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Rational Function Approximation of a Fundamental Fractional Order

Transfer Function

Djamel Boucherma1(&), Abdelfatah Charef2, and Hassene Nezzari1

1 Research Center in Industrial Technologies (CRTI), P.O. Box 64, 16014 Cheraga, Algiers, Algeria

djamelboucherma25@yahoo.com, nezzarih@yahoo.fr

2 Département d’Electronique, Universitédes frères Mentouri, Constantine, Algeria

afcharef@yahoo.com

Abstract. This paper introduces a rational function approximation of the fractional order transfer function HðsÞ ¼½1þ ðsðs0a

02a; for0\a0:5. This fractional order transfer function is one of the fundamental functions of the linear fractional system of commensurate order corresponding to pure complex conjugate poles or eigenvalues, in sa. Hence, the proposed approximation will be used in the solution of the linear fractional systems of commensurate order.

Illustrative examples are given to show the exactitude and the efficiency of the approximation method.

Keywords: Fractional power zero

Linear fractional system

Irrational

transfer function

Rational transfer function

1 Introduction

The theory of fractional order systems has gained some importance during the last decades (Miller et al.1993), (Podlubny1999), (Kilbas et al.2006), (Monje et al.2010), (Caponetto et al.2010). Therefore, active research work tofind accurate and efficient methods to solve linear fractional order differential equations is still underway to establish a clear linear fractional order system theory accessible to the general engi- neering community. More recently, a great deal of effort has been expended in the development of analytical techniques to solve them. The goal of these methods is to derive an explicit analytical expression for the general solution of the linear fractional differential equations (Charef2006a), (Bonilla et al.2007), (Oturançet al.2008), (Hu et al.2008), (Arikoglu et al. 2009), (Odibat2010), (Charef et al. 2011).

A linear single input single output (SISO) fractional system of commensurate order is described by the following linear fractional order differential equation:

XN

i¼0

aiDiayðtÞ ¼XM

j¼0

bjDjauðtÞ ð1Þ

©Springer International Publishing AG 2017

M. Chadli et al. (eds.),Recent Advances in Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 411, DOI 10.1007/978-3-319-48929-2_20

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where u(t) is the input, y(t) is the output,a is a real number such that 0 <a < 1, ai

(1 i N) andbi(0 j M) are constant real numbers withM N. With zero initial conditions, the fractional order system transfer function is given as:

GðsÞ ¼YðsÞ UðsÞ¼

PM

j¼0

bjð Þsa j PN

i¼0aið Þsa i

ð2Þ

This fractional transfer function can be decomposed into several elementary fun- damental functions corresponding to different types of poles, insa, as:

GðsÞ ¼XK

k¼1

HkðsÞ ð3Þ

where the functionsHk(s) are given, according to the poles of the fractional system, as:

• For a simple real pole:

HkðsÞ ¼ 1 saþp

ð Þ ð4Þ

• For a pair of complex poles with negative real part:

HkðsÞ ¼ x2n

½s2aþ2fxnsaþx2n ð5Þ

HkðsÞ ¼ saþxnf

½s2aþ2fxnsaþx2n ð6Þ

• For a pair of complex poles with null real part:

HkðsÞ ¼ x2n

½s2aþx2n ð7Þ

HkðsÞ ¼ xnsa

½s2aþx2n ð8Þ

In previous works (Charef2006a) and (Charef et al. 2011), (Nezzari et al. 2011), (Boucherma et al.2011), the elementary fundamental functions defined in (4), (5), (6) and (7) have been approximated by rational ones in order to represent them by linear

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time-invariant system models so as to derive their closed form impulse and step responses as well as their performance characteristics. Using these approximations, simple analog circuits have been also derived to represent the above irrational functions of the fractional order system. This paper gives a rational function approximation of the fundamental function represented by the irrational transfer function of (8) which cor- responds to pure complex conjugate poles or eigenvalues, insa. In (Boucherma et al.

