Journal of Electrical Engineering and Science, Vol. 1(1) 2015, pp. 1-10
RESEARCH ARTICLE
A New Model Order Reduction for Linear Continuous Time Interval Systems Gulshad Begum1, *M Siva Kumar2 1 PG Student, Department of Electrical and Electronics Engineering, Gudlavalleru Engineering College, Gudlavalleru, AP, India. 2 Department of Electrical and Electronics Engineering, Professor & Head of the Department, Gudlavalleru Engineering College, Gudlavalleru, AP, India Received-28 August 2015, Revised-28 September 2015, Accepted-12 October 2015, Published-12 October 2015
ABSTRACT This paper presents a new method for order reduction of linear continuous time interval systems. This method is based on Stability equation method, Pade approximation and Kharitonov’s theorem. The reduced order interval model is determined by using Kharitonov’s polynomials which makes use of Kharitonov’s theorem and general form of Stability equation method for denominator while Pade approximation is used for numerator coefficients. This method generates stable reduced order model if the original higher order system is stable. The proposed method is illustrated with the help of typical numerical example considered from the literature. Keywords: Interval system, Model order reduction, Stability equation method, Pade approximation, Kharitonov’s theorem.
1. INTRODUCTION The model of the original system is fairly complex and is of higher order. Higher order system analysis is both tedious and costly. The understanding of the behavior of the system is difficult due to complexity. To avoid the above problems order reduction implementation is necessary. Model Order Reduction (MOR) is a branch of systems and control theory, for reducing the complexity of a higher system, while preserving their inputoutput behaviour. Order reduction methods are broadly classified into two types. Frequency domain order reduction methods are for transfer function model. Time domain order reduction methods are for state space model. Several methods are available in the literature for the order reduction of linear continuous systems in time domain as well as frequency domain. The reduced order model obtained in the frequency domain gives better matching of the impulse response with its higher order system. Some of the most popularly used frequency domain order reduction methods are pade approximation
[11] and continued fraction methods. These are computationally fast and are able to exactly match the maximum number of system parameters (usually time moments or Markov parameters) to the reduced model. But these methods have a disadvantage that stability of the reduced model is not guaranteed for stable system. Efforts has been devoted to developing stability preserving methods such as Routh-Approximation, Mihailov criterion etc.The stability of these methods is achieved but the disadvantage of these methods is the cost of series loss of accuracy. The advantage of this method is that it preserves stability in the reduced model, if the original high-order system is stable, and retains the first two time-moments of the system. These methods are applicable for fixed-coefficients systems only. However, many systems the coefficients are constants but uncertain within a finite range. Such systems are classified as interval systems. The above methods are applicable for fixed system only.The reduced model of interval system is unstable even the original higher order
*Corresponding author. Tel.: +919701764651 Email address:
[email protected] (M.S.Kumar) Double blind peer review under responsibility of DJ Publications http://dx.doi.org/10.18831/djeee.org/2015011001 2455-1945 © 2016 DJ Publications by Dedicated Juncture Researcher’s Association. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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G.Begum and M.S.Kumar./ Journal of Electrical Engineering and Science, Vol. 1(1), 2015 pp. 1-10
interval system is stable. In [1] for interval system γ − δ Routh Approximation method for interval systems is proposed. An improvement is proposed [2,3] such that the resulting interval Routh approximant is robustly stable. To improve the effectiveness of model order reduction many mixed methods are proposed recently [4,5,6] based on interval arithmetic. Thus the stability of the reduced mode obtained using interval arithmetic is not guaranteed, even if the original interval system is stable. A reduction technique for linear interval systems using Kharitonov’s theorem are presented in [7] [8] to generate stable reduced technique for linear interval models. In [9] recently a reduction technique for linear interval systems using Kharitonov’s theorem interval and RouthApproximation is presented to generate stable reduced order interval system. In [10] the reduced order interval model is obtained using Kharitonov’s polynomials it retains stability and full impulse response energy of the higher order interval system in its reduced order interval model. Among these various model order reduction methods for stability preservation available in the literature, the stability equation method [13] is one of the most popular techniques. In this paper, model order reduction of interval systems is carried out by using Kharitonov’s theorem, stability equation method and Pade approximation method. The denominator of the reduced model is obtained by stability equation method and the numerator is obtained by Pade approximation method. Thus the stability is guaranteed if the original system is stable and the response is matching between original higher system and reduced order system
and Bi+ as lower and upper bounds of interval [Bi-,Bi+] respectively. It is proposed to obtain a reduced order interval model of the form ( )
2. PROBLEM FORMULATION Consider a higher order continuous interval system by the transfer function ( )
3.2. Reduction procedure Consider a family of real interval transfer function ( ) ( ) ( )
[
] [
[ ]
] [
[ ]
[ [
[ ] [
]
[
]
]
[
]
where [ai-,ai+] for i=0,1….,r are denominator coefficients of with Gr(s) ai-and a+i as lower and upper bounds of interval [a i,ai+] respectively, and [bi-,bi+] for i=0,1…,r-1 are numerator coefficients of Gr(s) with bi- and bi+ as lower and upper bounds of interval [b i,bi+] respectively. 3. PROPOSED METHOD 3.1. Kharitonov’s theorem An interval polynomial family ( ) ∑ [ ] with invariant degree is robustly stable if itsfour Kharitonov polynomials are stable. According to the Kharitonov theorem, every interval polynomial p(s) is associated with four following fixed parameter polynomials called Kharitonov polynomials. They are definedas
( ) ( ) ( ) ( ) The interval system is stable if and only if its four Kharitonov polynomials satisfies Routh Hurwitz stability criterion.
] [
]
]
[
] [
where for i=0,1….,n are denominator coefficients of Gn(s) with Ai-and Ai+ as lower and upper bounds of interval [A i,Ai+] respectively, and [Bi-,Bi+] for i=0,1…,n-1 are numerator coefficients of Gn(s) with Bi-
[ ]
] [
[ ]
] [
]
[Ai ,Ai+] -
The four fixed Kharitonov’s transfer functions associated withGn(s) are given as:
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G.Begum and M.S.Kumar./ Journal of Electrical Engineering and Science, Vol. 1(1), 2015 pp. 1-10
( )
( ) ( )
( ) ( )
( )
(
(
(
)
For stable first Kharitonov transfer function G1n(s), the denominator D1n(s) of the HOS is bifurcated in the even and odd parts in the form of stability equations as ( ) (
( )
)
( )
( )
∑ (
( ) ( )
( )
)
∑ 5
(
}
where zi is poles of even parts of the denominator polynomial and pi is poles of odd parts of the denominator polynomial, m1 and m2 are the integer parts of n/2 and (n-1)/2 respectively and z21
