new approach fro reduced ordee modelling of fractional order systems in delts domain
VivekKumar265461
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May 27, 2024
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About This Presentation
ffsdbjdjhd
Slide Content
End SemesterPresentation
on
Submittedby
Abhishek kumar
Reg.No.:2022EE07
NEWAPPROACHFORREDUCEDORDERMODELLINGOFFRACTIONAL
ORDERSYSTEMSINDELTADOMAIN
UnderTheSupervisionof
Dr. Deepak Kumar
Associate Professor,EED
MNNITAllahabad
DEPARTMENT OFELECTRICALENGINEERING
MOTILALNEHRU NATIONALINSTITUTEOFTECHNOLOGY ALLAHABAD
PRAYAGRAJ
on
Submittedby
Abhishek kumar
Reg.No.:2022EE07
NEWAPPROACHFORREDUCEDORDERMODELLINGOFFRACTIONAL
ORDERSYSTEMSINDELTADOMAIN
UnderTheSupervisionof
Dr. Deepak Kumar
Associate Professor,EED
MNNITAllahabad
DEPARTMENT OFELECTRICALENGINEERING
MOTILALNEHRU NATIONALINSTITUTEOFTECHNOLOGY ALLAHABAD
PRAYAGRAJ
CONTENTS
➢Motivation of work
➢Introduction
➢Mathematical Model
➢Simulations and Results
➢Proposed work
➢Conclusion
➢References
➢Motivation of work
➢Introduction
➢Mathematical Model
➢Simulations and Results
➢Proposed work
➢Conclusion
➢References
MOTIVATION OF WORK
➢Simplifyinganalysisbyprovidingmoremanageablemodelsandfacilitatingcontroldesign.
➢Itcanhelptoidentifythedominantmodesofasystemandreducethecomplexityoftheproblem.
➢Enableefficientuseofcomputationalresourcesbyreducingthedimensionalityofmodels,making
themmoresuitableforreal-timeapplicationsandimplementationonresource-constrainedplatforms.
➢EnhancecontrolstrategiesforFOSbyemployingreducedordermodels,facilitatingthedesignof
moreresponsiveandcomputationallyfeasiblecontrolsystems.
➢ExtendtheimpactofROMbyapplyingitacrossvariousdisciplines,suchascontroltheory,signal
processing,andscientificcomputing,addressingreal-worldchallengesinadiverserangeoffields
➢Simplifyinganalysisbyprovidingmoremanageablemodelsandfacilitatingcontroldesign.
➢Itcanhelptoidentifythedominantmodesofasystemandreducethecomplexityoftheproblem.
➢Enableefficientuseofcomputationalresourcesbyreducingthedimensionalityofmodels,making
themmoresuitableforreal-timeapplicationsandimplementationonresource-constrainedplatforms.
➢EnhancecontrolstrategiesforFOSbyemployingreducedordermodels,facilitatingthedesignof
moreresponsiveandcomputationallyfeasiblecontrolsystems.
➢ExtendtheimpactofROMbyapplyingitacrossvariousdisciplines,suchascontroltheory,signal
processing,andscientificcomputing,addressingreal-worldchallengesinadiverserangeoffields
INTRODUCTION
➢Fractional order systems are those systems where the order of differential equation
describing the system is fraction or non integer.
➢ Fractional-order systems introduce memory and more complex dynamics into the
modeling process compared to integer-order systems.
➢Some examples of fractional order system are : voltage-current relationship in semi-
infinite transmission line ,charging and discharging of lossy capacitor ,flow of fluid in
porous media etc.
➢Fractional order systems model complex dynamics accurately but pose computational
challenges due to non-integer derivatives.
FRACTIONAL ORDER SYSTEM
describing the system is fraction or non integer.
➢ Fractional-order systems introduce memory and more complex dynamics into the
modeling process compared to integer-order systems.
➢Some examples of fractional order system are : voltage-current relationship in semi-
infinite transmission line ,charging and discharging of lossy capacitor ,flow of fluid in
porous media etc.
➢Fractional order systems model complex dynamics accurately but pose computational
challenges due to non-integer derivatives.
FRACTIONAL ORDER SYSTEM
MATHEMATICALMODELLING OFTHEFOS
The transfer function form of the fractional order model is :
Where L[y(t)]=Y(s), L[u(t)]=U(s)
If the order of differentiations of the fractional differential equation (14) is a multiple of
single base order then the system is called commensurate order and it reduces to the
following form
(15)
Where 10
10
10
10
..........()
()
() .........
mm
nn
mm
a
nn
bsbs bsYs
Gs
Usasas as
−
−
−
−
+ + +
==
+ + +
(14) ,,
kk
kR
+
= 00
() ()
nm
kk
kk
kk
aDyt bDut
==
=
The transfer function form of the fractional order model is :
Where L[y(t)]=Y(s), L[u(t)]=U(s)
If the order of differentiations of the fractional differential equation (14) is a multiple of
single base order then the system is called commensurate order and it reduces to the
following form
(15)
Where 10
10
10
10
..........()
()
() .........
mm
nn
mm
a
nn
bsbs bsYs
Gs
Usasas as
−
−
−
−
+ + +
==
+ + +
(14) ,,
kk
kR
+
= 00
() ()
nm
kk
kk
kk
aDyt bDut
==
=
The state space representation of a generalized fractional order multiple input multiple
output system is given below,
Integer order system is stable if all roots lie in the left half of the complex plane. but for
fractional order system roots may lie on the right half of complex plane.
