Developing an innovative method to control the probability density function shape of the state response for nonlinear stochastic systems

2021 ◽  
Author(s):  
Lingzhi Wang ◽  
Guo Xie ◽  
Fucai Qian ◽  
Anqi Shangguan
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Stochastic Systems ◽  
The State ◽  
Nonlinear Stochastic Systems ◽  
State Response ◽  
Innovative Method ◽  
Function Shape

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.

Exact Stationary Response Solutions of Nine Classes of Nonlinear Stochastic Systems Under Stochastic Parametric and External Excitations

1999 ◽  
Author(s):  
Rubin Wang ◽  
Kimihiko Yasuda
Keyword(s):  
Probability Density Function ◽  
Nonlinear System ◽  
Probability Density ◽  
Density Function ◽  
Stochastic Systems ◽  
Gaussian White Noise ◽  
Nonlinear Function ◽  
Stationary Solutions ◽  
Stationary Probability ◽  
Nonlinear Stochastic Systems

Abstract In this paper, a systematic procedure is developed to obtain the stationary probability density function for the response of a general nonlinear system under parametric and external Gaussian white noise excitations. In reference [15], nonlinear function of system was expressed to the polynomial formula. The nonlinear system described here has the following form: x¨+g(x,x˙)=k1ξ1(t)+k2xξ2(t), where g(x,x˙)=∑i=0∞gi(x)x˙i and ξ1,ξ2 are Gaussian white noises. Thus, this paper is a generalization for the results studied in reference [15]. The reduced Fokker-Planck (FP) equation is employed to get the governing equation of the probability density function. Based on this procedure, the exact stationary probability densities of many nonlinear stochastic systems are obtained, and it is shown that some of the exact stationary solutions described in the literature are only particular cases of the presented generalized results.

A filter with a guaranteed asymptotic performance

1993 ◽  
Vol 441 (1911) ◽  
pp. 33-57
Keyword(s):  
Kalman Filter ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Extended Kalman Filter ◽  
Noise Level ◽  
Low Noise ◽  
The State ◽  
Optimal Probability ◽  
Expected Values

In the case of low noise levels the optimal probability density function summarizing the available information about the state of a system can be accurately approximated by the product of a gaussian function and a linear function. The approximation preserves the ability to estimate to an accuracy of O ( λ -2 ) the expected value of any twice continuously differentiable function defined on the state space. The parameter λ depends on the noise level. If the noise level in the system is low then λ is large. A new filtering method based on this approximation is described. The approximating function is updated recursively as the system evolves with time, and as new measurements of the system state are obtained. The updates preserve the ability to estimate the expected values of functions to an accuracy of O ( λ -2 ). The new filter does not store previous measurements or previous approximations to the optimal probability density function. The new filter is called the asymptotic filter, because the definition of the filter and the analysis of its properties are based on the theory of asymptotic expansion of integrals of Laplace type. An analysis of the state propagation equations shows that the asymptotic filter performs better than a particular widely used suboptimal approximation to the optimal filter, the extended Kalman filter. The extended Kalman filter does not, in general, preserve the ability to estimate expected values to an accuracy of O ( λ -2 ). The computational cost of the asymptotic filter is comparable to that of the iterated extended Kalman filter.

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