Exact Periodic Solution for Control System Containing Static Nonlinear Function

A solution of the periodic problem in a nonlinear system comprising a single-valued symmetric nonlinearity and linear dynamics is presented. The solution is designed as an iterative algorithm of refinement of the approximate solution obtained via application of the describing function (DF) method. The algorithm is based upon the transformation of the original nonlinear system into an equivalent nonlinear system and the concept of the periodic signal mapping applied to the latter. The solution is sought for as a fixed point of the periodic signal mapping. It is shown that the DF method can be viewed as a method of approximate calculation of the periodic signal mapping. It is proved via the exact approach that for the considered type of nonlinear systems, the necessary conditions of sliding mode existence, previously obtained via the DF method, are valid. The proposed approach is illustrated by examples of analysis of periodic motions in nonlinear systems