View of REDUCTION OF MULTIVARIABLE DISCRETE TIME SYSTEMS USING ROUTH APPROXIMATION AND STEP RESPONSE MATCHING TECHNIQUE

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REDUCTION OF MULTIVARIABLE DISCRETE TIME SYSTEMS USING ROUTH APPROXIMATION AND STEP RESPONSE MATCHING TECHNIQUE

Dr. Ashok Kumar Mittal

Associate Professor, Department of Mathematics,

Govt. Degree College, Chaubattakhal (Pauri-Garhwal) Uttarakhand 246 162 (India) Abstract - A new mixed method is presented for the reduction of multivariable discrete linear time invariant high order systems. This method combines the advantages of Routh approximation and error minimization technique. This paper is the extension of the work [11], in which the denominator polynomial of the reduced model is obtained by Routh approximation method and the numerator terms of the lower reduced model are obtained by minimizing the Generalized Integral Square Error (GISE) between the step responses of the original system and the reduced model using steady state equality condition. The error function is converted into the frequency domain and the minima operation is carried out in this domain to minimize the error function. The proposed method is illustrated by a concrete example.

1 INTRODUCTION

In the analysis and synthesis of a high order multivariable system, it is often necessary to obtain a low order model so that it may be used for an analogue or digital simulation of the system. A large number of publications [1-8] on model order reduction have been appeared.

Among these only a very few [9-11] are available to reduce discrete time systems. The author [4,8] introduced a mixed method to obtain reduced order models for continuous time systems using the advantages of Routh approximation method and step response error minimization technique. Recently in the paper [11], the method is extended to reduce SISO discrete time systems. The proposed paper is the extension of the work [11] for MIMO discrete time systems. 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:

Let an

n

^th order linear time-invariant multivariable (q input - p output) discrete time system described in frequency domain by

z

- transfer matrix

 















_n

i i i n

i i i uv pxq

z b

z A z

g z G

0

* 1

0

*

) ( )

(

(1)

where

^A

i^*

^   ^a

uv^i* _pxq

^{( 1  u  p ; 1  v  q ; 0  i  n  1 )}

are the known constant matrices of order

pxq

and

b

_i^*

( 0  i  n )

are constants.

Alternatively, let an

n

^th order MIMO (

q

input -

p

output) discrete time system be described in time-domain by the following state-space form

) ( )

(

) ( )

( )

1 (

k x C k y

k u B k x A k

x

d

d d









(2)

where

x

is

n

dimensional state vector,

u

is

q

dimensional control vector and

y

is

p

dimensional output vector with

p  n , q  n

. Also,

A

^d is

nxn

system matrix,

B

^d is

nxq

input matrix and

C

^d is

pxn

output matrix.

Irrespective of the descriptions (1) or (2) in frequency or time domain, the problem is to find an

r

^th(

r  n

) order reduced model in the following

z

- transfer matrix form

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 















_r

i i i r

i i i uv pxq

z d

z C z

r z R

0

* 1

0

*

) ( )

(

(3)

where

C

_i^*

   c

_uv^i* _pxq

^{( 1}  u  p ^{; 1}  v  q ^{; 0}  i  r  ^{1 )}

are the unknown constant matrices of the same order

pxq

and

d

_i^*

( 0  i  r )

are unknown constants.

If the original system is given in state-space form (2), then it may be represented in the form (1) by using Faddeeva-Leverrier algorithm [12,13] as

d d

d

zI A B

C z

G ( )  (  )

^1 (4) Now for the reduction of high order multivariable systems described by (1), the following steps are performed.

2 MODEL REDUCTION METHOD:

The Model Reduction Method consists of following steps : Step 1 : Transformation of G(z) into w -domain :

Firstly transform G(z) into w-domain by using bilinear transformation

z  ( 1  w ) ( 1  w )

as

( ) ( )

^*

( )

1

1

G w

z G w H

w

z w

 





 



(5)

Where



_

w

w

H ( )



(6)

and















_n

i i i n

i i i

w b

w A w

D w w N

G

0 1

* 0

) (

) ) (

(

(7)

where

A

_i

   a

_uvⁱ _pxq

^{( 1}  u  p ^{; 1}  v  q ^{; 0}  i  n  ^{1 )}

are the matrices of order

pxq

and

) 0

( i n

b

_i

 

are constants.

