Analysis of a control system containing two non-linearities†

Abstract

A control system containing two non-linear elements may be analysed in the parameter plane to determine the existence, stability and values of amplitude and frequency of limit cycles. Using describing functions to represent the non-linearities, the system characteristic equation may be obtained. Two adjustable parameters are selected, each containing one of the describing functions and possibly several control system gains. A correlation between these parameters and the roots of the characteristic equation is determined by mapping stability contours from the complex 8-plane onto the chosen parameter plane. A relationship between the inputs to the two non-linearities is determined next, and a locus representing the variation of the describing functions also is plotted on the parameter plane. If this locus and the stability contour associated with a pair of pure imaginary roots intersect, the existence and characteristics of a limit cycle are indicated. Particular emphasis is placed on organizing the equations in a manner that lends itself readily to solution by a digital computer.

Notes

†Communicated by the Author. A portion of this paper was presented at the Third Asilomar Conference on Circuits and Systems, Pacific Grove, California, 1969.