Uncertain Systems Order Reduction by Aggregation Method

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 7, No. 1, February 2017, pp. 244~252
ISSN: 2088-8708, DOI: 10.11591/ijece.v7i1.pp244-252  244
Journal homepage: http://iaesjournal.com/online/index.php/IJECE
Uncertain Systems Order Reduction by Aggregation Method
Bogapurapu Gayatri1
, Kalyana Kiran Kumar2
, Akella Venkata Santosh Lakshmi3
,
Vyakaranam Sai Karteek4
1,3,4
Department of Electrical and Electronics Engineering, PG Scholar, VITS College of Engineering
2
Professor and Head of EEE Department,VITS College of Engineering
Article Info ABSTRACT
Article history:
Received Jun 2, 2016
Revised Aug 10, 2016
Accepted Aug 18, 2016
In the field of control engineering, approximating the higher-order system
with its reduced model copes with more intricateproblems. These complex
problems are addressed due to the usage of computing technologies and
advanced algorithms. Reduction techniques enable the system from higher-
order to lower-order form retaining the properties of former even after
reduction. This document renders a method for demotion of uncertain
systems based on State Space Analysis. Numerical examples are illustrated to
show the accuracy of the proposed method.Keyword:
Aggregation method
Interval systems
Kharitonov‟s theorem
Order reduction
Stability Copyright © 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Kalyana Kiran Kumar,
Professor and Head of EEE Department,
VITS College of Engineering,
Sontyam, Anandapuram,
Visakhapatnam, Andhra Pradesh, India.
Email: kirankalyana1@gmail.com
1. INTRODUCTION
In present time'sengineers and scientists are often with the analysis, design and synthesis of real time
problems. The primary step to be followed is to develop a mathematical model which is an equivalent
representation for the real problem.
In general, these mathematical models possessedwith large dimensions and are named as large-scale
systems. On the other hand, a highly detailed model would lead to a great deal of unnecessary complications.
Hence forth a mechanism being applied to bring a compromisebetween the reduced model and original
system so as to preserve the properties of the original system in its reduced model. The large-scale systems
model reduction has two approaches like Time domain and Frequency domain. In frequency domain,
continued fraction expansion method has low computational efforts and is applicable to multivariable
systems, but the major drawback in this method is it does not preserve the stability after reduction [1].
Similarly in Time domain approach using routh approximation method, the steady state condition of the
system is preserved, but it does not preserve the transient state of system [2].
Reduction of continuous interval systems by routh approximation [3], and discrete interval systems
by retention of dominant poles and direct series expansion method [4]. Based on study of stability and
transient analysis of interval systems many methods have been proposed by the researchers on interval
systems [5-9]. In this note the author extends the paper by reduction of interval systems in to state space form
of realization using „Aggregation method‟.
The outline of this note consists of four sections. In Section 2, Problem Statement will be is
discussed. In Section 3, Procedural Steps for the proposed technique is explained. In Section 4, Error
Analysis is done. In Section 5, the performance of the proposed method is shownby a numerical example. In

IJECE ISSN: 2088-8708 
Uncertain Systems Order Reduction by Aggregation Method (Bogapurapu Gayatri)
245
Section 6, comparisons between the proposed and other methods are illustrated. At last, a conclusion is stated
in Section 7.
2. PROBLEM STATEMENT
Consider an original linear time invariant uncertain system in Controllable Canonical Form:
∑ {
̇( ) ( ) ( )
( ) ( ) ( )
∑ [ ] (1)
where
[
[ ] [ ] [ ]
]
[ ]
[[ ] [ ] [ ]]
n=no. of state variables; m=no. of inputs variables; q=no. of outputs variables
The Corresponding demoted order model of an uncertain system is represented in Controllable
Canonical Form (CCF) as follows:
∑ {
̇ ( ) ( ) ( )
( ) ( ) ( )
∑ [ ] (2)
where
[
[ ] [ ] [ ]
]
[ ]
[[ ] [ ] [ ]]
r=reduced order
Hansen. E [10] explainedthe fundamental arithmetic rules for an interval plant as follows:
Addition:
[i, j]+ [o, v] = [i+o, j+v] (3)

 ISSN: 2088-8708
IJECE Vol. 7, No. 1, February 2017 : 244 – 252
246
Subtraction:
[i, j]- [o, v] = [i-o, j-v] (4)
Multiplication:
[i,j]×[o,v]=[min(io,iv,jo,jv),max(io,iv,jo,jv)] (5)
Division:
[ ]
[ ]
[ ] [ ] (6)
3. PROCEDURAL STEPS
Step 1: The equivalent transfer function for the original uncertain plant expressed in Equation 1 is:
( )
( )
( )
(7)
where
( ) ∑ [ ]
( ) ∑ [ ]
Step 2: The above uncertain system is converted to four fixed transfer functions which carries the
coefficients of Equation 7 this can be represented in its general form using Kharitonov‟s theorem [3]
( )
∑
∑
(8)
where
e ≤ n-1; f ≤ n; p=1, 2, 3, 4
Step 3: The above four transfer functions are transformed into four fixed state models:
∑ [ ] (9)
p=1, 2, 3, 4
Step 4: Evaluate the Eigen values for the obtained four fixed state models represented in
Equation 9 individually
Step 5: Calculate the modal matrix ( ) is calculated for each individual state model:
[ ] (10)
where
p=1, 2, 3, 4 ; ( ) ( )
Step 6: Now the inverse of modal matrix(̅ ) is to be evaluated from

