Ef24836841

J.V.B.Jyothi, T.Narasimhulu / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.836-841

An Osculatory Rational Interpolation Method in Linear Systems
by Using Routh Model
*J.V.B.Jyothi * T.Narasimhulu
* Pursuing M. E (Control Systems), ANITS, Sangivalasa, Visakhapatnam – 531162, India.

Abstract:
In the present paper, theorem for osculatory This paper has six sections, in section II, the basic
rational interpolation was shown to establish a concept of Routh model reduction method is
new criterion of interpolation and Routh model explained. In section III, theorem and algorithm of
reduction method for linear systems was interpolation method is discussed. In section IV,
explained. On the basis of this conclusion a gives the further discussions of the present paper. In
practical algorithm was presented to get a section V, numerical example of the proposed
reduction model of the linear systems. A well- method is illustrated. Section VI gives the
known numerical example of the proposed conclusion.
method is presented to illustrate the correctness
effectively. II.ROUTH MODEL REDUCTION
METHOD
Keywords: Osculatory rational interpolation, Let the transfer function of a higher order
criterion of interpolation, Routh table, model system be represented by [6], [7]
reduction. D0  D1s  ...  Dk 1s k 1 D( s)
G ( s)   , (1)
I.INTRODUCTION e0  e1 s  ...  ek s k E ( s)
The transfer function of frequency-domain in a Where Di, i=0,1,…, k-1 are constant l x r matrices,
linear system is an expression method that describes and ei, i=0,1,…, k are scalar constants.
the system. In practical application, realistic models
are so high in dimension that a direct simulation or Assume that the reduced model R (s) of order n is
design would be neither computationally desirable required, and let it be of the form
nor physically possible in many cases. Hence, it is
significant to reduce the dimension of the system Dn (s) A0  A1s  ...  An 1s n 1
R( s )   , (2)
(i.e., the order of transfer function) on the basis of En (s) b0  b1s  ...  bn 1s n 1  bn s n
retaining the information of the original model of a
large extent. In recent decades, many methods [1- where the Ai, i=0,1,…, n -1 are constant l x r
5]had been introduced in scalar and interval systems. matrices, and bi, i=0,1,…., n are scalar constants.
The Pade approximation [6,7]was a classical method
among them, which made use of the series Algorithm 1
expansion at zero of the original transfer function Step 1:The denominator En (s) of reduced model
G(s), namely it only used the information of each transfer function can be constructed from the Routh
order derivative with G(s) at zero. In the paper the Stability array of the denominator of the system
Pade approximation method will be extended to
transfer function as follows.
osculatory rational interpolation. The denominator
polynomial of the reduced order model is obtained The Routh stability array is formed by the following
from Routh table and the numerator polynomial is
obtained from theinterpolation method. This Pade bi  2,1bi 1, j 1
Approximation method can reduce the original bi , j  bi  2, j 1  , (3)
transfer function by taking advantage of the bi 1,1
information of each order derivative with the
 (k  i  3) 
transfer function G(s) at many other points. where i  3 and 1  j   
 2 

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The routh table for the denominator of the system transfer pq  qp  g (s) f (s).
ˆ ˆ
function is given as

b11  e k b12  e k  2 b13  e k  4 b14  e k 6 ... Divide on the two sides of the above relation by q q
b21  e k 1 b22  e k 3 b23  e k 5 b24  e k 7 ... to get.

b31 b32 b33 ..... ˆ
p p g ( s) f ( s) f ( s)
   g ( s)
..... q q ˆ qqˆ qqˆ
bk 1,1 bk 1, 2 Thus
bk ,1 d (k )  p d (k )  p
ˆ
  si  ( k )
q  
q
bk 1,1 (4) ds ( k )   ds  ˆ  si
th
i  0,1, ...., n, k  0,1,...., mi  1
En (s) may be easily constructed from the (k+1-n)
th
and (k+2-n) rows of the above to give In the following proof we use the mathematical
induction. For k=0, obviously,
n
E n ( s)   b j s j ˆ
p(si ) p( si )
j 0
 ,
ˆ
q(si ) q( si )
 bk 1 n.1 s n  bk  2 n.1 s n 1  bk 1 n.2 s n  2  .... (5) and it follows that
p( s i )q( s i )  p( s i )q ( s i )  0.
ˆ ˆ
III.THEOREM AND ALGORITHMOF
INTERPOLATION METHOD Hence
In this section we first prove a primary theorem and
pq  pq  ( s  s i ) f 0 ( s ).
ˆ ˆ
then give a useful algorithm to reduce the linear system
models.
Where f0(s) is a polynomial.
ˆ ˆ
Theorem 1 Let p / q, p / q be two rational fractions. And let
Assume that it is true for k<n-1, that is,
q(si )  0, q(si )  0. then
ˆ
d (k )  p d (k )  p
  si  ( k )
q  
q
d (k )
 p d (k )
 p
ˆ ds ( k )   ds   si
 
