Rational Function Approximation of a Fundamental Fractional Order Transfer Function
Abstract—This paper introduces a rational function approximation of the fractional order transfer
function for0 05
s 1 s s
H 2
0
0 , .
] ) ( [
) ) (
(
. 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 s. 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.
I. INTRODUCTION
The theory of fractional order systems has gained some importance during the last decades (Miller et al, 1993), (Podlubny, 1999), (Kilbas et al, 2006), (Monje et al, 2010), (Caponetto et al, 2010). Therefore, active research work to find 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 engineering 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 (Charef, 2006a), (Bonilla et al, 2007), (Oturanç et al, 2008), (Hu et al, 2008), (Arikoglu et al, 2009), (Odibat, 2010), (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:
) ( )
(t b D u t
y D
a j
N
0 i
M
0 j
j i
i
(1) where u(t) is the input, y(t) is the output, is a real number such that 0 < < 1, ai (1 ≤ i ≤ N) and bi (0 ≤ j ≤ M) are constant real numbers with M ≤ N. With zero initial
conditions, the fractional order system transfer function is given as:
N
0 i
i i M
0 j
j j
s a
s b s
U s s Y G
) (
) ) (
( (2)
This fractional transfer function can be decomposed into several elementary fundamental functions corresponding to different types of poles, in s, as:
K
1 k
k s H s
G( ) ( ) (3) where the functions Hk(s) are given, according to the poles of the fractional system, as:
For a simple real pole:
s 1 p
s Hk
)
( (4)
For a pair of complex poles with negative real part:
] [
)
( 2
n n 2
2 n
k s s 2 s
H
(5)
] [
)
( 2
n n 2
n
k s 2 s
s s
H
(6)
For a pair of complex poles with null real part:
] [
)
( 2
n 2
2 n
k s s
H
(7)
] [
)
( 2
n 2
n
k s
s s
H
(8) In previous works (Charef, 2006a) 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 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 Djamel Boucherma
Welding and NDT Research Centre (CSC).
BP 64 CHERAGA – ALGERIA djamelboucherma25@yahoo.com
Abdelfatah Charef Département d’Electronique,
Université Constantine 1 CONSTANTINE- ALGERIA
afcharef@yahoo.com
Hassene Nezzari Welding and NDT Research
Centre (CSC).
BP 64 CHERAGA – ALGERIA nezzarih@yahoo.fr
function represented by the irrational transfer function of (8) which corresponds to pure complex conjugate poles or eigenvalues, in s. In (Boucherma et al, 2011), the approximation of (8) has been done for0.51. In this work the approximation of (8) will be done for00.5. 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.
II. RATIONALFUNCTIONAPPROXIMATION
For
n 0
1
, (8) can be rewritten as:
5 0 0 s
1 s s
E s s X
H 2
0
0 , .
] ) ( [
) ( ) (
) ) (
(
(9) The above irrational function is the transfer function of the linear fractional order system represented by the following fundamental linear fractional order differential equation:
dt t e t d
x dt
t x d
2 0 2 2
0 (10)
In this context, the transfer function of (9) has two pure complex conjugate poles, in s. 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 derivative .
A. Case 1: 0 < < 0.5
For this case, the function of (9) can be decomposed in two functions as follows:
20 0
2
1 1 s
x 1 s s H s H s
H (11)
where H1
s 0s and
0 22 1 s
s 1
H .