2011), the approximation of (8) has been done for 0:5\a\1. In this work the approximation of (8) will be done for 0\a0:5. First, the basic ideas and the derived formulations of the approximation technique are presented. Then, the impulse and step responses of this type of fractional system are derived. Finally, illustrative examples are presented to show the exactitude and the usefulness of the approximation method.

2 Rational Function Approximation

Forð Þs0 a¼x1

n, (8) can be rewritten as:

HðsÞ ¼XðsÞ

EðsÞ¼ ðs0a

½1þ ðs02a; 0\a0:5 ð9Þ The above irrational function is the transfer function of the linear fractional order system represented by the following fundamental linear fractional order differential equation:

s0

ð Þ2ad2ax tð Þ

dt2a þx tð Þ ¼ð Þs0 adae tð Þ

dta ð10Þ

In this context, the transfer function of (9) has two pure complex conjugate poles, in sa. To represent the linear fractional order system of (10) by a linear time-invariant system model so as to derive their closed form impulse and step responses, its irrational transfer function of (9) will be approximated by a rational function. To do so, we will consider two cases based on the fractional derivativea.

2.1 Case 1: 0 <a< 0.5

For this case, the function of (9) can be decomposed in two functions as follows:

H sð Þ ¼H1ð Þ s H2ð Þ ¼s ðs0a 1

1þðs02a ð11Þ whereH1ð Þ ¼s ðs0a andH2ð Þ ¼s 1þ 1

s0s ð Þ2a.

In a given frequency band of interest [xL,xH], around the frequencyx0= (1/s0), the fractional order differentiator H1ð Þ ¼s ðs0a can be approximated by a rational function as follows (b, 2006):

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H1ð Þ ¼s ðs0affisa0 KD

QN1

i¼0

zs

i

QN1

i¼0

ps

i

2 66 64

3 77

75 ð12Þ

where the poles pi and the zeros zi(0 i N1), the constant KDand the number N1 of the approximation are given by:

pi = p0 (ab)i;zi = z0 (ab)i;KD¼ ðxcÞa; N1¼ Integer logðxmax=z0Þ

logð Þab

þ1

ð13Þ

For some given real values y (dB),dandb, the approximation parameters a, b, p0, z0,xcand xmaxcan be calculated as:

a¼10

y 10 1að Þ

h i

;b¼10½ 10ay ;xc¼dxL; xmax¼bxH; z0¼xc ffiffiffi

pb

andp0¼az0

ð14Þ

By the decomposition of the rational function of (12), we will get:

H1ð Þ ¼s ðs0affisa0 KDþXN1

i¼0

kis 1þpsi

0

@

1

A ð15Þ

ki¼ KD

p0ðabÞi QN1

j¼0ð1aðabÞðijÞÞ QN1

j¼0;i6¼jð1 ðabÞðijÞÞ

;i¼0;1;. . .;N1 ð16Þ

Because 0 <a< 0.5 the number 2a is then 0 < 2a< 1; hence, in a given fre- quency band [0,xH], the fractional systemH2ð Þ ¼s 1þ 1

s0s

ð Þ2acan be approximated by a rational function as follows (Charef2006a):

H2ðsÞ ¼ 1

1þ ðs02a2NX21

j¼1

kkj

ppsj

ð17Þ

where the poles ppjand the residues kkj(for 1 j 2N2−1), and the number N2of the approximation are given, for some given real valueskand b, by:

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ppj¼ðkÞðjNÞ s0 kkj¼ 1

2p

sin½ð1aÞp cosh½a logðs1

0ppjÞ cos½ð1aÞp

" #

N2 ¼Integer log½s0bxH logð Þk

þ1

ð18Þ

Therefore, the function of (11) is approximated by a rational function as follows:

H sð Þ ffisa0 KDþXN1

i¼0

kis 1þps

i

0

@

1 A 2NX21

j¼1

kkj

pps

j

0

@

1

A ð19Þ

H sð Þ ffi 2NX21

j¼1

sa0KD

ppjkkj

sþppj

!

