# Describing Function Inversion: Theory and Computational Techniques

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## Abstract

In the last few years the study of nonlinear mechanics has received the attention of numerous investigators, either under the scope of pure mathematics or from the engineering point of view. Many of the recent developments are based on the early works of H. Poincare [1] and A. Liapunov [2] As examples can be cited the perturbation method, harmonic balance, the second method of Liapunov, etc. An approximate technique developed almost simultaneously by C. Goldfarb [3] in the USSR, A. Tustin [4] in England, R. Kochenburger [5] in the USA, W, Oppelt [6] in Germany and J. Dutilh [7] and C. Ecary [8] in France, known as the describing function technique, can be considered as the graphical solution of the first approximation of the method of the harmonic balance. The describing function technique has reached great popularity, principally because of the relative ease of computation involved and the general usefulness of the method in engineering problems. However, in the past, the describing function technique has been useful only in analysis. More exactly, it is a powerful tool for the investigation of the possible existence of limit cycles and their approximate amplitudes and frequencies. Several extensions have been developed from the original describing function technique. Among these can be cited the dual-input describing function, J. C. Douce et al. [9]; the Gaussian-input describing function, R. C, Booton [10]; and the root-mean-square describing function, J. E. Gibson and K. S. Prasanna-Kumar [11]. In a recent work which employs the describing function, C. M„ Shen [12] gives one example of stabilization of a nonlinear system by introducing a saturable feedback. However, Shen’s work cannot be qualified as a synthesis method since he fixes "a priori" the nonlinearity to be introduced in the feedback loop. A refinement of the same principle used by Shen has been proposed by R. Haussler [13] The goal of this new method of synthesis is to find the describing function of the element being synthesized. Therefore, for Haussler’s method to be useful, a way must be found to reconstruct the nonlinearity from its describing function. This is called the inverse-describing-function-problem and is essentially a synthesis problem. This is not the only ease in which the inverse-describing-function-problem can be useful. Sometimes, in order to find the input-output characteristic of a physical nonlinear element, a harmonic test can be easier to perform rather than a static one (which also may be insufficient). The purpose of this report is to present the results of research on a question which may then be concisely stated as; "If the describing function of a nonlinear element is known, what is the nonlinearity?" The question may be divided into two parts, the first part being the determination of the restrictions on the nonlinearity (or its describing function) necessary to insure that the question has an answer, and the second part the practical determination of that answer when it exists. Accordingly, the material in this report is presented in two parts. Part I is concerned with determining what types of nonlinearities are (and what types are not) uniquely determined by their conventional (fundamental) describing function. This is done by first showing the non-uniqueness in general of the describing function, and then constructing a class of null functions with respect to the describing function integral, i.e., a class of nonlinearities not identically zero whose describing functions are identically zero. The defining equations of the describing function are transformed in such a manner as to reduce the inverse describing function problem to the problem of solving a Volterra integral equation, an approach similar to that used by Zadeh [18]. The remainder of Part I presents the solution of the integral equations and studies the effect of including higher order harmonics in the description of the output ware shape. The point of interest here is that inclusion of the second harmonic may cause the describing function to become uniquely invertible in some cases. Part II presents practical numerical techniques for effecting the inversion of types of describing functions resulting from various engineering assumptions as to the probable form of the nonlinearities from which said describing functions were determined. The most general method is numerical evaluation of the solution to the Volterra integral equations developed in Part I, A second method, which is perhaps the easiest to apply, requires a least squares curve fit to the given describing function data. Then use is made of the fact that the describing function of a polynomial nonlinearity is itself a polynomial to calculate the coefficients in a polynomial approximation to the nonlinearity. This approach is indicated when one expects that the nonlinearity is a smooth curve, such as a cubic characteristic. The third method presented assumes that the nonlinearity can be approximated by a piecewise linear discontinuous function, and the slopes and y-axis intercepts of each linear segment are computed. This approach is indicated when one expects a nonlinearity with relatively sharp corners. It may toe remarked that the polynomial approximation and the piecewise linear approximation are derived independently of the material in Part I. All three methods presented in Part II are suited for use with experimental data as well as with analytic expressions for the describing functions involved. Indeed, an analytical expression must toe reduced to discrete data for the machine methods to the of use. To the best of the authors® knowledge, research in the area of describing function inversion has been nonexistent with the exception of Zadeh’s paper [18] in 1956. It seems that a larger effort in this area would toe desirable in the light of recent extensions of the describing function itself to signal stabilization of nonlinear control systems by Oldentourger and Sridhar [19] and Boyer [20] and the less restrictive study of dual-input describing functions for nonautonomous systems by Gibson and Sridhar [21]. There presently exist techniques for determining a desired describing function for use in avoiding limit cycle oscillations in an already nonlinear system (Haussler [13]), and the methods presented in this report now allow the exact synthesis of the nonlinear element from the describing function data.

12-1-1962

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