In a given frequency band of interest [ωL, ωH], around the frequency ω0 = (1/0), the fractional order differentiator
s sH1 0 can be approximated by a rational function as follows (b, 2006):
N1
0
i i
N1
0
i i
D 0 0
1
p 1 s
z 1 s K
s s
H
(12)
where the poles pi and the zeros zi (0 ≤ i ≤ N1), the constant KD and the number N1 of the approximation are given by:
ab 1
Integer z N
K (ab) z
= z , (ab) p
= p
0 1
c D i 0 i i 0 i
log log
, ) ( ,
max
(13)
For some given real values y (dB), and , the approximation parameters a, b, p0 , z0, ωc and ωmax can be calculated as:
0 0 c
0 H max
L c 10
y 1
10 y
z a p and ω b
z 10 b , 10
a
,
, ,
(14)
By the decomposition of the rational function of (12), we will get:
N1
0 i
i i D
0 0
1
p 1 s
s K k
s s
H (15)
N1
j i 0 j
j i N1
0 j
j i
i 0
D i
ab 1
ab a 1 ab
p k K
,
) ) ( (
) ) ( ( ) (
, i=0, 1, ..., N1 (16)
Because 0 < < 0.5 the number 2 is then 0 < 2 < 1;
hence, in a given frequency band [0, ωH], the fractional system
0 22 1 s
s 1
H can be approximated by a
rational function as follows (Charef, 2006a):
2N2 1
1 j
j j 2
0 2
pp 1 s
kk s
1 s 1
H
) ) (
( (17)
where the poles ppj and the residues kkj (for 1 ≤ j ≤ 2N2- 1), and the number N2 of the approximation are given, for some given real values and ,by:
1Integer N
pp 1 1
1 2
kk 1 pp
H 0 2
j 0 j
0 N j j
log ] log[
] ) cos[(
)]
log(
cosh[
] ) sin[(
)
( ( )
(18)
Therefore, the function of (11) is approximated by a rational function as follows:
2 1 N 2
1 j
j 1 j
N
0 i
i i D
0
pp 1 s
kk
p 1 s
s K k
s
H (19)
N1
0 i
2 1 N 2
1
j i j
j j i i 0 2 1
N 2
1
j j
j j D 0
pp s p s
s kk pp k p pp s
kk pp s K
H
(20)
By the decomposition of the rational function of (20), we will get:
N1
0 i
2 1 N 2
1
j j
ij i
ij 2 1
N 2
1
j j
j j D 0
pp s
B p
s A
pp s
kk pp s K
H
(21)
where the residues Aij and Bij (for 0 ≤ i ≤ N1 and 1 ≤ j ≤ 2N2-1) are given by:
i j
j 2 i j i 0 ij j
i
j j i 2 0 i
ij pp p
kk pp k p pp B
p
kk pp k A p
, (22)
Hence, we can write that:
N1 0
i i
2 1 N 2
1 j 1 ij
N2 2
1
j j
N1 0 i
ij 2 1
N 2
1
j j
j j D 0
p s
A pp
s B
pp s
kk pp K s
H
(23)
2 1 N 2
1
j j
1 j N
0
i i
i
pp s
B p
s s A
H (24)
where the residues Ai (0 ≤ i ≤ N1), and Bj(1 ≤ j ≤ 2N2-1), are given by:
N1
0 i
ij j
j D j 0
2 1 N 2
1 j
i Aij B K pp kk B
A , (25)
B. Case 2: = 0.5
For this case, the function of (9) can be rewritten as:
1
ss s H
0 5 0 0
. (26) From equation (15), the above function is approximated by a rational function as follows:
1 s
1 p
1 s s K k
s H
0 N1
0 i
i i D
5 0
0
. (27)
N1
0
i i 0
i i 5 0 0 0
D 5 0 0
1 s p s
s k p 1
s s K
H
/ /
. )
. (
(28) By the decomposition of the rational function of (28), we will get:
N1
0
i 0
i i
i 0
D 5 0 0
1 s
D p
s C 1
s s K
H
/ /
) . (
(29) where the residues Ci and Di (for 0 ≤ i ≤ N1) are given by:
i 0
i i 5 1 0 i 0 i
i 2 i 5 0 0
i 1 p
k D p
1 p
k p
C
, / /
) . . (
(30) Hence, we can write that:
0
N1 0
i i
i
1 s
D p
s s C
H / (31)
with the residues Ci Ci (for 0 ≤ i ≤ N1) and
N1
0 i
i D
5 0
0 K D
D ( .) .