þ XN1

i¼0

X

2N21

j¼1

sa0

ðpikiÞ ppjkkj s sþpi

ð Þ sþppj

! ð20Þ

By the decomposition of the rational function of (20), we will get:

H sð Þ ffi 2NX21

j¼1

sa0KD

ppjkkj

sþppj

!

þ XN1

i¼0

X

2N21

j¼1

Aij sþpi

ð Þþ Bij sþppj

! ð21Þ

where the residues Aijand Bij(for 0 i N1and 1 j 2N2−1) are given by:

Aij¼ sa0

p2

iki

ppjkkj

pippj

; Bij¼ sa0

ðpikiÞ pp2

jkkj

ppjpi

ð22Þ

Hence, we can write that:

H sð Þ ffi 2NX21

j¼1

sa0KD

ppjkkj

sþppj

!

þ 2NX21

j¼1

PN1

i¼0

Bij

sþppj

0

BB

@

1 CC

Aþ XN1

i¼0

P

2N21 j¼1

Aij

sþpi

ð Þ

0 BB B@

1 CC CA

ð23Þ

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H sð Þ ffi XN1

i¼0

Ai sþpi

ð Þ

!

þ 2NX21

j¼1

Bj sþppj

!

ð24Þ

where the residuesAi (0 i N1), andBj(1 j 2N2−1), are given by:

Ai¼2NX21

j¼1

Aij; Bj¼ sa0KD ppjkkj

þXN1

i¼0

Bij ð25Þ

2.2 Case 2:a= 0.5

For this case, the function of (9) can be rewritten as:

H sð Þ ¼ ðs00:5

1þðs0sÞ ð26Þ

From Eq. (15), the above function is approximated by a rational function as follows:

H sð Þ ffis00:5 KDþXN1

i¼0

kis 1þpsi

0

@

1

A 1

1þðs0

ð27Þ

H sð Þ ffi sð00 :KD sþ1=s0

ð Þ

!

þ XN1

i¼0

sð00:5Þ

piki

ð Þs sþpi

ð Þðsþ1=s0Þ 0

@

1

A ð28Þ

By the decomposition of the rational function of (28), we will get:

H sð Þ ffi sð00 :KD

sþ1=s0

ð Þ

! þ XN1

i¼0

Ci

sþpi

ð Þþ Di

sþ1=s0

ð Þ

!

ð29Þ

where the residuesCiand Di (for 0 i N1) are given by:

Ci¼ sð00:5Þ

p2iki

pi1=s0

; Di¼ sð0 1:5Þ

piki

ð Þ 1=s0pi

ð30Þ

Hence, we can write that:

H sð Þ ffi XN1

i¼0

Ci

sþpi

ð Þ

!

þ D

sþ1=s0

ð Þ

ð31Þ

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with the residuesCi¼Ci (for 0 i N1) and D¼ sð0 0:5ÞKD

h i

þ PN1

i¼0

Di

.

3 Time Responses

3.1 Case 1: 0 <a< 0.5 From Eq. (24), we have that:

H sð Þ ¼XðsÞ

EðsÞ¼ XN1

i¼0

Ai

sþpi

ð Þ

!

þ 2NX21

j¼1

Bj

sþppj

!

ð32Þ

then,

XðsÞ ¼ XN1

i¼0

Ai sþpi

ð Þ

!

þ 2NX21

j¼1

Bj sþppj

!

" #

EðsÞ ð33Þ

For e(t) =d(t) the unit impulse E(s) = 1, we will have:

XðsÞ ¼ XN1

i¼0

Ai

sþpi

ð Þ

!

þ 2NX21

j¼1

Bj

sþppj

!

" #

ð34Þ

Hence, the impulse response of (10) is:

x tð Þ ¼L1fX sð Þg ¼ XN1

i¼0

Aiexpðpi

!

