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Showing papers on "Describing function published in 1972"


Journal ArticleDOI
TL;DR: It is shown that a nonlinear microwave circuit can easily be represented by a feedback model, and this formalistic identity with nonlinear control systems suggests that methods and results can be borrowed from non linear control theory.
Abstract: It is shown that a nonlinear microwave circuit can easily be represented by a feedback model. This formalistic identity with nonlinear control systems suggests that methods and results can be borrowed from nonlinear control theory. The describing function technique, a concept that has been developed to a high degree of sophistication in control theory, is applied to the problem of phase-locking of microwave oscillators. The use of describing functions in the study of nonlinear microwave circuits may give a simple conceptual understanding of locking phenomena, for example, and thereby reduce computational efforts.

58 citations


Proceedings ArticleDOI
22 May 1972
TL;DR: In this article, an analysis for the transfer functions of a particular pulsewidth modulator and power switch subsystem that has been widely used in practical switching-mode d-c regulator systems is presented.
Abstract: An analysis is presented for the transfer functions of a particular pulse-width modulator and power switch subsystem that has been widely used in practical switching-mode d-c 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 a-c signals, the corresponding describing functions (DF) are derived. It is shown that current-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 d-c but not for a-c. Experimental confirmation of the analytical results for the control DF are presented.

21 citations


Journal ArticleDOI
01 Sep 1972
TL;DR: A nonlinear multiparameter human operator model is presented which considers the complex input information rate in a single display and shows a marked superiority of the nonlinear model.
Abstract: A systematic method is proposed for the development, optimization, and comparison of human controller models. The method is suitable for any model, including multiparameter systems. The evaluation criteria for assessing model quality are based on three separate components: 1) the cost or criterion function, 2) the comparison between the input/output functions of the human operator and those of the model, and 3) characteristic values and functions of statistical signal theory (mean values, auto- and crosscorrelation functions, power spectral density functions, and histograms of time function data). A nonlinear multiparameter human operator model is presented which considers the complex input information rate in a single display. The nonlinear features of the model are brought about by a modified threshold element and a decision algorithm. A random search technique is used for parameter optimization. Different display content arrangements as well as various transfer functions of the controlled element are explained by different optimized parameter combinations. The comparison with the well-known quasi-linear describing function for the human operator shows a marked superiority of the nonlinear model.

18 citations


Journal ArticleDOI
TL;DR: An optimal (suboptimal) periodic control for a single-input single-output time-invariant linear system is shown to be easily obtainable via a describing function approach.
Abstract: This paper deals with a periodic optimization problem. Specifically, an optimal (suboptimal) periodic control for a single-input single-output time-invariant linear system is shown to be easily obtainable via a describing function approach. This also allows pointing out some interesting features of the problem and the significant role played by the concept of resonance frequency.

9 citations


Journal ArticleDOI
TL;DR: In this article, an analysis based on the concepts of equivalent nonlinearity and describing function is used to derive an expression for the minimum amplitude of dither required to quench limit cycle oscillations in feedback systems containing a hysteresis-type non-linearity in the forward path.
Abstract: An analysis based on the concepts of equivalent non-linearity and describing function is used to derive an expression for the minimum amplitude of dither required to quench limit cycle oscillations in feedback systems containing a hysteresis-type nonlinearity in the forward path. The treatment is related to recent work by Manohar and Sellappan.

9 citations


Journal ArticleDOI
TL;DR: In this article, the incremental-input describing function (IDF) and variational approach are compared for general nonlinear second-and third-order systems, and situations in which the two stability techniques disagree are determined and studied in detail.
Abstract: Loeb's criterion and the incremental-input describing function (IDF) stability technique are compared for general nonlinear second- and third-order systems. Situations in which the two stability techniques disagree are determined and studied in detail. The IDF stability technique is related directly to the variational approach for a general system, and Loeb's criterion is related to that for second-order systems. The IDF stability technique is shown to involve the approximation of time averaging in addition to the inherent describing function (DF) approximation. Discrepancies between the two are found to coincide with relatively poor DF approximations. One technique cannot be considered better than the other in applications-they should be used together for proper interpretation.