III. TIMERESPONSES
A. Case 1: 0 < < 0.5
From Equation (24), we have that:
2 1 N 2
1
j j
1 j N
0
i i
i
pp s
B p
s A s
E s s X
H ( )
)
( (32)
then,
( ))
( E s
pp s
B p
s s A
X
2 1 N 2
1
j j
1 j N
0
i i
i
(33) For e(t) = (t) the unit impulse E(s)=1, we will have:
2 1 N 2
1
j j
1 j N
0
i i
i
pp s
B p
s s A
X( ) (34)
Hence, the impulse response of (10) is :
) exp(
) exp(
2 1 N 2
1 j
j j N1
0 i
i i 1
t pp B
t p A
s X L t x
(35)
For e(t) = u(t) the unit step E(s)=1/s, (33) will be:
s
1 pp s
B p
s s A
X
2 1 N 2
1
j j
1 j N
0
i i
) i
( (36)
2 1 N 2
1
j j
1 j N
0
i i
i
s 1 pp s
B s
1 p s s A
X( ) (37)
2 1 N 2
1
j j j
i N1
0
i i i
i
pp s
1 s 1 pp
B
p s
1 s 1 p s A
X( )
(38)
Hence, the step response of (10) is:
2 1 N 2
1 j
j j
j N1
0 i
i i
1 i
t pp pp 1
B
t p p 1
s A X L t x
) exp(
) exp(
)
( (39)
B. Case 2: = 0.5
From Equation (31), we have that:
0
N1 0
i i
i
1 s
D p
s C s
E s s X
H ( ) /
)
( (40)
then,
/
( ))
( E s
1 s
D p
s s C
X
0 N1
0
i i
i
(41)
For e(t) = (t) the unit impulse E(s)=1, we will have:
0
N1
0
i i
i
1 s
D p
s s C
X( ) / (42)
Hence, the impulse response of of (10), for = 0.5, is:
exp( / )
) exp(
0 N1
0 i
i i 1
t D
t p C
s X L t x
(43)
For e(t) = u(t) the unit step E(s)=1/s, (41) will be:
s
1 1
s D p
s s C
X
0 N1
0
i i
i
) /
( (44)
s
1 1
s D s
1 p s s C
X
0 N1
0
i i
i
) /
( (45)
0 0
N1 0
i i i
i
1 s
1 s D 1
p s
1 s 1 p s C
X
/ )
(
(46)
Hence, the step response of f (10), for = 0.5, is:
exp( / )
) exp(
) (
0 0
N1 0 i
i i
1 i
t 1
D
t p p 1
s C X L t x
(47)
IV. ILLUSTRATIVEEXAMPLE
Let us first consider the fractional system represented by the following fundamental linear fractional order differential equation with = 0.35 and = 2 as:
35 0 35 0 35 0 7
0 7 0 7 0
dt t e 2 d
t x dt
t x 2 d
. . . .
.
. (48)
its transfer function is given by:
0735 0
s 2 1
s s 2
H .
.
(49)
Its rational function approximation, in a given frequency band, is given as:
2 1 N 2
1
j j
1 j N
0
i i
i 7
0 35 0
pp s
B p
s A s
2 1
s s 2
H .
.
