A solution to the transorming a z-domain transfer function to s-domain.

There is a well-known transformation that is used to convert a continuous-time transfer function in $s$-domain into a discrete-time transfer function in $z$-domain. When engineers take their digital control course, they learn to use the bilinear transformation $s\approx\frac{2}{T}\frac{z-1}{z+1}$ and plug this into any $s$-domain transfer function to convert it to $z$-domain. For example, if we define $K=2/T$, then when $N\ge M$

\[\frac{Y(s)}{X(s)} = \frac{b_0+b_1s+b_2s^2+\cdots+b_Ms^M}{a_0+a_1s+a_2s^2+\cdots+a_Ns^N}\]

can be transformed to

\[\frac{Y(z)}{X(z)} = \frac{b_0K(z+1)^P+b_1K(z-1)(z+1)^{P-1}+b_2K^2(z-1)^2(z+1)^{P-2}+\cdots+b_MK^M(z-1)^M(z+1)^{P-M}}{a_0K(z+1)^P+a_1K(z-1)(z+1)^{P-1}+a_2K^2(z-1)^2(z+1)^{P-2}+\cdots+a_NK^N(z-1)^N(z+1)^{P-N}},\]

which can be regrouped to produce the polynomials in $z^{-k}$ for $k\in(0,P)$ needed to implement the transfer function in a simple state machine.

However, it is less common to find the description of the inverse transformation using $z\approx\frac{K+s}{K-s}$. A similar approach can be taken, i.e.,

\[\frac{Y(z)}{X(z)} = \frac{b_0z^{-M}+b_1z^{1-M}+b_2z^{2-M}+\cdots+b_{M-2}z^{-2}+b_{M-1}z^{-1}+b_M}{a_0z^{-N}+a_1z^{1-N}+a_2z^{2-N}+\cdots+a_{N-2}z^{-2}+a_{N-1}z^{-1}+a_N}\]

can be transformed to

\[\frac{Y(s)}{X(s)} = \frac{b_0(s+K)^P+b_1(s-K)(s+K)^{P-1}+b_2(s-K)^2(s+K)^{P-2}+\cdots+b_M(s-K)^M(s+K)^{P-M}}{a_0(s+K)^P+a_1(s-K)(s+K)^{P-1}+a_2(s-K)^2(s+K)^{P-2}+\cdots+a_N(s-K)^N(s+K)^{P-N}},\]

which can be regrouped to produce the polynomials in $s^k$ for $k\in(0,P)$ needed to implement the transfer function as a state-space model.