A view on limit cycle bifurcations in relay feedback systems

The paper is concerned with the study of oscillations in linear dynamic systems with relay feedback. The specific interest is about the bifurcations of these periodic solutions, with regard to phenomena which also occur in smooth systems and to others due to the relay discontinuity. The followed approach moves from the describing function method, leading to results which are approximate in nature but which express in a simple and correct way the essential mechanisms of the studied phenomena, as shown in the proposed examples.

[1]  Karl Henrik Johansson,et al.  Self-oscillations and sliding in Relay Feedback Systems: Symmetry and bifurcations , 2001, Int. J. Bifurc. Chaos.

[2]  Derek P. Atherton,et al.  Stability of nonlinear systems , 1981 .

[3]  Alberto Tesi,et al.  Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems , 1992, Autom..

[4]  I︠a︡. Z. T︠S︡ypkin Relay Control Systems , 1985 .

[5]  P. A. Cook Simple feedback systems with chaotic behaviour , 1985 .

[6]  Luigi Glielmo,et al.  Switchings, bifurcations, and chaos in DC/DC converters , 1998 .

[7]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[8]  Karl Johan Åström,et al.  Oscillations in Systems with Relay Feedback , 1993 .

[9]  Yury V. Kolokolov,et al.  Bifurcations and Chaotic oscillations in an Automatic Control Relay System with Hysteresis , 2001, Int. J. Bifurc. Chaos.

[10]  Alberto Tesi,et al.  Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics , 1996, Autom..

[11]  Derek P. Atherton,et al.  Designing autonomous relay systems with chaotic motion , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[12]  A. I. Mees,et al.  Dynamics of feedback systems , 1981 .

[13]  D. P. Atherton,et al.  Nonlinear Control Engineering-Describing Function Analysis and Design , 1977, IEEE Transactions on Systems, Man, and Cybernetics.

[14]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[15]  Austin Blaquière,et al.  Nonlinear System Analysis , 1966 .

[16]  A. Megretski Global Stability of Oscillations Induced by a Relay Feedback , 1996 .

[17]  Shlomo Engelberg,et al.  Limitations of the describing function for limit cycle prediction , 2002, IEEE Trans. Autom. Control..

[18]  Y. Kuznetsov Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[19]  Alberto Tesi,et al.  A Frequency Method for Predicting Limit Cycle Bifurcations , 1997 .

[20]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[21]  R. Seydel From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis , 1988 .

[22]  Karl Henrik Johansson,et al.  Stability of limit cycles with chattering in relay feedback systems , 2003, 2003 European Control Conference (ECC).

[23]  Karl Henrik Johansson,et al.  Fast switches in relay feedback systems , 1999, Autom..

[24]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .