Stochastic Nonlinear System Control Based On Probability Density Function Compensation

With the development of control theory,the design of controllers and state estimators under random environments has attracted more and more attention.Since the random environment will destroy the stability of the controlled system and reduce the accuracy of the system,it is of great significance to achieve high-precision tracking and state estimate.This paper is aimed at stochastic systems,using probability density functions to realize the design of controller,state estimator.For the design of discrete-time stochastic system controller with stochastic nonlinearity,a fusion control strategy is proposed,including probability density function(PDF)compensation and gain-scheduled control.Generally,it is difficult to strictly control the tracking error to zero in a random environment.Therefore,the proposed method aims to achieve stability in the exponential mean square sense of the closed-loop system,and stable in the distribution sense,that is,the distribution of the output error is as close as possible to the expected PDF.By solving the appropriate linear matrix inequality,the gain parameters of the gain scheduling controller can be obtained,and the parameters of the PDF compensator can be updated in real time according to the Kullback-Leibler difference between the output error PDF and the required error.This method has the characteristics of low conservativeness and easy implementationFor the state estimator design of discrete systems lacking measurement and Gaussian mixed noise,a hybrid state estimating strategy is proposed,including gain-scheduled state estimator and probability density function compensation.In a Gaussian mixture noise environment,the gain-scheduled state estimator is used to stabilize the random variable of the estimated error,and the linear matrix inequalities(LMIs)is constructed by the Lyapunov function to obtain the state estimator gain matrix parameters;the PDF compensator is used to make the error variable track the expected Gaussian distribution,whose mean is zero and variance is small.In order to further improve the real-time estimation speed,a PDF compensation optimization index based on Wassertein distance is proposed to dynamically update the gain of the compensator.The state estimator designed in this paper is applicable to both multiplicative noise and additive noise systems.Finally,the effectiveness of the proposed method is verified by simulation.