Exponential describing function in the analysis of nonlinear systems

In this paper, signals in
(L)_{2}(- \infty, t]
, a subspace of the space of square integrable signals defined on
(- \infty, t]
, are approximated by signals in
(L)_{2}^{1}(- \infty, t]
, the one-dimensional subspace of
(L)_{2}(- \infty, t]
spanned by the first function from the set of reversed time Laguerre functions. A system mapping
(L)_{2}(- \infty, t]
into itself is associated with a system mapping
(L)_{2}^{1}(- \infty t]
into itself; the latter system is characterized by a gain-exponential describing function. This type of describing function is developed as an analysis tool for studying the transient response of a large class of nonlinear feedback systems. The contraction-mapping fixed-point theorem is used to develop conditions for the existence of a solution prior to the use of the exponential describing function to obtain an approximate solution.