Describing-function series: a new means for nonlinear-control-system analysis

The general expressions of the describing-function series are determined. These are valid for any single-valued nonlinearities, both asymmetrical and symmetrical, continuous or discontinuous, as well as for combined continuous-discontinuous nonlinearities. The main characteristics of this means of calculation for nonlinear automatic systems are examined and compared with the conventional describing functions. Thus it is shown that, unlike the describing functions, which become imaginary on some parts of the nonlinearities, the describing-function series express the equivalent gain of the nonlinear element over the whole domain. Thus the describing-function series allow a decomposition of the complicated nonlinearities into simple nonlinearities, the series expansion coefficients of which can be found in tables. The expressions for these coefficients are established, forming a table with six primary nonlinearities, from which an extremely large number of nonlinearities can be calculated. The characteristics obtained through describing functions for particular nonlinearities are also compared with those obtained by means of describing-function series.