Abstract
The singularly perturbed relay control systems (SPRCS) are examined. The mathematical apparatus for investigation of the fast periodic oscillations of SPRCS is developed. The theorem about existence of fast periodic solution of SPRCS is proved. The theorem about averaging is given. It is proved that the slow motions in SPRCS with fast periodic solutions are approximately described by equations obtained from the equations for the slow variables of SPRCS by averaging along fast periodic motions. The algorithm of asymptotic representation for the fast periodic solution of SPRCS is suggested. The algorithm for correction of the averaged equation is given. The stability of the fast periodic solution is investigated.
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 example is given which shows 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 describe the motions in the sliding mode in the reduced system. The algorithm is proposed which allows to solve the problem of eigenvalue assignment for averaged equations using the additional dynamics of fast actuator.
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Fridman, L. (1999). The problem of chattering: an averaging approach. In: Young, K., Özgüner, Ü. (eds) Variable structure systems, sliding mode and nonlinear control. Lecture Notes in Control and Information Sciences, vol 247. Springer, London. https://doi.org/10.1007/BFb0109986
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DOI: https://doi.org/10.1007/BFb0109986
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