Research paperA unified approach to explain contrary effects of hysteresis and smoothing in nonsmooth systems

- It is known that there is no canonical way to define piecewise dynamical systems in the switching manifold.
- The most widely used definitions at the switching manifold are the Filippov and Utkin conventions.
- Hysteresis and smoothing parameter regularizations are defined and we prove that they give respectively, in the limit when the parameter goes to zero, the Filippov and the Utkin conventions.
- An unified approach is proposed to treat simultaniously both regularizations by its embedding in a higuer dimensional singular perturbed system.
- Estimations of the order of approximation to both conventions are derived in terms of the parameters involved.

Piecewise smooth dynamical systems make use of discontinuities to model switching between regions of smooth evolution. This introduces an ambiguity in prescribing dynamics at the discontinuity: should the dynamics be given by a limiting value on one side or other of the discontinuity, or a member of some set containing those values? One way to remove the ambiguity is to regularize the discontinuity, the most common being either to smooth it out, or to introduce a hysteresis between switching in one direction or the other across it. Here we show that the two can in general lead to qualitatively different dynamical outcomes. We then define a higher dimensional model with both smoothing and hysteresis, and study the competing limits in which hysteretic or smoothing effects dominate the behaviour, only the former of which correspond to Filippov's standard 'sliding modes'.