Research on PDF Shape Control for Nonlinear Stochastic System Using an Approximate Solution of FPK Equation

Abstract

In this paper, we developed a probability density function (PDF) shape control method for non-linear stochastic systems using a hybrid logistic function (HLF) as an approximate PDF of the state variable. First, the functional relationship between the hybrid logistic probability density function and the controller was established based on the Fokker–Planck–Kolmogorov (FPK) equation. Then, the optimal PDF shape controller derivation was completed using the optimization method and the inner product definition of Hilbert space. This approach is suitable for any non-linear stochastic system. To evaluate the effectiveness and performance of the proposed method, we conducted a comparison experiment with the multi-Gaussian closure (MGC) method and the exponential polynomial (EP) method. The experimental results show that, for different types of targeted PDFs (symmetric unimodal, asymmetric unimodal, bimodal, and trimodal shapes), the PDF shape controller obtained using the HLF approach can make the PDF shape of the state variable track the targeted PDF effectively. In particular, when the targeted PDF has an asymmetric or complex trimodal shape, the proposed technique has comparatively better control effects. Compared with the EP method, our method requires a much smaller number of parameters, greatly reducing the computational complexity while achieving the same control effects. This study provides another approach for controlling the PDF shape of state variables in non-linear stochastic systems, which has important research significance.