Model Order Reduction of Higher Order Discrete Time Systems using Modified Pole Clustering Technique

Model Order Reduction of Higher Order Discrete Time Systems using Modified Pole Clustering Technique

N.Sai Dinesh 1, Dr.M.Siva Kumar 2, D.Srinivasa Rao 3

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.

   1. INTRODUCTION

      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 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.

      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, 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 linear 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.

   2. PROBLEM FORMULATION

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

      Therefore, it is required to derive a kth order reduced model in P domain. It is given by

      2

      2

      = () = 0 +1 +2 2++11(2)

      () 0 +1+2 ++

      where ; 0 1 and ; 0 are scalar constants.

      By applying inverse transformation, the reduced order model in Z-domain is obtained.

      = |=1

   3. REDUCTION PROCEDURE

      The proposed method for getting the order reduced model, consists of the following two steps:

      Step1: Determination of the denominator polynomial for the order reduced model, using modified pole clustering technique.

      Step2: Determination of the numerator of 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.

      1. Separate clusters should be made for real poles and complex poles.

      2. Poles on the jw-axis have to be retained in

         N (Z )

         e e z e z 2 ……… e

         zn1

         G(Z )

         D(Z )

         0 1 2 n1

         2 n

         2 n

         f0 f1 z f 2 z ……….. fn z

         the reduced order model.

      3. Clusters of poles in the left half s-plane should not contain any pole of the right half s-plane and vice-versa.

         Convert the higher order discrete time system into P domain using linear transformation Z=P+1.

         = |=+1

         Therefore the transfer function of higher order original system of order n in P-domain is

         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

         1 , 2 , 3 ,then Inverse Distance Measure (IDM) identifies cluster center as

         = () = 0 +1 +2 2++ 1 1

         (1)

         () 0 +1 +2 2++

         1

         1

         Where ; 0 1 and ; 0 are scalar constants.

         = ÷

         =1

         (3)

         where 1 < 2 < 3 ,then modified cluster center can be obtained by using the algorithm.

         Step 1: Let r real poles in a cluster be 1 <

         2 < 3 . . .

         Step 2: Set j=1.

         = 1 1 2

         2 . . ( /2)( /2) (6)

         where 1 and 1 are complex conjugate cluster centers or 1 = + and 1 =

         Case 3: If (k-2) cluster centers are real and one pair of cluster center is complex conjugate, then the

         denominator polynomial of the order reduced

         Step 3: Find pole cluster centre

         =

         1 ÷ 1

         model can be obtained as

         Step 4: Set j=j+1

         =1

         = 1 2 . . ( (2))(

         1 )( 1 ) (7)

         Step 5: Find a modified cluster centre from

         1

         = 1 + 1 ÷ 2

         1 1

         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 cluster as =

         Let m pair of complex conjugate poles in the cluster be

         [ 1 ± 1 , 2 ± 2 , . . ± ]

         then the complex center is in the form of ± .

         Step 2: Determination of the numerator of order reduced model using Pade approximations.

         The original order system can be expanded in power series about p=0 as

         = = 0 +1 +2 2++ 1 1

         0 +1 +2 2++

         = 0 + 1 + 2 2 + (8)

         The coefficients of power series expansion are calculated as follows:

         0 = 0

         = 1 , > 0

         =

         =

         Whee

         =1

         1 ÷ 1

         0

         =1

         and

         =

         =

         =1

         1 ÷ 1 (4)

         = 0, > 1 (9)

         The reduced order model is written as

         One of the following cases may occur, for synthesizing the order denominator polynomial.

         Case 1: If all the modified cluster centers are real,

         =

         then reduced denominator polynomial of order k, can be taken as

         = 0 +1 +2 2++11

         0 +1+2 2++

         (10)

         = 1 2 . ( )

         where 1 , 2 , . are 1 , 2 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

         Here, (s) can be determined through equations (5-7).

         For of equation (10) to be Pade approximants of G(s) of equation (8), we have

         0 = 00

         1 = 0 1 + 10

         (11)

         .

         1 = 01 + 12 + + 1 1 + 0

         the coefficients ; j=0,1,2,3..k-1 can be found by solving above k linear equations.

