Describing function

About: Describing function is a research topic. Over the lifetime, 1742 publications have been published within this topic receiving 26702 citations.

Analysis of oscillations in nonlinear systems using multiple input describing functions

Abstract: T i t l e o f Thes is : Analys is o f O s c i l l a t i o n s in Nonlinear Systems Using M ul t ip le Input Describ ing Funct ions F red e r ic L. Swern, Doctor o f Engineering Sc ien ce , 1981 Thesis Direc ted by: Dr. A. U. Meyer, P ro fe s so r o f E l e c t r i c a l Engineer ing Dr. D. Blackmore, A sso c ia te P ro fe s so r o f Mathematics S u f f i c i e n t c o n d i t io n s f o r o s c i l l a t i o n as well as absence o f o s c i l ­ l a t i o n s a re p resen ted f o r a c l a s s o f systems c o n ta in in g one lumped l i n e a r element and a d i f f e r e n t i a b l e n o n l i n e a r i t y . The r e s u l t s a r e ob ta ined by e s t im a t in g th e e r r o r i n h e re n t in us ing a d e s c r ib in g f u n c t io n a n a l y s i s . C ontrac t ion type arguments a r e used to show common to p o lo g ic a l p r o p e r t i e s o f th e d e s c r ib in g fun c t io n s o l u t i o n and th e ba lance o f f i r s t harmonic terms o f t h e system. A f te r d i s c u s s in g th e d e s c r ib in g fu n c t io n method, two theorems a re p re sen ted rega rd ing e x i s t e n c e o r nonex is tence o f o s c i l l a t i o n s from homo­ t o p i c c o n s id e r a t i o n s . A g raph ica l method f o r examining systems with power law n o n l i n e a r i t i e s i s given us ing a paramete r p lane o f f requency and am pl i tude . As an example, th e method i s a p p l i e d to the Van der Pol o s c i l l a t o r when the l i n e a r e lement i s s u f f i c i e n t l y low pass . An a n a ly ­ t i c a l method i s de r ived t h a t i s p a r t i c u l a r l y easy to apply in th e design o f power law o s c i l l a t o r s . I t i s shown t h a t m u l t i p l e in p u t d e s c r ib in g fu n c t io n s may be used in some cases f o r which th e d e sc r ib in g fu n c t io n method i s in c o n c l u s iv e . The r e s u l t s ob ta ined in e s t im a t in g the ampli tude and frequency o f o s c i l ­ l a t i o n using dual i n p u t d e s c r ib in g func t ions a r e compared to t h e i r s i n g l e i n p u t c o u n te r p a r t s f o r a number o f examples. The c l a s s o f n o n l i n e a r i t i e s f o r which the methods may be ap p l ied inc ludes polynomial fu n c t io n s . I t i s shown t h a t one can a l so apply s i m i l a r techn iques to systems c o n ta in in g jump d i s c o n t i n u i t i e s when the n o n l in e a r e lement can be approximated a r b i t r a r i l y c l o s e l y by a co n t in u ­ ous f u n c t i o n . One can say t h a t such a fu n c t io n i s a lmost con t inuous . An idea l r e l a y , a r e l a y w i th deadzone and a s t a i r c a s e fu n c t io n a re analyzed in t h i s manner. In some systems, improved r e s u l t s a r e ob ta ined by r e p r e s e n t i n g the n o n l i n e a r i t y as the sum o f a bounded almost c o n t i n ­ uous fu n c t io n and a polynomial. All o f th e methods developed have been computer ized . Numerous examples a re p re sen ted to i l l u s t r a t e the a p p l i c a t i o n o f th e methods.

Inverse random describing function

Abstract: A solution to the problem of evaluating a nonlinearity that yields a given random describing function is given. Approximate methods are discussed for cases where the describing function is only known graphically.

Human Tracking Performance in Uncoupled and Coupled Two-Axis Systems

Abstract: This report presents tile results of an experimental and analytical study of human performance in uncoupled and coupled control systems. Human pilot performance in single and two-axis systems was mathematically modeled by linear second-order describing functions. Model parameters were determined using model matching techniques. Analysis of the models showed that the amplitude ratio and phase lead of the describing function increased with training indicating an increase in open loop bandwidth. The phase margin also decreased with training. Increasing the plant lag time constant resulted in an increase in the model lead time constant and a decrease in the zero frequency gain. No significant difference was found to exist in the normalized tracking error per axis between the two-axis tasks and the single-axis tasks. However tile model lead time constant was significantly greater in two-axis tracking. Manual tracking of two-axis systems with cross-coupling was studied experimentally and analytically. Approximate methods for modeling two-axis performance were developed and checked using a precise spectral analysis approach. Coupled and uncoupled, symmetrical and asymmetrical two-axis performance was compared. The results show that modeling of cross-coupled systems is feasible and that trained subjects are capable of decoupling the axes of some systems. A methodology study compared the identification performance of continuous, iterative, and extrapolation model matching techniques. An iterative technique employing sensitivity equations for the generation of influence coefficients was found to be the best technique due to its excellent identification accuracy and ease of implementation. Convergence in iterative techniques can be improved substantially by equalizing the parameter adjustment rates and limiting the maximum correction per iteration.

Limit Cycle Prediction of a Neural Vehicle Control System with Gain-Phase Margin Tester

Abstract: Based on some useful frequency domain methods, this paper proposes a systematic procedure to address the limit cycle prediction of a neural vehicle control system with adjustable parameters. A simple neurocontroller can be linearized by using describing function method firstly. According to the classical method of parameter plane, the stability of linearized system with adjustable parameters is then considered. In addition, gain margin and phase margin for limit cycle generation are also analyzed by adding a gain-phase margin tester into open loop system. Computer simulations show the efficiency of this approach.

Limit cycle predictions of nonlinear multivariable feedback control systems with large transportation lags

Abstract: A practical method is developed for limit-cycle predictions in the nonlinear multivariable feedback control systems with large transportation lags. All nonlinear elements considered are linear independent. It needs only to check maximal or minimal frequency points of root loci of equivalent gains for finding a stable limit-cycle. This reduces the computation effort dramatically. The information for stable limit-cycle checking can be shown in the parameter plane also. Sinusoidal input describing functions with fundamental components are used to find equivalent gains of nonlinearities. The proposed method is illustrated by a simple numerical example and applied to one 2×2 and two 3×3 complicated nonlinear multivariable feedback control systems. Considered systems have large transportation lags. Digital simulation verifications give calculated results provide accurate limit cycle predictions of considered systems. Comparisons are made also with other methods in the current literature.