Condition for stability
Where 0 < α < 1 and eig (A) is the eigen value of matrix A. Imj
α
Re
Stable
Region
0 ()()()
()()()
DtxtAxtBut
ytCxtDut
=+
=+ |arg(())|
2
eigA
(16)
(17)
Unstable
Region2
output system is given below,
Integer order system is stable if all roots lie in the left half of the complex plane. but for
fractional order system roots may lie on the right half of complex plane.
Condition for stability
Where 0 < α < 1 and eig (A) is the eigen value of matrix A. Imj
α
Re
Stable
Region
0 ()()()
()()()
DtxtAxtBut
ytCxtDut
=+
=+ |arg(())|
2
eigA
(16)
(17)
Unstable
Region2
PROBLEM STATEMENT
➢We use an approximation method to a fractional order system for obtain a integer order
transfer function.
➢Obtain higher integer order transfer function and then we modelled this in delta domain.
➢For better performances we used different approximation methods like Oustaloup
approximation, Differential Evolution Algorithm, Genetic Algorithm.
➢We use Approximate frequency fitting method for reduced order modelling of fractional
order system.
➢We use an approximation method to a fractional order system for obtain a integer order
transfer function.
➢Obtain higher integer order transfer function and then we modelled this in delta domain.
➢For better performances we used different approximation methods like Oustaloup
approximation, Differential Evolution Algorithm, Genetic Algorithm.
➢We use Approximate frequency fitting method for reduced order modelling of fractional
order system.
➢Rational approximation simplifies fractional-order system analysis by using higher-order
transfer functions while preserving essential system attributes for control design.
➢The most famous and used ones which are the approximation method are Oustaloup and
Singularity function method.
➢Oustaloup approximation is widely used in FOS. This approximation method based on
recursive distribution of poles and zero’s. It approximates the basic fractional order operator
to an integer order rational form within a chosen frequency band.
RATIONAL APPROXIMATION OF FRACTIONAL ORDER SYSTEM
[9] P. Sarkar, R. R. Shekh and A. Iqbal, "A unified approach for reduced order modeling of fractional order
system in delta domain," 2016 International Automatic Control Conference (CACS), Taichung, Taiwan,
2016, pp. 257-262, doi: 10.1109/CACS.2016.7973920
transfer functions while preserving essential system attributes for control design.
➢The most famous and used ones which are the approximation method are Oustaloup and
Singularity function method.
➢Oustaloup approximation is widely used in FOS. This approximation method based on
recursive distribution of poles and zero’s. It approximates the basic fractional order operator
to an integer order rational form within a chosen frequency band.
RATIONAL APPROXIMATION OF FRACTIONAL ORDER SYSTEM
[9] P. Sarkar, R. R. Shekh and A. Iqbal, "A unified approach for reduced order modeling of fractional order
system in delta domain," 2016 International Automatic Control Conference (CACS), Taichung, Taiwan,
2016, pp. 257-262, doi: 10.1109/CACS.2016.7973920
OUSTALOUP ‘S APPROXIMATION :
By a rational function
Where C is a gain adjusted both sides of shall have unit gain at 1 rad/s and N is the
number of poles and zeros related to the desired bandwidth and error criteria .
'
1
()
kN
k
app
k k
sw
HsC
sw
=
=
+
=
+
Where
w
u is the unity frequencies gain
Where w
h and w
b are respectively
upper and lower bounds
u bh
www= ()Hss
+
= ' (21)/
(21)/
kN
k bu
kN
k bu
h
www
www
Cw
−−
−+
=
=
=
(5)
(6)
By a rational function
Where C is a gain adjusted both sides of shall have unit gain at 1 rad/s and N is the
number of poles and zeros related to the desired bandwidth and error criteria .
'
1
()
kN
k
app
k k
sw
HsC
sw
=
=
+
=
+
Where
w
u is the unity frequencies gain
Where w
h and w
b are respectively
upper and lower bounds
u bh
www= ()Hss
+
= ' (21)/
(21)/
kN
k bu
kN
k bu
h
www
www
Cw
−−
−+
=
=
=
(5)
(6)
DIFFERENTIAL EVOLUTION METHOD
Original FOS : �??????=
�1??????
�1+�2??????
�2+�3??????
�3+⋯+��??????
��
�1??????
�1+�2??????
�2+�3??????
�3+⋯+��??????
��
(7)
We have to find 5
th
order approximated transfer function
�??????=
�4??????
4
+�3??????
3
+�2??????
2
+�1??????+�0
�5??????
5
+�4??????
4
+�3??????
3
+�2??????
