The PDF shape tracking control for nonlinear stochastic systems

Aiming at a class of nonlinear stochastic systems with additive Gaussian White noise and the polynomial nonlinear function we propose a shape control technique for probability density function (PDF) of the state variable. Controlling the PDF shape requires to design a controller making the PDF shape as close as possible to the desired PDF, and this actually is to determine the parameters of the controller. Firstly, we design the control law of the controller to be polynomial form, which is substituted into the dynamical equation of the nonlinear systems, obtaining the corresponding FPK equation. After a mount of derivation, we have the solution of the FPK equation, and the solution includes the parameters of the controller. Adopting the linear least square method, we find the exact solution of the FPK equation, and then solve out the parameters of the controller. Lastly, the algorithm is verified to be effective through the simulation.