þ 2NX21

j¼1

Bjexpðppj

! ð35Þ

For e(t) = u(t) the unit step E(s) = 1/s, (33) will be:

XðsÞ ¼ XN1

i¼0

Ai sþpi

ð Þ

!

þ 2NX21

j¼1

Bj sþppj

!

" #

1

s ð36Þ

XðsÞ ¼ XN1

i¼0

Ai

sþpi

ð Þ

1 s

!

þ 2NX21

j¼1

Bj

sþppj

1 s

!

ð37Þ

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XðsÞ ¼ XN1

i¼0

Ai pi

1 s 1

sþpi

ð Þ

!

þ 2NX21

j¼1

Bi

ppj

1

s 1

sþppj

!! ð38Þ

Hence, the step response of (10) is:

x tð Þ ¼L1fXðsÞg ¼ XN1

i¼0

Ai

pi

1 expðpi

½

!

þ 2NX21

j¼1

Bj

ppj

1expðppj

! ð39Þ

3.2 Case 2:a= 0.5 From Eq. (31), we have that:

H sð Þ ¼XðsÞ

EðsÞ¼ XN1

i¼0

Ci

sþpi

ð Þ

!

þ D

sþ1=s0

ð Þ

ð40Þ

then,

XðsÞ ¼ XN1

i¼0

Ci

sþpi

ð Þ

!

þ D

sþ1=s0

ð Þ

" #

EðsÞ ð41Þ

For e(t) =d(t) the unit impulse E(s) = 1, we will have:

XðsÞ ¼ XN1

i¼0

Ci sþpi

ð Þ

!

þ D

sþ1=s0

ð Þ

" #

ð42Þ

Hence, the impulse response of (10), fora = 0.5, is:

x tð Þ ¼L1fX sð Þg ¼ XN1

i¼0

Ciexpðpi

!

þ Dexpðt=s0Þ ð43Þ

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For e(t) = u(t) the unit step E(s) = 1/s, (41) will be:

XðsÞ ¼ XN1

i¼0

Ci

sþpi

ð Þ

!

þ D

sþ1=s0

ð Þ

" #

1

s ð44Þ

XðsÞ ¼ XN1

i¼0

Ci sþpi

ð Þ

1 s

!

þ D

sþ1=s0

ð Þ

1 s

ð45Þ

XðsÞ ¼ XN1

i¼0

Ci

pi 1 s 1

sþpi

ð Þ

!

þ Ds0 1

s 1

sþ1=s0

ð Þ

ð46Þ

Hence, the step response of f (10), fora = 0.5, is:

x tð Þ ¼L1fXðsÞg ¼ XN1

i¼0

Ci

pi ½1expðpi

!

þ Ds0½1expðt=s0Þ ð47Þ

4 Illustrative Example

Let usfirst consider the fractional system represented by the following fundamental linear fractional order differential equation witha= 0.35 ands0= 2 as:

ð Þ2 0:7d0:7x tð Þ

dt0:7 þx tð Þ ¼ð Þ2 0:35d0:35e tð Þ

dt0:35 ð48Þ

its transfer function is given by:

H sð Þ ¼ ð Þ2s0:35

1þð Þ2s 0:7 ð49Þ

Its rational function approximation, in a given frequency band, is given as:

H sð Þ ¼ ð Þ2s 0:35

1þð Þ2s0:7¼ XN1

i¼0

Ai sþpi

ð Þ

!

þ 2NX21

j¼1

Bj sþppj

!

