An Approach for the Qualitative Graphical Representation of the Describing Function in Nonlinear Systems Stability Analysis

TL;DR

This paper introduces a new qualitative approach for plotting the describing function in nonlinear systems, simplifying the process and aiding control education by enabling hand-drawn approximations for systems with discontinuities.

Contribution

It presents a novel qualitative plotting method for the describing function that reduces mathematical complexity and enhances understanding of nonlinear system limit cycles.

Findings

Qualitative plots match exact method results in limit cycle estimation

Simplifies the analysis process for nonlinear systems with discontinuities

Facilitates control education through hand-drawn plotting techniques

Abstract

The describing function method is a useful tool for the qualitative analysis of limit cycles in the stability analysis of nonlinear systems. This method is inherently approximate; therefore, it should be used for a fast qualitative analysis of the considered systems. However, plotting the exact describing function requires heavy mathematical calculations, reducing interest in this method especially from the point of view of control education. The objective of this paper is to enhance the describing function method by providing a new approach for the qualitative plotting of the describing function for piecewise nonlinearities involving discontinuities. Unlike the standard method, the proposed approach allows for a straightforward, hand-drawn plotting of the describing function using the rules introduced in this paper, simply by analyzing the shape of the nonlinearity. The proposed case studies show that the limit cycles estimation performed using the standard exact plotting of the describing function yields the same qualitative results as those obtained using the proposed qualitative method for plotting the describing function.