Step 2 : Order reduction of

G

^*

( w )

in

w

- domain

Now the system

G

^*

( w )

is reduced to an

r

^th order model















_r

i i i r

i i i

r r

w d

w C w

D w w N

R

0 1

0

*

*

) (

) ) (

(

in

w

-

domain using the method [4,8,11] in the following way –

2.1 Determination of Denominator

D

_r

(w )

using Routh Approximation Method Let

1 1 5

5 3 3 1

4 4 2 2 0

. ...

. ...

1 ) 1 (

 















 



n n

n n n

w b w

b w b w b

w b w

b w b w b

Q

,

n

is even

(8)

w w

w

n

1 . 1

1 . 1 1 1

1

2 1

















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Now, truncating (8) after the first

r

quotients, i.e.,

(9)

D

_r

(w )

is obtained by the denominator of the rational function obtained from

inverting the continued fraction (9).

Now, for alpha parameters, the alpha table [1,11] is constructed as Alpha Table

1 1 0

0 0 0

b b

b b





3 1 2

2 0 2

b b

b b





5 1 4

4 0 4

b b

b b





...

...

...

...

5 0 4 0 5

4 0 3 0 4

3 0 2 0 3

2 0 1 0 2

1 0 0 0 1

b b

b b

b b

b b

b b





















....

...

...

....

...

...

4 2 4 3 2 5 0

3 2 3 2 2 4 0

2 2 2 1 2 3 0

1 2 1 0 2 2 0

b b

b

b b

b

b b

b

b b

b

























....

...

...

....

...

...

2 4 2 1 4 3 2

1 4 1 0 4 2 2

b b

b

b b

b













....

...

...

....

...

...

The



_i

( 1  i  n )

parameters are the ratio of consecutive entries in the first column of alpha table. These are related by a set of recursive relations [1,11] :

...

...

...

...

...

...

...

...

1 ) (

) (

1 )

(

1 )

(

3 1 2 3 2 3 3 2 1 3

2 2 2 1 2

1 1





















w w

w w

D

w w

w D

w w

D























In general, for

r  1 , 2 , 3 ,... ...

) ( )

( )

( w wD

₁

w D

₂

w

D

_r

 

_{r r}



_r (10)

with

D

_1

( w )  D

₀

( w )  1

Thus, the obtained reduced denominator

D

_r

(w )

may be rewritten as







^r

i i i

r

w d w

D

0

)

(

(11)

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1) Determination of Numerator

N

_r^*

( w )

using Error Minimization Technique

The matching condition of steady state values of the reduced model

R

^*

( w )

and the system

)

*

( w

G

, i.e.,

Y

_r

(  )  Y (  )

leads to

0 0 0 0

0

K

b A d

C  

(say)

₀

0 0 0

0

uv uv

uv

k

b a d

c  



(12) where

k

_uv0 is invariant.

Now let

Y ( t )  [ y

_uv

( t )]

_pxq and

Y

_r

( t )  [ y

_uv^r

( t )]

_pxq be the step responses of the original system and the reduced model respectively, the Generalized Integral Square Error (GISE) function

E

between the step responses of the original system and the reduced model is given by:

dt t Y t Y t Y t Y trace

E ^ 

0^

[ ( ) ^

_r

( )][ ( ) ^

_r

( )]

^T

^ 

0^

trace [ H ( t ) ^ H

_r

( t )][ H ( t ) ^ H

_r

( t )]

^T

dt

h t h t dt

p

u q

v

r uv o



uv

 







1 1 0

)]

2

( ) (

[

(13) where,

H ( t )  [ h

_uv^o

( t )]

_pxq

 Y (  )  Y ( t )

and

H

_r

( t )  [ h

_uv^r

( t )]

_pxq

 Y

_r

(  )  Y

_r

( t )

are the transient parts of the step responses of the original system

G

^*

( w )

and the reduced model

R

^*

( w )

respectively.