(50) For the fractional order differentiator (2s)0.35, the frequency band of approximation is [ωL, ωH] = [10-4rad/s, 104 rad/s], around ω0 = (1/0) = 0.5 rad/s, y = 1dB, = 0.1, and
=100. For the fractional system
2s071 1
. , the
frequency band of approximation [0, ωH] = [0, 104 rad/s],
= 4 and =100. Then, the approximation parameters of H(s) are:
11 N and 25 N 0.0178 K
, 10
* 1.9802 p
10 10
1.9307 b
1.4251, a
2 1
D 5 - 0
6 max 5 c
, ,
, ,
,
Hence, poles pi, the residues Ai (0 ≤ i ≤ 25), the polesppj and the residues Bj(1 ≤ j ≤ 20), are given by:
21 1 j
j 11
i 5
- 11
j 25
j i 0 j
j i 25
0 j
j i
11 j i 7
-
i
i -5
i
3 0 4
35 0
3 0
(2.7514) 10
* 1.9802 4
5 0
2.7514 1
2.7514 1.4251 1
4 (2.7514) 10
* 1.1230
A
(2.7514) 10
* 1.9802 p
] ) . cos[(
)]
) log((
. cosh[
] ) . sin[(
* ) ( .
) ) ( (
) )(
(
*
) (
) ( )
( ,
) (
(51)
25
0 i
j 11
i 5
- 11
j
25
j i 0 j
j i 25
0 j
j i 11
j 2
j 11 11
j j
11 j j
3 0 4
35 0
3 0
(2.7514) 10
* 1.9802 4
5 0
2.7514 1
2.7514 1.4251 1
4 0.0029 -
3 0 4
35 0
3 4 0
0.0057 B
4 5 0 pp
] ) . cos[(
)]
) log((
. cosh[
] ) . sin[(
) ( .
) ) ( (
) )(
( )
(
] ) . cos[(
)]
) log((
. cosh[
] ) . sin[(
) ( ) ( .
) ( ) (
, ) (
) ( )
( ) (
(52) Fig. (1) and Fig. (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.
10-4 10-2 100 102 104 -70
-60 -50 -40 -30 -20 -10
Frequency (rad/sec)
Magnitude (dB)
Proposed method H (s)
Fig.(1): Magnitude Bode plots of (49) and of its proposed approximation
10-4 10-2 100 102 104
-40 -20 0 20 40
Frequency (rad/sec)
Phase (deg)
Proposed method H (s)
Fig. (2): Phase Bode plots of (49) and of its proposed approximation
Fig. (3) and Fig. (4) show, respectively, the impulse and the step responses of the fractional order system of (48).
0 1 2 3 4 5 6 7 8 9 10
-0.5 0 0.5 1 1.5 2 2.5
Time (sec)
Amplitude
Fig. (3): Impulse response of (48) from its proposed approximation
0 1 2 3 4 5 6 7 8 9 10
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65
Time (sec)
Amplitude
Fig. (4): Step response of (48) from its proposed approximation
As a second example, we will consider the fractional order system represented by the following fundamental linear fractional order differential equation with = 0.5 and = 0.16 as:
5 0 5 0 5 0
dt t e 16 d
0 t dt x
t x 16 d
0 .
.
. .
. (53)
its transfer function is given by:
016s
1 s 16 s 0
H
5 0
.
. .
(54) Its rational function approximation, in a given frequency band, is given as:
0
N1 0
i i
5 i 0
1 s
D p
s C s
16 0 1
s 16 s 0
H . /
. .
(55)
For the fractional differentiator (0.16s)0.5, the frequency band of approximation is [ωL, ωH] = [10-4 rad/s, 104 rad/s], around ω0 = (1/0) = 6.25 rad/s, y = 1dB, = 0.1, and = 100. Then, the approximation parameters of H(s) are:
28 N and 0.0032 K
, 10
* 1.9953 p
10 10
1.5849 b
1.5849, a
1 D
5 - 0
6 max -5 c
, , ,
Hence, poles
p
i, the residues Ci (0 ≤ i ≤ 25), and the residueD, are given by:
28
j i 0 j
j i 28
0 j
j i 5 i
- 7 i - i
-5 i i
2.5119 1
2.5119 1.5849
1
25 6 2.5119 10
* 1.9953
2.5119 10
* 1.5963 C
2.5119 10
* 1.9953 p
,
) (
) )(
(
*
. (56)