         FLOW CHART FOR ORDER REDUCTION

         = ()

         ()

         24 + 1.83 + 0.82 + 0.1 0.1

         = 4 1.23 + 0.32 + 0.1 + 0.02

         Substituting Z=P+1,

         = ()

         ()

         24 + 9.83 + 18.22 + 15.1 + 4.6

         = 4 + 2.83 + 2.72 + 1.1 + 0.22

         Start

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

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

         Convert the higher order discrete- time system into continuous using linear transformation Z=P+1

         The poles are : 1.1199 ± 0.1351,

         0.2801 ± 0.3073

         By using the above modified pole clustering algorithm, modified cluster center can be formed from the poles as

         1 = 0.6400 ± 0.1570

         The denominator polynomial () of reduced order model is obtained by using the equation (6) as

         = 1 ( 2 )

         = 2 + 1.28 + 0.43425

         Therefore, the 2 order reduced model is

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

         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 P=Z-1

         Convert the obtained continuous reduced order model into Discrete – Time System using inverse linear Transformation P=Z-1

         = 0 + 1

         0.2074435 + 0.53334 + 2

         Stop

   4. NUMERICAL EXAMPLES

      EXAMPLE 1

      Consider a 4th order discrete-time system given by Younseok Choo [15]:

      The numerator coefficients are obtained by pade approximations. By using equations (9) and (11),

      0 = 9.0797

      1 = 11.1702

      Therefore, finally 2 -order reduced model is obtained as

      2 2

      2 2

      = 9.0797 + 11.1702 0.43425 + 1.28 +

      By applying inverse transformation P = Z-1,

      we obtain the reduced order model of discrete time system as

      2 2

      2 2

      = 2.0905 + 11.1702 0.15425 0.72 +

      For comparison purposes, a second order approximant by Younseok Choo [15]and [16] is found to be

      = ()

      ()

      2.04 0.75 0.9423592 0.717487

      0.5195656 ( 0.5)

      0.3 0.5 0.75 0.85

      0.9 ( 0.95)

      Substituting Z=P+1,

      21 =

      211 =

      4.87768 2.55604

      2 1.50888 + 0.63166

      4.2 0.9023114

      = ()

      ()

      2.045 + 3.2039984 + 1.87709783 + 0.50015822

      +0.0577944 + 0.0019949

      2 1.442346 + 0.616743

      = 6 + 1.755 + 1.11254 + 0.3231253 + 0.452122

      +0.002893 + 0.0000656

      The poles are: -0.7,-0.5,-0.25,-0.15,-0.1,

      -0.05.

      Figure 1: Comparison of Step response of high order system and reduced order models for example1.

      It has been observed from Fig1 that the 2nd 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 6th order discrete-time system,

      By using the above modified pole clustering algorithm, modified cluster centers can be formed from the real poles as

      1 = 0.057144

      2 = 0.15939

      The denominator polynomial () of reduced order model is obtained by using the equation (5) as

      = 1 ( 2 )

      = + 0.057144 ( + 0.15939)

      = 2 + 0.216534 + 0.009108

      Therefore, the 2 order reduced model is

      2

      2

      = 0 + 1 0.009108 + 0.216534 +

      The numerator coefficients are obtained by pade approximations. By using equations (9) and (11),

      0 = 0.2769748

      1 = 2.39431

      Therefore, finally 2 -order reduced model is obtained as

      2 2

      2 2

      = 0.2769748 + 2.39431 0.009108 + 0.216534 +

      By applying inverse transformation P = Z-1,

      we obtain the reduced order model of discrete time system as

      2 2

      2 2

      = 2.1173352 + 2.39431 0.792574 1.783466 +

      Figure 2: Comparison of Step response of high order system and reduced order models for example2.

   5. CONCLUSION

      A new method is presented to determine the reduced order model of a higher order discrete time system. The denominator polynomial of the reduced order model is obtained by using modified pole clustering while the numerator coefficients are obtained by pade approximation. The effectiveness of proposed method is illustrated with the help of examples chosen from the literature and the responses of the original and reduced system are compared graphically as shown in fig.1

   6. REFERENCES

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      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.

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   7. BIOGRAPHIES

Dr Mangipudi Siva Kumar was born in Amalapuram, E. G. Dist, Andhra Pradesh, India, in 1971. He received bachelors 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 bachelors 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 and International Conferences held in India.

N.SaiDinesh 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.