An averaging approach to chattering

The singularly perturbed relay control systems (SPRCS) as mathematical mode ls of chattering in the small neighborhood of the switching surface in slid ing mode systems are examined. Sufficient conditions for existence and stab ility of fast periodic solutions to the SPRCS are found. It is shown that t he slow motions in such SPRCS are approximately described by equations deri ved from equations for the slow variables of SPRCS by averaging along fast periodic motions. It is shown that in the general case, when the equations of a plant contain relay control nonlinearly, the averaged equations do not coincide with the equivalent control equations or with the Filippov's defi nition for the sliding motions in the reduced system; however, in the linea r case, they coincide.