Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems

1992 ◽  
Vol 114 (1) ◽  
pp. 20-26 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Monte Carlo ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Stochastic System ◽  
Linearization Method ◽  
Weighted Sum ◽  
Gaussian Density ◽  
Parametric Noise ◽  
Non Gaussian ◽  
Parametric And External Excitations

A new practical non-Gaussian linearization method is developed for the problem of the dynamic response of a stable nonlinear system under both stochastic parametric and external excitations. The non-Gaussian linearization system is derived through a non-Gaussian density that is constructed as the weighted sum of undetermined Gaussian densities. The undetermined Gaussian parameters are then derived through solving a set of nonlinear algebraic moment relations. The method is illustrated by a Duffing-type stochastic system with/without parametric noise excited term. The accuracy in predicting the stationary and nonstationary variances by the present approach is compared with some exact solutions and Monte Carlo simulations.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

scholarly journals Online Health Management for Complex Nonlinear Systems Based on Hidden Semi-Markov Model Using Sequential Monte Carlo Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Qinming Liu ◽  
Ming Dong
Keyword(s):  
Monte Carlo ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Markov Model ◽  
Sequential Monte Carlo ◽  
Health Management ◽  
Health State ◽  
Health States ◽  
Complex Nonlinear System ◽  
Complex Nonlinear Systems

Health management for a complex nonlinear system is becoming more important for condition-based maintenance and minimizing the related risks and costs over its entire life. However, a complex nonlinear system often operates under dynamically operational and environmental conditions, and it subjects to high levels of uncertainty and unpredictability so that effective methods for online health management are still few now. This paper combines hidden semi-Markov model (HSMM) with sequential Monte Carlo (SMC) methods. HSMM is used to obtain the transition probabilities among health states and health state durations of a complex nonlinear system, while the SMC method is adopted to decrease the computational and space complexity, and describe the probability relationships between multiple health states and monitored observations of a complex nonlinear system. This paper proposes a novel method of multisteps ahead health recognition based on joint probability distribution for health management of a complex nonlinear system. Moreover, a new online health prognostic method is developed. A real case study is used to demonstrate the implementation and potential applications of the proposed methods for online health management of complex nonlinear systems.

A Practical Technique for Spectral Analysis of Nonlinear Systems Under Stochastic Parametric and External Excitations

1991 ◽  
Vol 113 (4) ◽  
pp. 516-522 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Spectral Response ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Matching Condition ◽  
External Noise ◽  
Equivalent Linearization ◽  
Parametric Noise ◽  
Parametric And External Excitations

A practical approach is developed for analyzing the spectral response of a nonlinear system subjected to both parametric and external Gaussian white noise excitations. The technique is implemented through the combined methods of equivalent external excitation and equivalent linearization to derive an equivalent linear system under equivalent external noise excitation. The spectral response is then obtained through utilizing the input/output spectral relation and covariance matching condition. A parametric noise excited linear system, Duffing oscillator, and nonlinear system with hysteretic nonlinearity are selected for investigation. The validity of the proposed method for analyzing spectral response is further supported by some analytical solutions and FFT technique through Monte Carlo simulations.

scholarly journals An Ensemble-Based Smoother with Retrospectively Updated Weights for Highly Nonlinear Systems

2007 ◽  
Vol 135 (1) ◽  
pp. 186-202 ◽  
Author(s):  
T. M. Chin ◽  
M. J. Turmon ◽  
J. B. Jewell ◽  
M. Ghil
Keyword(s):  
Monte Carlo ◽  
Nonlinear Systems ◽  
Ensemble Kalman Filter ◽  
Nonlinear Problems ◽  
Monte Carlo Algorithm ◽  
Operational Mode ◽  
Highly Nonlinear ◽  
Non Gaussian ◽  
Double Well Potential ◽  
Full Probability

Abstract Monte Carlo computational methods have been introduced into data assimilation for nonlinear systems in order to alleviate the computational burden of updating and propagating the full probability distribution. By propagating an ensemble of representative states, algorithms like the ensemble Kalman filter (EnKF) and the resampled particle filter (RPF) rely on the existing modeling infrastructure to approximate the distribution based on the evolution of this ensemble. This work presents an ensemble-based smoother that is applicable to the Monte Carlo filtering schemes like EnKF and RPF. At the minor cost of retrospectively updating a set of weights for ensemble members, this smoother has demonstrated superior capabilities in state tracking for two highly nonlinear problems: the double-well potential and trivariate Lorenz systems. The algorithm does not require retrospective adaptation of the ensemble members themselves, and it is thus suited to a streaming operational mode. The accuracy of the proposed backward-update scheme in estimating non-Gaussian distributions is evaluated by comparison to the more accurate estimates provided by a Markov chain Monte Carlo algorithm.

scholarly journals Non-Gaussian Stochastic Equivalent Linearization Method for Inelastic Nonlinear Systems with Softening Behaviour, under Seismic Ground Motions

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Francisco L. Silva-González ◽  
Sonia E. Ruiz ◽  
Alejandro Rodríguez-Castellanos
Keyword(s):  
Monte Carlo ◽  
Ground Motions ◽  
Joint Probability ◽  
Structural Response ◽  
Linearization Method ◽  
Degree Of Freedom ◽  
Structural Behavior ◽  
Equivalent Linearization ◽  
Equivalent Linearization Method ◽  
Non Gaussian

