收稿日期: 2023-09-28
网络出版日期: 2024-02-26
基金资助
国家自然科学基金资助项目(62073259)
Probability Density Function Shape Control Method for Nonlinear Stochastic Systems Based on Compactly Supported Multi-Variable Polynomials
Received date: 2023-09-28
Online published: 2024-02-26
Supported by
the National Natural Science Foundation of China(62073259)
针对非线性随机系统的概率密度函数（PDF）形状控制问题，文中以Fokker-Planck-Kolmogorov（FPK）方程为研究工具，提出了一种基于紧支撑多变量多项式（CSMP）函数的非线性随机系统PDF形状控制方法。当系统处于稳定状态时，系统的PDF被困在特定的紧凑子空间中，不需要对整个空间进行积分。而CSMP函数在一段连续的空间内非0，满足紧凑子空间的特征。因此，文中将CSMP的线性组合（CSMP-LC）作为FPK方程的稳态近似解逼近目标PDF。首先，采用飞蛾扑火优化（MFO）算法优化CSMP-LC函数的参数；然后，通过对多维稳态FPK方程的每一维状态变量进行积分，确保稳态FPK方程在整个空间中的积分为0；最后，求解出一维和二维非耦合的状态变量PDF形状控制器，并进行了仿真实验。结果表明，对于一维非线性随机系统，文中提出的方法能有效地实现对不同类型目标PDF形状（单峰形状、双峰形状、三峰形状）的控制，且在目标PDF形状为复杂的三峰时，文中方法的均值、方差、峰度和偏度误差均优于其他两种方法。文中方法扩展到二维状态变量非耦合的非线性随机系统时，也能较好地实现对PDF形状的控制，为多变量随机系统的PDF形状控制研究提供了新的思路。同时，CSMP函数可以减少积分计算的复杂性，降低了非线性随机系统的PDF形状控制器的求解难度。
王玲芝 , 张坤 , 钱富才 . 基于紧支撑多变量多项式函数的非线性随机系统概率密度函数形状控制方法[J]. 华南理工大学学报(自然科学版), 2024 , 52(7) : 39 -52 . DOI: 10.12141/j.issn.1000-565X.230610
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.
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