Cs36565569

N. Sai Dinesh et al Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.565-569

RESEARCH ARTICLE

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OPEN ACCESS

Order Reduction of Discrete Time Systems Using Modified Pole
Clustering Technique
N. Sai Dinesh1, Dr. M. Siva Kumar2, D. Srinivasa Rao3
1

(PG Student, Department of Electrical and Electronics Engineering, Gudlavalleru Engineering College,
Gudlavalleru , AP, India)
2
(Professor & Head of the Department, Department of Electrical and Electronics Engineering, Gudlavalleru
Engineering College, Gudlavalleru, AP, India)
3
(Associate Professor, Department of Electrical and Electronics Engineering, Gudlavalleru Engineering College,
Gudlavalleru,AP,India)

ABSTRACT
In this paper, a new method is presented to derive a reduced order model for a discrete time systems. This
method is based on modified pole clustering technique and pade approximations using bilinear transformations,
which are conceptually simple and computer oriented. The denominator polynomial of the reduced order model
is obtained by using modified pole clustering technique and numerator coefficients are obtained by Pade
approximations. This method generates stable reduced models if the original higher order system is stable. The
proposed method is illustrated with the help of typical numerical examples considered from the literature.
Keywords: Model Order Reduction, Modified Pole Clustering, Pade approximation, Cluster centre, Inverse
Distance Measure.

I.

INTRODUCTION

The modeling of a higher order system is
one of the most important subjects in engineering
and sciences. A model is often too complicated to be
used in real life problems. It is an un-debated
conclusion that, the development of mathematical
model of physical system made it feasible to analyze
and design. So the procedures based on the physical
considerations or mathematical models are used to
achieve simpler models than the original one.
Whenever a physical system is represented by a
mathematical model it may a transfer function of
very high order. Available methods for analysis and
design may become cumbersome when applied to a
system of higher order. At this juncture, application
of large scale order reduction methods is inevitable
to reduce computational effort and process time.
Efforts towards obtaining low order models from
high order systems are related to the aims of deriving
stable reduced order models from the stable original
ones and ensuring that reduced-order model matches
some quantities of the original one. Many methods
are available in the international literature that
addresses the main objective of the modeling of
large-scale systems. Several methods are available in
the literature for the order reduction of linear
continuous systems in time domain as well as
frequency domain [1]-[8].The methods belonging to
time domain are Lageurre polynomials [9] and
Krylov method [10] the methods belonging to the
frequency domain are Routh Approximation Method
suggested by Hutton and Fried Land [11], continued
fraction expansion method [12] given by Shamash,
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Moment Matching Method and Pade Approximation
methods. The reduced order model obtained in the
frequency domain gives better matching of the
impulse response with its high order system. Many
of these methods can be easily extended to discrete
time systems by applying simple transformations
[13,14].
In this paper, the authors proposed a method
for the order reduction of high order discrete time
systems using modified pole clustering technique.
The original higher order discrete system is
transformed to continuous time system by applying
bilinear transformation and the reduced order model
is derived for the continuous time system, by using
modified pole clustering technique and pade
approximation. And finally corresponding inverse
transformation yields reduced order model in
discrete time system.

II.

PROBLEM FORMULATION

Let the transfer function of higher order
original Discrete Time System of order „n‟ be

G( Z ) 


N ( Z ) e0  e1 z  e2 z 2  ......... en1 z n1

2
D( Z )
f 0  f1 z  f 2 z  .......... f n z n
.

Convert the higher order discrete time system into
1+𝑤
„w‟ domain using bilinear transformation 𝑍 =
.
1−𝑤
𝐺 𝑊 = 𝐺 𝑍 |𝑍=1+𝑤
1−𝑤

Therefore the transfer function of higher
order original system of order „n‟ in w-domain is
𝐺 𝑊 =

𝑁(𝑤 )
𝐷(𝑤 )

