Probability Density Function Shape Control Method for Nonlinear Stochastic Systems Based on Compactly Supported Multi-Variable Polynomials

doi:10.12141/j.issn.1000-565X.230610

Abstract

Abstract:

For the probability density function (PDF) shape control problem of nonlinear stochastic systems, this paper used the Fokker-Planck-Kolmogorov (FPK) equation as a tool and proposed a PDF shape control method based on the compactly supported multivariable polynomials (CSMP) function. When the system is in a steady state, the PDF of the system was trapped in a specific compact subspace and didn’t need to be integrated over the whole space. The CSMP function is non-zero in a continuous space, satisfying the compact subspace characteristic. Therefore, the linear combination of CSMP (CSMP-LC) was utilized as the steady-state approximate solution of FPK equation for approaching the target PDF. Firstly, the moth-flame optimization (MFO) algorithm was used for optimizing parameters of the CSMP-LC function. Then, by integrating each dimensional state variable of the multidimensional steady-state FPK equation, the integration of the steady-state FPK equation over the whole space was ensured to be zero. Finally, the solution of the one-dimensional and two-dimensional uncoupled state variable PDF shape controller was completed, and simulation experiments were conducted. The results demonstrate that the proposed method can achieve PDF shape control for different types of target PDFs (single-peaked shapes, double-peaked shapes and triple-peaked shapes) for one-dimensional nonlinear stochastic systems. In particular, for complex triple-peaked shapes, it has a significant advantage over the multi-Gaussian closure method and the exponential polynomial method. The method in the paper was extended to the nonlinear stochastic system with uncoupled two-dimensional state variables, which can better realize the control of PDF shape and provide a new research idea for the study of PDF shape control of multivariate stochastic systems. Moreover, the CSMP function can reduce the complexity of the integral computation and reduce the difficulty of solving PDF shape controllers for nonlinear stochastic systems.

Key words: Fokker-Planck-Kolmogorov equation, nonlinear stochastic system, probability density function, compactly supported multi-variable polynomials

CLC Number:

TP271⁺.62

WANG Lingzhi, ZHANG Kun, QIAN Fucai. Probability Density Function Shape Control Method for Nonlinear Stochastic Systems Based on Compactly Supported Multi-Variable Polynomials[J]. Journal of South China University of Technology(Natural Science Edition), 2024, 52(7): 39-52.

Figures/Tables 7

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目标PDF形状	控制误差
目标PDF形状	MGC	EP	CSMP
单峰	0.000 00	0.000 00	0.001 59
双峰	0.002 96	0.193 10	0.002 99
三峰	0.003 24	0.115 32	0.000 65

PDF形状	PDF	均值	方差	峰度	偏度
单峰	目标PDF	0	1.000 0	4.900 1	1.829 3
	MGC-PDF	0	1.000 0	4.900 1	1.829 3
	EP-PDF	0	0.999 9	4.900 1	1.829 5
	CSMP-PDF	0	0.842 3	4.405 4	1.730 4
双峰	目标PDF	0	1.500 0	5.060 9	1.878 9
	MGC-PDF	-5.000 0×10^-5	1.370 0	5.232 5	1.827 0
	EP-PDF	-1.387 8×10^-17	1.144 5	5.203 6	1.909 6
	CSMP-PDF	-5.554 5×10^-8	1.345 6	4.725 2	1.885 1
三峰	目标PDF	-4.475 5	12.069 1	5.434 8	1.969 0
	MGC-PDF	-3.164 7	12.435 4	6.082 1	2.071 7
	EP-PDF	8.437 0×10^-7	31.826 4	13.601 1	3.499 4
	CSMP-PDF	-4.425 4	12.053 3	5.061 6	1.905 2

状态变量	前四阶矩	目标PDF	CSMP-PDF	误差
x₁	均值	1.000 0	1.000 0	0.000 0
	方差	0.500 0	0.417 5	0.082 5
	峰度	4.217 4	3.912 1	0.305 2
	偏度	1.642 3	1.581 0	0.061 3
x₂	均值	0.500 0	0.500 0	0.000 0
	方差	1.000 0	0.835 2	0.164 8
	峰度	2.722 6	2.468 0	0.244 6
	偏度	1.123 9	1.053 5	0.070 4

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