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

TL;DR

The paper addresses the challenge of using the describing function method for qualitative stability analysis in nonlinear systems, especially with piecewise discontinuities. It introduces a qualitative graphical representation $\tilde{F}(X)$ and a decomposition into standard blocks $F_d(X)$ and $F_r(X)$, enabling fast hand-drawn plotting and a simplified algorithm for $F(X)$. Through two case studies, the authors demonstrate that the qualitative plots yield the same insights into limit-cycle existence as the exact $F(X)$, validating the approach. This work enhances control education by providing a rapid, intuitive tool for predicting oscillations without heavy calculations, supported by open resources like Matlab code.

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.

Figures (10)

• Figure 1: Considered nonlinear autonomous feedback system.
• Figure 2: (a) Dead zone function $y_d(x)$; (b) Describing function $F_d(X)$.
• Figure 3: (a) Relay with threshold $y_r(x)$; (b) Describing function $F_r(X)$.
• Figure 4: (a) Piecewise nonlinear function $y(x)$; (b) Components of $y(x)$ in \ref{['yx_d_r']}; (c) Describing function $F(X)$ in \ref{['yx_d_r_bis']} computed using Algorithm \ref{['Algorithm_optimal_eff']}.
• Figure 5: (a) Piecewise nonlinear function $y(x)$. (b) Describing function $F(X)$ of function $y(x)$, corresponding qualitative graphical representation $\tilde{F}(X)$, and exponential-like function $\tilde{F}_j(X)$ for $j \in \{0,\,1,\,2,\,3\}$.