International Journal of Advanced Scientific and Technical Research
Available online on http://www.rspublication.com/ijst/index.html
Issue 6 volume 1, Jan. –Feb. 2016
ISSN 2249-9954
Order Reduction of Linear Time Invariant System by Mixed Methods of Routh
Approximation and Factor Division Method.
Harendra singh*, VR singh**
Department of Electrical Engineering, Mewar University, Gangrar, Chittorgarh (Rajasthan)
Email:- *harendrasngh65 @gmail.com
**vrsinghieee@gmail.com
Abstract – A mixed methods for model reduction of linear time invariant system is proposed. In
this technique two methods are employed, one method is used with advantages of Routh
approximation and other is factor division method. Routh approximation is used to determine
the denominator of reduced model and numerator is determined by factor division method. This
technique guarantees the stability of reduced order model if original system is stable .This
method is illustrated with the help of numerical example.
Key words: - Model reduction, Routh approximation, Factor division method stability, Transfer
function.
1. INTRODUCTION
Approximation of original system (HOS) by reduced order models has been of concern for
engineers for long time. Reducing the high order system has been studied by several researchers
[1-9]In spite of these reduction no one researcher gives always best result. To overcome the
stability problem Hutton and Friedland[10],R.KAppiah[11]and chan [12]gave different method.
Model reduction by Routh approximation has been adopted by some researchers [13-15] and
factor division [16-18] by also some researchers. Factor division method is introduced by Lucas
[19 ],in which dominant poles are retained and initial time moments are preserved .This method
was extended by Lucas[20].
This present paper is organized by two methods for linear dynamic system. The denominator is
determined by Routh approximation and coefficients of numerator of reduced order model are
determined by using factor division method.
2.Methods Used
2.1 Statement of Approximation
Method.
Let nth order linear time invariant system represented by transfer function as
N (s) b21  b22 s  ......  b 2 n s n 1
H(s) =
=
D( s ) b11  b12 s  .......  b1n 1s n
(1)
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International Journal of Advanced Scientific and Technical Research
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Where b21, b22 …b2n and b11 ,b12 ……….b1,n+1
are constants.
Reduced order model is given by
N ( s ) d 21  d 22 s  .....  d 2 k s k 1
R( s)  k

Dk ( s ) d11  d12 s  ....  d1k 1s k
(2)
Where d21, d22, ……d2k and d11, d12……d1k+1 are constants.
2.2 Order Reduction Method.
This method consists of two steps
Step:- 1
Determination of the denominator coefficients of reduced order model:Routh array for denominator polynomial of original system H(s), is given ,as
b11 b12 b13 b14 ..
b21 b22 b23 b24 ....
b31 b32 b33 b34 .
..
..
bn.1
bn 1,1
The well known algorithm form the above array as
bi , j  bi 2, j 1   bi 2,1.bi 1, j 1  / bi 1,1)
Where 𝑖 ≥ 3 and 1 ≤ 𝑗 ≤ [(𝑛 − 𝑖 + 3) 2][. ] stands for the integral part of the quantity.
A Polynomial of lower order „k‟ may be easily constructed [13] with(𝑛 + 1 − 𝑘)𝑡ℎ and
(𝑛 + 2 − 𝑘)𝑡ℎ rows of the above array and gives
Dk  s   b( n1k ) s k  b( n2k ) s k 1  b( n3k ) s k 2 ..
 3
From Routh array of denominator polynomial of given transfer function starting with first array
as constant term. To obtain a reduced order model of order „k‟,a new Routh array is formed
where the first (k+1) terms of the above array forms the first column The remaining entries of the
array are now easily filled. Once array is completed, it will be noted that the last element in first
column moves two places up and one to the right at each step. The denominator of reduced
model Dk(s) can be written from first two rows of the array.
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Issue 6 volume 1, Jan. –Feb. 2016
ISSN 2249-9954
Step:-2
Determination of numerator coefficients of reduced model of „kth order by factor division method
as
(a) Product of numerator of original system (HOS) and denominator of reduced model to the
denominator of original system (HOS) given by
n
b
2i
si - 1
N (s)
 i k1
D( s)
d1i s i

Dk ( s )
i 1
Where D(s) is known.
