Stability of a Nonlinear Feedback System in the Presence of Gaussian Noise

1962 ◽  
Vol 84 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Rangasami Sridhar ◽  
Rufus Oldenburger
Keyword(s):  
Gaussian Noise ◽  
Natural Extension ◽  
Feedback System ◽  
Frequency Component ◽  
Analog Computer ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Describing Function ◽  
Power Spectral ◽  
Describing Function Method

A stability criterion for certain types of nonlinear feedback systems in the presence of Gaussian noise is established here. This criterion may be considered as a natural extension of the describing function method. It is assumed that the lowest frequency component in the power spectral density of the noise is at least ten times higher than the highest significant frequency of the system. The method developed here is applicable to feedback systems with just one instantaneous, nonmemory type nonlinearity in the loop. The results mentioned in this paper have been experimentally verified on an analog computer. The theory explained here may be used by the designer to predict the manner in which noise will affect the performance of a system.

Predicting the Autonomous Behaviour of Simple Nonlinear Feedback Systems

1990 ◽  
Vol 43 (10) ◽  
pp. 251-260
Author(s):  
D. P. Atherton
Keyword(s):  
Limit Cycle ◽  
Chaotic Motion ◽  
Function Method ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Describing Function ◽  
Relay Systems ◽  
Software Implementations ◽  
Describing Function Method ◽  
Nonlinear Feedback Systems

The paper examines in depth two approaches, namely the describing function and Tsypkin methods, for predicting the autonomous behaviour of simple nonlinear feedback systems. Both procedures are supported by software which, in the case of the describing function method, allows iteration to the exact limit cycle solution and, for both methods, enables display of resulting limit cycle waveforms. One advantage of the Tsypkin method, which is applicable primarily to relay systems, is that the exact stability of the limit cycle solution can be found. It is shown how this may be helpful in indicating the possibility of chaotic motion. Several examples are given to show the advantages and limitations of the software implementations of the methods.

Stability of Subharmonic Oscillations in Nonlinear Systems

1964 ◽  
Vol 86 (1) ◽  
pp. 116-120 ◽  
Author(s):  
Rufus Oldenburger ◽  
Robert E. Nicholls
Keyword(s):  
Nonlinear Systems ◽  
Necessary Conditions ◽  
Analog Computer ◽  
Higher Order ◽  
Nonlinear Element ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Describing Function ◽  
Subharmonic Oscillation ◽  
Nonlinear Feedback Systems

This paper presents a method for finding necessary conditions such that a subharmonic oscillation may exist in certain types of nonlinear feedback systems. The method is applicable to feedback systems with one, instantaneous, nonmemory-type, nonlinear element. Equations are derived giving the fundamental output of a nonlinear element when forced by two sine waves of integer ratio frequency. Normal describing-function assumptions are made with regard to attenuation of higher-order harmonics. An example of a system incorporating a perfect relay is presented. The results of the analysis have been verified experimentally on the analog computer.

Coexistent multiple-stability of a fractional-order delayed memristive Chua’s system based on describing function

2020 ◽  
Vol 34 (14) ◽  
pp. 2050146
Author(s):  
Dawei Ding ◽  
Jun Luo ◽  
Xiangyu Shan ◽  
Yongbing Hu ◽  
Zongli Yang ◽  
...  
Keyword(s):  
Fractional Order ◽  
Feedback System ◽  
Chua’S Circuit ◽  
Nonlinear Feedback ◽  
Describing Function ◽  
Chua's Circuit ◽  
System A ◽  
Dynamical Behaviors ◽  
Low Pass ◽  
Multiple Stability

In this paper, in order to analyze the coexistent multiple-stability of system, a fractional-order memristive Chua’s circuit with time delay is proposed, which is composed of a passive flux-controlled memristor and a negative conductance as a parallel combination. First, the Chua’s circuit can be considered as a nonlinear feedback system consisting of a nonlinear block and a linear block with low-pass properties. In the complex plane, the nonlinear element of the system can be approximated by a variable gain called a describing function. Second, compared with conventional computation, the describing function can accurately predict the hidden dynamics, fixed points, periodic orbits, unstable behaviors of the system. By using this method, the full mapping of the system dynamics in parameter spaces is presented, and the coexistent multiple-stability of the system is investigated in detail. Third, using bifurcation diagram, phase diagram, time domain diagram and power spectrum diagram, the dynamical behaviors of the system under different system parameters and initial values are discussed. Finally, based on Adams–Bashforth–Moulton (ABM) method, the correctness of theoretical analysis is verified by numerical simulation, which shows that the fractional-order delayed memristive Chua’s system has complex coexistent multiple-stability.

An Extension of the Describing-Function Method

1966 ◽  
Vol 88 (2) ◽  
pp. 469-474 ◽  
Author(s):  
Eugenio Sarti ◽  
Yasundo Takahashi
Keyword(s):  
State Space ◽  
Function Method ◽  
Block Diagram ◽  
Autonomous Systems ◽  
Analog Computer ◽  
Feedback Control System ◽  
Describing Function ◽  
Scalar Variable ◽  
Describing Function Method ◽  
Function Approximations

The use of the describing-function method has been generally limited to systems with nonlinearities which can be combined in a block diagram of a feedback control system which is represented by a scalar variable. This paper deals with an extension of the method to some autonomous systems with more than one nonlinearity. The paper also shows that nonlinear systems can have hypersurfaces or manifolds in state space which divide regions of different stability characteristics. Each of these manifolds can be regarded as a boundary element of a family of surfaces related to a Lyapunov function. The describing-function method is applied in such a way that the hypersurfaces are approximated by a quadratic form. Two cases in an example, for which data obtained by analog computer and describing function approximations are compared, show that the amount of error caused by the approximation can be acceptably small, particularly in the subspace of the state space in which a system exhibits nonlinearities.

Passivity-Based Tracking Control for Uncertain Nonlinear Feedback Systems

2016 ◽  
Vol 28 (6) ◽  
pp. 837-841 ◽  
Author(s):  
Ni Bu ◽  
◽  
Mingcong Deng ◽  
Keyword(s):  
Control Problem ◽  
Tracking Control ◽  
Feedback System ◽  
Factorization Method ◽  
Tracking Performance ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Control Scheme ◽  
Asymptotic Tracking ◽  
Nonlinear Feedback Systems

[abstFig src='/00280006/07.jpg' width='300' text='The asymptotic tracking performance and the passivity property' ] The tracking control problem for the uncertain nonlinear feedback systems is considered in this paper by using passivity-based robust right coprime factorization method. Concerned with the passivity for the nonlinear feedback system, two stable controllers are designed such that the nonlinear feedback system is robust stable and the plant output asymptotically tracks to the reference output. A numerical example is given to show the validity of the control scheme as well as the tracking performance.

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