Showing papers on "Describing function published in 1973"
62 citations
TL;DR: A new approach is taken to analyze stabilization of a general nonlinear system with a dither input and it is shown that if the relaxed system is stable, then the original system with dither control would be stable in the finite time.
37 citations
01 Jan 1973
TL;DR: Two graphical methods used for studying the stability of nonlinear systems, the circle criterion and the descibing function, are compared and shown to be complementary in that each tends to compensate for the deficiencies of the other.
Abstract: Two graphical methods used for studying the stability of nonlinear systems-the circle criterion and the descibing function-are compared and shown to be complementary in that each tends to compensate for the deficiencies of the other. A means of increasing the reliabilty of the describing-function method is given. The describing-function method is then extended to a fairly wide class of multivariable systems. The usual advantages to system designers of the describing function are retained in the suggested to produce qualitative changes from the behavior with the interactions removed.
28 citations
TL;DR: In this paper, an analysis for the transfer functions of a particular pulsewidth modulator and power-stage subsystem that has been widely used in practical switching-mode dc regulator systems is presented.
Abstract: An analysis is presented for the transfer functions of a particular pulsewidth modulator and power-stage subsystem that has been widely used in practical switching-mode dc regulator systems. The switch and filter are in a ?buck? configuration, and the switch is driven by a constant-frequency, variable duty-ratio, push-pull magnetic modulator employing square-loop cores. The two transfer functions considered are that with modulator control signal as input and that with line voltage as input. For ac signal,the corresponding describing functions (DF) are derived. It is shown thatcurrent-source drive to the modulator extends the control DF frequency response over that with voltage drive, and that complete cancellation of the effects of line variations can be obtained at dc, but not for ac. Experimental confirmation of the analytical results for the control DF are presented.
16 citations
TL;DR: New graphical techniques, based on the well-known describing function method, are introduced for obtaining first estimates of these responses and their stability, making it simple to find approximate solutions of high order.
Abstract: If a periodic force is applied to a non-linear feedback system, several different responses may be possible. This paper introduces new graphical techniques, based on the well-known describing function method, for obtaining first estimates of these responses and their stability. Once a solution has been found graphically, more information about the solution and its stability can be got by using algorithms derived from the properties of the describing function matrix. The algorithms make it simple to find approximate solutions of high order, thus resolving any doubts about the correctness of first-order predictions and also allowing solutions to be found as exactly as desired.
8 citations
01 Jan 1973
7 citations
01 Jan 1973
TL;DR: In this article, a new graphical procedure for the analysis of nonlinear multivariable systems is presented, which uses the eigenvalue loci of the linear part of the system and the describing function loci for the nonlinear parts.
Abstract: : A new graphical procedure for the analysis of nonlinear multivariable systems is presented. The method uses the eigenvalue loci of the linear part of the system and the describing function loci of the nonlinear parts, and is an extension of the eigenvalue methods first introduced by Bohn for the analysis of linear multivariable systems. For some special classes of systems, it is shown that very realistic stability estimates can be obtained in a simple manner. Examples are given to illustrate the use of the method and the stability bounds obtained are compared with those resulting from the use of other stability criteria. (Author)
5 citations
TL;DR: In this paper, a new describing function for a servomechanism which employs a motor driving an inertia load subject to combined Coulomb and viscous frictions, through gearing having backlash is presented.
Abstract: The paper presents a new describing function for a servomechanism which employs a motor driving an inertia load subject to combined Coulomb and viscous frictions, through gearing having backlash. It is shown that describing functions for various types of backlash phenomena reported in the literature can be considered as special cases of the describing function obtained in this paper.
4 citations
TL;DR: In this article, the existence of a free vibration in a given neighbourhood of the approximate vibration obtained by the describing function method is investigated and a theorem on the estimation of frequency of the free vibration is given.
Abstract: The most popular method of investigation of periodic oscillations in nonlinear systems is the describing function method. Two theorems on the existence of a free vibration in a given neighbourhood of the approximate vibration obtained by this method are presented in the paper. Moreover, for a typical transfer function K(jω) of the linear part of the system and a typical describing function of the non-linear element, the theorem on the estimation of frequency of the free vibration is given.