IJECE ISSN: 2088-8708 
247
̅ [
̅ ̅
̅ ̅
] (11)
where
p=1, 2, 3, 4; ̅ ( ) ̅ ( )
Step 7: Obtain the arbitrary matrix ( )followed by the Equation below:
[ ( )] (12)
Step 8: Determine the aggregation matrix „ ‟ using the Equation
̅ (13)
Step 9: Using the aggregation matrix the four reduced state models are obtained and are represented
in generalized form as
∑ [ ] (14)
where
[ ]
[ ]
p=1, 2, 3, 4; r = order of reduced system
Step 10: The corresponding four reduced rth
order transfer functions for the above obtained four
reduced rth
order state model expressed in Equation 14 in its general form:
( )
∑
∑
(15)
where
g≤ r-1; h ≤ r; p=1, 2, 3, 4; r = order of reduced system
Step 11: Now the equivalent transfer function for reduced interval system is equated below:
( )
[ ] [ ] [ ]
[ ] [ ] [ ]
(16)
Step 12: Finally the demoted model of the interval system is expressed in Controllable Canonical
Form (CCF):
∑
{
̇ ( ) [
[ ] [ ] [ ]
] ( )
[ ]
( )
( ) [[ ] [ ] [ ]] ( )
(17)

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4. INTEGRAL AND RELATIVE INTEGRAL SQUARE ERROR
The integral and relative integral square error between transient responses of original and reduced
systems is also determined as formulated below:
Relative ISE=∫ [ ( ) ( )] ∫ [ ( ) ( )] (18)
ISE=∫ [ ( ) ( )] dt
where ( ) and ( ) are the unit step responses of original Q(s) and reduced R(s) systems, ( ) final value of
original system
5. NUMERICAL ILLUSTRATION
5.1. Example 1
Let us consider an interval system having state model as followed below:
∑
{
̇( ) [
̇ ( )
̇ ( )
̇ ( )
] [
[ ] [ ] [ ]
][
( )
( )
( )
] [ ] ( )
( ) [ [ ] [ ] [ ]] [
( )
( )
( )
]
(19)
1. The equivalent transfer function of an uncertain is as follows:
( )
[ ] [ ] [ ]
[ ] [ ] [ ] [ ]
(20)
2. Evaluate the four 3rd
order transfer functions by using Kharitonov‟s theorem as expressed in Equation (8)
3. The above four transfer functions are converted into four state models by using Equation 9 are:
∑
[
[ ] [ ]
[ ] [ ] ]
(21)
∑
[
[ ] [ ]
[ ] [ ] ]
(22)
∑
[
[ ] [ ]
[ ] [ ] ]
(23)
∑
[
[ ] [ ]
] [ ] ]
(24)
4. Next Eigen values are to be determined individually for the above four state models
5. Then modal matrix, inverse of modal matrix and matrices are obtained for the four state models
individually by using Equations 10 to 13

IJECE ISSN: 2088-8708 
249
6. Four reduced order state models are obtained from Equation 14 as given below:
∑ [
[ ] [ ]
[ ] [ ]
] (25)
∑ [
[ ] [ ]
[ ] [ ]
] (26)
∑ [
[ ] [ ]
[ ] [ ]
] (27)
∑ [
[ ] [ ]
[ ] [ ]
] (28)
5. The corresponding demoted order transfer functions are obtained as expressed in Equation 15 using four
reduced state models from Equations 25 to 28
6. Thenthe equivalent reduced order transfer function for interval system is obtained as expressed in
Equation 16:
( )
[ ] [ ]
[ ] [ ] [ ]
(29)
Under steady state condition then
( )
[ ] [ ]
[ ] [ ] [ ]
(30)
7. The CCF of the reduced order interval system
∑ {
̇ ( ) [
[ ] [ ]
] ( ) [ ] ( )
( ) [[ ] [ ] ] ( )
(31)
Step responses of both original and reduced 3rd
order systems are shown in Figure 1 below
Figure 1. Step Response of Original and Reduced 3rd
Order System using Proposed Method
Step Response
Time (seconds)
Amplitude
0 0.5 1 1.5 2 2.5 3 3.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
ORG SYS LOWER LIMIT RESPONSE
RED SYS LOWER LIMIT RESPONSE
ORG SYS UPPER LIMIT RESPONSE
RED SYS UPPER LIMIT RESPONSE

