1321

or, through an integral permutation:

yðtÞ ¼Z t

0

1ac

Xk

pkeðt�sÞpk uðsÞdsþZ 1

0

sinðcpÞp

xc

a2 � 2axc cosðcpÞ þ x2c

Z t

0e�ðt�sÞxuðsÞds

� �dx: ð19Þ

Let

wðt; xÞ ¼Z t

0e�ðt�sÞxuðsÞds: ð20Þ

wðt; xÞ is thus solution of the differential equation

_wðt; xÞ ¼ �xwðt; xÞ þ uðtÞ: ð21Þ

Using relations (19) and (21), the following state space representation can be obtained for the system (15):

_w1ðtÞ...

_wkðtÞ_wðt; xÞ

266664

377775 ¼

p1 ð0Þ. .

.

pk

ð0Þ �x

266664

377775

w1ðtÞ...

wkðtÞwðt; xÞ

266664

377775þ

1...

11

266664

377775uðtÞ

yðtÞ ¼ y1ðtÞ þ y2ðtÞ ¼p1ac � � � pk

ac

� � w1ðtÞ...

wkðtÞ

2664

3775þ sinðcpÞ

p

Z 1

0

xcwðt; xÞac � 2axc cosðcpÞ þ x2c dx:

ð22Þ

Such a representation is connected to the diffusive representation introduced by Montseny and Matignon [1,11,2].

3.2. Generalization to a larger class of fractional systems

Representation (22) defined for system (15) can be generalized to a larger class of fractional systems defined by

GðsÞ ¼ BðsÞAðsÞ ð23Þ

with BðsÞ ¼Pr

l¼0qlsbl and AðsÞ ¼

Pmk¼0rksak where blþ1 P bl P 0 and akþ1 P ak P 0.

Let fp1; . . . ; pkg be the poles of the transfer function GðsÞ. Computation of these poles permits to define output y1ðtÞ of (22).Under commensurate order hypothesis and through a partial fraction decomposition, transfer function GðsÞ appears as a lin-ear combination of terms

1= sn � klð Þð Þqi ; ð24Þ

where kl and qi represent respectively the sn poles of GðsÞ and their order.Poles pk of sub-system (24) are then defined by pk ¼ jpkjejhk , with

jpkj ¼ jkljð Þ1=n;

hk ¼ argðklÞn þ 2kp

n ;

� n2�

argðklÞ2p < k < n

2�argðklÞ

2p :

8>><>>: ð25Þ

Then according to [2], output y2ðtÞ in (22) is defined by

y2ðtÞ ¼Z 1

0lðxÞwðt; xÞdx ð26Þ

with

lðxÞ ¼ 12ip G ð�xÞ�ð Þ � G ð�xÞþ

� � �¼ 1

p

Pmk¼0

Pql¼0akql sinððak � blÞpÞxakþblPm

k¼0a2k x2ak þ

P06k<l<m2akal cosððak � alÞpÞxakþal

: ð27Þ

3.3. Physical interpretation

Several mathematical and physical interpretations of fractional differentiation and of fractional systems exist in the lit-erature [12–18]. A demonstration of Montseny [1] for a fractional integrator is now adapted to deduce a physical interpre-tation of a fractional system. Consider a fractional system described by (exponential part omitted)

_wðtÞ ¼ �xwðt; xÞ þ uðtÞ;y2ðtÞ ¼

R10 lðxÞwðt; xÞdx;

(x 2 Rþ: ð28Þ

1322 J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326

Initial condition are defined for this system by wð0; xÞ ¼ qðxÞ.System (28) can also be written as

_wðt; xÞ ¼ �xwðt; xÞ þ uðtÞ;y2ðtÞ ¼ 1

2

R10 lðxÞwðt; xÞdxþ 1

2

R10 lðxÞwðt; xÞdx:

(ð29Þ

Using the following changes of variable z ¼ffiffiffixp

=ð2pÞ and z ¼ �ffiffiffixp

=ð2pÞ applied respectively to the first and the secondintegral of (29), system (28) is also equivalent to

_wðt; zÞ ¼ �4p2z2wðt; zÞ þ uðtÞy2ðtÞ ¼

R1�1 4p2zlð4pz2Þwðt; zÞdz

(ð30Þ

with wðt; zÞ ¼ wðt;4p2z2Þ and wð0; zÞ ¼ qð4p2z2Þ; z 2 R.If Wðt; zÞ denotes spatial Fourier transform of function /ðt; fÞ such that

wðt; zÞ ¼Ff/ðt; fÞg ¼Z 1

�1/ðt; fÞe�izf df; ð31Þ

system (30) is also given by

F @/ðt;fÞ@t

n o¼ �4p2z2Ff/ðt; fÞg þ uðtÞ;

y2ðtÞ ¼R1�1 4p2zlð4pz2ÞFf/ðt; fÞgdz:

