Showing papers on "Describing function published in 1970"

On the phase-plane analysis of relay control system with dead time†

Abstract: An exact method of analysis of non-linear systems containing a two-position relay element and a dead-time element is given using the phase-plane concept. Equations of the switching lines for both unsymmetrical and symmetrical relay characteristics are obtained under consideration of the dead time. Expressions to evaluate the period and amplitude of the self-oscillation of the system are derived from the stable limit cycle in the phase plane. This method is developed for a second-order system with a zero in its transfer function. For such a system it is indicated with a help of an example that approximation methods as the describing function yield large errors because the low-pass condition is not satisfied. The phase-plane method will be developed for other relay characteristics.

An investigation into certain aspects of the describing function of a human operator controlling a system of one degree of freedom

Abstract: : An experimental investigation was carried out of the 'remnant' portion of the mathematical model of the human operator. This model consists of the combination of a quasilinear describing function and a remnant term. A single-axis tracking task, with random forcing functions and a compensatory display, was used to investigate the effect of the type of manipulator and the bandwidth of the forcing function on the remnant. Pressure and free-moving manipulator were employed with rate-control vehicle dynamics and filtered-white-noise forcing functions similar in spectral shape to those used in previous work. Data is presented which show the effects of the manipulator and of the forcing function on the describing function, on the performance measures of the system and on the power spectrum and the amplitude probability distribution of the remnant. (Author)

Stability properties of certain thyristor-bridge control systems. Part 2: The interrelationships of discrete- and continuous-design methods

Abstract: Part 1 of the paper developed discrete models of thyristor-bridge control systems, and introduced some simple design roles for ensuring stability. The present paper investigates the common features of continuous-system design methods for stable thyristor-bridge control systems, and the implications of applying these techniques to a discrete control system are discussed. This is followed by a detailed discussion of the existing describing-function method. Results for describing functions obtained by using crosscorrelation techniques are given. The critical conditions for 1st- and 2nd-order systems are used to show that the Nyquist conditions for stability must take into account critical points other than the point (− 1, 0). A derivation of the equation for the general translation of the critical point is given using discrete-system methods, and a discussion on the translation for low- and high-bandwidth systems follows. An example shows the use of the equation in synthetising the minimum sampling frequency for a given continuous system. An example of a halfsection filter is investigated using discrete-system and root-locus techniques, and the results are used to show that unstable systems exhibit some interesting random characteristics.

Application of a describing function technique to memory-less non-linearities in tandem†

Abstract: A method for the analysis of the single-loop feedback systems with multiple memory-loss non-linearities, using a describing function technique, is presented. This method, which can be extended to any number of non-linearities, is very simple, and immediately allows qualitative, or quantitative answers, subject to the usual errors and restrictions of the describing function technique, regarding regions of stability instability, and existence of limit cycles, etc.

Stabilization of relay-type discrete systems using triangular and square wave dither

Abstract: In a previous correspondence, stabilization of relay-type discrete systems using high-frequency sinusoidal dither, by the method of a dual input discrete describing function was discussed. The dual input discrete describing function reduces to the ordinary describing function of a saturating amplifier and a relay with dead zone when triangular and square wave stabilizing signals are used.

A comment on the discrete describing function method

Abstract: This correspondence points out the inadequacy of the discrete describing function method in predicting certain modes of oscillations in a class of relay sampled-data systems. These modes, however, do not exist in this class of systems. This must be ascertained before the previously mentioned method could be applied to determine all the possible modes of a system. Furthermore, the necessary change in critical regions is suggested for this class of systems to predict the correct mode of oscillation.

Describing-function series: a new means for nonlinear-control-system analysis

Abstract: The general expressions of the describing-function series are determined. These are valid for any single-valued nonlinearities, both asymmetrical and symmetrical, continuous or discontinuous, as well as for combined continuous-discontinuous nonlinearities. The main characteristics of this means of calculation for nonlinear automatic systems are examined and compared with the conventional describing functions. Thus it is shown that, unlike the describing functions, which become imaginary on some parts of the nonlinearities, the describing-function series express the equivalent gain of the nonlinear element over the whole domain. Thus the describing-function series allow a decomposition of the complicated nonlinearities into simple nonlinearities, the series expansion coefficients of which can be found in tables. The expressions for these coefficients are established, forming a table with six primary nonlinearities, from which an extremely large number of nonlinearities can be calculated. The characteristics obtained through describing functions for particular nonlinearities are also compared with those obtained by means of describing-function series.