2
+�1??????+�0
(8)
For 5
th
order transfer function we have required 11 real coefficients
θ = [�
5 �
4 �
3 �
2 �
1 �
0 �
4 �
3 �
2 �
1 �
0]
T (9)
By using DE method we can obtain an adequate solution for the given approximation problem if
the fitness function is modelled in an adequate way .
Measuring the the error b/w bode diagrams of original HOFS and the ROM
??????
�(�)= ȁȁȁ�(??????)
d??????−ȁ�(??????)ȁ
d??????∣
??????
??????(�)=ȁarg(�(??????))−arg(�(??????))ȁ
(10)
Let us define the combined error function � as,
�(�)=�⋅??????
�(�)+�.??????
??????(�) (11)
where �∈[0,1] and �=(1−�)/10.
Original FOS : �??????=
�1??????
�1+�2??????
�2+�3??????
�3+⋯+��??????
��
�1??????
�1+�2??????
�2+�3??????
�3+⋯+��??????
��
(7)
We have to find 5
th
order approximated transfer function
�??????=
�4??????
4
+�3??????
3
+�2??????
2
+�1??????+�0
�5??????
5
+�4??????
4
+�3??????
3
+�2??????
2
+�1??????+�0
(8)
For 5
th
order transfer function we have required 11 real coefficients
θ = [�
5 �
4 �
3 �
2 �
1 �
0 �
4 �
3 �
2 �
1 �
0]
T (9)
By using DE method we can obtain an adequate solution for the given approximation problem if
the fitness function is modelled in an adequate way .
Measuring the the error b/w bode diagrams of original HOFS and the ROM
??????
�(�)= ȁȁȁ�(??????)
d??????−ȁ�(??????)ȁ
d??????∣
??????
??????(�)=ȁarg(�(??????))−arg(�(??????))ȁ
(10)
Let us define the combined error function � as,
�(�)=�⋅??????
�(�)+�.??????
??????(�) (11)
where �∈[0,1] and �=(1−�)/10.
Performance criterion in the frequency domain J is proposed for evaluating the reduced integer-
order model using the integral squared error (ISE) as follows:
�(??????)=
0
∞
�(�)d� (12)
set of constraints for the parameters ?????? as �⊂ℜ
11
Find ??????
⋆
∈� such that,
�??????
⋆
=min
??????∈�
�(??????) (13)
DE approximation procedure
1.Intilialize models with random value.
2.For each model evaluate an error between Bode diagram of original high-order FOM and ROM.
3.Update parameters models.
4.Choose the best model and go to step 2.
5.End the max iteration, giving optimal model.
[13]Bourouba, B.,Ladaci, S. & Chaabi, A. Reduced-Order Model Approximation of Fractional-Order
Systems Using Differential Evolution Algorithm.J Control Autom Electr Syst29, 32–43 (2018).
https://doi.org/10.1007/s40313-017-0356-5
order model using the integral squared error (ISE) as follows:
�(??????)=
0
∞
�(�)d� (12)
set of constraints for the parameters ?????? as �⊂ℜ
11
Find ??????
⋆
∈� such that,
�??????
⋆
=min
??????∈�
�(??????) (13)
DE approximation procedure
1.Intilialize models with random value.
2.For each model evaluate an error between Bode diagram of original high-order FOM and ROM.
3.Update parameters models.
4.Choose the best model and go to step 2.
5.End the max iteration, giving optimal model.
[13]Bourouba, B.,Ladaci, S. & Chaabi, A. Reduced-Order Model Approximation of Fractional-Order
Systems Using Differential Evolution Algorithm.J Control Autom Electr Syst29, 32–43 (2018).
https://doi.org/10.1007/s40313-017-0356-5
DELTA OPERATOR MODELLING
The state-space representation using shift operator is:
q[x(k)] =A
qx[k] + B
qu[k]
y[k] = C
qx[k] + D
qu[k] (18)
Where q x(k) =x(k+1) is the forward shift and x(k) is signal
(19)
Delta operator is defined as:
and the corresponding complex variable is defined as:
(20)
Where , s = σ + jω
When σ =0 ,the imaginary axis in s-plane
is mapped to the unit circle with centre at
origin in z plane and also mapped to circle -1/T
s
and radius 1/T
s in δ-plane 11
ssT
ss
Ze
TT
−−
== ()
1
sjT
s
e
T
+
−
= 1
s
q
T
−
=
The state-space representation using shift operator is:
q[x(k)] =A
qx[k] + B
qu[k]
y[k] = C
qx[k] + D
qu[k] (18)
Where q x(k) =x(k+1) is the forward shift and x(k) is signal
(19)
Delta operator is defined as:
and the corresponding complex variable is defined as:
(20)
Where , s = σ + jω
When σ =0 ,the imaginary axis in s-plane
is mapped to the unit circle with centre at
origin in z plane and also mapped to circle -1/T
s
and radius 1/T
s in δ-plane 11
ssT
ss
Ze
TT
−−
== ()
1
sjT
s
e
T
+
−
= 1
s
q
T
−
=
The state space representation using delta operator are as :
(21)
It relates with the continuous –time and shift operator model as :
(22)
(23)
(24)
(25)
where A
c, B
c, C
c, & D
c are the state space matrices of continuous –time state space
representation.
