Practical stability and limit cycles of dithered relay feedback systems

Abstract

A new constructive way for choosing dither amplitude and frequency in order to effectively stabilize relay feedback systems by triangular dither is proposed. It is argued that the proposed averaging approach is not easily extended to square dither signals. Trajectories composed by smooth and switching parts are also analyzed and their relationship with limit cycles exhibited by the smoothed system is investigated.

Dither for smoothing relay feedback systems

“…A detailed analysis of such behaviors is out of the scope of this paper. We have done some preliminary work in this direction [27], [30] and we have verified that these type of phenomena also can appear in laboratory experiments [31]. However, further investigation on the behavior of relay feedback systems with various dither signals is needed.…”

“…It can be shown that the corresponding smoothed nonlinearity is discontinuous in and but is linear in the region [27], [30], [31]. The smoothed and the dithered systems have output waveforms that are highly different in time, see Fig.…”

Effects of dither shapes in nonsmooth feedback systems: experimental results and theoretical insight

“…For dither signals that have zero-slope over non-vanishing intervals (such as square wave dither), it has been shown that the smoothed system can behave significantly different from the dithered system [14]. In this paper we will illustrate these phenomena by new experimental results followed by a theoretical explanation that the authors have recently rigorously proved and generalised to a wider class of nonlinear systems in [20].…”

“…Some other techniques for improving these results by exploiting a stability condition based on an LMI have been proposed in [14], [15]. In these papers we indicated through simulations that the considered averaging analysis carried out for triangular dither does not extend to square dither signals.…”