=

𝑒 0 +𝑒1 𝑤 +𝑒 2 𝑤 2 +⋯+𝑒 𝑛 −1 𝑤 𝑛 −1
𝑓0 +𝑓1 𝑤 +𝑓2 𝑤 2 +⋯+𝑓 𝑛 𝑤 𝑛

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(1)

Where 𝑒 𝑖 ; 0≤ 𝑖 ≤ 𝑛 − 1 and 𝑓𝑖 ; 0 ≤ 𝑖 ≤ 𝑛 are scalar
constants.
Therefore, it is required to derive a „kth‟ order
reduced model in w domain. It is given by
𝑅𝑘 𝑤 =

𝑁 𝑘 (𝑤 ) 𝑎 0 +𝑎 1 𝑤 +𝑎 2 𝑤 2 +⋯+𝑎 𝑘 −1 𝑤 𝑘 −1
𝐷 𝑘 (𝑤 )
𝑏 0 +𝑏1 𝑤 +𝑏 2 𝑤 2 +⋯+𝑏 𝑘 𝑤 𝑘

(2)
where 𝑎 𝑖 ; 0≤ 𝑖 ≤ 𝑘 − 1 and 𝑏 𝑖 ; 0 ≤ 𝑖 ≤ 𝑘 are
scalar constants.
By applying inverse transformation, the reduced
order model in Z-domain is obtained.
𝑅 𝑘 𝑍 = 𝑅 𝑘 𝑤 |𝑤 = 𝑧−1
𝑧+1

III.

REDUCTION PROCEDURE

The proposed method for getting the 𝑘 𝑡ℎ
order reduced model, consists of the following two
steps:
Step1: Determination of the denominator polynomial
for the 𝑘 𝑡ℎ order reduced model, using modified pole
clustering technique.
Step2: Determination of the numerator of 𝑘 𝑡ℎ order
reduced model using Pade approximation.
The following rules are used for clustering the poles
of the original system to get the denominator
polynomial for reduced order models.
a. Separate clusters should be made for real
poles and complex poles.
b. Poles on the jw-axis have to be retained in
the reduced order model.
c. Clusters of poles in the left half s-plane
should not contain any pole of the right half
s-plane and vice-versa.
By using a simple method, “Inverse
Distance Measure”, the cluster center can be formed
as follows:
Let there be a r real poles in one cluster are
𝑝1 , 𝑝2 , 𝑝3 … … … … 𝑝 𝑟 ,then Inverse Distance Measure
(IDM) identifies cluster center as
𝑟

−1

1
𝑝𝑖

𝑝𝑢 =
𝑖=1

÷ 𝑟

… (3)

where 𝑝1 < 𝑝2 < 𝑝3 … … 𝑝 𝑟 ,then modified
cluster center can be obtained by using the
algorithm.
Step 1: Let r real poles in a cluster be 𝑝1 <
𝑝2 < 𝑝3 … … … … . . 𝑝 𝑟 .
Step 2: Set j=1.
Step
3:
Find
pole
cluster
centre
𝑐
−1
−1
𝑟
𝑗=

𝑖=1

𝑝𝑖

÷𝑟

Step 4: Set j=j+1
Step 5: Find a modified cluster centre from
𝑐𝑗 =

−1
𝑝1

+

−1
𝑐 𝑗 −1

−1

÷2

Step 6: Is r=j? if No, and then go to step 4,
otherwise go to step 7
Step 7: Modified cluster centre of the 𝑘 𝑡ℎ
cluster as 𝑝 𝑢𝑘 = 𝑐𝑗
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Let m pair of complex conjugate poles in the cluster
be
[ 𝛼1 ± 𝑗𝛽1 , 𝛼2 ± 𝑗𝛽2 , … . . 𝛼 𝑚 ± 𝑗𝛽 𝑚 ]
then the complex center is in the form of 𝐴 𝑢 ± 𝑗𝐵 𝑢 .
𝑚
𝑖=1