[()][][]
[][][]
xkAxkBuk
ykCxkDuk
=+
=+ 0
1
c
T
A
s
q
c
s
cq
cq
ed
T
AI
AA
T
CCC
DDD
=
−
==
==
==
(21)
It relates with the continuous –time and shift operator model as :
(22)
(23)
(24)
(25)
where A
c, B
c, C
c, & D
c are the state space matrices of continuous –time state space
representation.
[()][][]
[][][]
xkAxkBuk
ykCxkDuk
=+
=+ 0
1
c
T
A
s
q
c
s
cq
cq
ed
T
AI
AA
T
CCC
DDD
=
−
==
==
==
REDUCTION METHOD
➢The approximate frequency fitting (AFF) method so far used in reduced order
modelling in delta domain has been extended for developing reduced model of FOS
in delta domain.
➢ The transfer function of the HOM and the ROM are matched at a number of
frequency points in the low frequency range and the resultant linear algebraic
equations are solved to arrive at a low order model.
➢This method is general in the sense that the frequency points of interest may be
chosen quite arbitrarily without resort to any search procedure and does not involve
choosing some critical frequency points.
[9] P. Sarkar, R. R. Shekh and A. Iqbal, "A unified approach for reduced order modeling of
fractional order system in delta domain," 2016 International Automatic Control Conference
(CACS), Taichung, Taiwan, 2016, pp. 257-262, doi: 10.1109/CACS.2016.7973920
➢The approximate frequency fitting (AFF) method so far used in reduced order
modelling in delta domain has been extended for developing reduced model of FOS
in delta domain.
➢ The transfer function of the HOM and the ROM are matched at a number of
frequency points in the low frequency range and the resultant linear algebraic
equations are solved to arrive at a low order model.
➢This method is general in the sense that the frequency points of interest may be
chosen quite arbitrarily without resort to any search procedure and does not involve
choosing some critical frequency points.
[9] P. Sarkar, R. R. Shekh and A. Iqbal, "A unified approach for reduced order modeling of
fractional order system in delta domain," 2016 International Automatic Control Conference
(CACS), Taichung, Taiwan, 2016, pp. 257-262, doi: 10.1109/CACS.2016.7973920
Let Transfer function of single (SISO) higher order model (HOM) in Delta domain
�(�)=
�(??????)
??????(??????)
=�
1+�
1??????+�
2??????
2
+⋯+�
�??????
�
1+�
1??????+�
2??????
2
+⋯+�
�??????
�
(26)
where �≤�, it is assumed that �(�) is irreducible.
Let transfer function of (ROM) is represented by:
(27)
�(�)=
??????(�)
??????(�)
=�
1+�
1�+�
2�
2
+⋯+�
��
�
1+�
1�+�
2�
2
+⋯+�
��
�
where �<<�,�≤�. The order of the reduced model is assumed as q, this
necessitates computation of at least 2�−1 free parameters of the reduced
order model.
�(�)=
�(??????)
??????(??????)
=�
1+�
1??????+�
2??????
2
+⋯+�
�??????
�
1+�
1??????+�
2??????
2
+⋯+�
�??????
�
(26)
where �≤�, it is assumed that �(�) is irreducible.
Let transfer function of (ROM) is represented by:
(27)
�(�)=
??????(�)
??????(�)
=�
1+�
1�+�
2�
2
+⋯+�
��
�
1+�
1�+�
2�
2
+⋯+�
��
�
where �<<�,�≤�. The order of the reduced model is assumed as q, this
necessitates computation of at least 2�−1 free parameters of the reduced
order model.
In the AFF, the frequency response of the HOM and that of the ROM are approximately
matched, such that:
�(�)≈�(�) (28)
Writing the ROM in eqn. (17) in its structural form
(29)
�=1
??????−1
�
��
�
−��
�=1
??????
�
��
�
=��−1
writing eqn. (18) in polar form
(30)
�=1
??????−1
�
�ȁ�ȁ
�
�
�??????�
−ȁ�(�)ȁ�
�??????
�=1
??????
�
�ȁ�ȁ
�
�
�??????�
=ȁ�(�)ȁ�
�??????
−1
where �=ȁ�ȁ�
�??????
Equating real and imag. part of eqn(18) the expressions are obtained
matched, such that:
�(�)≈�(�) (28)
Writing the ROM in eqn. (17) in its structural form
(29)
�=1
??????−1
�
��
�
−��
�=1
??????
�
��
�
=��−1
writing eqn. (18) in polar form
(30)
�=1
??????−1
�
�ȁ�ȁ
�
�
�??????�
−ȁ�(�)ȁ�
�??????
�=1
??????
�
�ȁ�ȁ
�
�
�??????�
=ȁ�(�)ȁ�
�??????
−1
where �=ȁ�ȁ�
�??????
