On joint stationary probability density function of nonlinear dynamic systems

1998 ◽  
Vol 130 (1-2) ◽  
pp. 29-39 ◽  
Author(s):  
Z. Zhang ◽  
R. Wang ◽  
K. Yasuda
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Nonlinear Dynamic Systems ◽  
Stationary Probability ◽  
Stationary Probability Density ◽  
Stationary Probability Density Function

Reshaping of the probability density function of nonlinear stochastic systems against abrupt changes

2019 ◽  
Vol 26 (7-8) ◽  
pp. 532-539
Author(s):  
Lei Xia ◽  
Ronghua Huan ◽  
Weiqiu Zhu ◽  
Chenxuan Zhu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Stochastic Systems ◽  
Stationary Probability ◽  
Markov Jump ◽  
Nonlinear Stochastic Systems ◽  
Stationary Probability Density ◽  
Abrupt Changes ◽  
Stationary Probability Density Function

The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.

scholarly journals On the Uncertainty Identification for Linear Dynamic Systems Using Stochastic Embedding Approach with Gaussian Mixture Models

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3837
Author(s):  
Rafael Orellana ◽  
Rodrigo Carvajal ◽  
Pedro Escárate ◽  
Juan C. Agüero
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Dynamic Systems ◽  
Gaussian Mixture ◽  
Error Model ◽  
Fault Detection And Diagnosis ◽  
Deterministic Models ◽  
Linear Dynamic Systems ◽  
Linear Dynamic

In control and monitoring of manufacturing processes, it is key to understand model uncertainty in order to achieve the required levels of consistency, quality, and economy, among others. In aerospace applications, models need to be very precise and able to describe the entire dynamics of an aircraft. In addition, the complexity of modern real systems has turned deterministic models impractical, since they cannot adequately represent the behavior of disturbances in sensors and actuators, and tool and machine wear, to name a few. Thus, it is necessary to deal with model uncertainties in the dynamics of the plant by incorporating a stochastic behavior. These uncertainties could also affect the effectiveness of fault diagnosis methodologies used to increment the safety and reliability in real-world systems. Determining suitable dynamic system models of real processes is essential to obtain effective process control strategies and accurate fault detection and diagnosis methodologies that deliver good performance. In this paper, a maximum likelihood estimation algorithm for the uncertainty modeling in linear dynamic systems is developed utilizing a stochastic embedding approach. In this approach, system uncertainties are accounted for as a stochastic error term in a transfer function. In this paper, we model the error-model probability density function as a finite Gaussian mixture model. For the estimation of the nominal model and the probability density function of the parameters of the error-model, we develop an iterative algorithm based on the Expectation-Maximization algorithm using the data from independent experiments. The benefits of our proposal are illustrated via numerical simulations.

Nonlinear Dynamic Characteristics of Giant Magnetostrictive Nanofilm-SMA Composite Beam in Axial Stochastic Excitation

2013 ◽  
Vol 284-287 ◽  
pp. 173-177
Author(s):  
Zhi Wen Zhu ◽  
Wei Guo ◽  
Jia Xu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Dynamic Characteristics ◽  
Nonlinear Dynamic ◽  
Composite Beam ◽  
Stochastic Bifurcation ◽  
Hysteresis Phenomenon ◽  
Stochastic Excitation ◽  
Nonlinear Dynamic Characteristics

In this paper, nonlinear dynamic characteristics of giant magnetostructive nanofilm-shape memory alloy (SMA) composite beam in axial stochastic excitation were studied. Von del Pol nonlinear difference item was introduced to interpret the hysteresis phenomenon of the strain-stress curve of SMA, and the hysteretic nonlinear dynamic model of giant magnetostructive nanofilm-SMA composite beam in axial stochastic excitation was developed. The steady-state probability density function and the joint probability density function of the system were obtained in quasi-nonintegrable Hamiltonian system theory. The result of simulation shows that the stability of the trivial solution varies with bifurcation parameter, and stochastic Hopf bifurcation appears in the process. The result is helpful to stochastic bifurcation control to giant magnetostructive nanofilm-SMA composite beam.

Sign in / Sign up

Export Citation Format

Share Document