Control design to shape the stationary probability density function

2005 ◽  
Vol 27 (5) ◽  
pp. 331-346 ◽  
Author(s):  
Michael G. Forbes ◽  
J. Fraser Forbes ◽  
Martin Guay
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Control Design ◽  
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 Stochastic distribution tracking control for stochastic non-linear systems via probability density function vectorisation

2021 ◽  
pp. 014233122110169
Author(s):  
Yefeng Liu ◽  
Qichun Zhang ◽  
Hong Yue
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Real Time ◽  
Linear Systems ◽  
Density Function ◽  
Control Design ◽  
Data Driven ◽  
System Output ◽  
Non Linear Systems ◽  
Stochastic Distribution

This paper presents a new control strategy for stochastic distribution shape tracking regarding non-Gaussian stochastic non-linear systems. The objective can be summarised as adjusting the probability density function (PDF) of the system output to any given desired distribution. In order to achieve this objective, the system output PDF has first been formulated analytically, which is time-variant. Then, the PDF vectorisation has been implemented to simplify the model description. Using the vector-based representation, the system identification and control design have been performed to achieve the PDF tracking. In practice, the PDF evolution is difficult to implement in real-time, thus a data-driven extension has also been discussed in this paper, where the vector-based model can be obtained using kernel density estimation (KDE) with the real-time data. Furthermore, the stability of the presented control design has been analysed, which is validated by a numerical example. As an extension, the multi-output stochastic systems have also been discussed for joint PDF tracking using the proposed algorithm, and the perspectives of advanced controller have been discussed. The main contribution of this paper is to propose: (1) a new sampling-based PDF transformation to reduce the modelling complexity, (2) a data-driven approach for online implementation without model pre-training, and (3) a feasible framework to integrate the existing control methods.

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.

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