Equating real and imag. part of eqn(18) the expressions are obtained
(i) Real Part
(31)
�=1
??????−1
�
��
�(ψ)−
�=1
??????
�
��
�(??????)≈�(�)
(ii) Imaginary part
(32)
�=1
??????−1
�
��
�(�)−
�=1
??????
�
��
�(�)=�(�)
Where
�
�(�)=ȁ�ȁ
�
cos??????
�
�
�(�)=ȁ�(�)ȁȁȁ
�
cos??????
�+??????
�
�(??????)=ȁ�ȁ
�
sin??????
�
�
�(�??????)=ȁ�(�)ȁȁȁ
�
sin??????
�+??????
�(�)=ȁ�(�)ȁcos??????−1
�(�)=ȁ�(�)ȁsin??????
(33)
Left hand side of equation (31) and (32) are real function of ψ with unknown coefficients
�
� ,�
�.
�
�(�) and �
�(�) are two real (known) functions of ψ such that
(31)
�=1
??????−1
�
��
�(ψ)−
�=1
??????
�
��
�(??????)≈�(�)
(ii) Imaginary part
(32)
�=1
??????−1
�
��
�(�)−
�=1
??????
�
��
�(�)=�(�)
Where
�
�(�)=ȁ�ȁ
�
cos??????
�
�
�(�)=ȁ�(�)ȁȁȁ
�
cos??????
�+??????
�
�(??????)=ȁ�ȁ
�
sin??????
�
�
�(�??????)=ȁ�(�)ȁȁȁ
�
sin??????
�+??????
�(�)=ȁ�(�)ȁcos??????−1
�(�)=ȁ�(�)ȁsin??????
(33)
Left hand side of equation (31) and (32) are real function of ψ with unknown coefficients
�
� ,�
�.
�
�(�) and �
�(�) are two real (known) functions of ψ such that
ȁΦ
??????(�)
??????=??????�= ȁ�(�)
??????=??????�;??????∈[0,??????−1]
ȁΦ
??????(�)
??????=??????�= ȁ�(�)
??????=??????�;??????∈[0,??????−1]
(34)
where N is the number of linear equations separately for the real and imaginary parts. These may be
written in matrix form as:
??????�=� (35)
where is a matrix of dimension given by
??????=
�
1,1 �
1,2… �
1,??????−1 �
1,1 �
1,??????
�
1,1 �
1,2⋯ �
1,??????−1 �
1,1 … �
1,??????
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
�
�,1 �
�,2… �
�,??????−1 �
�,1 … �
�,??????
�
�,1 �
�,2… �
�,??????−1 �
�,1…�
�,??????
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
�
??????−1,1�
??????−1,2⋯ �
??????−1,??????−1 �
??????−1,1�
??????−1,??????
�
??????−1,1�
??????−1,2⋯�
??????−1,??????−1�
??????−1,1�
??????−1,??????
(36)
�=�
1,�
2,…�
�,…�
??????−1,�
1,�
2,⋯�
�,…�
??????−1
�
(37)
Finally, the parameter vector x is computed as
X=(A
T
A)
-1
A
T
b (38)
Parameter estimation of ROM is depend on the choice of ψ .
??????(�)
??????=??????�= ȁ�(�)
??????=??????�;??????∈[0,??????−1]
ȁΦ
??????(�)
??????=??????�= ȁ�(�)
??????=??????�;??????∈[0,??????−1]
(34)
where N is the number of linear equations separately for the real and imaginary parts. These may be
written in matrix form as:
??????�=� (35)
where is a matrix of dimension given by
??????=
�
1,1 �
1,2… �
1,??????−1 �
1,1 �
1,??????
�
1,1 �
1,2⋯ �
1,??????−1 �
1,1 … �
1,??????
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
�
�,1 �
�,2… �
�,??????−1 �
�,1 … �
�,??????
�
�,1 �
�,2… �
�,??????−1 �
�,1…�
�,??????
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
�
??????−1,1�
??????−1,2⋯ �
??????−1,??????−1 �
??????−1,1�
??????−1,??????
�
??????−1,1�
??????−1,2⋯�
??????−1,??????−1�
??????−1,1�
??????−1,??????
(36)
�=�
1,�
2,…�
�,…�
??????−1,�
1,�
2,⋯�
�,…�
??????−1
�
(37)
Finally, the parameter vector x is computed as
X=(A
T
A)
-1
A
T
b (38)
Parameter estimation of ROM is depend on the choice of ψ .
SIMULATIONS & RESULTS:
Example 1. A stable fractional order model [1], is taken as example are
�(??????)=
−2??????
0.63
+4
2??????
3.501
+3.8??????
2.42
+2.6??????
1.798
+2.5??????
1.31
+1.5
(39)
Using oustaloup approximation, the integer order model thus obtained is given as under
−32.39??????
30
−1.237×10
4
??????
29
−2.018×10
6
??????
28
−1.85×10
8
??????
27
−1.056×10
10
??????
26
−3.96×10
11
??????
25
−1.003×10
13
??????
24
−1.74×10
14
??????