ð50Þ

For the fractional order differentiator (2s)0.35, the frequency band of approximation is [xL, xH] = [104rad/s, 104rad/s], around x0= (1/s0) = 0.5 rad/s, y= 1 dB, d= 0.1, and b= 100. For the fractional system 1

1þð Þ2s0:7, the frequency band of

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approximation [0,xH] = [0, 104rad/s], k= 4 andb = 100. Then, the approximation parameters of H(s) are:

a¼1:4251; b¼1:9307; xc¼105; xmax¼106; p0¼1:9802105; KD¼0:0178; N1¼25; and N2¼11

Hence, polespi, the residuesAi (0 i 25), the polesppj and the residues Bj

(1 j 20), are given by:

pi¼ 1:9802105

ð2:7514Þi

Ai¼X21

j¼1

1:1230107

ð2:7514Þið4Þðj11Þ

Q25

j¼0

1 ð 1:4251Þð2:7514ÞðijÞ

h i

Q25

j¼0;i6¼jð1 ð2:7514ÞðijÞÞ 2

66 66 66 66 4

3 77 77 77 77 5 phð0:5Þð4Þðj11Þð1:9802105Þð2:7514Þii

h i

sin½ð0:3Þp

cosh½ð0:35Þlogðð4Þð11jÞÞ cos½ð0:3Þp

" #

ð51Þ

ppj¼ð0:5Þð4Þðj11Þ Bj¼ð0:0057Þ

p ð4Þðj11Þ sin½ð0:3Þp

cosh½ð0:35Þlogðð4Þð11jÞÞ cos½ð0:3Þp

" #

þX25

i¼0

ð0:0029Þð4Þ2ðj11Þ Q25

j¼0

1ð1:4251Þð2:7514ÞðijÞ

½

Q25

j¼0;i6¼j

ð1ð2:7514ÞðijÞÞ

phð0:5Þð4Þðj11Þð1:9802105Þð2:7514Þii

sin½ð0:3Þp

cosh½ð0:35Þlogðð4Þð11jÞÞ cos½ð0:3Þp

" #

8>

>>

>>

>>

>>

>>

<

>>

>>

>>

>>

>>

>:

9>

>>

>>

>>

>>

>>

=

>>

>>

>>

>>

>>

>;

ð52Þ

Figures1 and 2 show the bode plots of the fundamental linear fractional order system transfer function of (49) and of its proposed rational function approximation of (50). We can easily see that they are all quite overlapping in the frequency band of interest.

Figures3 and 4 show, respectively, the impulse and the step responses of the fractional order system of (48).

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Fig. 1. Magnitude bode plots of (49) and of its proposed approximation.

Fig. 2. Phase bode plots of (49) and of its proposed approximation

(12)

Fig. 3. Impulse response of (48) from its proposed approximation.

Fig. 4. Step response of (48) from its proposed approximation.

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As a second example, we will consider the fractional order system represented by the following fundamental linear fractional order differential equation witha= 0.5 and s0= 0.16 as:

0:16 ð Þd x tð Þ

dt þx tð Þ ¼ð0:16Þ0:5d0:5e tð Þ

dt0:5 ð53Þ

its transfer function is given by:

H sð Þ ¼ ð0:16sÞ0:5

1þð0:16sÞ ð54Þ

Its rational function approximation, in a given frequency band, is given as:

H sð Þ ¼ ð0:16sÞ0:5

1þð0:16sÞ¼ XN1

i¼0

Ci

sþpi

ð Þ

!

þ D

sþ1=s0

ð Þ

ð55Þ

For the fractional differentiator (0.16s)0.5, the frequency band of approximation is [xL, xH] = [10−4rad/s, 104rad/s], around x0= (1/s0) = 6.25 rad/s, y= 1 dB, d= 0.1, and b= 100. Then, the approximation parameters of H(s) are:

a¼1:5849; b¼1:5849; xc¼105; xmax ¼106; p0 ¼1:9953105;KD¼0:0032and N1¼28

Hence, polespi, the residues Ci(0 i 25), and the residueD, are given by:

pi¼ 1:9953105 2:5119 ð Þi Ci¼ ð1:5963107Þð2:5119Þi

1:9953105

ð Þð2:5119Þi6:25

!

