Research On PDF Control Algorithm For Stochastic Systems

Stochastic systems exist widely in aerospace,maritime transport,industrial production and other fields.Stochastic control theory has been made great achievements over the past decades.Among these achievements,the objectives of designing controllers for stochastic systems are mostly functional mean and variance.This kind of controller design method is just suitable for linear stochastic systems.However,actual systems usually have nonlinear characteristics,so the control methods based on mean and variance have great limitations in both theory and application.In order to overcome the shortcomings of existing control methods,a probability density function(PDF)control strategy is developed.In this paper,three PDF control algorithms for nonlinear stochastic systems based on the Fokker-Planck-Kolmogrov(FPK)equation are studied,and the application of PDF control in ship's parametric excitation rolling motion is analyzed.In addition,PDF control and its application are simulated and validated.The specific research work includes the following aspects:(1)The PDF control algorithm based on Gaussian form.Firstly,Gaussian PDF or multi-Gaussian PDF is used to approximate the given target PDF.Then,the approximate PDF obtained by the global version artificial fish swarm algorithm is substituted into the FPK equation to solve and determine the control function.The simulation results show that the determined control function can effectively control the PDF of the state variables of the systems affected by the external or parametric excitation.(2)The PDF control algorithm based on exponential polynomial form.The one-to-one relationship between the expansion coefficients of PDF exponential polynomial and the coefficients of the controlled system equation is obtained by the steady state analysis of the FPK equation.Then the control function is determined by the corresponding relationship.And different control strategies are considered according to the different conditions of the target PDF.The simulation results show that for systems affected by external excitation,the determined control function can enable the controlled PDF to track the target PDF with simple shape effectively.(3)The PDF control algorithm based on multi-exponential polynomial form.For PDFs with complex shapes,it is difficult to achieve ideal tracking accuracy by using the PDF control algorithm based on exponential polynomial form.Therefore,the concept of basis function and the functional form of exponential polynomial are considered comprehensively.Firstly,the target PDF is approximated by the multi-exponential polynomial PDF.Then,the control function is determined by the approximate solution of the FPK equation.Effective tracking of the complex target PDF is realized.(4)PDF control is applied to ship's rolling motion due to parametric excitation.Firstly,the Ito differential equation of ship's parametric excitation nonlinear rolling motion in stochastic longitudinal waves is set up.Then,the corresponding FPK equation is derived.Finally,the control function is determined by the PDF control algorithm based on Gaussian form,so that the PDF of stable stochastic rolling amplitude can reach the desired shape,thus avoiding the capsizing of the ship.Finally,some concluding remarks are given,and the future research works are pointed out.