WIAS Preprint No. 358, (1997)

Fast periodic oscillations in singularly perturbed relay control systems and sliding modes


  • Fridman, Leonid

2010 Mathematics Subject Classification

  • 34C15 34E10




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 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. The algorithm is proposed which allows to solve the problem of eigenvalues assignment for averaged equations using the additional dynamics of fast actuator.

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