(Authors: Prof. Navdeep M. Singh, VJTI, University of Mumbai and Nalin Pithwa, 1992).
Abstract: The bilinear transformation can be achieved by using the method of synthetic division. A good deal of simplification is obtained when the process is implemented as a sequence of matrix operations. Besides, the matrices are found to have simple structures suitable for algorithmic implementation.
I) INTRODUCTION:
Davies [1] proposed a method for bilinear transformation using synthetic division. This method can be quite simplified when the synthetic division is carried out as a set of matrix operations because the operator matrices are found to have simple structures and so can be easily generated.
II) THE ALGORITHM:
Given a discrete time transfer function , it is transformed to
in the s-plane under the transformation:
This can be sequentially achieved as :-
The first step is to represent the given characteristic polynomial in the standard companion form. Since the companion form represents a monic polynomial appropriate scaling is required in the course of the algorithm to ensure that the polynomial generated is monic after each step of transformation.
The method is developed for a third degree polynomial and then generalized.
Step 1:
(decreasing of roots by one)
Given (a_{3}=1) (monic)
Then, where
,
,
.
In the companion form, obviously the following transformation is sought:
and
and
Performing elementary row and column transformations on A using matrix operators, the final row operator and column operator matrices, and
, respectively are found to be
and
.
Thus, .
In general, for a polynomial of degree n,
and
, where both the matrices are
.
and , and B is also
.
Where , where
.
Now, the transformation is sought respectively. The row and column operator matrices
and
respectively are :
and
.
and
are found to have the following general structures:
and
, both general matrices
and
, being of dimensions
.
is lower triangular and can be generated as :
, where
and so we get
, where
, and
.
Similarly, is lower triangular and can be generated as :
, where
.
Thus, when A is the companion form of a polynomial of any degree n, then gives
in the companion form.
Step 2:
(scaled inversion).
Let where
,
,
.
The scaling of the entire polynomial by ensures that the polynomial generated is monic and hence, can be represented in the companion form.
The following transformation is sought:
The row and column operator matrices and
respectively are:
and
In general, and
, where both the general matrices
and
are of dimensions
.
So, we get , which is also a matrix of dimension
.
Step 3:
, with
(scaling of roots)
If , then
.
The following transformation is sought:
and
.
The row and column operators, and
, respectively are:
, and
In general, , and
, where the general matrices
and
are both of dimensions
and
Step 4:
(increasing of roots by one)
For the third degree case, the following transformation is sought:
and
and
,
where ,
,
.
The row and column operators and
are :
and
In general, , and
, both the general matrices
and
being of dimensions
.
Where and
, where
and
is a lower triangular matrix where
.
In general, we have
where
where
where
and so on
Now, the transformation is to be achieved. The row and column operators,
and
, respectively are:
and
In general, and
, where both the general matrices
and
are of dimensions
.
, a lower triangular matrix can be easily generated as:
;
.
and , also a lower triangular matrix can be easily generated as:
; and
when
; and
.
Thus, finishes the process of bilinear transformation. Steps 2 and 3 can be combined so that the algorithm reduces to three steps only.
If the original polynomial is non-monic (that is, ), then multiplying the final tranformed polynomial by
restores it to the standard form.
III. Stability considerations:
In the plane, the Schwarz canonical approach can be applied as an algorithm directly to the canonical form of bilinear transformation of the polynomial obtained previously because a companion form is a non-derogatory matrix.
IV. An Example:
Similarly, and
Hence,
Steps 2 and 3:
Step 4:
,
,
,
.
,
,
,
, so we get
The final monic polynomial is , and multiplying it by
, that is,
, restores it to the non-monic form:
V. Conclusion:
Since the operator matrices have lesser non-zero elements, storage requirements are lesser. The computational complexity should reduce for higher-order systems because the non-zero elements lesser manipulations are also lesser, besides lesser storage requirements. Additionally, the second and third steps can be combined giving a three step method only. Thus, the algorithm easily achieves bilinear transformation, especially, for higher systems compared to other available methods hitherto.
VI. References:
- Davies, A.C., “Bilinear Transformation of Polynomials”, IEEE Automatic Control, Nov. 1974.
- Barnett and Storey, “Matrix Methods in Stability Theory”, Thomas Nelson and Sons Ltd.
- Datta, B. N., “A Solution of the Unit Circle Problem via the Schwarz Canonical Form”, IEEE Automatic Control, Volume AC 27, No. 3, June 1982.
- Parthsarthy R., and Jaysimha, K. N., “Bilinear Transformation by Synthetic Division”, IEEE Automatic Control, Volume AC 29, No. 6, June 1986.
- Jury, E.I., “Theory and Applications of the z-Transform Method”, John Wiley and Sons Inc., 1984.