View of REDUCTION OF LINEAR DYNAMIC SYSTEMS USING ROUTH HURWITZ ARRAY AND ERROR MINIMIZATION TECHNIQUE

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REDUCTION OF LINEAR DYNAMIC SYSTEMS USING ROUTH HURWITZ ARRAY AND ERROR MINIMIZATION TECHNIQUE

Dr. Ashok Kumar Mittal

Assistant Professor, Department of Mathematics,

Govt. P. G. College, Kotdwara (Garhwal) Uttarakhand 246 149 (India)

Abstract: A new combined method for model reduction of a continuous linear time invariant system is introduced, in which the denominator of the reduced model is obtained by using Routh-Hurwitz array method and then the numerator of the reduced model is obtained by minimizing the step response error function between the original system and the reduced model, while also satisfying steady state constraint. The proposed method is supported by a concrete example and the result is compared to other model reduction methods.

1 INTRODUCTION

It is widely recognized that the analysis and design of complex systems can easily be achieved by using low order models at a reduced cost. Various model reduction methods [1-4] are available in the literature for the purpose of reduction of high order linear time invariant continuous systems. Recently investigation of combined methods [5-8] for the model reduction has been subjected to increasing attention. In these combined methods, the denominator of the reduced model is obtained by some stability preservation method (Routh approximation, Routh Hurwitz array methods etc.) and then fixing the denominator of the reduced model its numerator is obtained by some other model reduction method.

Recently, some error minimization techniques [6-9] have also been suggested in the area of model reduction.

In the proposed method, firstly the denominator of the reduced model is obtained by Routh Hurwitz array method [4] and then the numerator coefficients are determined by minimizing the Integral Square Error (ISE) between the step responses of the original system and the reduced model while the steady state equality condition is satisfied [9]. The error function is converted into the frequency domain and then the minima operation is carried out in this domain.

The proposed method is illustrated by an example and the unit step responses of original and reduced systems have been compared using MATLAB Software Package.

1.1 Statement of Problem:

Consider an n^th order linear time invariant dynamic SISO (single input single output) system described in frequency domain by the transfer function











 





 



_n

u u u n

n n n

n n

s b

s a s

b s b s

b s b b

s a s

a s a a s

D s s N

G

0 1

1 2

2 1 0

1 1 2

2 1 0

...

. ...

...

) (

) ) (

(

(1)

where

a

_u

( u  0 , 1 , 2 ,... .., n  1 )

and

b

_u

( u  0 , 1 , 2 ,... , n  1 , n )

are scalar constants with

b

_n

 1

.

The corresponding

r

^th(

r  n

) order reduced model is of the form









 







 



_r

u u u r

r r r

r

s d

s c s

d s d s

d s d d

s c s

c s c c s

D s s N

R

0 1

1 2

2 1 0

1 1 2

2 1 0

...

) (

) ) (

(

(2)

where

c

_u

( u  0 , 1 , 2 ,... .., r  1 )

and

d

_u

( u  0 , 1 , 2 ,... , r  1 , r )

are scalar constants with

 1

d

r .

(2)

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In this paper, assuming the original system described by (1), our problem is to find a reduced order model of the form (2) such that the reduced model retains the important characteristics of the original system and approximates its response as closely as possible for the same type of inputs.

1.2 The Proposed Model Reduction Method:

Using Routh-Hurwitz array [4], we obtain the denominator of the reduced model by constructing denominator stability array

Denominator Stability Array

1 1 , 2

1 , 1







n n

b b

3 2 , 2

2 2 , 1





n n

b b

5 3 , 2

4 3 , 1





n n

b b

7 4 , 2

6 4 , 1





n n

b b

1 , 1 1 ,

1 , 6

1 , 5

1 , 4

1 , 3

...

 n n

b b b b b b

...

2 , 6

2 , 5

2 , 4

2 , 3

b b b b

...

3 , 4

3 , 3

b b

...

The above array is completed in the conventional way by computing the coefficients of succeeding rows by the algorithm

1 , 1

1 , 1 , 2 1 , 2 ,













i j i j i j i j

i

b

b b b

b

], 2 / ) 3 [(

1

1 3











i n j

n

i

(3)

where [x] stands for the integral part of the quantity x.

The

r

^th order reduced denominator

D

_r

(s )

is then given by

k s

b s b s b s b s

D

_r

( ) 

_n_₁__r_,₁ ^r



_n_₂__r_,₁ ^r^¹



_n_₁__r_,₂ ^r^²



_n_₂__r_,₂ ^r^³

 ... ... 

(4) where

 

 











odd is r if b

even is r if k b

r r n

1 2 ) 1 ( , 2

1 2 , 1

After normalizing (4),

D

_r

(s )

may be rewritten in the form

r r r

r

s d d s d s d s s

D ( ) 

₀



₁



₂ ²

 ... 

_₁ ^¹



(5) In terms of the poles of

G (s )

and

R (s )

, (1) and (2) may be rewritten as



^

 



 

 



¹

0 1

' '

1 2

' 1 1 '

0

... ...