Now using Parseval's theorem [14], the step response error function

E

can be expressed as:

3 2

1

I 2I

I

E   

(14) where,

 

 





  







^p

u q

v j

j

o uv o

uv p

u q

v

uv

H w H w dw

I j I

1 1

1 1

1

1

( ) ( )

2 1



(14a)





 





  







^p

u q

v j

j

r uv r

uv p

u q

v

uv

H w H w dw

I j I

1 1

1 1

2

2

( ) ( )

2 1



, (14b) and





 





  







^p

u q

v j

j

r uv o

uv p

u q

v

uv

H w H w dw

I j I

1 1

1 1

3

3

( ) ( )

2 1



. (14c) with

 

n n

i i i n

i i i o

uv

w w b

w P w

w G w t Y

H L w H





 



 









 1

0 1

0

*

( ) )

) ( ( )

(

(15)

where

n n

i i i

b K P

n i A

b K P

0 1

1 1

0

0 2

















 (16)

and

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 

r r

i i i r

i i i r

r r

uv

w w d

w Q w

w R w t Y

H L w H





 



 









 1

0 1

0

*

( ) )

) ( ( )

(

(17)

where

r r

i i i

d K Q

r i C

d K Q

0 1

1 1

0

0 2

















 (18) Following Newton et al. [14], we observe that each integral

I

_uv1 evaluates in terms of the coefficients of the original system and hence is a constant, i.e.,

I

_uv1= constant =

k

_uv(say),

So

I

₁= constant =

k

(say), (19) The integral

I

_uv2 equals to

) 1 (

2

ˆ



_uvr

uv

b

I

, (20) where,

ˆ

_{( 1)}

 r

b

uv is obtained separately for the two cases, namely

(i )

when

r

is even and

)

(ii r

is odd, from the matrix equations :

YB

_uv

 Q

_uv, (21) where

B

_uv and

P

_uv are the following unknown

r

rowed column matrices

 

 

 

 

 

 

 







 

 

 

 

 

 

 











 









) 1 (

) 2 (

2 1 0

r uv

r uv

uv uv uv

uv

b b b b b

B











&

 

 

 

 

 

 

 





 

 

 

 

 

 

 



































2 ) 1 ( 1

2 ) 2 ( 2 )

1 ( ) 3 ( 3

2 2 3 1 4

0

2 1 2 0

2 0

) 1 (

) 1 ( )

1 ( 2

2 2

2

r uv r

r uv r r

uv r uv r

uv uv uv uv

uv

uv uv uv

uv

uv

q

q q

q

q q q q

q

q q q

q

Q











and the known

r

rowed square matrix

Y

for the two cases is given by

 

 

 

 

 

 

 





 

 

 

 

 

 

 































































1 3 2

1

3 4

5 6

1

1 2

3 4

3 2

1

0 1

2 5

4 3

2

1 2

3 4

0 1

2 0

2 2

0 0

0 0

0 0

2 2

2 2

0 0

0 0

2 2

2 2

2 2

0 0

2 2

2 2

2 2

2 2

0 2

2 2

2 2

2 2

0 0

0 0

2 2

2 2

0 0

0 0

0 2

2 2

0 0

0 0

0 0

0 2

r r

r r

r r

r r

r r

r r

r r

r r

d d

d d

d d

d d

d d

d d

d d

d d

d d

d d

d d

d d

d d

d

d d

d d

d d

d d

Y









































































Vol. 04, Issue 04,April 2019 Available Online: www.ajeee.co.in/index.php/AJEEE

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(for even

r

)

and

 

 

 

 

 

 

 





 

 

 

 

 

 

 





































































1 3 2

1

2 3

4 5 2

1

0 1

2 3 4

3 2

1

0 1 6

5 4

3

1 2

3 4

0 1

2 0

2 2 0

0 0

0 0

0

2 2

2 2 0

0 0

0

2 2

2 2 2

2 2 0

2 2

2 2 2

2 2

2

0 0

2 2 2

2 2

2

0 0

0 0

2 2

2 2

0 0

0 0

0 2

2 2

0 0

0 0

0 0

0 2

r r

r r

r r r

r r

r r

r r

r r

r r

d d

d d

d d

d d

d d d

d d

d d

d d d

d d

d

d d d

d d

d

d d

d d

d d

d d

Y









































































(for odd

r

)

From the equation (21), ( 1)

 r

b

uv can explicitly be obtained as

uv r r r

uv

y y y y Q

b

(1)

[

_{1 2} ₁

].