Where 𝐴 𝑢 =
and 𝛽 𝑢 =

𝑚
𝑖=1

1
𝛽𝑖

1
𝛼𝑖

÷ 𝑚
÷ 𝑚

−1

−1

(4)

One of the following cases may occur, for
synthesizing the 𝑘 𝑡ℎ order denominator polynomial.
Case 1: If all the modified cluster centers are real,
then reduced denominator polynomial of order „k‟,
can be taken as
𝐷 𝑘 𝑤 = 𝑤 − 𝑝 𝑢1 𝑤 − 𝑝 𝑢2 . … (𝑤 − 𝑝 𝑢𝑘 )
where
𝑝 𝑢1 , 𝑝 𝑢2 , … . 𝑝 𝑢𝑘
are
1 𝑠𝑡 , 2 𝑛𝑑 … … 𝑘 𝑡ℎ
modified cluster center respectively.
(5)
Case 2:If all the modified cluster centers are
complex conjugate, then the reduced denominator
polynomial of order „k‟ can be taken as
𝐷 𝑘 𝑤 = 𝑤 − 𝑝 𝑢1 𝑤 − 𝑝 𝑢1 𝑤 − 𝑝 𝑢2 𝑤 −
𝑝 𝑢2 … . . (𝑤 − 𝑝 𝑢𝑘 /2 )(𝑤 − 𝑝 𝑢𝑘 /2 )
(6)
where 𝑝 𝑢1 and 𝑝 𝑢1 are complex conjugate cluster
centers or 𝑝 𝑢1 = 𝐴 𝑢 + 𝑗𝐵 𝑢 and 𝑝 𝑢1 = 𝐴 𝑢 − 𝑗𝐵 𝑢
Case 3: If (k-2) cluster centers are real and one pair
of cluster center is complex conjugate, then the
denominator polynomial of the 𝑘 𝑡ℎ order reduced
model can be obtained as
𝐷 𝑘 𝑤 = 𝑤 − 𝑝 𝑢1 𝑤 − 𝑝 𝑢2 … . . (𝑤 −
𝑝 𝑢 (𝑘−2) )(𝑤 − 𝑝 𝑢1 )(𝑤 − 𝑝 𝑢1 )
(7)
Step 2: Determination of the numerator of 𝑘 𝑡ℎ
order reduced model using Pade approximations.
The original 𝑛 𝑡ℎ order system can be expanded in
power series about w=0 as
𝐺 𝑤 =

𝑁 𝑤
𝐷 𝑤

=

𝑒0 +𝑒1 𝑤 +𝑒2 𝑤 2 +⋯+𝑒 𝑛 −1 𝑤 𝑛 −1
𝑓0 +𝑓1 𝑤 +𝑓2 𝑤 2 +⋯+𝑓 𝑛 𝑤 𝑛

= 𝑐0 + 𝑐1 𝑤 + 𝑐2 𝑤 2 + ⋯ … …
(8)
The coefficients of power series expansion are
calculated as follows:
𝑐0 = 𝑒0
1
𝑖
𝑐𝑖 =
𝑒 𝑖 − 𝑗 =1 𝑓𝑗 𝑐 𝑖−𝑗 , 𝑖 > 0𝑒 𝑖 = 0, 𝑖 > 𝑛 − 1
𝑓0

(9)
The reduced 𝑘 𝑡ℎ order model is written as
𝑁𝑘 𝑤
𝑅𝑘 𝑤 = 𝐷 𝑤
𝑘
𝑎0 + 𝑎1 𝑤 + 𝑎2 𝑤 2 + ⋯ + 𝑎 𝑘−1 𝑤 𝑘 −1
=
𝑏0 + 𝑏1 𝑤 + 𝑏2 𝑤 2 + ⋯ + 𝑏 𝑘 𝑤 𝑘
(10)
Here, 𝐷 𝑘 (w) can be determined through equations(57)
For 𝑅 𝑘 𝑤 of equation (10) to be Pade approximants
of G(w) of equation (8), we have
𝑎0 = 𝑏0 𝑐0 𝑎1 = 𝑏0 𝑐1 + 𝑏1 𝑐0
(11)
…………………….
𝑎 𝑘−1 = 𝑏0 𝑐 𝑘−1 + 𝑏1 𝑐 𝑘−2 + ⋯ + 𝑏 𝑘−1 𝑐1 + 𝑏 𝑘 𝑐0
the coefficients 𝑎 𝑗 ; j=0,1,2,3…..k-1 can be found by
solving above k linear equations.