8 citations


Journal ArticleDOI
TL;DR: The network analyst will find the first chapters on graph theory and networks familiar whereas control engineers will be very conversant with the chapters containing the Nyquist approach to stability analysis via the describing function and the limited methods of phase plane analysis.
Abstract: R Clay Chichester: Wiley 1971 pp x + 284 price £655 Anyone looking for a powerful unified approach to nonlinear systems analysis which they can absorb as bedtime reading had better not attempt this book. The network analyst will find the first chapters on graph theory and networks familiar whereas control engineers will be very conversant with the chapters containing the Nyquist approach to stability analysis via the describing function and the limited methods of phase plane analysis including the Li£nard construction.

7 citations


Journal ArticleDOI
TL;DR: In this article, a generalized procedure is presented by which the existence and stability of limit cycles can be examined in a single-loop feedback system with multiple memory-less non-linear characteristics.
Abstract: In this paper a generalized procedure is presented by which the existence and stability of limit cycles can be examined in a single-loop feedback system with multiple memory-less non-linear characteristics. The method is based upon the describing Function technique and has been programmed for the digital computer to handle complex and high-order systems. Nevertheless, the method is equally applicable in the absence of computational facilities. Several examples illustrate the proposed technique.

6 citations


01 Jan 1972
TL;DR: In this paper, the automobile driver describing function for lateral position control was estimated for three subjects from frequency response analysis of straight road test results, and the measurement procedure employed an instrumented full size sedan with known steering response characteristics, and equipped with a lateral lane position measuring device based on video detection of white stripe lane markings.
Abstract: The automobile-driver describing function for lateral position control was estimated for three subjects from frequency response analysis of straight road test results The measurement procedure employed an instrumented full size sedan with known steering response characteristics, and equipped with a lateral lane position measuring device based on video detection of white stripe lane markings Forcing functions were inserted through a servo driven double steering wheel coupling the driver to the steering system proper Random appearing, Gaussian, and transient time functions were used The quasi-linear models fitted to the random appearing input frequency response characterized the driver as compensating for lateral position error in a proportional, derivative, and integral manner Similar parameters were fitted to the Gabor transformed frequency response of the driver to transient functions A fourth term corresponding to response to lateral acceleration was determined by matching the time response histories of the model to the experimental results The time histories show evidence of pulse-like nonlinear behavior during extended response to step transients which appear as high frequency remnant power

5 citations


Journal ArticleDOI
TL;DR: In this article, the switching-type double-valued nonlinearities are defined and general relations concerning their modified characteristics and their dual-input describing functions (DIDF) for two sinusoidal inputs with unrelated frequencies are derived.
Abstract: The ‘switching-type’ double-valued non-linearities are defined and general relations concerning their modified characteristics and their dual-input describing functions (DIDF) for two sinusoidal inputs with unrelated frequencies are derived. An illustrative example is also presented.

5 citations


01 Jan 1972
TL;DR: In this article, an optimal control model is used to predict pilot performance in a series of longitudinal hovering tasks, where configurational changes are considered that alter significantly the system response to both control and disturbance inputs.
Abstract: An optimal control model is used to predict pilot performance in a series of longitudinal hovering tasks Configurational changes are considered that alter significantly the system response to both control and disturbance inputs Model predictions of mean-squared performance are compared with measurements obtained in an independent experimental study of the task In addition, the optimal control model is used to predict describing functions that correspond to the loop closing pilot transfer functions frequently employed in classical multiloop manual control analyses

Journal ArticleDOI
TL;DR: In this article, a method of describing function is applied both to the linear problem and to the stirred tank, where the hysteresis element is separable and the assumption is made that the signals to it are of sinusoidal form when the reactor is operating with a limit cycle.

