Sliding modes and fast periodic oscillations of singularly perturbed relay control systems

As a mathematical model of chattering in the small neighbourhood of switching surface in the sliding mode systems, we examine the singularly perturbed relay control systems (SPRCS). The sufficient conditions for existence of fast periodic solutions in such systems are found. Their stability is investigated. It is proved that the slow motions in such SPRCS are approximately described using equations obtained from the equations for the slow variables of SPRCS by averaging along fast periodic motions. It is shown that in the case when the original SPRCS contains the relay control linearly the averaged equations and equations describe the motions of the reduced system in the sliding mode