N k (s) 
(4)
D( s )
i.e put
Dk ( s )
the coefficients of D(s) in first rows and coefficients of Dk(s) in 2nd row. Second time put N(s)
D( s )
in first row and expansion of
in 2nd row.Therefore numerator Nk(s) of reduced order
Dk ( s )
model R(s) in eqn(2) will be series expansion of
(b) By using factor division, first to find the terms up to sk-1 in the expansion of
k 1
b2i s i
N (s) 
 ik11
(5)
D( s)
i
b1i s
Dk ( s ) 
i 1
about s=0 up to term of order sk-1 .
This can be obtained by modifying the moment generating and using Routh recurrence formulae
to generate the third, fifth, and seventh etc rows as
0 
b11 b11 b12 b13 ......
d11 d11 d12 d13 ......
1 
c11 c11 c12 c13 ......
d11 d11 d12 d13 ......
………………………………..
………………………………….
 k 2 
p11 p11 p12 p13 ......
d11 d11 d12 d13. .....
.
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International Journal of Advanced Scientific and Technical Research
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 k 1 
v11 v11 v12 v13. .....
d11 d11 d12 d13 ......
Where
c11  b1i 1   0 d1 i 1
i = 0,1,2….
……………………….
……………………….
v11  p12   k  2 d12
Therefore the numerator Nk(s) of eqn (2) is given by
k 1
N k (s)    i si
i 0
3. Numerical Analysis.
Let consider 4th order transfer function given by [21]
H 4 ( s) 
N (s)
2s 3  12s 2  18s  8
 4
D(s) s  6s 3  14s 2  16s  8
(6)
Step-1
Using first step form the Routh table for the denominator of original system as:
8
14 1
16 6
11 1
4.55 0
1
Using first three entries of first column to form a new array for reduced order model as
8
16
11
Thus denominator of reduced model is written as
Dk(s) = 11s2 +16s +8
Normalizing Dk(s), yields
s 2  1.454s  0.727
(7)
Step-2
Numerator coefficients of reduced model are determined as
D4 / D2 results
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Issue 6 volume 1, Jan. –Feb. 2016
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International Journal of Advanced Scientific and Technical Research
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 0  11.004
1  0
8
0.727
16
1.454
0
0.727
Now consider N4 / D4 / D2
 0  0.727
1  1.636
8
18
11.004
0
18
11.004
Thus , Numerator of reduced model is written as
N2  s   1.636s  0.727
Thus, transfer function of reduced model is given
1.636s  0.727
R( s)  2
s  1.454s  0.727
(8)
(9)
Fig-1.presents step response of original high order system H(s) and reduced order model H r(s)
and fig-2 shows the frequency response of original high order system and reduced model. It may
be seen that steady state response of original system and reduced order model are exactly match
close to the original system.
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Step Response
1.4
1.2
Amplitude
1
....... original system
- - - - proposed method
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
10
Time (sec)
Fig.1
Step response
Bode Diagram
10
....... original system
- - - - proposed ethod
Magnitude (dB)
0
-10
-20
-30
Phase (deg)
-40
45
0
-45
-90
-1
10
0
1
10
10
2
10
Frequency (rad/sec)
Fig.2 Frequency response
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Issue 6 volume 1, Jan. –Feb. 2016
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Comparison of Reduced model with Original System
Characteristics of Original system
Rise Time : 0.4471
Settling Time : 4.4829
Settling min : 0.9601
Settling max : 1.3588
Overshoot
: 35.8752
Undershoot :0
Peak
:1.3588
Peak time
: 0.4599
Characteristics of Reduced Model
Rise Time
: 0.7781
Settling Time :6.1341
Settling min :0.9154
Settling max :1.2301
Overshoot
:23.0085
Undershoot :0
Peak
:1.2301
Peak Time
:2.2369
5. Conclusion: - An algorithm
which combines the advantages of Routh approximation and
factor division methods has been presented. The denominator of reduced order model of linear
dynamic system is obtained by method of Routh approximation and numerator of reduced model
is determined by factor division method . These proposed methods assure the stability of reduced
model if original system (HOS) is stable. These methods are also extended for MIMO. It is clear
that the proposed methods are simple and in quality with other existing techniques of reduced
model. These methods preserve the stability and avoid errors in between initial and final value.
These methods are tested with numerical example and time and frequency responses of original
system and reduced order model are compared graphically.
6. References:
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automatic control. Vol 19, pp-615-616. 1974
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Issue 6 volume 1, Jan. –Feb. 2016
ISSN 2249-9954
5. Bai-Wu Wan “Linear model reduction using Mihailov criterion and Pade approximation”
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ISSN 2249-9954
20.T.N Lucas “ Factor division: A useful algorithm in model reduction”.IEE Proceedings130(6),
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