Limit cycles with chattering in relay feedback systems

Relay feedback has a large variety of applications in control engineering. Several interesting phenomena occur in simple relay systems. In this paper, scalar linear systems with relay feedback are analyzed. It is shown that a limit cycle where part of the limit cycle consists of fast relay switchings can occur. This chattering is analyzed in detail and conditions for approximating it by a sliding mode are derived. A result on existence of limit cycles with chattering is given, and it is shown that the limit cycles can have arbitrarily many relay switchings each period. Limit cycles with regular sliding modes are also discussed. Examples illustrate the results.