4 citations
TL;DR: In this article, the stability problem of a closed-loop regulator system with a linear transfer function and a non-linear element is studied, and the interrelationship of Popov's theorem, Aizerman's hypothesis and the describing function is obtained for a single-valued nonlinear element, confined to a finite sector.
Abstract: This paper deals with the stability problem of a closed-loop regulator system which contains in its open loop a linear transfer function and a non-linear element. It is concerned with the interrelationship of Popov's theorem, Aizerman's hypothesis, and the describing function. The interrelationship is obtained for a single-valued non-linear element, confined to a finite sector.
3 citations
01 Jul 1973
TL;DR: In this paper, a review on the method for the calculation of the describing function of double valued characteristics is presented with extension to the inverse describing function problem, which leads to analytical forms from which the inverse problem, of finding special characteristics from their given describing functions, can be solved.
Abstract: A review on the method for the calculation of the describing function of double valued characteristics is presented here with extension to the inverse describing function problem. With this method the calculation of the DF of a characteristic is obtained by considering the function of the characteristic alone without the need of expanding the output waveform into Fourier series. In addition the method Leads to analytical forms from which the inverse problem, of finding special characteristics from their given describing functions, can be solved.
TL;DR: A generalized method for the limit cycle analysis of a class of nth-order systems is proposed and the critical gain for self-sustained oscillations is found without explicitly solving the system characteristic equation.
Abstract: A generalized method for the limit cycle analysis of a class of nth-order systems is proposed. The method utilizes describing function technique and is applicable to dynamical systems with both memoryless and with memory non-linearities. The critical gain for self-sustained oscillations is found without explicitly solving the system characteristic equation. Relatively simple expressions are derived for the evaluation of limit cycle frequency. The results reduce to the conventional Routh equation for memoryless non-linearities. Application of the proposed method is demonstrated by several examples.
TL;DR: In this paper, an asymmetrical saturation type nonlinearity in series with a linear block whose factors depend upon the number of pulses of the rectifier and the order of subharmonic in question is used.
Abstract: Often subharmonic oscillations of firing angle of thyristor converters with large magnitude in negative feedback control systems are a kind of jump phenomena. For their investigation rectifiers may be modeled by an asymmetrical saturation type nonlinearity in series with a linear block whose factors depend upon the number of pulses of the rectifier and the order of subharmonic in question. Application of the incremental describing function technique results in a set of curves, which is useful for the determination of the critical gain of the system and its design. Results of digital computer simulated systems are in good agreement with the predicted values
01 Sep 1973
TL;DR: In this article, the effect of inherent finite delay which is not an integral multiple of the sampling period on the character of limit cycles in a second-order feedback control system with a relay type nonlinearity is analyzed.
Abstract: The effect of inherent finite delay which is not an integral multiple of the sampling period on the character of limit cycles in a second-order feedback control system with a relay type non-linearity is analysed. It is found that the addition of transportation log alters the character of the modes sustained by the system. All the modes that can be sustained for each fractional value of the delay which lies between zero and one and two and three sampling periods are determined by the discrete describing function method. In this connection the critical regions developed by the authors for periods ranging from 9T to 20T have been found to be very useful. The different sots of initial conditions which give rise to the various modes are also established via the state transition technique. It is observed that an increase in the transportation lag usually results in an increase in the amplitude as well as the period of oscillation. Square-wave dither is found to produce a large steady-state error for a step inpu...
01 Nov 1973
TL;DR: In this paper, the authors derived a describing function of the control moment gyros (CMG) frictional nonlinearity using the analytic torque equation, along with the transfer functions of the sampled-data LST system, and the discrete description function of GMC frictionality.
Abstract: A describing function of the control moment gyros (CMG) frictional nonlinearity is derived using the analytic torque equation. Computer simulation of the simplified Large Space Telescope (LST) system with the analytic torque expression is discussed along with the transfer functions of the sampled-data LST system, and the discrete describing function of the GMC frictionality.
01 Jan 1973
TL;DR: In this paper, two examples for the measurement of driver/vehicle multiloop response properties using a single disturbance input are provided for a single-input single-output (SISO) controller.
