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Nonlinear Vibration
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
The describing function consists of both magnitude and phase, where the magnitude is the ratio of the fundamental component of the output to the input amplitude. In general, it is a complex-valued function of the input amplitude e0 and frequency. Using the describing function method we can approximate the nonlinear control system as a linear control system shown in Figure 11.24.
Stability of Non-Linear Systems
Published in T. Thyagarajan, D. Kalpana, Linear and Non-Linear System Theory, 2020
The describing function approach can be employed to perform stability analysis of non-linear systems. Consider a non-linear system with unity feedback as shown in Figure 5.2a, where ‘N’ represents the non-linear element. The non-linear element can be linearized by replacing it with a describing function, KN (X,ω) or KN, as shown in Figure 5.2b.
Nonlinear Stability Analysis
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
For some nonlinear systems, with a few reasonable assumptions, the frequency response method, called the describing function method, can be used to approximately analyze and predict nonlinear behavior. The main purpose of this technique is for the prediction of limit cycles in nonlinear systems. This part presents a brief introduction to the describing function analysis of nonlinear systems.
Optimum Setting Algorithm Based PI Controller Tuning for SRF-PLL Used Grid Synchronization System
Published in Electric Power Components and Systems, 2023
S. A. Lakshmanan, Bharat Singh Rajpurohit, Amit Jain
An ideal on-off relay with certain heights and is placed in the feedback loop, the output lags behind the input by radians and limit cycle exists in the loop. Hence, the closed loop system shows oscillatory output with amplitude and time period Therefore, crossover frequency of the relay [20] feedback experiment is [30] Describing function analysis offers a tool for analyzing a frequency domain of a non-linear system. DF is obtained by considering only the principal harmonic of the relay output signal.
Multi-jump resonance systems
Published in International Journal of Control, 2020
Arturo Buscarino, Carlo Famoso, Luigi Fortuna, Mattia Frasca
Several methods to obtain approximate solutions for the study of the frequency response of nonlinear systems subjected to a sinusoidal input have been presented in the literature (Mitropolski & Mitropolski, 1961). Here, we apply the method based on the describing function (Atherton, 1982), which, similarly to other classical methods (Mitropolski & Mitropolski, 1961) assumes that each dynamical variable of the system can be approximated as a finite sum of sinusoidal terms. Under this assumption and the further condition of the presence of filtering effect in the loop, the nonlinear elements of the system are then represented by blocks that are functions of the input amplitude and frequency, as their action on each sinusoidal term were linear. The describing function approach, thus, introduces a quasi-linear approximation of the system under the assumption of the periodicity of the involved signals. The approach, however, can be further generalised to the case of almost-periodic signals with arbitrary, also incommensurable, frequencies (Cook, 1994). It is known, in fact, that although a rigorous justification of the describing function method is a very delicate question, it provides a reasonably accurate approximation of the behaviour of many classical nonlinear systems. Even in presence of more harmonics, the describing function approximation is useful to predict the onset of complex behaviour, such as chaos (Genesio & Tesi, 1992).
Why do nonlinearities matter? The repercussions of linear assumptions on the dynamic behaviour of assemble-to-order systems
Published in International Journal of Production Research, 2019
Note that a fundamental requirement for the system is that CL must be at least larger than averaged demand due to the accumulative errors driven by the feedback integrator (1/s). In other word, the DORATESA will increase exponentially if manufacturing capacity is less than the averaged demand rate and the system will become unstable. Under the assumption, the output function, ORATESA, can be represented by three linear piecewise equations as follow: To analyse the discontinuous nonlinearities in the ATO system, the describing function method can be applied (Spiegler et al. 2016b; Spiegler and Naim 2017). This method is a quasi-linear representation for a nonlinear element subjected to specific input signal forms such as Bias, Sinusoid and Gaussian process and system's low-pass filter property (Vander and Wallace 1968). The principle advantage of using the describing function method is it enables the aid of analytically designing nonlinear systems. The basic idea is to replace the nonlinear component by a type of transfer function, or a gain derived from the effect of input (e.g. sinusoidal input). For an asymmetric saturation, as illustrated in Figure 5, DORATESA is smaller than zero or greater than CL, at least two terms need to be identified: one term describes the change in amplitude (NA(CA)) in relation to the input amplitude and the other defines the change in mean (NB(CA)) in relation to the input mean. Furthermore, output phase angle (φ) in relation to the input angle may also be changed.