597

Abstract— The authors present a method for order reduction of linear dynamic systems with real distinct eigen values, using the combined advantages of Eigen spectrum analysis and modified Cauer continued fraction method. Pole centroid and system stiffness of both original and reduced order systems remain same in this method to determine the poles, whereas zeros are synthesized by modified Cauer continued fraction method. The proposed method guarantees stability of the reduced model if the original high order system is stable. The proposed method has also been extended for the order reduction of linear multivariable systems. Two numerical examples are solved to illustrate the superiority of the method over some existing ones including one example of multivariable system. Index Terms—Cauer continued fraction, Eigen spectrum, Integral square error, Multivariable systems, Order reduction, Stability.

I. INTRODUCTION very physical system can be translated into mathematical model. The mathematical procedure of system modeling

often leads to comprehensive description of a process in the form of high order differential equations which are difficult to use either for analysis or controller synthesis. It is hence useful, and sometimes necessary, to find the possibility of finding some equation of the same type but of lower order that may be considered to adequately reflect the dominant characteristics of the system under consideration. Some of the reasons for using reduced order models of high order linear systems could be : (i) To have a better understanding of the system. (ii) To reduce computational complexity. (iii) To reduce hardware complexity. (iv) To make feasible designs.

A large number of methods are available in the literature for order-reduction of linear continuous systems in time domain as well as in frequency domain [1-6]. Further, several methods have also been suggested by combining the features of two different methods [7-11]. Some extensions of single input single output (SISO) methods to reduce multi-input multi- Department of Electronics Engineering, Rajasthan Technical University, Kota – 324 022 (India) E-mail: * girish_parmar2002@yahoo.com: # mb_eck@yahoo.com.

output (MIMO) systems have also been carried out in [12-15]. Each of these methods has both advantages and disadvantages when tried on a particular system. In spite of several methods available, no approach always gives the best results for all systems.

The problem of overcoming the instability of reduced order models derived through continued fraction technique has been investigated in [16-18]. In [16], the denominator of the reduced order model is formed by Routh array, while in [17, 18] the stability equation method is used for the same purpose. Then the numerator dynamics is chosen to fit a given number of continued fraction quotients.

The present attempt is towards evolving a method for order reduction of linear dynamic systems in which both the pole centroid and system stiffness of the original and reduced order systems are kept exactly same to obtain the reduced order system poles while the zeros are synthesized by modified Cauer continued fraction method. The method has also been extended for order reduction of linear multivariable systems. The method is developed only for the systems with real distinct poles and is illustrated with the help of two numerical examples.

The paper is organized as follows. In the second section, the method is described in detail using the mathematical aspects of Eigen spectrum analysis [10] and modified Cauer continued fraction method [18]. In the third section, method has been extended for multivariable systems. Two numerical examples are presented in fourth section and finally, the conclusions are made in the fifth section.

II. DESCRIPTION OF THE METHOD Let, the transfer function of high order system (HOS) of

order 'n' is: 2 1

11 12 13 1,2 1

11 12 13 1,

......( )

......

nn

n n nn

b b s b s b sG s

a a s a s a s s

−

−

+ + + +=

+ + + + + (1)

or,

2 1

11 12 13 1,

1 2

......( )

( ) ( )...... ( )

nn

nn

b b s b s b sG s

s s sλ λ λ

−+ + + +=

+ + + (2)

where, - λ1 < - λ2 < …….< - λn are poles of the HOS. Let, the transfer function of low order system (LOS) of

order 'r' to be synthesized is:

Girish Parmar* and Manisha Bhandari #

REDUCED ORDER MODELLING OF LINEAR DYNAMIC SYSTEMS USING EIGEN SPECTRUM ANALYSIS AND

MODIFIED CAUER CONTINUED FRACTION

E

XXXII NATIONAL SYSTEMS CONFERENCE, NSC 2008, December 17-19, 2008

598

111 12 1,

111 12 1,

......( )

......

rr

r r rr

q q s q sG s

p p s p s s

−

−

+ + +=

+ + + + (3)

or,

1

11 12 1,' ' '

1 2

......( )

( ) ( ) ......( )

rr

rr

q q s q sG s

s s sλ λ λ

−+ + +=

+ + + (4)

where, - '1λ < - '

2λ < …….< - 'rλ are the poles of the LOS then

steps are as under : Step-1: Fixing of the eigen spectrum zone (ESZ) of the HOS as shown in Fig. 1: If poles -λi (i = 1,…,n) are located at – ( Re iλ Im iλ± ) (i = 1,…,p) within the ESZ, then the two lines passing through the nearest (Reλ1) and farthest (Reλp) real poles when cut by two lines passing through the farthest imaginary pole pairs ( Im (max)± ) form the ESZ. Step-2: Quantification of pole centroid and stiffness of HOS : Pole centroid is defined as the mean of real parts of the poles and is expressed as :