Application of a Modified Fast Fourier Transform to Calculate Human Operator Describing Function

Abstract: A modified fast Fourier transform (FFT) is used in a hybrid computer program to permit processing of tracking data during a run to yield the human operator's describing function almost immediately after the data-taking period. The computer processing time is substantially reduced at no cost in accuracy.

D.C. gain of the non-linear element and the describing function†

Abstract: In the present paper it has been shown that the maximum value of the describing function is related to the d.c. gain of the non-linearity, and the describing function analysis, where applicable, gives more information regarding system stability than is available from Aizerman's conjecture.

Application of Popov's criterion to signal stabilization in discrete systems

Abstract: In a previous paper [1], a new method called the dual input discrete describing function (DIDDF) was presented, which enables one to determine the minimum amplitude of sinusoidal dither to be applied, to stabilize a relay type sampled-data control system. In this correspondence Popov's method is used to find the minimum amplitude of the dither to be applied to quench the limit cycles occurring in relay type nonlinear control systems. Furthermore, the dependence of the minimum value of the dither on sampling time is explicitly seen in Popov's method.

A Note on the Limit Cycle Stability

Abstract: It is shown that for non-linearities with real describing functions incorporated in a feedback system, the limit cycle stability of the system can be ascertained from the negative slope of the describing function plot at its intersection point with a describing function, independent plot of the system. It is also shown that amplitude of stable oscillation depends both on the non-linearity and the linear part and is determined from the intersection point, whereas the frequency is independent of the non-linearity and can be determined from the associated linear part parameters alone. The analysis is confined to second order systems with appropriate constraints.

An extended criterion for statistical linearization

Abstract: The random-input describing function is the best linear approximation to an instantaneous nonlinearity in the sense of minimum mean-squared error. It is shown here that the describing function also satisfies a more comprehensive criterion which is important to the accuracy of approximate analysis and synthesis techniques for systems with nonlinear elements.

Some Examples of Pilot/Vehicle Dynamics Identified from Flight Test Records

Abstract: This communication illustrates a simple technique for identifying the pilot/vehicle describing functions from routine flight test records. This technique provides a straightforward method of analyzing and comparing the dynamics of closed-loop attitude control tasks from actual flight test operations. Some results from the Gemini and X-15 flight test records tend to confirm previous findings that the pilot tries to adjust his dynamics such that the combined pilot/vehicle describing function can be described by a simple crossover model.

A note on the dual input discrete describing function

Abstract: The exact method of calculating the amplitude of the dither signal from the dual input discrete describing function (DIDDF) portrait is pointed out. It is shown that a previously established relationship between φ and the system under consideration considerably simplifies its application. Further, it is established that only half the range of φ need be considered for a particular period of oscillation. Thus the utility of the DIDDF is considerably enhanced.

Modified application of describing functions in the determination of limit cycles for relay systems

Abstract: A modified method for the study of limit cycles in relay systems using a describing-function approach is presented. Results of applying the method in three systems are quoted. The method is appropriate for situations in which the normal describing function approach fails to predict a limit cycle.

Analytical and experimental studies in signal stabilization of discrete systems

Abstract: This paper describes the definition, development and application of a dual input discrete describing function (D.I.D.D.F.) for the analytical study of non-linear sampled-data systems. An analytical procedure is described which aims at deriving the actual D.I.D.D.F. for an ideal relay. The limit cycles that are ordinarily found in non-linear sampled-data systems can be completely quenched by injecting a high-frequency dither signal at the input to the non-linear element. By applying the D.I.D.D.F. technique, the magnitude of the dither signal required for complete suppression of the limit cycles can be obtained analytically. This technique can be applied to the study of a problem utilizing sinusoidal, triangular or square-wave dither signals. The latter half of the paper presents the results obtained from a comprehensive experimental study of the problem on a model built in our laboratory.