8<: ð32Þ

Given that

y2ðtÞ ¼Z 1

�14p2zlð4p2z2Þ

Z 1

�1/ðt; fÞe�izf df

� �dz ¼

Z 1

�1

Z 1

�14p2zlð4p2z2Þe�izf dz

� �/ðt; fÞdf ð33Þ

inverse spatial Fourier transform applied to (32), leads to

@/ðt;fÞ@t ¼

@2/ðt;fÞ@f2 þ uðtÞdðfÞ

y2ðtÞ ¼R1�1 mðfÞ/ðt; fÞdf

(ð34Þ

with mðfÞ ¼F�1f4p2flð4p2f2Þg; /ðf;0Þ ¼F�1fqð4p2z2Þg; f 2 R.If now the exponential part (whose output is y1ðtÞ in (22)) is taken into account, then any fractional order system can be

described by

_w1ðtÞ...

_wkðtÞ

2664

3775 ¼

p1 ð0Þ. .

.

ð0Þ pk

2664

3775

w1ðtÞ...

wkðtÞ

2664

3775þ

1...

1

264

375uðtÞ; @/ðt; fÞ

@t¼ @

2/ðt; fÞ@f2 þ uðtÞdðfÞ;

y1ðtÞ þp1ac

1� � � pk

ack

h i w1ðtÞ...

wkðtÞ

2664

3775; y2ðtÞ ¼

Z 1

�1mðfÞ/ðt; fÞdf; yðtÞ ¼ y1ðtÞ þ y2ðtÞ:

ð35Þ

Any fractional system can thus be seen as an infinite dimensional system described by a diffusion equation (parabolicdifferential equation [19]) associated to a classical linear (exponential) system.

Fig. 2 is a representation of the interpretation previously given for the system whose output is y2ðtÞ: /ðf; tÞ is the systemstate (of infinite dimension), and uðtÞ is both the input of a classical linear system and of the infinite dimensional systemapplied at f ¼ 0.

To conclude, it can be said that a fractional order system is at the boundary of two classes of systems: classical linear‘‘exponential” rational systems and distributed parameters systems.

Fig. 2. Representation of system whose output is y2ðtÞ.

J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326 1323

4. Interest for the initialization problem

The representation introduced in the previous section permits to take into account initial condition in a coherent waywith system physics.

4.1. Initial conditions compatible with a physical system

In Section 2, it was demonstrated that neither Riemann–Liouville nor Caputo definitions, allow taking into account cor-rectly the initial conditions. Using representation (22) it is proposed now to take into account the initial conditions in thefollowing way:

yðtÞ ¼X

k

pk

ac wkð0Þepkt þZ t

0epkðt�sÞuðsÞ

� �þZ 1

0lðxÞ wð0; xÞe�xt þ

Z t

0e�xðt�sÞuðsÞds

� �dx; ð36Þ

In the case of system (1) and given that no pole is generated, relation (36) becomes

yðtÞ ¼Z 1

0lðxÞ wð0; xÞe�xt þ

Z t

0e�xðt�sÞuðsÞds

� �dx ð37Þ

with lðxÞ ¼ sinðcpÞ=pxc. Given (21), wð0; xÞ is defined by

wð0; xÞ ¼ ð1� e�2xt0 Þ � ð1� e�xt0Þ ¼ e�xt0 � e�2xt0 ð38Þ

and yðsÞ (see (4) for a definition of s) is

yðsÞ ¼ sinðcpÞp

Z 1

0

ðe�xt0 � e�2xt0 Þxc e�xt dx: ð39Þ

Fig. 3 compares the responses given by (39) and by (3). Such a comparison reveals that (39) permits a correct initializationof the system.

4.2. Initial conditions estimation

Obviously, an infinite number of initial conditions wð0; xÞ (or /ðn;0Þ in the interpretation of Fig. 2), is required for initial-izing the system. Suppose that the problem is to find an approximation of these initial conditions. Using relation (36), thesystem output is defined by

yðtÞ ¼X

k

pk

ac wkð0Þepkt þZ 1

0lðxÞwð0; xÞe�xt dxþ

Xk

pk

ac

Z t

0epkðt�sÞuðsÞdsþ

Z 1

0lðxÞ

Z t

0e�xðt�sÞuðsÞdsdx: ð40Þ

Using the change of variable x ¼ e�z, and thus dx ¼ �e�z dz, Eq. (40) becomes

yðtÞ ¼X

k

pk

ac wkð0Þepkt þZ 1

�1lðe�zÞwð0; e�zÞe�e�zte�z dzþ

Xk

pk

ac

Z t

0epkðt�sÞuðsÞdsþ

Z 1

�1lðe�zÞ

Z t

0e�e�zðt�sÞuðsÞdse�z dz:

ð41Þ

-20 -15 -10 -5 0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Res

pons

e co

mpa

rison

s

InputExact solutionPhysical Interpretation

Fig. 3. Comparison of the exact response of system (1) with the responses obtained with relation (39).