1Re

p

ii

m p

λλ

∆==∑

(5)

System stiffness is defined as the ratio of the nearest to the farthest pole of a system in terms of real parts only and is put as :

1ReRes

p

λλλ

∆

= (6)

Step-3: Determination of eigen spectral points of LOS : If '

mλ and 'sλ are pole centroid and system stiffness of

LOS such that 'mλ = λm and '

sλ = λs then following situation arise :

1

'

Re ''Re 's s

p

λλ λ

λ= =

1 2 'Re ' Re ' ....... Re ''

'p

m mpλ λ λ

λ λ+ + +

= = (7)

where, 'iλ (i = 1,…,r) are the poles of LOS located at

– ( 'Re iλ 'Im iλ± ) i = 1,…, p' . Now if,

' 1Re ' Re '' 1

p Mp

λ λ−=

− (8)

i.e., Reλ'1 + M = Reλ'2, Reλ'2 + M = Reλ'3 and so on till Reλ'p'-1 + M = Reλ'p' then (7) can be put as :

1 ' 1 2

' 2

Re ' Re ' (Re ' ) (Re ' )

... (Re ' )'

p

pm

M M

Mp

λ λ λ λ

λλ −

+ + + + + +

+ +=

Fig. 1. Eigen spectrum zones and points of system.

or, λmp' = 1 ' 1Re ' Re ' (Re ' )p Mλ λ λ+ + + 1(Re ' 2 ) ........+ + +Mλ 1..... (Re ' ( ' 2) )p Mλ+ + − = 1 'Re ' Re 'pλ λ+ + Reλ'1 (p'-2) + (M+2M+……+ (p' - 2)M)

or, N = Reλ'1 (p'-1) + Reλ'p' + QM (9) where, N = λmp' and QM = M +2M +…+ (p' - 2) M. By putting Reλ'1 = λs Reλ'p', (8) and (9) will be as under : Reλ'p' - λs Reλ'p' = M (p' - 1) (10)

Reλp

+ Im (max)

Reλ1

jw

σ

-Im (max)

(a) eigen spectrum zone (ESZ) of HOS

Reλ'p'

+ Im (mean)

jw

σ

-Im (mean)

(b) eigen spectrum zone (ESZ) of LOS

M M

X X X

M M

XReλ'p'

Reλ'3 Reλ'2

Reλ'1

(c) eigen spectral points (ESP) of LOS

jw

σ

X X

X

X X XReλ'3 Reλ'2 Reλ'1

599

λs Reλ'p' (p' - 1) + Reλ'p' + QM = N (11) Equations (10) and (11) can be put as : Reλ'p' (1 - λs) + M (1 - p') = 0

Reλ'p' [λs (p' - 1) + 1] + MQ = N or,

'Re '( ' 1) 1(1 ) (1 ') 0

ps

s

p Q Np M

λλλ− +

= − − (12)

Equation (12) can be solved for Reλ'p' and M enabling thereby to locate the eigen spectral points (ESP) as shown in Fig. 1.

Therefore, the denominator polynomial in (4) is now known, which is given by :

( )rD s = 1

11 12 1,...... r rrp p s p s s−+ + + + (13)

Step-4: By applying the algorithm [19], the first ‘r’ quotients of modified Cauer form of continued fraction, viz. h1, H1, h2, H2,….are evaluated. Step-5: Now a modified Routh array for r = 6 is built as given below :

(14)

where, the first two rows are formed from the denominator and numerator coefficients of Gr(s) in (3) and the remaining entries in the array are obtained by the algorithm given in [16]. The sequence of computation is indicated by the arrows.

III. EXTENSION TO MULTIVARIABLE SYSTEMS Let, the transfer matrix of the HOS of order 'n' having ‘p’

inputs and ‘m’ outputs is:

11 12 13 1

21 22 23 2

1 2 3

( ) ( ) ( ) ... ( )( ) ( ) ( ) ... ( )1[ ( )]

...( )( ) ( ) ( ) ... ( )

p

p

n

m m m mp

a s a s a s a sa s a s a s a s

G sD s

a s a s a s a s

=

M M M M

or, [ ( )] [ ( ) ], 1, 2, ......, ; 1, 2, ......,ijG s g s i m j p= = = (15) is a m p× transfer matrix. The general form of ( )ijg s of [ ( )]G s in (15) is taken as :

( )( )

( )ij

ijn

a sg s

D s=

2 1

11 12 13 1,2 1

11 12 13 1,

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

nnn n

n

b b s b s b sa a s a s a s s

−

−

+ + + +=

+ + + + + (16)

or,

2 1

11 12 13 1,

1 2

......( )