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250
6. COMPARSION OF METHODS
The demoted order models of proposed method are compared with other methods are shown in
Figure 2, 3 and 4.
Order System using Mihailov and Cauer Second Form
Figure 3. Step Response of Original and REDuced 3rd
Order System using Routh and Factor Division
Method
Order System using Mihailov and Factor Division
Method
0 1 2 3 4 5 6 7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Step Response
Time (seconds)
Amplitude
Step Response
Time (seconds)
Amplitude
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (seconds)
Amplitude

IJECE ISSN: 2088-8708 
251
Table 1 shows Comparison of Reduced Order Models for 3rd
order system.
Table 1. Comparison of Reduced Order Models for 3rd
Order System
S.no. Methods Reduced Order Systems
Step Response of
Lower Limit
Step Response of
Higher Limit
ISE
Values
Relative
ISE
Values
ISE
Values
Relative
ISE
Values
1.
Proposed
Method
( )
[ ] [ ]
[ ] [ ] [ ]
0.012 0.070 0.009 0.066
2.
Mihailov
and cauer
second
form
( )
[ ] [ ]
[ ] [ ] [ ]
0.014 0.086 0.025 0.157
3.
Routh and
Factor
Division
Method
( )
[ ] [ ]
[ ] [ ] [ ]
0.344 2.152 0.007 0.042
4.
Mihailov
and Factor
Division
Method
( )
[ ] [ ]
[ ] [ ] [ ]
0.457 2.833 0.623 3.856
7. CONCLUSION
To decrease the complexity of the system order reduction is done. In this note the order reduction by
proposed method is numerically solved. The proposed method implemented for order reduction represented
the uncertain systems in state model. The demoted model obtained by proposed method is compared with
other methods, and the ISE & Relative ISE values of step response are also compared. Hence the proposed
method maintains stability with low ISE values compared to other existing methods.
REFERENCES
[1] L. S. Shieh and M. J. Goldman, “Continued Fraction Expansion and Inversion of cauer third Form,” IEEE
Trans.On Circuits and systems, vol/issue: CAS-21(3), pp. 341-345, 1974.
[2] M. Chand, “Reducing Model Ordering using Routh Approximation Method,” International Journal of Emerging
Technology and Advanced Engineering, vol/issue: 4(8), 2014.
[3] Bandyopadhyay B., et al., “Routh pade approximation for interval systems,” IEEE Trans Autom Control, vol. 39,
pp. 2454–2456, 1994.
[4] C. Younseok, “A note on discrete interval system reduction via retention of dominant poles,” Int J Control Autom
Syst, vol/issue: 5(2), pp. 208–211, 2007.
[5] Saraswathi G., “A mixed method for order reduction of interval systems,” Int Conf IntelAdv. Syst, pp. 1042–1046,
2007.
[6] Ismail O. and Bandyopadhyay B., “Model order reduction of linear interval systems using pade approximation,”
IEEE Int Symp Circ Syst, 1995.
[7] Singh V. P. and Chandra D., “Routh approximation based model reduction using series expansion of interval
systems,” IEEE Int Conf Power Control Embed Syst (ICPCES), vol. 1, pp. 1–4, 2010.
[8] Singh V. P. and Chandra D., “Model reduction of discrete interval system using dominant poles retention and direct
series expansion method,” in Proceedings of the IEEE 5th International power engineering and optimization
conference (PEOCO), vol 1. pp. 27–30, 2011.
[9] K. Kumar D, et al.. “Model order reduction of interval systems using modiﬁed routh approximation and factor
division method,” in Proceedings of 35th national system conference (NSC), IIT Bhubaneswar, India, 2011.
[10] Hansen E., “Interval arithmetic in matrix computations,” Part I Siam J Numer Anal, pp. 308–320, 1965.

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252
BIOGRAPHIES OF AUTHORS
Gayatri Bogapurapu received B.Tech degree in Electrical and Electronics Engineering in the
year 2014 from Simhadhri College of Engineering. She is now pursuing M.tech degree in the
specialization of Power System Automation and Control in VITS College of Engineering,
Visakhapatnam, India.
Kiran Kumar Kalyana received B.Tech degree in Electrical and Electronics Engineering in the
year 2001 from GMRIT Rajam, M.E degree in the specialization of Control Systems in the year
2004 from AUCE, Andhra University, Ph.D degree in Electrical Engineering from A.U,
Visakhapatnam in 2015. His research areas are control systems,interval systems,Non-linear
systems,Adaptive control systems. He is a member of ISTE, IAENG(HK), IETE.
Venkata Santosh Lakshmi Akella received B.Tech degree in Electrical and Electronics
Engineering in the year 2013 from Nightingale Engineering College for Women. She is now
pursuing M.tech degree in the specialization of Power System Automation and Control in VITS
College of Engineering, Visakhapatnam, India.
Sai Karteek Vyakaranam received B.Tech degree in Electrical and Electronics Engineering in
the year 2014 from Sri Chaitanya College of Engineering. He is now pursuing M.tech degree in
the specialization of Power System Automation and Control in VITS College of Engineering,
Visakhapatnam, India.

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