Find a Continuous Time model from Discrete time model

Question

When we assume that the sampling time period is $h$ and the hold time is also $h$, how can we transfer the following discrete time system to a continuous time system? $$(1-q^{-1}) y_{k} = (bq^{-1}+cq^{-2})u_{k}$$

Answer 1

According to Bilinear transform

$$s={2(z-1)\over T(z+1)}$$

$$z={-(sT+2)\over (sT-2)}$$

where $T$ is the sample time.

So,

$$(1-{-(sT-2)\over (sT+2)})y(s)=(b{-(sT-2)\over (sT+2)}+c({-(sT-2)\over (sT+2)})^2)u(s)$$

It is a lot of mess. If I have not made any mistake, it will be

$${y(s)\over u(s)}={(c-b)T^2s^2-4cTs+4b \over 2Ts(Ts+2)}$$