( ) ( ) ......( )

nn

ijn

b b s b s b sg s

s s sλ λ λ

−+ + + +=

+ + + (17)

where, 1 2 ...... nλ λ λ− <− < < − are poles of the HOS. Let, the transfer matrix of the LOS of order 'r' having ‘p’

inputs and ‘m’ outputs to be synthesized is :

11 12 13 1

21 22 23 2

1 2 3

( ) ( ) ( ) ... ( )( ) ( ) ( ) ... ( )1[ ( )]

...( )( ) ( ) ( ) ... ( )

p

p

r

m m m mp

b s b s b s b sb s b s b s b s

R sD s

b s b s b s b s

=

M M M M

or, [ ( )] [ ( ) ], 1, 2, ......, ; 1, 2, ......,ijR s r s i m j p= = = (18) is a m p× transfer matrix. The general form of ( )ijr s of [ ( )]R s in (18) is taken as :

( )

( )( )

ijij

r

b sr s

D s=

1

11 12 1,' ' '

1 2

......( ) ( ) ......( )

rr

r

q q s q ss s sλ λ λ

−+ + +=

+ + + (19)

where, ' ' '1 2 ...... rλ λ λ− < − < < − are the poles of the LOS.

The proposed method consists of pole synthesis of the LOS by Eigen spectrum analysis where zeros are synthesized by using the modified Cauer continued fraction method (MCF). Basically; the method starts with fixation of the denominator of the LOS by Eigen spectrum analysis followed by the determination of coefficients of the numerator polynomials of each element of the LOS transfer matrix by matching the quotients of modified Cauer continued fraction method.

IV. NUMERICAL EXAMPLES

Two numerical examples are chosen from the literature for the comparison of the LOS with the original HOS. The proposed method is described in detail for one example while only the result of the other example is given.

An error index ISE [11] known as integral square error in between the transient parts of original and reduced order systems is calculated to measure the goodness/quality of the LOS (i.e. the smaller the ISE, the closer is ( )rG s to ( )nG s ), which is given by :

h1

p11 p12 p13 p14 p15 p16 1

q11 q12 q13 q14 q15 q16

h2

p21 p22 . p24 p25 1

q21 . . q24 q25

h3

p31 . . p34 1

. . . q34

H1

1. .

H2

H3

600

2

0ISE = [ ( ) ( )]ry t y t dt

∞−∫ (20)

where, ( )y t and ( )ry t are the unit step responses of original and reduced order systems, respectively.

Also, the impulse response energy (IRE) [8] is calculated for original and various reduced order models, which is given by :

2

0IRE = ( )g t dt

∞

∫ (21)

where, ( )g t is the impulse response of the system. Example-1. Consider a 4th order system taken from Mukherjee & Mishra [20] and Mittal et al. [21] :

3 2

4 4 3 2

7 24 24( )10 35 50 24s s sG s

s s s s+ + +

=+ + + +

(22)

The poles of the above system are all real and given by : λ1 = -1, λ2 = -2, λ3 = -3, λ4 = -4.

The system is having IRE of 4.66251 x 10-1. If a 2nd order model 2 ( )G s is to be synthesized using this algorithm, steps to be followed are as under : Step-1: Fixing of ESZ of HOS :

Since all poles are real, it will be a line joining the nearest and farthest poles. Step-2: Quantification of pole centroid and stiffness of HOS :

4

1

1

4

2.54

0.25

ii

m

s

λλ

λλ

λ

== =

= =

∑

Step-3: Determination of eigen spectral points of LOS : Equation (12) can be formed as under :

'Re '1.25 0 50.75 -1 0

=

p

M

λ (23)

where, the values of sλ , Q, p' and N are to be put as 0.25, 0, 2, 5 respectively. Solution of (23) gives the location of the farthest pole '

'Re pλ and M. where, M = (Farthest pole-

Nearest pole) / (p'-1), and since p' = 2; Reλ'p' = 4; M = 3, ESPs of LOS are its two poles as λ'1 = 1 and λ'2 = 4. Therefore, 2

2 ( ) 5 4D s s s= + + . Step-4: Evaluate the MCF quotients by forming the array : (24)

Step-5 : Construct the modified Routh array as in (14) : (25)

Therefore, the reduced 2nd order model is given by :

2 2

4( )5 4

sG ss s

+=

+ + (26)

with an ISE of 3.57134 x 10-3 . The unit step and frequency responses of original and reduced order models are shown in Fig. 2 (a)-(b) and a comparison of the proposed method with some other existing methods, for a 2nd order reduced model, ( 2 ( )G s ) is given in Table I. It can be seen in Table I that, the proposed method gives low value of the ISE in comparison to the other existing methods. Further, the value of IRE for the proposed method is close to that of original 4th order system, which is 4.66251 x 10-1. Also, the unit step and frequency responses of original and reduced order models are comparable as shown in Fig. 2.

