Order Reduction of MIMO System using Modified Cauer...

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ISSN: 2321-7782 (Online) e-ISJN: A4372-3114 Impact Factor: 6.047

Volume 5, Issue 9, September 2017

International Journal of Advance Research in Computer Science and Management Studies

Research Article / Survey Paper / Case Study Available online at: www.ijarcsms.com

Order Reduction of MIMO System using Modified Cauer Form

and Factor Division Method Harshita Mittal

1

Shobhit University, Gangoh

Saharanpur – India

Dr. Jasvir Singh Rana2

Shobhit University, Gangoh

Saharanpur – India

R. P Agarwal3

Shobhit University

Meerut – India

Abstract: In this paper, a combination of modified Cauer form and factor division method has been applied for high order

linear dynamic multi-input-multi-output (MIMO) system. In this technique, the denominator polynomial of the reduced

order model is derived by using the modified Cauer form while the numerator coefficients are obtained by using factor

division method The reduction procedure is simple computer oriented and the approach is comparable in quality with the

other well-known reduction techniques. Also, the proposed method guarantees stability of the reduced model if the original

high-order MIMO system is stable. The proposed approach of MIMO system order reduction is illustrated with the help of

an example and the results are compared with the recently published other well-known reduction techniques to show its

superiority.

Keywords: Multi-input-multi-output (MIMO) system Modified Cauer Form; Order Reduction, Stability, Transfer Function.

I. INTRODUCTION

In present day technology, there are a large number of problems which are highly complex and large in dimension. In

system theory the approximation of higher order system by lower order models is one of the important challenges. Because by

using the reduced lower order model, the implementation of analysis, simulation and various system designs become easier.

Various order-reduction methods are available for linear continuous time domain system as well as systems in frequency

domain. Further, the extension of single input single-output (SISO) methods to reduce multi-input multi-output (MIMO)

systems has also been carried out in [1]-[5].

II. STATEMENT OF THE PROBLEM

Consider the nth order linear time invariant general MIMO system described in frequency domain by the rectangular

function matrix

[G(s)]

[

]

Where

(1)

Where matrices of are appropriate dimension and are scalar constants. The elements of the transfer function matrix

[G(s)] are rational functions of„s‟ of the form

(2)

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The corresponding rth(k order model is of the form

[R(s)]=

=

(3)

Where are square matrices and are scalar constants

III. REDUCTION METHOD

The reduction procedure for getting the kth

-order reduced models consists of the following two steps:

Step-1: Determination of the denominator polynomial for the kth

-order reduced model using modified Cauer form by the

following procedure

By continuing the above sequence of expansion we get the following form

the quotients h1, h2, …………. H2, H1…….. are evaluated from the modified array

A reduced model of order k, is obtained by truncating equation (4) after the first 2kth terms and inverting it to yield the

transfer function formed out of the values of the quotients hj and Hj.

The reduced order denominator obtain by considering first 2k coefficients and inversion table

Step-2:

Determination of the numerator of kth order reduced model using Factor Division algorithm [12]

After obtaining the reduced denominator, the numerator of the reduced model is determined as follows

⁄ (5)

Where Dk(s) is reduced order denominator

By expressing N(s)Dk(s)/D(s) as N(s)/[D(s)/Dk(s)] and using factor division algorithm twice; the first time to find the term

up to sk-1

in the expansion of D(s)/Dk(s)(i.e. put D(s) in the first row and Dk(s) in the second row, using only terms up to sk-1

),

and second time with N(s) in the first row and the expansion [D(s)/Dk(s)] in the second row.

Therefore the numerator Nk(s) of the reduced order model (Rk(s)) in eq.(5) will be the series expansion of

∑

∑

About s=0 up to term of order sk-1

.

This is easily obtained by modifing the moment generating[14].which uses the familiar routh recurrence formulae to

generate the third, fifth, and seventh etc rows as,




..................................................

Where

i=0,1,2,........................... i=0,1,2.......................

Therefore, the numerator of eq.(3) is given by

(s)=∑

IV. METHOD OF COMPARISON

An integral square error (ISE) in between the transient parts of the original and reduced model is calculated using

MATLAB to measure the goodness of the reduced order model i.e. lower the ISE, closer the RK(s) to Gn(s) , which is given by

ISE=∫ | |

dt

Where, y(t) and yK(t) are the unit step responses of original and reduced system respectively.

V. NUMERICAL EXAMPLE

Consider the sixth order two input two output MIMO system taken from[85] described by the transfer function matrix[G(s)]

[G(s)]=[

]=[

] =

[

]

Where the common denominator D(s) is

D(s) =(s+1)(s+2)(s+3)(s+5)(s+10)(s+20) =s6+41s

5+571s

4+3491s

3+10060s

2+13100s+6000

And numerators are

,

,

Now considering

By using modified Routh array & the quotients h1, h2, h3, ,h5, h6, and H1, H2, H3 , , H5,H6 are obtain as

With the knowledge of the first four quotients (k=2), h1=1, H1=2, h2=-1.125 H2=-8 and with the help of the inversion table

denominator is obtain as

Now by using the factor division algorithm the following coefficients of numerator rn11(s) of reduced order model are

calculated

Consider D6(s)/D2(s) gives




6 131

1 11

65

1

,

= Now considering

,

Thus Reduced Numerator is given as Rn11(s) = 10+2s Thus the Reduced model is given as

