Dither for smoothing relay feedback systems: an averaging approach

Dither signals are commonly used for compensating nonlinearities in feedback systems in electronics and mechanics. The seminal works by Zames and Shneydor and more recently by Mossaheb present rigorous tools for systematic design of dithered systems. Their results rely however on a Lipschitz assumption on the nonlinearity and thus do not cover important applications with discontinuities. The aim of this thesis is to provide some ideas and tools on how to analyse and design dither in nonsmooth systems. In particular, it is shown that a dithered relay feedback system can be approximated by a smoothed system. Guidelines are given for tuning the amplitude and the period time of the dither signal, in order to stabilize the nonsmooth system. Stability results based on Popov-like and Zames-Falb criteria jointly with some Linear Matrix Inequalities are proposed. Moreover it is argued that in dithered relay feedback systems the shape of dither signals is relevant for stabilization. Some peculiar behaviours of relay feedback systems dithered with a particular class of dither signals are presented. When the dither signal is a square wave, the dithered system can exhibit an asymmetric periodic orbit, though the smoothed system is asymptotically stable. We even show an example in which, by using a trapezoidal dither signal, both systems have a stable oscillation, but the period time for the oscillation of the smoothed system is different from the one of the dithered system. Finally some engineering applications are presented in order to show the usefulness of techniques and results discussed in the thesis. Thesis Supervisor: Franco Garofalo, Professor of Automatic Control Thesis Supervisor: Francesco Vasca, Associate Professor of Automatic Control