This paper presents a complex order filter developed and subsequently integrated into a PID-based controller design. The nonlinear filter is designed with reset elements to have describing function based frequency response similar to that of a linear (practically non-implementable) complex order filter. This allows for a design which has a negative gain slope and a corresponding positive phase slope as desired from a loopshaping controller-design perspective. This approach enables improvement in precision tracking without compromising the bandwidth or stability requirements. The proposed designs are tested on a planar precision positioning stage and performance compared with PID and other state-of-the-art reset based controllers to showcase the advantages of this filter.

A controller with the frequency response of a complex order derivative may have a gain that decreases with frequency, while the phase increases. This behaviour may be desirable to ensure simultaneous rejection of high-frequency noise and robustness to variations of the open-loop gain. Implementations of such complex order controllers found in the literature are unsatisfactory for several reasons: the desired behaviour of the gain may be difficult or impossible to obtain, or non-minimum phase zeros may appear, or even unstable open-loop poles. We propose an alternative nonlinear approximation, combining a CRONE approximation of a fractional derivative with reset control, which does not suffer from these problems. An experimental proof of concept confirms the good results of this approximation and shows that nonlinear effects do not preclude the desired performance.