Similarly other reduced order models are as follows

Fig. Comparison of step response and frequency response of original and reduced order system

TABLE I COMPARISON OF REDUCTION METHODS Reduction method R11 R12 R21 R22

Proposed method 2.0750x10-4 3.451x10-5 2.242x10-4 4.63 x10-3

Vishwakarma and Prasad [14] 0.001515 7.845x10-5 2.9984x10-4 0.0047

Parmar et al. [15] 0.014498 0.00874 0.002538 0.01574

Prasad and Pal [16] 0.136484 0.00244 0.040291 0.06790

Safonov and Chiang [17] 0.590617 0.03712 0.007328 1.06612

Prasad et al. [18] 0.030689 0.00025 0.261963 0.021683

VI. CONCLUSION

The authors presented an order reduction method for the multi-input multi-output (MIMO) linear dynamic system of high

order systems. The modified Cauer form for determination of denominator polynomial of the reduced model while Factor

division algorithm is used for calculation of the numerators coefficients. The advantages of proposed method are stable,

simplicity, efficient and computer oriented. The proposed method has been explained with an example taken from the literature.

The comparison step responses and Bode plots of the original and reduced system are shown in the Figure-1. A quantitative

comparison of reduced order model obtain by proposed method with the original system is shown in the Table-I from which we

can conclude that proposed method is comparable in quality.

References

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Vol. 325, No. 2 , pp. 235-245,1998.

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No. 6, pp. 999-1009,1988.

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2004.

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Comparision of Step response between g11 and r11

Time (sec)

Am

plit

ud

e

g11

r11

-60

-50

-40

-30

-20

-10

0

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

103

-90

-45

0

Ph

as

e (

de

g)

Comparision of frequency response between g11 and r11

Frequency (rad/sec)

g11

r11

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Time (sec)

Am

plit

ud

e

g12

r12

-40

-30

-20

-10

0

Ma

gn

itu

de

(d

B)

10-1

100

101

102

-90

-45

0

Ph

as

e (

de

g)


Frequency (rad/sec)

g12

r12

0 1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Time (sec)

Am

plit

ud

e

g21

r21

-80

-60

-40

-20

0

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

103

-90

-45

0

Ph

as

e (

de

g)


Frequency (rad/sec)

g21

r21

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Time (sec)

Am

plit

ud

eg22

r22

-40

-30

-20

-10

0

Ma

gn

itu

de

(d

B)

10-2

10-1

100

101

102

-135

-90

-45

0

Ph

as

e (

de

g)


Frequency (rad/sec)

g22

r22




7. Sastry G.V.K.R Raja Rao G. and Mallikarjuna Rao P., “Large scale interval system modeling using Routh approximants”, Electronic Letters, 36(8),

pp.768-769, 2000.

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pp.341-345, 1974.

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863.

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12. T.N. Lucas,” Factor division: A useful algorithm in model reduction”2761D, 22nd August 1982.

13. Vishwakarma and Prasad, “MIMO System Reduction Using Modified Pole Clustering and Genetic Algorithm”, Hindawi Pub. Corp. Mod. And Sim. in

Engineering,

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AUTHOR(S) PROFILE

Ms. Harshita Mittal, was born in Nakur (Saharanpur), U.P. India, in 1994. She received her B.Tech

degree with honor from Shobhit Institute of Technology Gangoh, Saharanpur U.P. India, in Electronics

and communication in 2014 with honor. Now currently she is pursuing her M.Tech degree from

Shobhit University Gangoh, Saharanpur U.P. India, She is always a brilliant student in her whole

academics. This paper is a part of her M.Tech degree work. Her research interest is „„Model Order

Reduction Of Multi Input and Multi output of Higher Order LTI System”.

Dr. J. S. Rana, was born in Wajirpur (Saharanpur) U.P, India, in 1976. He received B.Sc. degree from

Meerut University, India, in 1997, B.Tech. degree in Electronics & Instrumentation Engineering from

V.B.S Purvanchal University, Jaunpur India, in 2001 and M.Tech in Instrumentation Engineering from

the School of Instrumentation at Devi Ahilya University, Indore India, in 2006. He received his Ph.D.

degree from Shobhit University, Meerut in 2017. Presently, he is working as Assistant Professor in

Shobhit University, Gangoh (Saharanpur), India.

Dr. R. P. Agarwal, is a man of vast experience in Research, Development, Teaching and

Administration for more than years. He Born on March 4th, 1945 at Bina (MP, India), Dr. Agarwal

obtained his B.Sc. from Agra University, B.E.(Hons.) in Electronics & Telecommunication

Engineering from Jabalpur University (Govt. Engineering College) and M.E. in Electronics &

Telecommunication Engineering from Pune University (College of Engineering) in 1964,1967,1970

respectively.He joined Department of Electronics & Computer Engineering, UOR (Now IIT Roorkee)

as a Lecturer in 1970. In 1974, he was awarded Commonwealth Scholarship for pursuing higher

studies. He joined University of Newcastle upon Tyne, UK and completed PhD in 1977. He has a

wider experience of teaching, research, development, administration and of organization of

professional events of International and National levels of over 32 years. He has published over 80

technical papers in International and National journals at repute and conferences. He has guided 4

Ph.D. thesis and is currently guiding 3 more, apart from a number of B.Tech., AMIE, M.Phil. and

M.Tech. projects, seminars and dissertation. He is the author of 2 books and edited 5 proceedings of

National seminar.

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