Steady state analysis of non-linear systems and multiple input describing functions (M.I.D.F.)

Abstract

The describing function technique is considered as a special case of the more general problem of determining the amplitude of any frequency component in the output of a single valued non-linearity with a multiple frequency input. By using a power series, Fourier transform, and Fourier series, respectively, to represent the non-linear input-output characteristic, three formulas for the amplitude of any output frequency component are derived. The formula based on the Fourier series representation of the non-linearity is presented in detail with a discussion of numerical techniques for very rapid computation of output amplitudes. These methods are generalized for cases when the input frequencies are commensurate. It is shown that the formula for any output amplitude in this case is given by a series which converges rapidly, allowing efficient numerical computation. As an important application, in automatic control theory, of the above methods, both the exact representation and an efficient numerical approximation for the Multiple Input Describing Function (M.I.D.F.) are presented for the case of commensurate as well as incommensurate frequencies. To illustrate these methods, a subharmonic Dual Input Describing Function (D.I.D.F.) is considered, and its relationship to the D.I.D.F. for incommensurate input frequencies is established.