Q28

j¼0

1 ð1:5849Þð2:5119ÞðijÞ

Q28

j¼0;i6¼j1 ð2:5119ÞðijÞ 0

BB B@

1 CC CA

ð56Þ

D¼ð0:008Þ

þX28

i¼0

ð0:05Þ

6:25ð1:9953105Þð2:5119Þi

!

Q28

j¼01 ð1:5849Þð2:5119ÞðijÞ Q28

j¼0;i6¼j

1 ð2:5119ÞðijÞ

0 BB B@

1 CC CA 2

66 66 66 66 64

3 77 77 77 77 75

ð57Þ

(14)

Fig. 5. Magnitude bode plot of (54) and of its proposed approximation.

Fig. 6. Phase bode plot of (54) and of its proposed approximation.

(15)

Fig. 7. Impulse response of (53) from its proposed approximation.

Fig. 8. Step response of (53) from its proposed approximation.

(16)

Figures5 and 6 show the bode plots of the fundamental linear fractional system transfer function of (54) and of its proposed rational function approximation of (55).

We note that they are overlapping in the frequency band of interest.

Figures7 and 8 show, respectively, the impulse and the step responses of the fractional order system of (53).

5 Conclusion

In this paper, we have presented a rational function approximation of the fractional order transfer function HðsÞ ¼½1þ ðsðs0a

02a; for0\a0:5. This fractional order transfer function is one of the fundamental functions of the linear fractional system of commensurate order, represented by the linear fractional state-spaceDax tð Þ ¼A x tð Þ.

H(s) corresponds to pure complex conjugate eigenvalues of the A matrix. First, closed form of the approximation technique has been derived. Then, the impulse and step responses of this type of fractional system have been obtained. Finally, illustrative examples are presented to show the exactitude and the usefulness of the approximation method.

References

Arikoglu, A., Ozkol, I.: Solution of fractional differential equations by using differential transform method. Chaos Soliton Frac.40(2), 521–529 (2009)

Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput.187, 68–78 (2007)

Boucherma, D., Charef, A.: Approximation d’une fonction fondamentale d’ordre fractionnaire.

In: Proceedings de la seconde conférence sur les systèmes d’ordre fractionnaire et leur applications, SOFA 2011, Tizi-ouzou, Algeria, 24–26 October 2011

Caponetto, R., Dongola, G., Fortuna, L., Petráš, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific Publishing Co. Pte. Ltd, Singapore (2010)

Charef, A.: Modeling and analog realization of the fundamental linear fractional order differential equation. Nonlinear Dyn.46, 195–210 (2006a)

Charef, A.: Analogue realisation of fractional order integrator, differentiator and fractional PIkDl controllers. IEE Proc. Control Theory Appl.153(6), 714–720 (2006b)

Charef, A., Nezzari, H.: On the fundamental linear fractional order differential equation.

Nonlinear Dyn.65(3), 335–348 (2011)

Hu, Y., Luo, Y., Lu, Z.: Analytical solution of the linear fractional differential equation by Adomian decomposition method. J. Comput. Appl. Math.215(1), 220–229 (2008)

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley Inc., New York (1993)

Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D., Feliu, V.: Fractional-Order Systems and Controls Fundamentals and Applications. Springer, England (2010)

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Nezzari, H., Charef, A.: Analog realization of generalized fractional order damped sine and cosine functions. In: Proceedings of the Symposium on Fractional Signals and Systems, FSS 2011, Coimbra, Portugal, 4–5 November 2011

Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math Appl.59, 1171–1183 (2010)

Oturanç, G., Kurnaz, A., Keskin, Y.: A new analytical approximate method for the solution of fractional differential equations. Int. J. Comput. Math.85(1), 131–142 (2008)

Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

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Based on the fact that the H ∞ norm of an an- alytic transfer function is defined as the supremum of its maximal singular value on the imaginary axis, the matching of the rational

Chorin et al ont montré dans leur étude portant sur 84 patients recevant le même traitement que l'intervalle QTc s’était sévèrement allongé à plus de 500 ms chez neuf (11%) de