The Use of Piecewise Continuous Expansions in the Identification of Nonlinear Systems

Abstract

This paper describes an experimental method of establishing an ordinary differential equation to represent, or describe, a nonlinear physical system. Each nonlinear function that appears in the differential equation is approximated by a piecewise continuous collection of simple power series expansions. A given elementary expansion is used to represent a function over only a small fraction of the total range of the corresponding argument. These expansions are characterized by a few coefficients which are determined, during the identification process, by a steep descent adjustment procedure. It is assumed that the system may be excited by a specified input and that neither the input nor the output is significantly corrupted by noise. Systems of any order, and with any number of unknown constant coefficients and continuous, single-valued nonlinear functions may be identified with this procedure. Prior knowledge required to implement the method includes the form and order of a differential equation to describe the unknown system, and the location and arguments of functions in this equation assumed or known to be nonlinear. Examples are given to illustrate the efficacy of the method.