  The ever-growing demands on speed and precision from the precision motion
industry have pushed control requirements to reach the limitations of linear
control theory. Nonlinear controllers like reset provide a viable alternative
since they can be easily integrated into the existing linear controller
structure and designed using industry-preferred loop-shaping techniques.
However, currently, loop-shaping is achieved using the describing function (DF)
and performance analysed using linear control sensitivity functions not
applicable for reset control systems, resulting in a significant deviation
between expected and practical results. We overcome this major bottleneck to
the wider adaptation of reset control with two contributions in this paper.
First, we present the extension of frequency-domain tools for reset controllers
in the form of higher-order sinusoidal-input describing functions (HOSIDFs)
providing greater insight into their behaviour. Second, we propose a novel
method which uses the DF and HOSIDFs of the open-loop reset control system for
the estimation of the closed-loop sensitivity functions, establishing for the
first time - the relation between open-loop and closed-loop behaviour of reset
control systems in the frequency domain. The accuracy of the proposed solution
is verified in both simulation and practice on a precision positioning stage
and these results are further analysed to obtain insights into the tuning
considerations for reset controllers.
