Transfer Function Pole Decomposition

It can be useful to decompose a transfer function into the contributions made by its poles.

Example:

$$P(s)=\frac{5(s+1{)}^{12}}{(s+3{)}^2(s+2{)}^3(s+5{)}^7}$$

We have:

• Poles at $p_i=\{-3,-2,-5\}$.
• Multiplicities at $n_i\in \{2,3,7\}$.

Then:

$$P(s)=\frac{a}{s+3}+\frac{b}{(s+3{)}^2}+\frac{c}{s+2}+\frac{d}{(s+2{)}^2}+\frac{e}{(s+2{)}^3}+\frac{f}{s+5}\cdots +\frac{h}{(s+5{)}^7}+L$$

Writing this down in summation form:

$$P(s)=P(\infty )+\sum _{i=1}^3\sum _{j=1}^{n_i}\frac{c_{i,j}}{(s-p_i{)}^j}$$

Note that:

$$\underset{s\to \infty }{\lim }P(s)=P(\infty )=L$$

In our case:

$$P(\infty )=\underset{s\to \infty }{\lim }P(s)=\underset{s\to \infty }{\lim }\frac{5s^{12}}{s^{12}}=5$$

Lemma: Partial fraction decomposition

Let $P(s)$/$G[z]$ be real, rational, and proper. Let $\{{\pi }_i{\}}_{i=1}^n$ be the poles of $P(s)$/$G[z]$, and let $n_i$ be the multiplicity of $p_i\forall i$.

Then there exist unique constants $c_{i,j}\forall i,j$ such that

$$P(s)=P(\infty )+\sum _{i=1}^n\sum _{j=1}^{n_i}\frac{c_{i,j}}{(s-p_i{)}^j}$$

or

$$G[z]=G[\infty ]+\sum _{i=1}^n\sum _{j=1}^{n_i}\frac{c_{i,j}}{(z-p_i{)}^j}$$