q   
q
 ˆ  si
si
ds ( k )   ds ( k ) k  0,1,...., n  1. (6)
i  0.1, .....n, k  0.1,....., mi  1. Then pq  pq  ( s  s i ) n f 0 ( s ).
ˆ ˆ
If and only if there exists a polynomial f(s), such that
Let a linear time-invariant system in the state-space
pq  qp
ˆ ˆ form [4] be
 [(s  s 0 ) m0 ( s  s1 ) m1 ....( s  s n ) mn ] f ( s)
 x(t )  Ax (t )  Bu (t ).

 g ( s) f ( s), 
 y (t )  Cx(t ).
Where
Where x, u and y are n-, m-, and r-dimensional
g(s) = (s-s0) m0
(s-s1) m1
…(s-sn) mn state, control, and output vectors, respectively. Using
the Laplace transformation, the above system can be
represented in the frequency domain as
ˆ ˆ
herep,q and p, q are polynomials.
Y ( s)  G( s)U ( s),
Proof Let  1
G( s)  C ( sI  A) B,

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Where G(s) is the (r x m) - dimensional transfer Thus
function matrix whose elements are rational
functions of s, i.e., ˆ
( pq) ( k ) si  f (k )  (qp) ( k )
ˆ si  h (k ) si
si
n 1
p( s) bn 1 S  ...  b0 j
G( s)   (7 ) i  0,1,..., j, k  0,1,..., mi  1.  mi  2m
q( s) a n s n  ....  a 0 i 0
By using the basic theorem of algebra, it is obtained
that
Which is a transfer function of the original system.
Now seek its reduction model f (s)  h(s). (11)
ˆ m 1 ˆ
ˆ ( s)  p( s)  bm 1 s  ....  b0
G
ˆ
(8) It is found that the coefficient of each term in

ˆ
q( s) a s m  ....  a ˆ ˆ ˆ ˆ
f(s) in (9) is the linear combination of a0, a1 ...., am
m 0

m  n 1 and the coefficient of each term in h(s) in (10) is the
ˆ ˆ ˆ
linear combination of b0 , b1 ..., bm1 .
that satisfies the following conditions
By means of the relation (11), a linear system
ˆ
G ( k ) ( si )  G ( k ) ( si ), with 2m+1 unknowns and 2m equations is formed.
j Because the coefficients of a rational fraction as in
i  0,1,..., j, k  0,1,...mi 1,  mi  2m (8) have one dependent variable, without losing
i 0 generality, it can be assumed a0  1. if the
ˆ
coefficient matrix of the above linear system is
In terms of Theorem 1, it is equivalent to
nonsingular, then its solution
ˆ ˆ ˆ ˆ ˆ
ˆ
( pq) ( k ) si  (qp) ( k ) ,
ˆ b0 , b1 ..., bm1 , a1 ..., a m can be uniquely determined
si
by using the Cramer rule.

i  0,1,..., j, k  0,1,..., mi 1. On the basis of the above discussion an
algorithm to obtain the reduction model (8) will be
Now take the following steps respectively: presented as follows.

Algorithm 2: Seek the coefficients of the reduction
ˆ
(1) Divided pq by g(s)=(s-s0)m 0 (s-s1)m 1 …(s-
model.
sj)m j to get the quotient e(s) and the
Step 1 Choose 2m points s0,s1,…, s2m-1,si  C
remainder f(s). (they can be multiple) and satisfy G(si)  0,
ˆ
(2) Divided pq by g(s)=(s-s0)m 0 (s-s1)m 1 …(s- then compute
sj)m j to get the quotient l (s) and the
g ( s)  ( s  s 0 )( s  s1 )....(s  s 2 m 1 )
remainder h(s). It is held that
 ( s  s 0 ) m0 ( s  s1 ) m1 ....( s  s j )
mj

pq  g ( s)e( s)  f ( s),
ˆ (9)
 s 2 m  g 2 m 1 s 2 m 1  ....  g1 s  g 0
qp  g ( s)l ( s)  h( s),
ˆ (10)

Where both f(s) and h(s) are polynomials of degree
at most (2m-1).