) (

n

u u

u n

n

p s

k p

s k p

s k p s s k

G

(6)



^

 



 

 



¹

0 1

' '

1 2

' 1 1 '

0

... ...

) (

r

u u

u r

r

q s

l q

s l q

s l q s s l

R

(7)

where we have assumed that the original system has non repeating distinct simple poles

 p

₁

,  p

₂

,... ..,  p

_n on the real axis in the left half of complex plane while the reduced model has

 q

₁

,  q

₂

,... ..,  q

_r as its distinct simple poles in the complex plane.

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Now for the step responses

y (t )

and

y

_r

(t )

of the original system and that of reduced model, the step response error function is

    

0 2 2

)]

( ) ( [ ) ( )

( t y t y t y t dt

y

E

_r _r , which may be expressed as

dt t g t g

E  0  r

)]

2

( ) (

[

(8) where,

g ( t )  y (  )  y ( t )

and

g

_r

( t )  y

_r

(  )  y

_r

( t )

are the transient parts of the step responses of the original system and the reduced model respectively.

The matching condition of steady state values of the original system and the reduced model, i.e.,

y (  )  y

_r

(  )

leads to

r r n

n

q l q

l q l p k p

k p

k

^' ₁

2 ' 1 1 ' 0 '

1 2

' 1 1 '

0

  ... .. 

^

   ... .. 

^ (9)

Now,

 

n n

o

s p

k p

s k p s

k s

s G s t y

g L s

G   

 

 

 



^1

2 1 1

0

... .

) ( ) ) (

( )

(



^







¹

0 1

n

u u

u

p s

k

where

1 '





u u

u

p

k k

(10)

Similarly, in view of equations (7) and (9), we have

 

r r r

r

s q

l q

s l q s

l s

s R s t y

g L s

G   

 

 

 



^1

2 1 1

0

... .

) ( ) ) (

( )

(



^







¹

0 1

r

u u

u

q s

l

where

1 '





u u

u

q

l l

(11)

Using (10) and (11) in (9), we get

1 1

0 1 1

0

 k  ... .  k

_n_

 l  l  ... .  l

_r_

k

(12)

Now using Parseval's theorem [10], we see that the step response error function

E

can be expressed as

3 2

1

I 2I

I

E   

(13)

where,

 j 

j

G

o

s G

o

s ds

I j ( ) ( )

2 1

1



, (13a)

 j 

j

G

r

s G

r

s ds

I j ( ) ( )

2 1

2



, (13b) and

 j 

j

G

o

s G

r

s ds

I j ( ) ( )

2 1

3



. (13c)

Following Newton et al. [10] the integrals

I

₁,

I

₂ and

I

₃ may be evaluated and be put in summation form as



^





 





¹

0 1

0 1 1

1 n

u n

v u v

v u

p p

k

I k

= constant =

k

(say), (14)

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

^





 





¹

0 1

0 1 1

2 r

u r

v u v

v u

q q

l

I l

, (15)

and



^





 





¹

0 1

0 1 1

3 n

u r

v u v

v u

q p

l

I k

. (16)

It is to be noted here that

k

_u

, p

_u

& p

_v are known constants that is why

I

₁ equals to some constant

k

.

Thus in view of equations ((13)-(16)), the step response error function can be expressed as a function of the variables

l

₀

, l

₁

,... .., l

_r_₁ in the form



^





  







  

 

 



¹

0 1

0 1 1

1

0 1

0 1 1

2

n

u r

v u v

v u r

u r

v u v

v u

q p

l k q

q l k l

E

(17)

where

k

_u

, p

_u

, p

_v

, q

_u

& q

_v are already known constants. Now the minimization conditions for the step response error function

E

, namely,

 0



 l

u

E

u  0 , 1 , 2 ,... ...., r  2 .

(18) yield a set of

( r  1 )

linear equations

0 ...

2 2

...

2 2

...

0 ...

2 2

2 ...

2

0 ..

...

2 2

2 ...

2

1 1 1

2 1 1

1 0 1

1 1

2 2

1 1 1

1 0

2 1 2

1 0 2

1 1

2 2 2

1 1 2

0

1 1 1

2 1 1

1 0 1

1 1

1 2 2

1 1 1

0

 



 





 

 



 

 



 





 

 

 



 

 



 





 

 

 



 





 



r n

n r

r r

r r r

r r

r

n n r

r r

r

n n r

r r

r

q p

k q

p k q

q l q

l q

q l q

q l

q p

k q

p k q

q l q

l q q

l

q p

k q

p k q p

k q

q l q

l

(19) in

l

₀

, l

₁

, l

₂

,... .., l

_r_₁.

Now equations (19) together with (12) give the values of

l

₀

, l

₁

, l

₂

,... .., l

_r_₁, which in view of (11) yield the unknown constants

l

₀^'

, l

₁^'

, l

₂^'

,... .., l

_r^'_₁. Thus, finally we get the required

r

^thorder reduced model R(s) as given by (7) or (2).

