Eigenvalue assignment in sliding mode systems with the help of averaging technique

As a mathematical model of chattering in the small neighbourhood of the switching surface in sliding mode systems we examine 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 by 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 the equations which describe the motions of the reduced system in the sliding mode coincide. An algorithm is proposed which solves the problem of eigenvalue assignment for the averaged equations using the additional dynamics of the fast actuator.