Feedback control of nonlinear stochastic dynamic systems for accurately tracking a specified stationary probability density function

Limited by the difficulty of obtaining the exact stationary solution of Fokker-Planck-Kolmogorov (FPK) equation, the general control strategy for tracking a specified stationary probability density function have not been obtained till now. However, the feedback control for tracking a pre-specified stationary probability density function plays an important role in engineering or industrial systems. In this paper, a feedback control strategy of nonlinear stochastic dynamic system for accurately tracking a specified stationary probability density function without the requirement of exact stationary solution of Fokker-Planck-Kolmogorov equation is proposed. According to the probability conservation form of the Fokker-Planck-Kolmogorov equation, the stationary Fokker-Planck-Kolmogorov equation is split into a probability circulation flow (PCF) equation and a probability potential flow (PPF) equation. The control force is divided into probability circulation flow part and probability potential flow part accordingly. The probability circulation flow part of the control force is determined to satisfy the probability circulation flow term of the controlled system constructed from the target stationary probability density function. The probability potential flow part of the control force is obtained by solving the probability potential flow equation. A two-dimensional nonlinear stochastic system is carried out as an example. The control force is designed to track different types of target stationary probability density functions. Numerical results show that the proposed control strategy can accurately track the stationary probability density functions without the requirement of the exact solution of Fokker-Planck-Kolmogorov equation. The control efficiency can be regulated by the control parameter C.