Example:

As an illustration to the above method we consider the fourth order system described by the transfer function [11]

) (

) ( 4 4 3 6 2 2 1 ) 1

( D s

s N s

s s

G 

 

 

 

 

(20)

where

N ( s )  24  24 s  7 s

²

 s

³

and

D ( s )  24  50 s  35 s

²

 10 s

³

 s

⁴ with its real roots -1,-2,-3,-4.

Applying the algorithm (3) to construct the denominator stability array

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Denominator Stability Array 1

10

35 50

24

30 42 24

24

and the second order reduced denominator is obtained as:

8 . 0 4 . 1 )

(

²

2

s  s  s 

D

Now following the above method, the second order reduced model is given by

¹ ₂

4 . 1 8 . 0

625263 .

0 8 . ) 0

2

( s s

s s

R  

 

and further the third order reduced model is given by

₂ ₃

2 1

3 5 4 . 2

601565 .

0 224063 .

2 4 . ) 2

3

(

s s s

R   



 

2 COMPARISON OF METHODS

In order to test the effectiveness of the proposed method, the above given fourth order system [11] is also reduced by other existing methods of model reduction and for the purpose of comparison of variously reduced models, the cumulative error index

' J '

[12,13],

defined as

  







^N

i

i r

i

Y t

t Y J

0

)

2

( )

(

where

Y ( t

_i

)

and

Y

_r

( t

_i

)

are the outputs of the original system and the reduced model respectively at the

i

^thsampling instant and

N

is the number of sampling periods, is calculated for the different reduced models. The corresponding output and cumulative error

'

' J

at

t  5

sec. with a sampling period of 0.25 sec. for variously reduced models is given in table 1.

The MATLAB software package has been used to obtain the unit step responses of the original system and the reduced models as obtained by proposed method and by Routh- Hurwitz array method [4] and are shown in fig.1.

Table 1: Comparison of methods Method of

Reduction Reduced model Cumulative error

'

' J

at 5 sec.

Proposed

method ²

1

2

0 . 8 1 . 4

625263 .

0 8 . ) 0

( s s

s s

R  

 

0.29567430E-01

Proposed

method ² ³

2 1

3

2 . 4 5 3

601565 .

0 224063 .

2 4 . ) 2

( s s s

s s s

R   



 

0.69560000E-02

Krishnamurthy

& Seshadri [4] ²

2

0 . 8 1 . 4

685714 .

0 8 . ) 0

( s s

s s

R  

 

0.36141730E-01

Lamba, Gorez

& Bandyo –

padhyay [6] ²

3

2

2 3

799803 .

0 ) 2

( s s

s s

R  

 

0.91209550E-03

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Fig. 1: Comparison of unit step responses 3 CONCLUSIONS

A new method for the model reduction of single input single output linear time invariant systems has been presented. This method is computationally simple and efficient. It gives the stable reduced system if the original one is stable. As compared to the method [6], the method proposed in this paper is a generalized one and can be used to obtain a reduced model of any order

r

, which may be even or odd for a given system of any order

n

. This method has also been successfully tried to reduce the multivariable systems.

REFERENCES:

1. Genesio, R., and Milanese, M., “A note on the derivation and use of reduced order models”, I.E.E.E Trans.

Autom. Control, vol. 21, pp. 118-122, 1976.

2. Elrazaz, Z., and Sinha, N. K., “Review of some model reduction techniques”, Canadian Elect. Eng .J., vol.6, pp. 34-40, 1981.

3. Fortuna, L., Nunnari. G., and Gallo. A., Model order reduction techniques with applications in Electrical Engineering, Springer-Verlag, London, 1992.

4. Krishnamurthy, V., and Seshadri, V., “Model reduction using the Routh stability criterion”, IEEE Trans.

Autom. Control, vol. 23, pp. 729-731, 1978.

5. Pal, J., “Stable reduced-order pade approximants using the Routh-Hurwitz array”, Electron. Lett., vol. 15, pp. 225-226, 1979.

6. Lamba, S. S, Gorez, R., and Bandyopadhyay, B., “New reduction technique by step error minimization for multivariable systems”, Int. J. Syst. Sci., vol. 19, pp. 999-1009, 1988.

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8. Mittal, A. K., “Reduction of multivariable systems using Routh approximation and error minimization technique”, Int. J. Sci., Tech. & Mgt., vol. 10, pp. 37-46, Jan. 2017.

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10. Newton, G. C., Gould, L. A., and Kaiser, J. F., Analytical design of linear feedback controls, Wiley, London, 1964.

11. Shamash, Y., “Linear model reduction using Pade approximation to allow retention of dominant modes”, Int. J. Control, vol. 21, pp. 257-272, 1975.

12. Pal, J., “An algorithmic method for the simplification of linear dynamic scalar systems”, Int. J. Control, vol. 43, pp. 257-269, 1986.

13. Sinha, A. K., and Pal, J., “Simulation based reduced order modelling using a clustering technique”, Comp. Elect. Engng., vol. 16, pp. 159-169, 1990.