838 | P a g e

Step 2: Compute p q and q p , respectively
 
ci(1)  ci( 0 )  cm0)n1 g i mn1,
(

i  0,1,..., m  n  2,
n 1
pq  (bn 1 s
ˆ  ...  b0 ) (a m s  ...  a 0 )
ˆ ˆ m
ci( 2 )  ci(1)  cm1n2 g i mn 2,
( )

 c m0)n 1 s m  n 1  c m0)n  2 s m  n  2  ...
( (
i  0,1,..., m  n  3,
c s  c ,
(0)
1
(0)
0 ci(3)  ci( 2 )  cm2)n3 g i mn3,
(

c00 )  a 0 b0 ,
(
ˆ i  0,1,..., m  n  4,
c (0)
1  a 0 b1  a1b0 ,
ˆ ˆ .......
.... ci(l )  ci(l 1)  cml1)l g i mnl ,
( 
n

c m0)n  2  a m bn  2  a m 1bn 1 ,
(
ˆ ˆ
i=0,1,…, m+n- l -1,
c (0)
m  n 1  a m bn 1,
ˆ
…..
and
ˆ ˆ ci( nm )  ci( nm1)  c2nm1) g i ,
(

qp  (a n s n  ...  a 0 ) (bm 1 s m 1  ...  b0 )
ˆ m

i  0,1,..., 2m  1,
 d m0)n 1 s m  n 1  d m0)n  2 s m  n  2  ...
( (

 d1( 0 ) s  d 00 ) ,
( When k <0, let g k =0. In the above the recursive
(n)
relations, the superscript n in c represent
ˆ
d (0)  b a , i
0 0 0 thecoefficientswhich are obtained after carrying out
the algorithm n steps, and the subscript i in ci(n )
d (0) ˆ ˆ
 b0 a1  b1 a 0 ,
1 represents the corresponding degree about the
.... variables.
ˆ ˆ
d m0)n  2  bm 1 a n 1  bm  2 a n,
(

(2)Divide q p by g ( s) to get h( s).
ˆ
d m0)n 1  bm 1 a n.
(

Step 4According to f ( s)  h( s), get a linear
Step 3: (1) Divide by g(s) to get f(s): system with (2m+1) unknowns and 2m equations. let
ˆ ˆ ˆ ˆ
a0  1and solveb0 ...., bm1, a1 ,..., am by
ˆ using
cm0) n 1s n  m 1  cm1) n  2 s n  m  2  ...
( (

the Cramer rule.
s 2m  g 2m 1s 2m 1  ...  g1s  g 0 cm0) n 1s m  n 1  cm0) n  2 s m  n  2
( (
 ...  cn(0)m 1s n  m 1  ...  c0(0)
cm0) n 1s m  n 1  g 2m 1cm0) n 1s m  n  2  ...  g 0 cm0) n 1s n  m 1
( ( (
IV.FURTHER DISCUSSION
mn2 nm3
c (1)
mn2 s  ...  c (1)
nm2 s  ...  c (1)
0
mn2 nm2
(1) Stabilization
c (1)
mn2 s  ...  g c (1)
0 mn2 s
m n3
c ( 2)
m n3 s  ...  c ( 2)
0
The above method can ensure the reduction model
stable. In order to produce the dominator
... ... ... ...
polynomial, the following famous method: the
c2(nmm) s 2m 1  ...  c0(n  m)
1 retaining dominant poles will be introduced. Then its
numerator can be produced by using the above
method, and at this time, the number of points is
Thus get the recursive relations:
( p)  1.
ˆ

839 | P a g e

(2) Choice of the interpolation points pq  ( s 4  13s3  63s 2  133s  102)(a2 s 2  a1s  a0 )
ˆ
By the way, the above method is similar to the
classicalPade approximation when all the pq  a2 s 6  (133a2  a1 ) s5  (63a2  133a1  a0 ) s 4  (133a2  63a1  133a0 ) s3
ˆ
interpolation points are zeros, it only takes
advantage of information of G(s) at zero.  (102a2  133a1  63a0 ) s 2  (102a1  133a0 ) s  102a0
pq  c6 s 6  c5 s 5  c4 s 4  c3 s 3  c2 s 2  c10 s  c0
ˆ 0 0 0 0 0 0
Usually, interpolation points chosen had
better reflect the features of the original model G(s) pq  g ( s)e( s)  f ( s )
ˆ
well. According to experience, we can choose the
points which are located in the disk centered at the
origin with radius: the distance between origin and f ( s )  pq  g ( s )e( s )
ˆ
the furthest poles; or besides some dominant poles;
or besides origin.
f ( s )  c5 s 5  c1 s 4  c3 s 3  c1 s 2  c1 s  c0
1
4
1
2
1 1

