Mapping the system function from the s-plane to the z-plane in the presence of multiple order poles.

April 22, 2010 Compiled on January 29, 2024 at 2:53am

Given

H (s)

of order

N

with all its poles

p_{i}

being distinct, it can be expressed in terms of partial fraction expansion in the form of

H (s) = \sum_{k = 1}^{N} \frac{A_{k}}{s - p_{k}}

and the resulting

H (z)

can be found to be

\sum_{k = 1}^{N} \frac{z A_{k}}{z - e^{p_{k} T}}

where

T

is the sampling period.

In the case when

H (s)

contains a pole

q

of order

2

, then

H (s)

can be written as

(\sum_{k = 1}^{N - 2} \frac{A_{k}}{s - p_{k}}) + \frac{A_{q}}{{(s - q)}^{2}}

and the resulting

H (z)

can be found to be

(\sum_{k = 1}^{N - 2} \frac{z A_{k}}{z - e^{p_{k} T}}) + \frac{T z e^{q T}}{{(e^{q T} - z)}^{2}}

In the case when

H (s)

contains a pole

q

of order

3

, then

H (s)

can be written as

(\sum_{k = 1}^{N - 3} \frac{A_{k}}{s - p_{k}}) + \frac{A_{q}}{{(s - q)}^{3}}

and the resulting

H (z)

can be found to be

(\sum_{k = 1}^{N - 3} \frac{z A_{k}}{z - e^{p_{k} T}}) + (- \frac{e^{2 q T} T^{2} z + e^{q T} T^{2} z^{2}}{2 {(e^{q T} - z)}^{3}})

The following table was generated in order to obtain the general formula. This table below shows only the part of

H (z)

due to the multiple order pole.


$n$ pole order	$H (z)$

$2$	$\frac{T z e^{q T}}{{(e^{q T} - z)}^{2}}$

$3$	$- \frac{e^{2 q T} T^{2} z + e^{q T} T^{2} z^{2}}{2 {(e^{q T} - z)}^{3}}$

$4$	$\frac{e^{3 q T} T^{3} z + 4 e^{2 q T} T^{3} z^{2} + e^{q T} T^{3} z^{3}}{6 {(e^{q T} - z)}^{4}}$

$5$	$\frac{- e^{4 q T} T^{4} z - 11 e^{3 q T} T^{4} z^{2} - 11 e^{2 q T} T^{4} z^{3} - e^{q T} T^{4} z^{4}}{24 {(e^{q T} - z)}^{5}}$

$6$	$\frac{e^{5 q T} T^{5} z + 26 e^{4 q T} T^{5} z^{2} + 66 e^{3 q T} T^{5} z^{3} + 26 e^{2 q T} T^{5} z^{4} + e^{q T} T^{5} z^{5}}{120 {(e^{q T} - z)}^{6}}$

It is easy to see that the denominator of

H (z)

has the general form

(n - 1)! {(e^{q T} - z)}^{n}

where

n

is the pole order, the hard part is to ﬁnd the general formula for the numerator. The following table is a rewrite of the above table, where only the numerator is show, and

e^{q T}

was written as

A

to make it easier to see the general pattern


$n$ pole order	numerator of $H (z)$

$2$	${(- 1)}^{n} (A T) z$

$3$	${(- 1)}^{n} [{(A T)}^{2} z - A {(T z)}^{2}]$

$4$	${(- 1)}^{n} [{(A T)}^{3} z + 4 A^{2} T^{3} z^{2} + A {(T z)}^{3}]$

$5$	${(- 1)}^{n} [{(A T)}^{4} z - 11 A^{3} T^{4} z^{2} - 11 A^{2} T^{4} z^{3} - A {(T z)}^{4}]$

$6$	${(- 1)}^{n} [{(A T)}^{5} z + 26 A^{4} T^{5} z^{2} + 66 A^{3} T^{5} z^{3} + 26 A^{2} T^{5} z^{4} + A {(T z)}^{5}]$

I am trying to determine the general formula to generate the above. This seems to involve some combination of binomial coeﬃcient. But so far, I did not ﬁnd the general formula.

Mapping the system function from the s-plane to the z-plane in the presence of multiple order poles.

1 References