23
−2.089×10
15
??????
22
−1.708×10
16
??????
21
−9.25×10
16
??????
20
−3.017×10
17
??????
19
−3.565×10
17
??????
18
−1.548×10
18
??????
17
+9.128×10
18
??????
16
+2.31×10
19
??????
15
+3.909×10
19
??????
14
+4.327×10
19
??????
13
+3.348×10
19
??????
12
1.838×10
19
??????
11
+7.204×10
18
??????
10
+2.02×10
18
??????
9
+4.055×10
17
??????
8
+5.806×10
16
??????
7
+5.889×10
15
??????
6
+4.179×10
14
??????
5
+2.035×10
13
??????
4
+6.604×10
11
??????
3
+1.357×10
10
??????
2
+1.591×10
8
??????
�
�?????? =
+8.09×10
5
20.09??????
33
+8105??????
32
+1.408×10
6
??????
31
+1.387×10
8
??????
30
+8.633×10
9
??????
29
+3.58×10
11
??????
28
+1.028×10
13
??????
27
+2.085×10
14
??????
26
+3.039×10
15
??????
25
+3.227×10
16
??????
24
+2.526×10
17
??????
23
+1.474×10
18
??????
22
+6.481×10
18
??????
21
+2.171×10
19
??????
20
+5.591×10
19
??????
19
+1.116×10
20
??????
18
+1.737×10
20
??????
17
+2.118×10
20
??????
16
+2.031×10
20
??????
15
+1.542×10
20
??????
14
+9.31×10
19
??????
13
+4.486×10
19
??????
12
1.718×10
19
??????
11
+5.171×10
18
??????
10
+1.202×10
18
??????
9
+2.113×10
17
??????
8
+2.759×10
16
??????
7
+2.623×10
15
??????
6
+1.779×10
14
??????
5
+8.383×10
12
??????
4
+2.656×10
11
??????
3
+5.361×10
9
??????
2
+6.2×10
7
??????
+3.12×10
5
Example 1. A stable fractional order model [1], is taken as example are
�(??????)=
−2??????
0.63
+4
2??????
3.501
+3.8??????
2.42
+2.6??????
1.798
+2.5??????
1.31
+1.5
(39)
Using oustaloup approximation, the integer order model thus obtained is given as under
−32.39??????
30
−1.237×10
4
??????
29
−2.018×10
6
??????
28
−1.85×10
8
??????
27
−1.056×10
10
??????
26
−3.96×10
11
??????
25
−1.003×10
13
??????
24
−1.74×10
14
??????
23
−2.089×10
15
??????
22
−1.708×10
16
??????
21
−9.25×10
16
??????
20
−3.017×10
17
??????
19
−3.565×10
17
??????
18
−1.548×10
18
??????
17
+9.128×10
18
??????
16
+2.31×10
19
??????
15
+3.909×10
19
??????
14
+4.327×10
19
??????
13
+3.348×10
19
??????
12
1.838×10
19
??????
11
+7.204×10
18
??????
10
+2.02×10
18
??????
9
+4.055×10
17
??????
8
+5.806×10
16
??????
7
+5.889×10
15
??????
6
+4.179×10
14
??????
5
+2.035×10
13
??????
4
+6.604×10
11
??????
3
+1.357×10
10
??????
2
+1.591×10
8
??????
�
�?????? =
+8.09×10
5
20.09??????
33
+8105??????
32
+1.408×10
6
??????
31
+1.387×10
8
??????
30
+8.633×10
9
??????
29
+3.58×10
11
??????
28
+1.028×10
13
??????
27
+2.085×10
14
??????
26
+3.039×10
15
??????
25
+3.227×10
16
??????
24
+2.526×10
17
??????
23
+1.474×10
18
??????
22
+6.481×10
18
??????
21
+2.171×10
19
??????
20
+5.591×10
19
??????
19
+1.116×10
20
??????
18
+1.737×10
20
??????
17
+2.118×10
20
??????
16
+2.031×10
20
??????
15
+1.542×10
20
??????
14
+9.31×10
19
??????
13
+4.486×10
19
??????
12
1.718×10
19
??????
11
+5.171×10
18
??????
10
+1.202×10
18
??????
9
+2.113×10
17
??????
8
+2.759×10
16
??????
7
+2.623×10
15
??????
6
+1.779×10
14
??????
5
+8.383×10
12
??????
4
+2.656×10
11
??????
3
+5.361×10
9
??????
2
+6.2×10
7
??????
