I have tried everything to identify the type of PID controller used but I think I'm missing something. The denominator of (s+10) really throws me off. Otherwise it would be standard system from below. The PID I know is standart Kc+Kds+Ki/s and half the transfer function fits it except for the abovementioned denominator
The extensive question is: i.e. which actions and/or other components are present in this controller?
As far as I understand the controller + actuator combo comes before disturbance and after the disturbance comes the plant. Standard closed loop system. I have done a lot of searching and tried to fit in similar first/second order systems but none of them looked the same. Below is what I got as a complete transfer function of the system, albeit rearanged
$$ \frac{(d s + 10 d + 6 s + 20)(r - z)}{s (s + 10)} = z $$
If you refer to the PID Controller article here, yours should be the ideal form of the PD controller structure:
$$G_{c}(s) = K_{p} \left[1 + \tau_{d} \frac{N s}{s + N}\right]$$
where $N$ is known as the Derivative filter coefficient.
When compared with the given controller
$$G_{c}(s) = 2 \left[1 + \frac{2 s}{s + 10}\right]$$
we know that $N = 10$, and since $\tau_{d} N = 2$, then $\tau_{d} = \frac{1}{5}$.
Sometimes, it is also expressed in this standard form as shown here:
$$G_{c}(s) = K_{p} \left[1 + \frac{\tau_{d} s}{\frac{1}{N} s + 1}\right]$$
I personally prefer to write the PD controller in this form:
$$G_{c}(s) = K_{p} + K_{d} \frac{s}{\tau_{f} s + 1}$$