Stochastic Dynamics And Control Of Inverted Pendulum System Under Bounded Noise Excitation

The inverted pendulum system is a typical controlled object and is widely used in the field of control theory research.The study of inverted pendulum has not only theoretical significance,but also great engineering significance.Many systems in the industrial field and aerospace field need to use inverted pendulum system for research.For the solution of nonlinear systems,the method of linearizing it is usually adopted,but when the influence of nonlinear factors is large,this method can no longer meet the requirements.Therefore,the stochastic averaging method is used to solve the nonlinear system.The stochastic averaging method can reduce the system dimension and simplify the solution.In this paper,the random response and stability of the inverted pendulum nonlinear system under bounded noise excitation are studied.The differential equations of the inverted pendulum are established,and the method of direct separation motions is used to divide the motion into two parts;fast and slow components,and the equivalent inverted pendulum system is obtained.Then the motion equations are reduced to Ito differential equations for response amplitude and phase difference by using the stochastic averaging method,and the corresponding FPK equations are solved.The random average method is used to analyze and solve the FPK equation,and the stochastic dynamic response and stability of the inverted pendulum nonlinear system under bounded noise excitation are analyzed.The analysis results show that the theoretical results obtained by the stochastic averaging method are basically consistent with the numerical simulation results.Therefore,the stochastic averaging method can predict the random response and stability of the inverted pendulum system under bounded noise excitation environment.Then introduce the principle and characteristics of sliding mode variable structure control.For the inverted pendulum system studied in this paper,the sliding mode variable structure controller is designed which minimizes the system's response.The simulation analysis of the system under sliding mode control is carried out,and analyze the influence of controller parameters on the system response.Compared with Bang-Bang control,the analysis shows that the control effect of sliding mode control is better than Bang-Bang control and the system can reach the steady state more quickly with no large oscillations.The sliding mode variable structure control method is applied to control the inverted pendulum under bounded noise excitation,and its response and stability are analyzed.The results show that the analytical solution is in good agreement with the Monte Carlo digital simulation,indicating the approximation method of this paper.It can accurately predict the random response and stability of the controlled object,and can be effectively applied to the study of the system response of the inverted pendulum in nonlinear stochastic systems.