+3.12×10
5
Fig.2. Step response of continuous -time system with fractional order system
Fig.3. Bode plot of FOS and Approximated HIOS
The corresponding delta domain model with T
s =0.01 sec is obtained as:
−8.717×10
14
�
28
−5.384×10
18
�
27
−2.304×10
22
�
26
−6.89×10
25
�
25
−1.453×10
29
�
24
−2.17×10
32
�
23
−2.289×10
35
�
22
−1.675×10
38
�
21
−8.179×10
40
�
20
−2.395×10
43
�
19
−3.226×10
45
�
18
−5.51×10
47
�
17
+8.142×10
50
�
16
+1.944×10
53
�
15
+3.148×10
55
�
14
+3.39×10
57
�
13
+2.57×10
59
�
12
+1.394×10
61
�
11
+5.408×10
62
�
10
+1.504×10
64
�
9
+3.001×10
65
�
8
+4.278×10
66
�
7
+4.323×10
67
�
6
+3.06×10
68
�
5
+1.487×10
70
�
4
+4.817×10
69
�
3
+9.887×10
69
�
2
Ga(γ) =
+1.158×10
70
??????+5.884×10
69
??????
33
+3123??????
32
+4.28×10
8
??????
31
+3.399∗10^12??????30 + 1.74×10
16
??????
29
+6.127×10
19
�28+1.522×10
23
�
27
+2.729×10
26
�
26
+3.587×10
29
�
25
+3.49×10
32
�
24
+2.55×10
35
�
23
+1.40×10
38
�
22
+5.89×10
40
�
21
+1.9×10
43
�
20
+4.743×10
45
�
19
+9.225×10
47
�
18
+1.405×10
50
�
17
+1.68×10
52
�
16
+1.591×10
54
�
15
+1.19×10
56
�
14
+7.132×10
57
�
13
+3.407×10
59
�
12
+1.295×10
61
�
11
+3.87×10
63
�
10
+8.941×10
65
�
9
+1.564×10
67
�
8
+2.033×10
66
�
7
+1.926×10
68
�
6
+1.303×10
68
�
5
+6.126×10
68
�
4
+1.938×10
69
�
3
+3.906×69�
2
+4.513×10
69
�+2.269×10
69
(40)
s =0.01 sec is obtained as:
−8.717×10
14
�
28
−5.384×10
18
�
27
−2.304×10
22
�
26
−6.89×10
25
�
25
−1.453×10
29
�
24
−2.17×10
32
�
23
−2.289×10
35
�
22
−1.675×10
38
�
21
−8.179×10
40
�
20
−2.395×10
43
�
19
−3.226×10
45
�
18
−5.51×10
47
�
17
+8.142×10
50
�
16
+1.944×10
53
�
15
+3.148×10
55
�
14
+3.39×10
57
�
13
+2.57×10
59
�
12
+1.394×10
61
�
11
+5.408×10
62
�
10
+1.504×10
64
�
9
+3.001×10
65
�
8
+4.278×10
66
�
7
+4.323×10
67
�
6
+3.06×10
68
�
5
+1.487×10
70
�
4
+4.817×10
69
�
3
+9.887×10
69
�
2
Ga(γ) =
+1.158×10
70
??????+5.884×10
69
??????
33
+3123??????
32
+4.28×10
8
??????
31
+3.399∗10^12??????30 + 1.74×10
16
??????
29
+6.127×10
19
�28+1.522×10
23
�
27
+2.729×10
26
�
26
+3.587×10
29
�
25
+3.49×10
32
�
24
+2.55×10
35
�
23
+1.40×10
38
�
22
+5.89×10
40
�
21
+1.9×10
43
�
20
+4.743×10
45
�
19
+9.225×10
47
�
18
+1.405×10
50
�
17
+1.68×10
52
�
16
+1.591×10
54
�
15
+1.19×10
56
�
14
+7.132×10
57
�
13
+3.407×10
59
�
12
+1.295×10
61
�
11
+3.87×10
63
�
10
+8.941×10
65
�
9
+1.564×10
67
�
8
+2.033×10
66
�
7
+1.926×10
68
�
6
+1.303×10
68
�
5
+6.126×10
68
�
4
+1.938×10
69
�
3
+3.906×69�
2
+4.513×10
69
�+2.269×10
69
(40)
Fig.4. Step response of continuos time system in delta domain
Fig.5.Bode Diagram of higher order delta domain model
Example 2 Consider a linear fractional order system whose transfer function is:
�??????=
1
0.8??????
2.2
+0.5 ??????
0.9
+1
(41)
Ous ??????
�
ap
(??????)=
??????
�
+����??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????+����
�.���??????
��
+����??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+����
Searching by DE algorithm for all the optimum parameters enables us to obtain the following fifth-order
rational approximating transfer function of the form with the parameter’s vector.
Performance indexes
Upper limit of iterations 100
Population Size 50
F 0.6
C
R
0.5
DE parameters’ setting
Indexes IAE ITAE ISE ITSE
DE method 1.309 18.202 0.543 0.169
Oustaloup’s method 2.1587 21.5876 0.7574 0.719
�??????=
1
0.8??????
2.2
+0.5 ??????
0.9
+1
(41)
Ous ??????
�
ap
(??????)=
??????
�
+����??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????+����
�.���??????
��
+����??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+�.�����
�
??????
�
+����
Searching by DE algorithm for all the optimum parameters enables us to obtain the following fifth-order
rational approximating transfer function of the form with the parameter’s vector.