TABLE I COMPARISON OF REDUCED ORDER MODELS

Method of order reduction

Reduced models;

2 ( )G s ISE IRE

Proposed Method 2

45 4

ss s

++ +

3.57134 x 10-3 5.31139 x 10-1

Pal [9] 2

16.0008 2430 42 24

ss s

++ +

1.1688 x 10-2 3.97301 x 10-1

Chidambara [22]

2

2

23 2

ss s

− ++ +

220.2379 x 10-3 3.07899

Davison [23] 2

2

23 2

ss s

− ++ +

220.2379 x 10-3 3.07899

Gutman et al.[24] 2

2[48 144]70 300 288

ss s

++ +

4.5593 x 10-2 7.54249 x 10-1

Krishnamurthy and Seshadri [25]

2

20.5714 2430 42 24

ss s

++ +

9.5891 x 10-3 4.69766 x 10-1

Prasad and Pal [26] 2

34.2465239.8082 34.2465

ss s

++ +

1.53427 6.14777 x 10-2

Shieh and Wei [27] 2

2.30145.7946 2.3014

ss s

++ +

142.5607 x 10-3 3.04667 x 10-1

h1 = 1

24 50 35 10 1

24 24 7 1

h2 = -13

26 28 9 1

-2 .

. .

H1 = 1

.

.

h1 = 1

4 5 1

4 1

1H1 = 1

601

(a)

(b)

Fig. 2. (a) Step responses (b) Frequency responses of original and reduced order models.

Example-2. Consider a 6th order two input two output system [15, 28] described by the transfer matrix :

2( 5) ( 4)

( 1) ( 10) ( 2) ( 5)[ ( )]

( 10) ( 6)( 1) ( 20) ( 2) ( 3)

s ss s s s

G ss s

s s s s

+ + + + + + =

+ + + + + +

= 11 12

21 226

( ) ( )1( ) ( )( )

a s a sa s a sD s

(27)

where, the common denominator )s(D6 is given by :

6 ( ) ( 1)( 2)( 3)( 5)( 10)( 20)D s s s s s s s= + + + + + + = 2 36000 13100 10060 3491s s s+ + + 4 5 6571 41s s s+ + +

and 2 3 4 5

11( ) 6000 7000 3610 762 70 2a s s s s s s= + + + + + 2 3 4 5

12 ( ) 2400 4160 2182 459 38a s s s s s s= + + + + + 2 3 4 5

21( ) 3000 3700 1650 331 30a s s s s s s= + + + + + 2 3 4 5

22 ( ) 6000 9100 3660 601 42a s s s s s s= + + + + + The proposed method is applied to the above multivariable

system and the reduced order models ( )ijr s of the LOS

[ ( )]R s are obtained. The general form of second order model is taken as :

11 12

21 222

( ) ( )1[ ( )]( ) ( )( )

b s b sR s

b s b sD s

=

(28)

where, 2

2 ( ) 13.6666 8.4707D s s s= + + and 11( ) 2 8.4707b s s= + , 12 ( ) 3.3883b s s= + 21( ) 4.2354b s s= + , 22 ( ) 8.4707b s s= +

A comparison of the proposed method with Prasad and Pal

[29] (for 1 22; 0r r= = ) is given in Table II, in terms of ISE for each element of the transfer function matrix, which is given by :

ISE = 2

0[ ( ) ( )] , 1, 2; 1, 2ij ijg t r t dt i j

∞− = =∫ (29)

where, )t(g ij and )t(rij are the unit step responses of original

and reduced order systems.

TABLE II COMPARISON OF REDUCED ORDER MODELS

ijr ISE

(By Proposed method)

ISE

(By Prasad and Pal [29])

11r 0.038713 0.135505

12r 0.028153 0.002446

21r 0.007419 0.040013

22r 0.144096 0.067897

602

V. CONCLUSIONS A method which combines the advantages of the Eigen

spectrum analysis and the modified Cauer continued fraction method has been presented, to derive stable reduced order models for linear dynamic systems.

In this method the poles of the LOS are determined by exact matching of pole centroid and system stiffness of both the original and reduced order systems while the zeros are synthesized by using the modified Cauer continued fraction technique. The method has also been extended for order reduction of linear multivariable systems. The proposed method has been tried on two numerical examples having real poles only. The method is simple, rugged and takes little computational time. It is being extended for systems with imaginary poles. The comparison in between the proposed and the other well known existing order reduction techniques is also shown as given in Tables I and II, from which it is clear that the proposed method compares well with other techniques of model order reduction. The method preserves model stability and avoids any error in between the initial or final values of the responses of original and reduced order models. The numerical examples show that the proposed method can be tried on many systems as it preserves both time domain and frequency domain characteristics of the original system.

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