Journal ArticleDOI
TL;DR: In this paper, the authors developed a technique to analyze the stability of networks containing linear and nonlinear elements interconnected in multifeed-back structures by extending the concept of describing function to include networks containing nonlinearities with frequency-dependent linear feedback.
Abstract: Active filters have recently acquired widespread use in the realization of frequency-selective networks. Unlike their passive counterparts, active filters have the potential of oscillating. Furthermore, it has been observed that the onset of oscillations in biquad active filters is dependent upon signal level. This led to the recognition that nonlinear stability theory would be necessary to comprehend this behavior. This paper develops a technique to analyze the stability of networks containing linear and nonlinear elements interconnected in multifeedback structures. This is accomplished by extending the concept of the “Describing Function” to include networks containing nonlinearities with frequency-dependent linear feedback. The technique is then applied to explain qualitatively and quantitatively nonlinear effects in op-amps and their relation to the stability of frequency-selective networks containing them (e.g., the Multiple Amplifier Biquad, MAB, and the Single Amplifier Biquad, SAB). The technique is also applied to explain frequency shifts in amplitude-limited oscillators. The most valuable result of this analysis is the discovery of nonlinear feedback circuits which circumvent the conditional stability of high-frequency biquads. This has allowed us to obtain Q's of 50 at 100 kHz in a MAB employing 709 op amps. Similarly, a MAB employing 702 op amps was made to operate at 2 MHz with a Q of 10.

Journal ArticleDOI
TL;DR: In this article, two methods are presented for finding steady-state solutions of differential equations of any order governing certain systems that are acted upon by a harmonic force and have one nonlinear element with hysteresis represented by piecewise linearization.
Abstract: Two methods are presented for finding steady-state solutions of differential equations of any order governing certain systems that are acted upon by a harmonic force and have one nonlinear element with hysteresis represented by piecewise linearization. Both methods need the solution of a set of linear algebraic equations. In the first method, the unknowns are the Fourier coefficients of the steady-state solution, while in the second method, the unknowns are the values of the derivatives of the steady-state solution at a break point of the piecewise linearized characteristic. In both methods, the unknowns have to be calculated for different values of the time angle at the break point, yielding different corresponding values of the amplitude of the forcing term. The required solution is that consistent with the given amplitude of this forcing term. In the first method, the parameters involved in the multiple-input describing functions of the nonlinear element are unified by normalization. Comparison of the two methods is given, and the advantages of the piecewise linearization of characteristics is discussed.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the stability of limit cycles in third and higher-order systems with non-linearities can be determined from the negative slope of the describing function at its point of intersection with a locus dependent on the coefficients of the numerator and denominator polynomials of the transfer function.
Abstract: The purpose of this paper is to show how the stability of limit cycles in third and higher-order systems having non-linearities with a real describing function can be ascertained from the negative slope of the describing function at its point of intersection with a locus dependent on the coefficients of the numerator and denominator polynomials of the transfer function. It is also shown that the amplitude of stable oscillations depends both on the non-linearity and the linear part of the plant and can be determined from the point of intersection whereas the frequency is independent of the non-linearity and can be computed from the coefficients of the linear plant alone. In the case of an all-pass transfer function and for a specific non-linearity the amplitude depends on the gain but is independent of the coefficients of the linear plant. The maximum gain giving stable limit cycles can be related to the non-linearity and the coefficients of the linear plant.

Journal ArticleDOI
TL;DR: In this article, the dual-input describing function technique is applied to investigate the self-oscillations of a low-power nuclear reactor with external feedback, and the existence of an unstable limit-cycle is shown.
Abstract: The dual-input describing function concept is discussed with reference to a low-power nuclear reactor. Results of an earlier analysis, where the conrventional describing function of a low-power nuclear reactor was derived by using the Wentzel-Kramers-Brillouin solution of the kinetics equations, are used to obtain the dual-input describing function of a low-power nuclear reactor. The dual-input describing function technique is applied to investigate the self-oscillations of a low-power nuclear reactor with external feedback. For a suitable frequency characteristic of the feedback, existence of an unstable limit-cycle is shown.