A non-Gaussian stochastic equivalent linearization (NSEL) method for estimating the non-Gaussian response of inelastic non-linear structural systems subjected to seismic ground motions represented as nonstationary random processes is presented. Based on a model that represents the time evolution of the joint probability density function (PDF) of the structural response, mathematical expressions of equivalent linearization coefficients are derived. The displacement and velocity are assumed jointly Gaussian and the marginal PDF of the hysteretic component of the displacement is modeled by a mixed PDF which is Gaussian when the structural behavior is linear and turns into a bimodal PDF when the structural behavior is hysteretic. The proposed NSEL method is applied to calculate the response of hysteretic single-degree-of-freedom systems with different vibration periods and different design displacement ductility values. The results corresponding to the proposed method are compared with those calculated by means of Monte Carlo simulation, as well as by a Gaussian equivalent linearization method. It is verified that the NSEL approach proposed herein leads to maximum structural response standard deviations similar to those obtained with Monte Carlo technique. In addition, a brief discussion about the extension of the method to muti-degree-of-freedom systems is presented.

Data-driven model-free sliding mode learning control for a class of discrete-time nonlinear systems

2020 ◽  
Vol 42 (13) ◽  
pp. 2533-2547
Author(s):  
Lei Cao ◽  
Shouli Gao ◽  
Dongya Zhao
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Discrete Time ◽  
Sliding Mode ◽  
Learning Control ◽  
Linearization Method ◽  
Data Driven ◽  
Controlled System ◽  
Model Free ◽  
The Mathematical Model

This paper proposes a data-driven model-free sliding mode learning control (MFSMLC) for a class of discrete-time nonlinear systems. In this scheme, the control design does not depend on the mathematical model of the controlled system. The nonlinear system can be transformed into a dynamic linear data system by a novel dynamic linearization method. A recursive learning control algorithm is designed for the nonlinear system that can drive the sliding variable reach and remain on the sliding surface only by using output and input data. Moreover, the chattering is reduced because there is no non-smooth term in MFSMLC. After the strict stability analysis, the effectiveness of MFSMLC is validated by MATLAB simulations.

Methods and Gaussian Criterion for Statistical Linearization of Stochastic Parametrically and Externally Excited Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 179-185 ◽  
Author(s):  
R. J. Chang ◽  
G. E. Young
Keyword(s):  
Nonlinear Systems ◽  
Goodness Of Fit ◽  
Duffing Oscillator ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Statistical Linearization ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Non Gaussian ◽  
Parametric And External Excitations

The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.

scholarly journals Stability and Controllability of Nonlinear Systems

2020 ◽  
pp. 51-60
Author(s):  
S. T. Cotterell ◽  
I. Davies ◽  
L. C. Abraham
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Equilibrium Solution ◽  
Indirect Method ◽  
Real Life ◽  
Linearization Method ◽  
Controlled System ◽  
The Stability ◽  
Theoretical Results

This paper is aimed at establishing new stability and controllability results for nonlinear systems. The approach is to use the Lyapunov indirect method to obtain the stability of the equilibrium solution of the uncontrolled nonlinear system by applying the Jacobi’s linearization method and the controllability of the controlled system obtained by the rank criterion for properness. Example is given with a real-life application to illustrate the effectiveness of the theoretical results.

Stationary Response of States-Constrained Nonlinear Systems Under Stochastic Parametric and External Excitations

1991 ◽  
Vol 113 (4) ◽  
pp. 575-581 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Fourier Series ◽  
Robot Manipulator ◽  
External Noise ◽  
Algebraic Equations ◽  
Working Space ◽  
Closure Scheme ◽  
Gaussian Density ◽  
The Moment ◽  
Non Gaussian ◽  
Parametric And External Excitations

A Fourier-series closure scheme is developed for the prediction of the stationary stochastic response of a stochastic parametrically and externally excited oscillator with a nonpolynomial type nonlinearity and under states constraint. The technique is implemented by deriving the moment relations and employing the Fourier series as the expansion of a non-Gaussian density for constructing and solving a set of algebraic equations with unknown Fourier coefficients. A single-arm robot manipulator operated in a constrained working space and subjected to parametric and/ or external noise excitations is selected to illustrate the present approach. The validity of the present scheme is further supported by some exact solutions and Monte Carlo simulations.

scholarly journals Output Tracking of Some Class Non-Minimum Phase Nonlinear Systems via linearization Input-Output

2021 ◽  
Vol 2123 (1) ◽  
pp. 012014
Author(s):  
Firman
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Control Design ◽  
Linearization Method ◽  
Minimum Phase ◽  
Tracking Problem ◽  
Input Output ◽  
Output Tracking ◽  
Input Control ◽  
Input Output Linearization

Abstract We present an output tracking problem for a non-minimum phase nonlinear system. In this paper, the input control design to solve the output tracking problem is to use the input output linearization method. The use of the input output linearization method cannot be initiated from output causing the system to be non-minimum phase. Therefore the output of the system will be redefined such that the system will become minimum phase with respect to a new output.

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