Analysis of chattering in sliding mode control systems with fast actuators via averaging

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 or motions in such SPRCS are approximately described with 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 which describe the motions of the reduced system in the sliding mode are coincide. It is also shown that in the general case when the original SPCSC contains the relay control nonlinearly, the averaging equations do not coincide with the equivalent control equations or the Filippov extension definition which describes the motions in the sliding mode in the reduced system