Well, I have no idea whether this is correct, but the only thing that came to my mind after the night's sleep and a morning shower is the following:
My inverse Laplace is wrong, because I did not account for the error input $e(t)$ that is time variant. Therefore, if we look at the derivative part of the transfer function of the controller, we can write
$$C_\mathrm{D}(s) = \frac{sN}{s + N}.$$
Now we take whatever the error input there is and recreate the output as
$$y_\mathrm{D}(s) = \frac{sN}{s + N}e(s) \implies sy_\mathrm{D}(s) + Ny_\mathrm{D}(s) = Nse(s),$$
which can be transformed into time domain as
$$\dot{y}\mathrm{D}(t) + Ny\mathrm{D}(t) = N\dot{e}(t).$$
The solution for this differential equation is
$$y_\mathrm{D}(t) = \mathrm{constant} \times e^{-Nt} + e^{-Nt}\int_0^tN\dot{e}(\tau)e^{N\tau}d\tau, \quad y_\mathrm{D}(0) = 0, \quad e(0) = e_0.$$
This could be implemented into the code, but I don't fancy the absolute time $t$ being there, because we are talking embedded system with finite memory and about 120 MHz clock.
The other option is to discretize the differential equation as
$$\frac{y_\mathrm{D}^n - y_\mathrm{D}^{n - 1}}{\Delta t} + Ny_\mathrm{D}^{n} = N\frac{e^n - e^{n - 1}}{\Delta t}, \quad n \in \mathbb{N},$$
therefore
$$y_\mathrm{D}^n(1 + N\Delta t) = N\left( e^n - e^{n - 1} \right) + y_\mathrm{D}^{n - 1}, \quad y_\mathrm{D}^0 = 0, \quad e^0 = e_0,$$
where $n$ is the time-step, not power to the $n$. 394
From this, we are able to compute the $y_\mathrm{D}^n$, sum it to the other two constants of the controller and feed that to the input of the actuators.
Well, the problem is that I am currently facing the fact that an embedded system will have to solve a differential equation, real-time. I don't know whether this is the correct approach in the industry, so I would be glad if anyone could point out a different solution. The problems I see with this are many - discretized equations are basically forward Euler, may this cause some problems? Shouldn't Runge-Kutta be used? 935