V. NUMERICAL EXAMPLE
In this section we give one numerical f ( s)  (133a2  a1 ) s 5  (74.6a2  133a1  a0 ) s 4
example to illustrate algorithm 1
 (90.4a2  63a1  133a0 ) s 3  (159.6a2  133a1  63a0 ) s 2
Example
 (102a1  133a0  21.6a2 ) s  102a0
For the interpolation points 0, 0.6,2, 3, and qp  (s 6  14.5s 5  81s 4  223s 3  318s 2  212.5s  50)(b1 s  b0 )
ˆ
6. Seek a reduction model of type [1/2] for
qp  b1 s 7  (14.5b1  b0 ) s 6  (81b1  14.5b0 ) s 5  (223b1  81b0 ) s 4
ˆ
s 4  13s 3  63s 2  133s  102  (318b1  223b0 ) s 3  (212.5b1  318b0 ) s 2
G( s) 
s 6  14.5s 5  81s 4  223s 3  318s 2  212.5s  50
 (50b1  21.5b0 ) s  50b0
qp  d 7 s 7  d 6 s 6  d 5 s 5  d 4 s 4  d 3 s 3  d 2 s 2  d10 s  d 0
ˆ 0 0 0 0 0 0 0
Solution:
qp  g ( s)l ( s)  h( s)
ˆ
Routh method applying to the denominator h( s)  qp  g ( s)l ( s)
ˆ
E(s)  s 6  14.5s 5  81s 4  223s 3  318s 2  212.5s  50 h( s )  d 6 s 6  d 5 s 5  d 4 s 4  d 3 s 3  d 2 s 2  d 1 s  d 0
1 1 1 1 1 1 1

Routh Table h( s)  (14.5b1  b0 ) s 6  (14.5b0  80b1 ) s 5  (81b0  234b1 ) s 4
s6 1 81 318 50  (223b0  275.4b1 ) s 3  (318b0  270.1b1 ) s 2
s5 14.5 223 212.5  (212.5b0  28.4b1 ) s  50b0
s 4
65.62 303.3 50 According to f ( s)  h( s) the linear system
s3 155.9 201.2
0 0 50 0  a1  102
 
2
s 218.6 50  102 21.6 212.5 28.4  a 2  133
   
s1 165.5  133  159.6 318 270.1  b0  63 
s 0
50     
 63  90.4 223 275.4 b1  13 
Thus the reduced order denominator is
Is formed
E2 (s)  218.6s 2  165.5s  50 Solve the systems to obtain the reduction model (see
fig.1)
Using algorithm 1 to get the numerator of reduced
order model 0.665s  2.04
R2 ( s) 
218.6s 2  165.5s  50
g ( s)  s( s  0.6)( s  2)( s  3)( s  6)
 s( s 2  9s  18)( s 2  2.6s  1.2) The below figure1 shows the simulation result of
g ( s)  s  11.6s  42.6s  57.6s  21.6s
5 4 3 2 comparison of step response for original and reduced
order transfer functions.

840 | P a g e

7. GuChuan-qing. Thiele-type and Lagrange-
Step Response type generalized inverse rational
interpolation for rectangular complex
matrices [J]. Linear Algebra and Its
ORIGINAL (6TH ORDER) Application, 1999, 295: 7-30.
ND
REDUCED (2 ORDER) 8. GU Chuan-quig, ZHANG Ying, “
AnOsculatory rational interpolation method
of model reduction in linear system”,
Amplitude

Journal of Shanghai University(English
Edition),2007,11(4):365-369.
9. WU Bei-bei, GU Chuan-quig , “ A matrix
pade-type routh model reduction method
for multivariable linear system”,[J].
2006,10(5):377-380.

0 2 4 6 8 10 12 14 16 18 20
Time (sec)

Fig1. Comparison of step response of original and
reduced order systems.

VI.CONCLUSION
In this paper, we have observed osculatory
rational interpolation to establish a new criterion of
interpolation. And the Routh model reduction was
introduced to obtain the denominator polynomial of
the reduced order transfer function. This Routh
method ensures the stability. This method is simple
and can be applied to practical control engineering.

REFERENCES

1. Ismail O, Bandyopadhyay B, Gorez R.
“Discrete interval system reduction using
Pade approximation to allow retention of
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1078.
2. Ismail O. On multipoint Pade
approximation for discrete interval systems
[C]//Proceedings of the Twenty Eighth
Southeastern Symposium on System Theory
Washington DC, USA: IEEE Computer
Society, 1996: 497-501.
3. Bandyopadhyay B, Ismail O, Gorez R.
Routh-Padeapproximation for interval
systems [J]. IEEE Transaction Automatic
Control, 1994, 39: 2454-2456.
4. Mohammad J. Large –scale Systems:
Modeling and Control [M]. New York:
Elsevier Science Publishing Press, 1983.
5. Hutton M F, Friedland B. Routh
approximations for reducing order of linear
time – invariant system [J]. IEEE
Transaction Automatic Control, 1975, 20:
329-337.
6. Baker G A, Jr, Graves – Morris PR. Pade
Approximants [M]. 2nd ed. New York:
Cambridge University Press, 1995.

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