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FLOW CHART FOR ORDER REDUCTION

Start

Convert the higher order discrete- time
system into continuous using bilinear
transformation

1+𝑤

𝑍 = 1−𝑤

Determine the reduced order
denominator polynomial by using
modified pole clustering Technique.

Determine the numerator coefficients
of reduced order model by Pade
approximation.

Convert the obtained continuous
reduced order model into Discrete Time System using inverse linear
Transformation

𝑤=

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𝐷 𝑘 𝑤 = 𝑤 2 + 0.77334𝑤 + 0.23391
Therefore, the 2 𝑛𝑑 order reduced model is
𝑎0 + 𝑎1 𝑤
𝑅𝑘 𝑤 =
0.23391 + 0.77334𝑤 + 𝑤 2
The numerator coefficients are obtained by
pade
approximations. By using equations (9) and (11),
𝑎0 = 4.89103
𝑎1 = −0.629655
Therefore, finally 2 𝑛𝑑 -order reduced model is
obtained as
4.89103 − 0.629655𝑤
𝑅2 𝑤 =
0.23391 + 0.77334𝑤 + 𝑤 2
By applying inverse transformation
𝑧−1
𝑤=
,
𝑧+1
we obtain the reduced order model of discrete time
system as
5.520685 + 9.78206𝑍 + 4.261375𝑍 2
𝑅2 𝑍 =
0.46057 − 1.53218𝑍 + 2.00725𝑍 2
For comparison purposes, a second order
approximant by Younseok Choo [15]and [16] is
found to be
4.87768𝑍 − 2.55604
𝐺2 1 𝑍 = 2
𝑍 − 1.50888𝑍 + 0.63166
4.2𝑍 − 0.9023114
𝐺2 11 𝑍 = 2
𝑍 − 1.442346𝑍 + 0.616743

𝑧−1
𝑧+1

Stop
IV.

NUMERICAL EXAMPLES

EXAMPLE 1
Consider a 4th order discrete-time system given by
Younseok Choo [15]:
𝑁(𝑍)
𝐺 𝑍 =
𝐷(𝑍)

2𝑍 4 + 1.8𝑍 3 + 0.8𝑍 2 + 0.1𝑍 − 0.1
𝑍 4 − 1.2𝑍 3 + 0.3𝑍 2 + 0.1𝑍 + 0.02
1+𝑤
Substituting 𝑍 =
,
1−𝑤
𝑁(𝑤)
𝐺 𝑤 =
𝐷(𝑤)
0.8𝑤 4 + 5𝑤 3 + 9.8𝑤 2 + 11.8𝑤 + 4.6
=
2.42𝑤 4 + 6.52𝑤 3 + 5.52𝑤 2 + 1.32𝑤 + 0.22
=

The poles are : −1.2203 ± 𝑗0.3408,
−0.1268 ± 𝑗0.2014
By using the above modified pole clustering
algorithm, modified cluster center can be formed
from the poles as
𝑝 𝑢1 = −0.38667 ± 𝑗0.29053
The denominator polynomial 𝐷 𝑘 (𝑤) of reduced
order model is obtained by using the equation (6) as
𝐷 𝑘 𝑤 = 𝑤 − 𝑝 𝑢1 (𝑤 − 𝑝 𝑢2 )
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Fig 1: Comparison of Step response of
high
order system and reduced order models for
example1.
It has been observed from Fig1 that the 2 nd
order reduced model obtained by the proposed
method gives good step response that the models
obtained from the methods given in [15] and [16].
EXAMPLE 2
Consider a 4th order discrete-time system
𝑁(𝑍)
𝐺 𝑍 =
𝐷(𝑍)