1324 J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326

An approximation of (41) is thus

yðtÞ �X

k

pk

ac wkð0Þepkt þXN2

j¼�N1

lðe�jDzÞwð0; e�jDzÞe�e�jDzte�jDzDzþX

k

pk

ac

Z t

0epkðt�sÞuðsÞds

þXN2

j¼�N1

Z t

0lðe�jDzÞe�e�jDzðt�sÞe�jDzDzuðsÞds ð42Þ

In the interpretation of Fig. 2, such an approximation of the initialization consists in considering a limited number of areasin which the initial conditions are constant (initial conditions being considered null elsewhere). This initialization is repre-sented in Fig. 4.

If the system model, its input and its output are known, a least square problem can thus be defined for the computation ofwkð0Þ and wð0; e�jDzÞ.

4.3. Approximation of the initialization function

Suppose that the problem is now to compute an initialization function in order to restore the system state if initial con-ditions defined in Section 3.1 are taken into account.

Let yðtÞ denotes the response of the system (at rest before t ¼ 0) to the input uðtÞ. Let yt0ðtÞ be the response of the systemto the input uðtÞHðt � t0Þ. The problem now is to find an initialization function as in [6] denoted wðtÞ such thatyðtÞ ¼ yt0ðtÞ þ wðtÞ with t > t0 (Fig. 5).

An approximation of wðtÞ was proposed for a fractional order integrator by [20] using a recursive RC network.For a general fractional system, the approximation of such an initialization function can be obtained using the initializa-

tion introduced in Section 3.1.Using relation (40) and the change of variable introduced before (41), the system output with zero initial conditions can

be approximated by

yðtÞ �X

k

pk

ac

Z t

0epkðt�sÞuðsÞdsþ

XN2

j¼�N1

Z t

0lðe�jDzÞe�e�jDzðt�sÞe�jDzDzuðsÞds: ð43Þ

Using Laplace transform relation (43) becomes

YðsÞ �X

k

pk

acUðsÞ

sþ pkþXN2

j¼�N1

lðe�jDzÞDzs

xkþ 1

UðsÞ ð44Þ

with xk ¼ e�jDz. One can note that the change of variable defined before relation (41) permits an approximation of yðtÞinvolving a recursive distribution of poles. A recursive ratio e�Dz indeed exists between two consecutive poles.

Fig. 4. Approximation of the initialization.

0t

( )ty ( )tyt0

( )tΨ

t0

Fig. 5. Initialization function definition.

Fig. 6. Approximation of yðtÞ; yt0ðtÞ and of the initialization function wðtÞ.

J. Sabatier et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1318–1326 1325

Now consider a time t0 such that 0 < t0 < t, then Eq. (44) can be written as

yðtÞ �X

k

pk

ac

Z t0

0epkðt�sÞuðsÞdsþ

XN2

j¼�N1

Z t0

0lðe�jDzÞe�e�jDzðt�sÞe�jDzDzuðsÞdsþ

Xk

pk

ac

Z t

t0

epkðt�sÞuðsÞds

þXN2

j¼�N1

Z t

t0

lðe�jDzÞe�e�jDzðt�sÞe�jDzDzuðsÞds: ð45Þ

Thus it can be written that

yt0�X

k

pk

ac

Z t

t0

epkðt�sÞuðsÞdsþXN2

j¼�N1

Z t

t0

lðe�jDzÞe�e�jDzðt�sÞe�jDzDzuðsÞds ð46Þ

and that

wðtÞ �X

k

pk

ac

Z t0

0epkðt�sÞuðsÞdsþ

XN

j¼�N

Z t0

0lðe�jDzÞe�e�jDzðt�sÞe�jDzDzuðsÞds: ð47Þ

Using a software such as MATLAB/SIMULINK, an approximation of yðtÞ; yt0ðtÞ and wðtÞ can be obtained with the diagramsrepresented by Fig. 6.

5. Conclusion

This paper demonstrates that neither the Riemann–Liouville nor the Caputo definitions permit to obtain a physicallyacceptable initialization of a fractional system. The diffusive representation introduced in [1,11,6] is then used to proposea physically acceptable solution for the initialization problem. Methods are also proposed to obtain an approximation ofthe initialization function defined in [6] and to estimate an approximation of the system initial state if the system inputand output are known. A physical interpretation of a fractional system is also proposed that demonstrates that any fractionalsystem can be viewed as an infinite dimensional system described by a diffusion equation (parabolic differential equation)associated to a classical rational linear (exponential) system.

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