Performance indexes
Upper limit of iterations 100
Population Size 50
F 0.6
C
R
0.5
DE parameters’ setting
Indexes IAE ITAE ISE ITSE
DE method 1.309 18.202 0.543 0.169
Oustaloup’s method 2.1587 21.5876 0.7574 0.719
Fig.5. Step Response of True fractional model, Oustaloup’s, and DE approximating models
Fig.6. Bode diagrams of the true fractional model, Oustaloup’s and DE approximating models
CONCLUSION
➢It has been seen in figure2 that the step response of the original FOS and its
Oustaloup approximation of 33th order integer approximation gives a very
good matching in the time domain.
➢The similarity in response characteristics in frequency domain as shown in
figure 3 exhibits that the Oustaloup approximation is a very powerful
technique in developing rational integer order approximation of FOS.
➢The discrete transfer function in the delta domain of the 33th order
continuous-time model at sampling time Ts =0.01 sec is shown in figure 4.
That is also shows the same response like original FOS.
➢Further we used DE optimizer for approximation of FOS in lower integer
system. Then we see some deviation in DE step response that have correct.
➢Then we use approximate frequency fitting method for reduced the order
and get similar results with all essential characteristics.
➢It has been seen in figure2 that the step response of the original FOS and its
Oustaloup approximation of 33th order integer approximation gives a very
good matching in the time domain.
➢The similarity in response characteristics in frequency domain as shown in
figure 3 exhibits that the Oustaloup approximation is a very powerful
technique in developing rational integer order approximation of FOS.
➢The discrete transfer function in the delta domain of the 33th order
continuous-time model at sampling time Ts =0.01 sec is shown in figure 4.
That is also shows the same response like original FOS.
➢Further we used DE optimizer for approximation of FOS in lower integer
system. Then we see some deviation in DE step response that have correct.
➢Then we use approximate frequency fitting method for reduced the order
and get similar results with all essential characteristics.
REFERENCES
[1] M. Khanra, J. Pal and K. Biswas, "Rational Approximation and Analog Realization of
Fractional Order Transfer Function with Multiple Fractional Powered Terms," Asian Journal of
Control, Vol. 15, No. 3, pp. 1-13, May 2013.
[2] D. Xue, C. Zhao and Y. Q. Chen,"A Modified Approximation Method of Fractional Order
System," IEEE International Conference on Mechatronics and Automation, Luoyang, China,
pp. 1043-1048, June 25-28, 2006
[3] Feliu, V., Hernandez, A., Vinagre, B. M., and Podlubny, I., "Some approximations of
fractional-order operators used in control theory and applications," Journal on Fractional
calculus and applied analysis, vol. 3, no. 3,pp. 231-248, 2000
[4] J. Pal, "An algorithmic method for the simplification of linear dynamic scalar systems",
Int. J. Control, vol. 43, no. 1, pp. 257-269, 1986.
[5] J. Pal, "A new method for model order reduction", J. Inst. Electron. Telecommun. Eng.
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Fractional Order Transfer Function with Multiple Fractional Powered Terms," Asian Journal of
Control, Vol. 15, No. 3, pp. 1-13, May 2013.
[2] D. Xue, C. Zhao and Y. Q. Chen,"A Modified Approximation Method of Fractional Order
System," IEEE International Conference on Mechatronics and Automation, Luoyang, China,
pp. 1043-1048, June 25-28, 2006
[3] Feliu, V., Hernandez, A., Vinagre, B. M., and Podlubny, I., "Some approximations of
fractional-order operators used in control theory and applications," Journal on Fractional
calculus and applied analysis, vol. 3, no. 3,pp. 231-248, 2000
[4] J. Pal, "An algorithmic method for the simplification of linear dynamic scalar systems",
Int. J. Control, vol. 43, no. 1, pp. 257-269, 1986.
[5] J. Pal, "A new method for model order reduction", J. Inst. Electron. Telecommun. Eng.
(India), vol. 41, no. 5, pp. 305-311, 1995.
[6] M. Rachid, B. Maamar and D. Said, "Comparison between two approximation methods of
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[7] Y. Bistritz and U. Shaked, "Discrete multivariable system approximations by minimal
Pade-type stable models", IEEE Trans. Circuits Syst., vol. 31, no. 4, pp. 382-390, Apr. 1984.
[8] D. Xue, C. Zhao, and Y. Q. Chen, “A modified approximation method of fractional order
system,” in Proc. 2006 IEEE Int. Conf. Mechatron. Autom., Jun. 2006, pp. 1043–1048
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[12]A.B.H.Adamou-MiticheandL.Mitiche,"MultivariableSystemsModelReductionBased
ontheDominantModesandGeneticAlgorithm,"inIEEETransactionsonIndustrial
Electronics,vol.64,no.2,pp.1617-1619,Feb.2017,doi:10.1109/TIE.2016.2618783.
[13]Bourouba,B.,Ladaci,S.&Chaabi,A.Reduced-OrderModelApproximationofFractional-
OrderSystemsUsingDifferentialEvolutionAlgorithm.JControlAutomElectrSyst29,32–43
(2018).https://doi.org/10.1007/s40313-017-0356-5
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