  

(22) where,

y

₁

, y

₂

,..., y

_r1

, y

_r are the elements of the

r

^th row of

Y

^1.

The third integral

I

_uv3 reduces to



( 1) ( 1)



3

ˆ ˆ

2 1









_{uvn uvr}

uv

d e

I

(23) where,

ˆ

_{( 1)}

& ˆ

_{( 1)}



 uvr n

uv

e

d

are obtained from the matrix equation

ZC

_uv

 R

_uv (24)

Here,

Z

is a known square matrix of order

( n  r )

and

C

_uv

& R

_uv are the

( n  r )

rowed unknown column matrices and are given by:























































































































 

 



 

 

 

 

















































 

 









 



 







n r r

r

n r n r r

r r r

n r n

r r

r r r r r

r n r r n r n

n

r n r r n r n

n n

r n r r

n r n

n n

r n r r n r n

n n

r n r r n r n

n n

b d

b b

d d

b b

d d

d

b b

b b d

d d

b b

b b b d d

d

b b

b b b d d

d

b b

b b b d

d

b b

b b b d

b b b d

d d

b b d

d

b d

Z

1 1 1 2

1 1

2 1 1

2 2

2 1 1

2 1 3 2 1

2 3

4

1 1 2 2 2

1 1

2 3

1 1

2 3

2 1 0

1 2

1 1 2

4 3 2 0

1

2 1 1 2 5

4 3 0

0 1 2 0

1 2

0 1 0

1

0 0

) 1 ( 0

0 0 0 )

1 ( 0 0

0 0 0

) 1 ( )

1 ( 0

0 0 )

1 ( ) 1 ( 0 0

0 0

) 1 ( )

1 ( 0 0 0 )

1 ( ) 1 ( ) 1 ( 0 0 0

) 1 ( )

1 ( 0

0 0 0

) 1 ( )

1 ( 0

0 0

) 1 ( )

( 0

0 0

) 1 ( )

1 ( 0

0 0 0

) 1 ( )

1 ( 0

0 0

0 0

0 0

0 0

0

0 0

0 0

0 0

0

0 0

0 0 0

0 0

0 0

































































































Vol. 04, Issue 04,April 2019 Available Online: www.ajeee.co.in/index.php/AJEEE

7











































































) 1 (

) 2 (

) 3 (

1 0 ) 1 (

) 2 (

) 3 (

2 1 0

ˆ ˆ ˆ ˆ ˆ

r uv

r uv

r uv

uv uv n uv

n uv

n uv

uv uv uv

uv

e e e e e d d d d d d

C





and

 

 

 

 

 

 

 

 

 





 

 

 

 

 

 

 

 

 







































0 ) 1 (

) 1 ( )

1 (

) 1 ( ) 1 ( 1

) 1 ( ) 2 ( 1 )

2 ( ) 1 ( 2

2 0 1 1 0 2

1 0 0 1

0 0

r uv n uv r

r uv n uv r r

uv n uv r

uv uv uv uv uv uv

uv uv uv uv

uv uv

uv

q p

q p q

p

q p q p q p

q p q p

q p

R















In view of the equation (24),

ˆ

_{( 1)}

& ˆ

_{( 1)}



 uvr n

uv

e

d

can explicitly be written as

uv r n r n r

uv

uv r n r n n

uv

R v v

v v e

R u u

u u d

].

ˆ [

].

[

1 2

1 ) 1 (

1 2

) 1 1 (













 















(25)

where

u

_i &

v

_i

( 1  i  n  r )

are the elements of the

n

^th and

( n  r )

^th rows of the matrix

1

Z

respectively.

