Self-Oscillations and Sliding in Relay Feedback Systems: Symmetry and Bifurcations

Abstract

This paper is concerned with the bifurcation analysis of linear dynamical systems with relay feedback. The emphasis is on the bifurcations of the system periodic solutions and their symmetry. It is shown that, despite what has been conjectured in the literature, a symmetric and unforced relay feedback system can exhibit asymmetric periodic solutions. Moreover, the occurrence of periodic solutions characterized by one or more sections lying within the system discontinuity set is outlined. The mechanisms underlying their formation are carefully studied and shown to be due to an interesting, novel class of local bifurcations.

On PERIODICAL OSCILLATIONS of LURIE SYSTEMS with DISCONTINUOUS NONLINEARITY

“…In papers (Aguilar, et al, 2007;Bliman, et al, 2000;Loh and Vasnani, 1994) authors applied describing functions like approaches for limit cycle existence substantiation (in paper (Bliman, et al, 2000) continuous nonlinearity was considered only), that provides easily verified practical frequency domain conditions of oscillations existence. However, describing function approach (Atherton, 1975) does not provide an exact answer on question about periodical oscillations existence since the approach starts with assumption that solutions of the system are harmonic functions of time, that is not a correct assumption for the nonlinear Lurie system (the possibility of asymmetric oscillations existence for Lurie system with sign nonlinearity was analyzed in (Di Bernardo, et al, 2001), see also (Kuznetsov, 1995)). Existence of limit cycles for Lurie system with sign nonlinearity was considered in paper (Di Bernardo, et al, 2001) applying Poincaré map approach (that is also a local technique).…”

“…However, describing function approach (Atherton, 1975) does not provide an exact answer on question about periodical oscillations existence since the approach starts with assumption that solutions of the system are harmonic functions of time, that is not a correct assumption for the nonlinear Lurie system (the possibility of asymmetric oscillations existence for Lurie system with sign nonlinearity was analyzed in (Di Bernardo, et al, 2001), see also (Kuznetsov, 1995)). Existence of limit cycles for Lurie system with sign nonlinearity was considered in paper (Di Bernardo, et al, 2001) applying Poincaré map approach (that is also a local technique). In paper frequency of oscillations were derived for the second order system (existence of limit cycles for second order systems follows from Poincaré-Bendixson theorem and its extensions (Mallet-Paret and Sell, 1996)).…”

Limit cycles with chattering in relay feedback systems

“…Moreover, denotes the initial point of the (nonsliding) part of the limit cycle outside , the final point of this part, the final point of the sliding mode part, and . Sliding limit cycles are further analyzed in [28], where it is shown that limit cycles with several first-order sliding segments exist and can be analyzed similarly as previously shown.…”

Micro-chaotic dynamics due to digital sampling in hybrid systems of Filippov type

“…System (40) is given in so-called observer canonical form [14]. The dynamics of (40) has been extensively studied in [15]. There, stable self-sustained oscillations as well as chaotic dynamics have been observed.…”