Real and complex-exponential describing functions for transient analysis of non-linear control systems†

†Communicated by the Authors.

Abstract

A real exponential describing function as an analysis tool for studying the transient response of a class of non-linear feedback systems was developed by Bickart (1966). In this paper describing functions are developed for similar analysis more suitable to higher-order non-linear feedback systems. The two classes of describing functions developed are identified as real exponential or complex exponential.

Here, signals in ℒ₂ ( − ∞, t] a space of the space of square integrable signals defined on (−∞, t ], are approximated by the sum of n signals in ℒ₂ ^{1, m} (−∞, t)one-dimensional sub-spaccs of ℒ₂ ( − ∞, t] having the mth function from the set of reversed time orthogonalized real or eponential functions as a basis. A system mapping ℒ₂ ^{1, m}(−∞, t] into itself is associated with a system mapping ℒ₂ ^{1, m}(−∞, t] into itself; the latter system is characterized by a gain—real or exponential describing function. These multiple one-dimensional system mappings give rise to approximation components of the response whoso addition will represent a better approximation to the actual response than each component by itself. The contraction-mapping fixed point theorem is also used to determine conditions for the existence of a solution prior to the use of the exponential describing functions for obtaining on approximate response.

Notes

†Communicated by the Authors.