Showing papers on "Describing function published in 1971"

Detectors, bandpass nonlinearities, and their optimization: Inversion of the Chebyshev transform

Abstract: From the voltage-response characteristic of a memoryless nonlinearity the output amplitude in any harmonic zone is easily found as a function of the amplitude of a narrow-band input, but no general method has been known for inverting this (Chebyshev) transformation. The inversion is of interest because the best detector, bandpass non-linearity, or harmonic generator for various purposes, e.g., maximization of the output signal-to-noise ratio, is most readily described in terms of its amplitude-response characteristic for the desired harmonic zone. A simple integration over the amplitude-response characteristic (or describing function) is found to yield the required voltage-response curve.

Justification of the Describing Function Method

Abstract: Explicit conditions are given under which the use of the describing function method to investigate the nature of oscillations in autonomous nonlinear feedback systems is justified. When these conditions are satisfied, bounds are given for the frequency, fundamental magnitude, and higher harmonics of the oscillation based on the describing function approximation.The feedback systems considered are those which can be decomposed into a linear time-invariant subsystem, not necessarily causal, stable, or finite-dimensional, and a nonlinear frequency independent subsystem, possibly containing hysteresis.The approach taken is to consider the describing function equation as an approximation to a determining equation for periodic solutions of the autonomous system’s operator equation, and to use local degree theory to guarantee the existence of a solution to the determining equation.

A describing function technique for multiple nonlinearities in a single-loop feedback system

Abstract: A graphical technique for determining the existence of limit cycles, their amplitude, frequency, and stability when they exist, and the stability of a single-loop feedback system with n - 1 memoryless nonlinear elements and one nonlinear element with memory is considered. The approach here is to assume an input to a nonlinear element and then apply the Nyquist stability condition to the linear system resulting after the nonlinear elements have been approximated by their describing functions. The method requires no trial and error procedure, is noniterative in nature, and is especially easy to apply. The method is subject to the usual errors and restrictions of the describing function method. An extension of the method to include n non-linear elements with memory and n nonlinear elements in parallel is also included. Three numerical examples are included to illustrate the method.

Control of distillation columns with sharp temperature profiles

Abstract: Temperature control problems are frequently encountered in distillation columns that, as a result of high purity products and an easy separation, have large temperature changes over a few trays. The describing function technique is used in this paper to analyze the stability of this type of column in which a nonlinear feedback control loop for column temperature control manipulates vapor boilup. The cycling usually experienced in these systems is shown to be unavoidable in many cases because the process transfer function has such a high gain that it acts like an on-off switch. The process also displays nonlinear dynamics: system time constants vary with the magnitude of change in vapor boilup. To overcome these difficulties an adaptive, nonlinear feedback controller is proposed that keeps the loop in a small-amplitude limit cycle by adaptively changing controller gain. The system is evaluated by digital simulation of a nonlinear model of the column.

An analytic and computer study of the jump phenomenon in ferroresonant regulators

Abstract: At certain ac input levels, discontinuous jumps appear in the ferroresonant regulator's steady-state output-versus-input characteristic. The conditions are derived under which this action, known as jump resonance, can occur. The describing function method of nonlinear system analysis is used to derive the range of circuit parameters in which the jump can occur. It is found that the conditions for the jump can be expressed in terms of the values of the linear energy storage elements and the character of the non-linearity, and that there exists a unit circle in the Nyquist plane which must be avoided to assure the nonoccurrence of the jump. The theoretical analysis is supplemented with a computer study of the circuit using an analog computer simulation program.

Oscillations in a two-dimensional non-linear system†

Abstract: The conditions necessary to obtain stable oscillations in a two-dimensional non-linear symmetrical system are found using the describing function technique, and the results are applied to two examples.

Analysis of Limited Authority Manual Control System

Abstract: : Systematic procedures for predicting pilot-vehicle-flight control system performance and proneness to pilot induced oscillations and instabilities are here developed and applied to examples. The systems analyzed have very limited maximum control surface rates and deflections. Performance analysis is by means of applying random input describing function theory to predict the root-mean-square level of key system variables as a function of the control surface rate and deflection limit levels. Acceptable limit levels are only two to three times the root mean square value of the variable at the point in the system where each limiter nonlinearity occurs. Results of analyzing three minimum back-up manual flight control system modifications for the F-4C are compared with data from piloted fixed-base simulator experiments for the same system configurations.

Highlights of a Brushless Direct-Drive Solar Array Control System Design

Abstract: This paper summarizes a drive system design for controlling the position and rate of solar power arrays on orbiting spacecraft. There are no gears or sliding contact elements used anywhere in the system and only low-speed bearings are needed. Such mechanization is particularly well suited to solid lubrication techniques, and wear rates are very low, so that the drive system can operate directly in the space environment for long periods of time. Three major components were developed for implementation of this design concept. They are: 1) a brushless dc torque motor; 2) a rotary power transformer; and 3) an offset-tooth shaft position and rate sensor. These components are combined in a hybrid system configuration in which the signal processing and logic functions are performed by digital and linear integrated circuits. A root contour and describing function analysis, confirmed by experimentation, shows that several modes of limit cycle generation can occur in the vicinity of null. Compensation circuits are given that inhibit or suppress limit cycling and provide controlled electronic damping of the system. The system offers relatively high stiffness and can be operated at indefinitely low angular rates with minimum power consumption.

A new formulation of the describing function method for non-linear systems subject to stochastic input †

Abstract: A new statement of the describing function method for non-linear systems subject to stochastic input is proposed. This approach is considered as an improvement on the deterministic case, and uses a mathematical formulation which is explained directly by means of the characteristic f(x) of the non-linearity. Firstly, we bear in mind this formulation, and then we describe successively a class of statistical linearization, a class of non-linear statistical optimization, and lastly we consider the self-oacillating problem in the presence of noise.

Response of non-linear systems to inputs with any number of frequency components†

Abstract: A general series expansion is derived for the amplitude of any modulation component in the output of a single-valued non-linearity subjected to an input which is the sum of any number of cosine waves of different frequencies An important property of this expansion is that it expresses the response of the non-linearity to a multiple cosine wave input in terms of its response to a single cosine wave input; thus an inter-modulation component or a Multiple Input Describing Function (MIDF) can be obtained by using the coefficients of the Fourier series expansion of the output of the non-linearity when the input is a single cosine wave of appropriate amplitude Since these coefficients can also be obtained by measurements, the method may be applied directly to empirical non-linearities without resorting to piecewise linear or polynomial approx imations The expansion is suitable for rapid computation on the digital computer, and because of its generality can be efficiently used for solving intermodulation prob