Abstract: Multiloop response properties of controllers are, in general, very difficult to obtain because an independent forcing function is needed for each describing function to be measured, and interpolation procedures may be required to obtain intermediate describing functions at common frequencies. Two examples are provided for the measurement of driver/vehicle multiloop response properties using a single disturbance input. The validity of the procedure is based on current multiloop operator adjustment rules and is made plausible by comparison with experimental data.
TL;DR: In this article, general relations concerning the dual input describing functions (DIDF) and the modified characteristics of the backlash type nonlinearities are derived, and the calculation of the DIDF of the given characteristic is based on the mathematical expression of the characteristic alone without involving the output waveform.
Abstract: In the present, paper general relations concerning the Dual Input Describing Functions (DIDF) and the modified characteristics of the ‘ backlash type ’ non-linearities are derived. The calculation of the DIDF of the given characteristic is based on the mathematical expression of the characteristic alone without involving the output waveform
TL;DR: In this article, a new limit cycle stability criterion for non-linear Lur'e-type systems is derived via the approach of transient oscillations analysis and the main approximation of the method consists in the use of the describing function relative to the nonlinear part of the system.
Abstract: A new limit cycle stability criterion for non-linear Lur'e-type systems is derived in this paper. The result is achieved via the approach of transient oscillations analysis and the main approximation of the method consists in the use of the describing function relative to the non-linear part of the system.
01 Jun 1973
TL;DR: The approach taken is to replace the nonlinear element by a mean square error minimizing approximation, and then use linear frequency domain synthesis techniques subject to additional constraints imposed by the limit cycle and the approximator.
Abstract: A synthesis theory is developed which allows system design to proceed from practical specifications on system command and/or disturbance response to a design which is very nearly optimal in terms of feedback sensor noise effects. The approach taken is to replace the nonlinear element by a mean square error minimizing approximation (dual-input describing function), and then use linear frequency domain synthesis techniques subject to additional constraints imposed by the limit cycle and the approximator. Synthesis techniques are also developed for a similar system using an externally excited oscillating signal with the above approach. The results remove the design of the systems considered from the realm of simulation and experimentation, permitting true synthesis and the optimization that accompanies it.
TL;DR: In this article, the authors present a describing function for a servomechanism which employs a motor driving a load through gearing having saturation, backlash and combined Coulomb and viscous frictions plus inertia on the load.
Abstract: This paper presents a describing function for a servomechanism which employs a motor driving a load through gearing having saturation, backlash and combined Coulomb and viscous frictions plus inertia on the load. In addition, the paper gives a classification of multiple-gear systems. For each class of the multiple-gear systems, the backlash phenomenon described by the derived describing function is investigated in detail and the application of the describing function technique to multiple-gear systems is discussed. Finally, the effect of saturation on the describing function for a multiple-gear system is illustrated by an example.
TL;DR: In this article, a Taylor-series expansion for a single-valued nonlinear characteristic is proposed, and expressions for its two input describing functions can be obtained as infinite series, which is applicable to sinusoidal or Gaussian inputs in the presence of any other independent signal with a known probability density function.
Abstract: The letter shows that, by using a Taylor-series expansion for a single-valued nonlinear characteristic, expressions for its two input describing functions can be obtained as infinite series. The method is applicable to sinusoidal or Gaussian inputs in the presence of any other independent signal with a known probability-density function.
TL;DR: In this paper, a universal curve for determining the oscillation amplitude in a negative resistance (NR) oscillator is proposed. But quite often when Negative Impedance Converters are used to simulate NR, a piecewise linear and symmetric υ-i characteristic results are obtained.
Abstract: Oscillation amplitude in a negative resistance (NR) oscillator is usually evaluated by the describing function approach in which the non-linear υ—i characteristic of the NR device is approximated by a few terms of a power series expanison But quite often when Negative Impedance Converters are used to simulate NR, a piece-wise linear and symmetric υ—i characteristic results The present paper exploits the above mentioned linearity and symmetry to obtain, to a first-order approximation, a universal curve for determining the oscillation amplitude