A contribution to the theory of nonlinear systems

✍ Scribed by L.A. Zadeh

• Publisher: Elsevier Science
• Year: 1953
• Tongue: English
• Weight: 984 KB
• Volume: 255
• Category: Article
• ISSN: 0016-0032
• DOI: 10.1016/0016-0032(53)90343-3

✦ Synopsis

A system of classification for nonlinear two-poles is introduced and some of the basic properties of two-poles of various classes are established. The classes are designated as ~)~1, 0~2, ~)~3, "" and are such that each class in the sequence is a subclass of all the classes following it. Furthermore, the class of linear two-poles is a subclass of every class in the sequence.

It is shown that a general nonlinear two-pole of class ~F~I is completely characterized by its responses to a family of step functions with amplitudes ranging over all real values. In an analogous mannez, the dual of a general two-pole of class ff~x is completely characterized by its responses to a family of complex exponentials whose amplitudes and frequencies range over all real values.

Several possible modes of realization of two-poles of class ~)' ~1 are indicated. One canonical form consists of a parallel combination of pairs of linear two-poles and zero-memory (non-inertlal) nonlinear two-poles. For this canonical form an explicit expression is developed for the response of an initially excited two-pole of class ~1 to a specified input.

A canonical realization for two-poles of class ~ in the form of an n-dimensional delay-line filter is indicated. It is shown that certain two-poles of class ~)~/~, which is a subclass of ~)~n, are completely characterized by their responses to n-tuplets of rectangular pulses and are additive with respect to such n-tuplets.

Finally, an extension of Wiener's theory of prediction and filtering to filters of class ~c£1 is briefly outlined.