Describing function

About: Describing function is a research topic. Over the lifetime, 1742 publications have been published within this topic receiving 26702 citations.

Response of certain nonlinearities to modulated inputs

Abstract: A method of evaluating the response of zero-memory frequency-independent nonlinearities to a modulated input is presented by representing the response by a Fourier series of the input variable. The special case of a limiter subject to a sinusoidally amplitude-modulated carrier-suppressed input is treated in some detail, and an envelope describing function evaluated. The application to the frequency-domain analysis of a.c. nonlinear carrier-control systems is mentioned.

Analysis and compensation of nonlinear friction effect on frequency identification

Abstract: This paper considers the issue of nonlinear friction effect on frequency identification. Without loss of generality, a mass-spring-damping system with nonlinear Coulomb friction is adopted to examine the mechanism of this nonlinear distortion. Based on the describing function (DF) analysis method, the gain reduction and phase lead-lag phenomena seen in actual frequency identification are examined in detail, which leads to a new understanding of the nonlinear friction effect. Built-upon the analysis results, a new identification method with nonlinear friction compensation is also proposed. Theoretical analysis shows that this novel identification method has two freedoms to guarantee a good identification result in practice, which makes it possible to obtain an accurate frequency response of the linear dynamics. In addition, this method is less sensitive to the limitations of amplifier and the stroke and the type of excitation signals. Both the simulation and experiment results verify the effectiveness of the proposed DF based analysis and the new identification method.

Adaptive control of tonal disturbance in mechanical systems with nonlinear damping

Abstract: This paper describes an adaptive system for controlling the tonal vibration of a single-degree-of-freedom system with nonlinear damping. The adaptive control system consists of a force actuator in parallel with the suspension, which includes the nonlinear damper, and a velocity sensor mounted on the mass. The adaptation of the controller is done once every period of the excitation. Because the response of the nonlinear system changes with excitation level, conventional adaptive algorithms, with a linear model of the plant, can be slow to converge and may not achieve the desired performance. An on-line observer is used to obtain a describing function model of the plant, which can vary with the excitation level. This allows the adaptive control algorithm to converge more quickly than using a fixed plant model, although care has to be taken to ensure that the dynamics of the observer do not interfere with the dynamics of the adaptive controller.

A novel framework to design and compare limit cycle minimizing controllers: demonstration with integer and fractional-order controllers

Abstract: In this paper, a constrained optimization problem is formulated to tune the limit cycle minimizing controllers meeting additional loop-shaping performances such as phase margin and gain crossover frequency. A graphical approach is proposed so as to determine the superior controller in terms of better limit-cycle suppression. The framework is illustrated with a suitable case of elementary servo plant which has separable static backlash nonlinearity in its model. For this plant, integer-order controllers and their fractional counterparts (PI and $$ PI ^\alpha , [ PI ]^\alpha $$ ; PID and $$ PI ^\alpha D^\beta $$ ) are designed and compared. Interestingly, it is found that the fractional controllers produce better limit-cycle responses than their integer counterparts while both meeting the rest of the specifications. Correspondingly, the better sustained oscillations in the plant output response are obtained with fractional controllers. Such a ‘fractional superiority’ is further verified with the closed-loop nonlinear simulation.