Topic

# Describing function

About: Describing function is a(n) research topic. Over the lifetime, 1742 publication(s) have been published within this topic receiving 26702 citation(s).

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01 Jan 1968

TL;DR: The theory of automatic control has been advanced in important ways during recent years, particularly with respect to stability and optimal control, but these theories do not, however, lay to rest all questions of importance to the control engineer.

Abstract: ABRAMSON Information theory and coding BATTIN Astronautical guidance BLACHMAN Noise and its effect on communication BREMER Superconductive devices BROXMEYER Inertial navigation systems GELB AND VANDER VELDE Multiple-input describing functions and nonlinear system design GILL Introduction to the theory of finite-state machines HANCOCK AND WINTZ Signal detection theory HUELSMAN Circuits, matrices, and linear vector spaces KELSO Radio ray propagation in the ionosphere MERRIAM Optimization theory and the design of feedback control systems MUUM Biological control systems analysis NEWCOMB Linear multiport synthesis PAPOULIS The fourier integral and its applications R. N. BRACEWELL) STEINBERG AND LEQUEUX (TRANSLATOR Radio astronomy WEEKS Antenna engineering PREFACE The theory of automatic control has been advanced in important ways during recent years, particularly with respect to stability and optimal control. These are significant contributions which appeal to many workers, including the writers, because they answer important questions and are both theoretically elegant and practically useful. These theories do not, however, lay to rest all questions of importance to the control engineer. The designer of the attitude control system for a space vehicle booster which, for simplicity, utilizes a rate-switched engine gimbal drive, must know the characteristics of the limit cycle oscillation that the system will sustain and must have some idea of how the system will respond to attitude commands while continuing to limit-cycle. The designer of a chemical process control system must be able to predict the transient oscillations the process may experience during start-up due to the limited magnitudes of important variables in the system. The designer of a radar antenna pointing system with limited torque capability must be able to predict the rms pointing error due to random wind disturbances on the antenna, and must understand how these random disturbances will influence the behavior of the system in its response to command inputs. But more important than just being able to evaluate how a given system will behave in a postulated situation is the fact that these control engineers must design their systems to meet specifications on important characteristics. Thus a complicated exact analytical tool, if one existed, would be of less value to the designer than an approximate tool which is simple enough in application to give insight into the trends in system behavior as a function of system parameter values or possible compensations, hence providing the basis for system design. As an analytical tool to answer questions such as these in a way …

1,227 citations

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TL;DR: The model is described in detail and is used to predict experimentally measured quantities for three simple, but basic, compensatory tracking tasks and is applied to study a complex VTOL hovering task.

Abstract: Application of optimal control and estimation theory is made to a wide class of problems in manual control. The situation considered is that for which the dynamical system being tracked is linear and is perturbed by an external white noise input. By assuming that the human behaves ''optimally'' in some sense, subject to his inherent psychophysical limitations, a quantitative model is developed for the response characteristics of the human operator. The resultant suboptimal model can be used to predict task performance, measured human controller describing functions, remnant and power spectra. The model is described in detail and is used to predict experimentally measured quantities for three simple, but basic, compensatory tracking tasks. In a companion paper the model is applied to study a complex VTOL hovering task.

508 citations

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TL;DR: Analysis of chattering in such systems with unmodeled based on the Lyapunov theory and the describing function method and various approaches to reduce chattering are described including methods based on relay control gain adaptation.

Abstract: The implementation of sliding mode control is often irritated by high frequency oscillations known as “chattering” in system outputs issued by dynamics from actuators and sensors ignored in system modeling. This paper provides analysis of chattering in such systems with unmodeled based on the Lyapunov theory and the describing function method. It also describes various approaches to reduce chattering including methods based on relay control gain adaptation. And for those systems to which the methods are not applicable, chattering frequency control using hysteresis loop will be provided.

446 citations

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Abstract: Self-excited oscillations of a confined flame, burning in the wake of a bluff-body flame-holder, are considered. These oscillations occur due to interaction between unsteady combustion and acoustic waves. According to linear theory, flow disturbances grow exponentially with time. A theory for nonlinear oscillations is developed, exploiting the fact that the main nonlinearity is in the heat release rate, which essentially ‘saturates’. The amplitudes of the pressure fluctuations are sufficiently small that the acoustic waves remain linear. The time evolution of the oscillations is determined by numerical integration and inclusion of nonlinear effects is found to lead to limit cycles of finite amplitude. The predicted limit cycles are compared with results from experiments and from linear theory. The amplitudes and spectra of the limit-cycle oscillations are in reasonable agreement with experiment. Linear theory is found to predict the frequency and mode shape of the nonlinear oscillations remarkably well. Moreover, we find that, for this type of nonlinearity, describing function analysis enables a good estimate of the limit-cycle amplitude to be obtained from linear theory.Active control has been successfully applied to eliminate these oscillations. We demonstrate the same effect by adding a feedback control system to our nonlinear model. This theory is used to explain why any linear controller capable of stabilizing the linear flow disturbances is also able to stabilize finite-amplitude oscillations in the nonlinear limit cycles.

434 citations

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Abstract: Analysis of combustion instabilities relies in most cases on linear analysis but most observations of these processes are carried out in the nonlinear regime where the system oscillates at a limit cycle. The objective of this paper is to deal with these two manifestations of combustion instabilities in a unified framework. The flame is recognized as the main nonlinear element in the system and its response to perturbations is characterized in terms of generalized transfer functions which assume that the gain and phase depend on the amplitude level of the input. This 'describing function' framework implies that the fundamental frequency is predominant and that the higher harmonics generated in the nonlinear element are weak because the higher frequencies are filtered out by the other components of the system. Based on this idea, a methodology is proposed to investigate the nonlinear stability of burners by associating the flame describing function with a frequency-domain analysis of the burner acoustics. These elements yield a nonlinear dispersion relation which can be solved, yielding growth rates and eigenfrequencies, which depend on the amplitude level of perturbations impinging on the flame. This method is used to investigate the regimes of oscillation of a well-controlled experiment. The system includes a resonant upstream manifold formed by a duct having a continuously adjustable length and a combustion region comprising a large number of flames stabilized on a multipoint injection system. The growth rates and eigenfrequencies are determined for a wide range of duct lengths. For certain values of this parameter we find a positive growth rate for vanishingly small amplitude levels, indicating that the system is linearly unstable. The growth rate then changes as the amplitude is increased and eventually vanishes for a finite amplitude, indicating the existence of a limit cycle. For other values of the length, the growth rate is initially negative, becomes positive for a finite amplitude and drops to zero for a higher value. This indicates that the system is linearly stable but nonlinearly unstable. Using calculated growth rates it is possible to predict amplitudes of oscillation when the system operates on a limit cycle. Mode switching and instability triggering may also be anticipated by comparing the growth rate curves. Theoretical results are found to be in excellent agreement with measurements, indicating that the flame describing function (FDF) methodology constitutes a suitable framework for nonlinear instability analysis.

415 citations