Abstract
The shape control of probability density function (PDF) of the system state is an important topic in stochastic systems. In this paper, we propose a control technique for PDF shape of the state variable in nonlinear stochastic systems. Firstly, we derive and prove the form of the controller by investigating the Fokker-Planck-Kolmogorov (FPK) equation arising from the stochastic system. Secondly, an approach for getting approximate solution of the FPK equation is provided. A special function including some parameters is taken as the approximate stationary solution of the FPK equation. We use nonlinear least square method to solve the parameters in the function, and capture the approximate solution of the FPK equation. Substituting the approximate solution into the form of the controller, we can acquire the PDF shape controller. Lastly, some example simulations are conducted to verify the algorithm.
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Foundation item: the National Natural Science Foundation of China (No. 61273127), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20116118110008), the Scientific Research Plan Projects of Shaanxi Education Department (No. 12JK0524)
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Wang, Lz., Qian, Fc. Technique of probability density function shape control for nonlinear stochastic systems. J. Shanghai Jiaotong Univ. (Sci.) 20, 129–134 (2015). https://doi.org/10.1007/s12204-015-1600-4
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DOI: https://doi.org/10.1007/s12204-015-1600-4
Key words
- nonlinear stochastic systems
- probability density function (PDF)
- shape control
- Fokker-Planck-Kolmogorov (FPK) equation