(**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.