CRONE has several generations of controllers. Kindly see the book in https://fomcon.net/fo/overview-of-literature/ for details.
For instance, the first generation CRONE controller has the following form
\[
C(s)=C_0 s^{\alpha}\tag{1},
\]
with $\alpha, C_0\in\mathbb{R}$. This can be seen as one of the dynamical components of a fractional-order PID controller, e.g., the integral component
\[
C_I(s)=K_i s^{-\lambda},
\]
where $C_0=K_i$ and $\alpha=-\lambda$ in (1).