Journal ArticleDOI
TL;DR: In this article, existence and uniqueness theorems were developed both in non-autonomous and autonomous cases, and two new algorithms were developed on the basis of the theoretical results, which form a rigorous mathematical foundation to the harmonic linearization and also to dual-and multiple-input describing function methods.


Journal ArticleDOI
TL;DR: This lecture considers "automatic control" in general, built to relieve the human being from any dull and repetitive operation, may be to supervise a chemical process or to determine the path of a rocket, a missile and finally to let satellites fulfill their task without manual interference.
Abstract: T lecture is given in memory of Theodore von Karman. There are two reason which make me happy to be this year's lecturer. I have known Dr. von Karman personally. I met him the first time at a conference in England in 1934. The other reason is the admiration I have had for his work since I started working in Gb'ttingen's Aerodynamics Research Institute in 1929, where von Karman spent time working with L. Prandtl and where he was well remembered. In 1940 von Karman delivered the fifteenth J. W. Gibbs lecture at a meeting of the American Mathematical Society, titled "The engineer grapples with nonlinear problems." He explained that in many fields of engineering nonlinearities could no longer be neglected, and that a deeper mathematical treatment of some problems had led to particularly interesting results. Since in my lecture nonlinear control theory and its growth will play a role, it seems to me that this topic is fit to remind us of von Karman and his far-reaching ideas. Speaking before an audience of aeronautical engineers, I cannot forget those times when "control" meant stabilization of an airplane with the help of control surfaces at the tail or ailerons at the wing. The pilot used them to fly a desired path and to counteract gusts, or adverse winds. There came a step forward when the pilot's task was eased by building the "automatic pilot" to which the human pilot would essentially give commands. My talk today shall not be restricted to this specific task, but consider "automatic control" in general, built to relieve the human being from any dull and repetitive operation, may be to supervise a chemical process or may be to determine the path of a rocket, a missile and finally to let satellites fulfill their task without manual interference. After World War II automatic controls have spread into many regions of technology. The consequence was that barely an engineer in any field could escape coming in contact with them. Some mechanisms and regulators were a long time in use, (for instance, the centrifugal regulator of steam engines), but complicated automatic control systems were still little used. Beginning with the 1950s, the mathematicians' interest awoke and this was the start of a tremendously fast development surpassing the linear control theory which was presented in a number of books written for engineers with relatively minor mathematical background. Let us go back to the airplane. One understood the basic behavior of the dynamic system, and the controls tended to keep the plane on its desired path and in its desired attitude by removing the undesired deviations which could be described by linear differential equations. Linear control elements were mostly used, however, at the same time the airplane was a wonderful example to show that linearization was not always suitable for describing the behavior of the system, and controls designed on that assumption could not be satisfactory. Maybe nonlinear controls had to be introduced. Among others, Kochenburger and MacCollhad attacked the problem of designing stable nonlinear control systems. I remember a conference at which two wind tunnel designers were enthusiastic about their success in designing a wind tunnel with special flow characteristics. After linear controls did not yield the desired airstream quality, they finally used Kochenburger's paper and, with the help of the describing function they designed a suitable control of the propulsion system. I could not agree only with their conclusions that Kochenburger's rule had solved all nonlinear control problems. The describing function allows us to handle certain nonlinear controls in the same manner as linear ones by way of approximation. J. G. Truxal's book on Control System Synthesis contains many historically interesting details. The essential assumptions for its use is: the system contains only one nonlinear element; the output of this element depends only on the present value and the past history of the input; if the input is sinusoidal only the fundamental output of the nonlinear element influences the system. One particular nonlinear type of control used was the discontinuous control; often designed with the help of the describing function, or by trial and error method. In this case, each of the control inputs has only two or three settings. In wartime, the controls were employed for steering missiles. Such controls are simple from the design standpoint, they are rugged and inexpensive. These qualities are appreciated in controlling objects which