2𝑍 4 + 1.84𝑍 3 + 0.64𝑍 2 + 0.008𝑍 − 0.096
= 4
𝑍 − 1.2203𝑍 3 + 0.2267𝑍 2 + 0.1162𝑍 + 0.0249
1+𝑤
Substituting 𝑍 =
,
1−𝑤
𝑁(𝑤)
𝐺 𝑤 =
𝐷(𝑤)
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=

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obtained by pade approximation. The effectiveness
0.696𝑤 4 + 4.72𝑤 3 + 10.144𝑤 2 + 12.048𝑤 + 4.392
2.2395𝑤 4 + 6.5734𝑤 3 + 5.696𝑤 2 + 1.2274𝑤 + 0.1475 of proposed method is illustrated with the help of
examples chosen from the literature and the
responses of the original and reduced system are
The poles are : −1.3454 ± 𝑗0.2000,
compared graphically as shown in fig.1
−0.1222 ± 𝑗0.1437
By using the above modified pole clustering
REFERENCES
algorithm, modified cluster center can be formed
[1]
A.K.Sinha, J. Pal, Simulation based
from the poles as
reduced order modeling using clustering
𝑝 𝑢1 = −0.38413 ± 𝑗0.18216
technique, Computer and Electrical Engg. ,
The denominator polynomial 𝐷 𝑘 (𝑤) of reduced
16(3), 1990 159-169.
order model is obtained by using the equation (6) as
[2]
S.K.Nagar, and S.K.Singh, An algorithmic
𝐷 𝑘 𝑤 = 𝑤 − 𝑝 𝑢1 (𝑤 − 𝑝 𝑢2 )
approach for system decomposition and
𝐷 𝑘 𝑤 = 𝑤 2 + 0.76826𝑤 + 0.18073
balanced realized model reduction, Journal
Therefore, the 2 𝑛𝑑 order reduced model is
of Franklin Inst., Vol.341, pp. 615𝑎0 + 𝑎1 𝑤
𝑅𝑘 𝑤 =
2
630,2004.
0.18073 + 0.76826𝑤 + 𝑤
[3]
S.Mukherjee,
Satakshi
and
The numerator coefficients are obtained by
pade
R.C.Mittal,”Model order reduction using
approximations. By using equations (9) and (11),
response-matching technique”, Journal of
𝑎0 = 5.38147
Franklin Inst.,Vol. 342, pp. 503-519,2005.
𝑎1 = −7.14291
[4]
G.Parmar, S.Mukherjee and R.Prasad,
Therefore, finally 2 𝑛𝑑 -order reduced model is
Relative mapping errors of linear time
obtained as
invariant systems caused by Particle swarm
5.38147 − 7.14291𝑤
𝑅2 𝑤 =
optimized reduced model,Int. J. Computer,
2
0.18073 + 0.76826𝑤 + 𝑤
Information and systems science and
By applying inverse transformation
𝑧−1
Engineering, Vol. 1, No. 1, pp. 83-89,2007.
𝑤=
,
𝑧+1
[5]
V.Singh, D.Chandra and H.Kar, Improved
we obtain the reduced order model of discrete time
Routh Pade approximants: A Computer
system as
aided approach, IEEE Trans. Autom.
Control, 49(2), pp .292-296,2004.
2
12.52438 + 10.76294𝑍 − 1.76144𝑍
[6]
C.P.Therapos, Balanced minimal realization
𝑅2 𝑍 =
0.41247 − 1.63854𝑍 + 1.94899𝑍 2
of SISO systems, Electronics letters,
Vol.19, No.11, pp. 242-2-426, 1983.
[7]
Y.Shamash, Stable reduced order models
using Pade type approximations, IEEE
Trans. Vol. AC-19, pp.615-616, 1974.
[8] Sastry G.V.K.R Raja Rao and Mallikarjuna
Rao P., Large scale interval system
modeling using Routh approximants,
Electrical letters, 36(8), pp.768-769,2000.
[9]
Model order reduction in the Time Domain
using Laguerre Polynomials and Krylov
Methods, Y.Chen, V.Balakrishnan, C-K.
Koh and K.Roy,Proceedings of the 2002
Design, Automation and Test in Europe
Conference and Exhibition DATE.02 2002
IEEE.
[10] Practical issues of Model Order Reduction
with Krylov-subspace method, Pieter Heres,
Wil Schilders.
Fig 2: Comparison of Step response of
high
[11] Hutton, M.F and Friedland,B,”Routh
order system and reduced order models for
Approximation for reducing order of linear
example2.
Time invariant system”,IEEE Trans, On
Automatic Control, Vol.20, 1975, PP.329V.
CONCLUSION
337.
A new method is presented to determine the
[12] Shieh, L.S and Goldman, M.J, Continued
reduced order model of a higher order discrete time
Fraction Expansion and Inversion of cauer
system. The denominator polynomial of the reduced
Third Form, IEEE Trans.On Circuits and
order model is obtained by using modified pole
systems, Vol.CAS-21, No.3 May 1974,
clustering while the numerator coefficients are
pp.341-345.
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[13]