Thus repeating the process of evaluating the integrals (equations (20)-(25)) for each particular

u

and

v

, the GISE function is expressed as

) ,...,

,

( _{0 1 ( 2)}





k q_uv q_uv q_uvr

E (26) where  is a function of the variables

q

_uv0

, q

_uv1

,... .., q

_uv(r2)

( 1  u  p ; 1  v  q )

. Now the minimization condition for the step response error

E

, namely,

 0



 q

uvi

E

1  u  p ; 1  v  q ; 0  i  r  2

(27)

yields a set of

pq ( r  1 )

linear equations in

q

_uv0

, q

_uv1

,... .., q

_uv(r2) and solving these equations we get the values of

q

_uv0

, q

_uv1

,... .., q

_uv(r2),i.e.,

Q

₀

, Q

₁

,... .., Q

_r2uniquely.

Thus in view of equations (12) and (18), unknown matrices

C

₀

, C

₁

,... .., C

_r1 are evaluated and and thus the numerator

N

_r^*

( w )

of

r

^th order reduced model

R

^*

( w )

is obtained.

Consequently the

r

^th order model

H

^*

( w )

in

w

- domain is given by

) (

) ) (

( )

(

*

*

*

w D

w w N

R w

H

r



r





  

(28)

Step 3: Retransformation of

H

^*

( w )

into

z

- domain

Now

H

^*

( w )

is retransformed back into

z

- domain by using inverse bilinear transformation,

w  ( z  1 ) ( z  1 )

to yield

) ( )

( )

(

^#

1 1

† *

z R w

H z R

z

w z

 





 



(29) where

  R

^†

( z )

_z (30)

Vol. 04, Issue 04,April 2019 Available Online: www.ajeee.co.in/index.php/AJEEE

8

and

) (

) ) (

(

#

#

z D

z z N

R

r



r (31)

To match the rank of original and reduced systems, the constant



is neglected. It is done because due to the nature of the bilinear transformation, the initial value of the step response of the reduced order model may not be zero even though the initial value of the step response of the original system is zero.

Step 4: Matching of steady state values

In the process of transforming from

z

- domain to

w

- domain and then back from

w

- domain to

z

- plane, there arises steady state error between the original system

G (z )

and its

r

^th order reduced model

R

^#

( z )

. To remove this error, the gain correction factor K is calculated as

1

#

( ) ) (







z

z

R z G SSR

K SSO

(32)

where SSO and SSR are the steady state values of

G (z )

and

R

^#

( z )

respectively.

Now multiplying

N

_r^#

( z )

by

K

, the numerator

N

_r

(z )

of the reduced order model

R (z )

is obtained and thus the

r

^th order reduced model

R ( z )  N

_r

( z ) D

_r

( z )

is obtained in the form of (3).

Example:

As an illustration to the above method consider a fourth order multivariable (2 input-2 output) system [15,16] in the following state space form.

) ( )

(

) ( )

( )

1 (

k x C k y

k u B k x A k

x

d

d d









where

 

 



 

 

 





 

 







 

 





 

 







 

0 1 0 0

0 0 0 , 1

4 5 . 0

0 2

8 . 0 25 . 0

2 1 ,

1 25 . 0 6 / 5 3 / 1

1 0 0

0

0 0 65 . 0 1 . 0

0 0 1

0

d d

d

B C

A

Using equation (4), this example can be expressed in the form of

z

- transfer matrix

) (

) ( ) (

) ( ) (

) ( ) (

) ( ) ) (

(

^{21 22}

12 11

22 21

12 11

z D

z n z n

z n z n

z g z g

z g z z g

G

 

 





 



 



 

where

1 . 0 65 . 0 4 . 1 )

(

^{3 2}

11

z  z  z  z 

n

125 . 0 5

. 2 2 )

(

^{3 2}

12

z  z  z  z 

n

366667 .

0 716667 .

1 8 . 2 2 )

(

^{3 2}

21

z  z  z  z 

n

066667 .

0 266667 .

1 4 )

(

²

22

z  z  z 

n

and the common denominator

025 . 0 2625 . 0 65

. 1 )

( z  z

⁴

 z

³

 z

²

 z  D

Applying the bilinear transformation,

z  ( 1  w ) ( 1  w )

on

G (z )

we get

) ( )

( )

(

^*

1

1

G w

z G w H

w

z w

 