Y.P.Shih, and W.T.Wu.,“Simplification of
z-tranfer functions by continued fraction”
Int.J.Contr.,Vo.17, pp.1089-1094,1973.
[14] Chuang C, “Homographic transformation
for the simplification of discrete-time
functions by Pade approximation”,Int.J.
Contr.,Vo.22, pp.721-729,1975.
[15] Y. Choo, “Suboptimal Bilinear Routh
Approximant for Discrete Systems,”ASME
Trans.,Dyn.Sys.Meas. Contr., 128, 2006
,pp. 742-745.
[16] VEGA Based Routh-Pade Approximants
For Discrete Time Systems: A Computer
Aided Approac, IACSIT International
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Vol.1, No.5, December 2009, ISSN:17938236.

BIOGRAPHIES
Dr Mangipudi Siva Kumar was born in
Amalapuram, E. G. Dist, Andhra Pradesh,
India, in 1971. He received bachelor‟s
degree in Electrical & Electronics
Engineering from JNTU College of Engineering,
Kakinada and M.E and Ph.D degree in control
systems from Andhra University College of
Engineering, Visakhapatnam, in 2002 and 2010
respectively. His research interests include model
order reduction, interval system analysis, design of
PI/PID controllers for Interval systems, sliding mode
control, Power system protection and control.
Presently he is working as Professor & H.O.D of
Electrical Engineering department, Gudlavalleru
Engineering College, Gudlavalleru, A.P, India. He
received best paper awards in several national
conferences held in India.
D.Srinivasa Rao was born in Nara kodur,
Guntur Dist, Andhra Pradesh, India, in
1969. He received bachelor‟s degree in
Electrical & Electronics Engineering from
JNTU College of Engineering, Kakinada and
M.Tech in Electrical Machines & Industrial Drives
from NIT Warangal in 2003.Now doing Ph.D in the
area of control systems in JNTU Kakinada.His
research interests include model order reduction,
interval system analysis, design of PI/PID controllers
for Interval systems, Multi-level Inverters, Presently
he is working as Associate Professor in Electrical
Engineering department, Gudlavalleru Engineering
College, Gudlavalleru, A.P, India. He presented
various papers in national c and International
Conferences held in India.
N.Sai Dinesh, a PG Student completed
B.Tech in Sri Sunflower College of
Engineering and Technology, and
pursuing M.Tech.control systems in
Gudlavalleru Engineering college, Gudlavalleru

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