Research On The Optimal Control Of Planar Inverted Pendulum Based On Kalman Filter

Inverted pendulum is a device with characteristics of nonlinear, multivariable, strong coupling, underactuated, unstable etc. It can directly reflect many key problems of control theory such as stability, controllability, robustness etc. Inverted pendulum is not only a typical experiment platform to study and verify the correctness and effectiveness of control algorithms, its theoretical system has been widely used and developed in robot steady walk, industrial process control, rocket vertical launch and satellite flight attitude control.In this paper, taking planar inverted pendulum as the research object, the trajectory tracking control of single planar inverted pendulum and the stability control of the two-stage planar inverted pendulum are studied. Compared with single planar inverted pendulum, two-stage planar inverted pendulum just add a pendulum, but the control difficulty is much bigger, because there the serious coupling between the first pendulum and the second pendulum, and the number of variables associated with pendulum is also twice of the single planar inverted pendulum. Planar inverted pendulum is subjected to a certain of noise interference, especially the two-stage planar inverted pendulum, if the lower pendulum angle detection is not accurate, the upper pendulum will be unstable, besides, the upper pendulum angle also has measurement noise, by differential operation, the noise is amplified, the output of the controller is not stable, and mixes with process noise, ultimately affect the stability of the planar inverted pendulum. In order to solve the problems of coupling, multivariable and noise interference of the planar inverted pendulum, the system model need to be established accurately, the controller which can optimize the control problem and the filter whice can suppress noise interference need to be designed.This paper establishes the nonlinear mathematical model by using Lagrange equation, linearizes the nonlinear model around the equilibrium position, the same decoupled and linear model in its orthogonal directions is obtained. Then, according to the linear quadratic optimal control theory, LQR controller of the planar inverted pendulum is designed. And then, the noise parameters of the planar inverted pendulum is obtained, and steady-state Kalman filter is designed to suppress the interference caused by noise. Finally, the LQR optimal controller based on steady-state Kalman filter is used, the stability control of the two-stage planar inverted pendulum and the trajectory tracking control of single planar inverted pendulum are realized. The experimental results show that the LQR optimal controller based on steady-state Kalman filter can effectively suppress the noise interference, improve the stability and robustness of system.The main work and innovation of this paper are as follows:Firstly, this paper introduces the background, significance and current situation of the research on the planar inverted pendulum. On the basis of analyzing the planar inverted pendulum, the nonlinear mathematical model of single and two-stage planar inverted pendulum are established by using Lagrange equation, then the nonlinear model around the equilibrium position is linearized by using limit and infinitesimal theorem, the decoupled and linear model of single and two-stage planar inverted pendulum in their orthogonal directions are obtained. Being different from other researchers who use Taylor series to linearize the nonlinear model, this paper uses limit and infinitesimal theorem to linearize the nonlinear model, simplify the modeling process.Then, this paper proposes the trajectory of planar inverted pendulum. For single planar inverted pendulum, the LQR optimal controller which base cart positioning prior to pendulum stabilization is designed, the trajectory tracking control simulation of single planar inverted pendulum is realized by MATLAB. For two-stage planar inverted pendulum, the LQR optimal controller which pendulum stabilization prior to cart positioning is designed, the pendulum stability control simulation of single planar inverted pendulum is also realized by MATLAB.And then, based on a large number of planar inverted pendulum real-time data, a statistical analysis method is proposed to obtain the off-line noise parameters of the planar inverted pendulum, based on the noise parameter, the steady-state Kalman filter of single and two-stage planar inverted pendulum are designed respectively, the LQR optimal controller of planar inverted pendulum based on steady-state Kalman filter is obtained, realizing the control simulation of the single and two-stage planar inverted pendulum based on steady-state Kalman, the steady-state Kalman filter can effectively suppress the noise of the planar inverted pendulum. The method to obtain the noise parameters is different from other researchers who assume the noise parameter, based on the real-time data of planar inverted pendulum, this paper uses statistical analysis method, the method in this paper is more reasonable and more targeted.Finally, the LQR optimal controller based on steady-state Kalman filter is used, the trajectory tracking real-time control of single planar inverted pendulum and the stability real-time control of the two-stage planar inverted pendulum are realized. Comparative experimental results show that the LQR optimal controller based on steady-state Kalman filter proposed in this paper can effectively suppress the noise interference and simplify the control problem, finally improve the stability and robustness of the planar inverted pendulum.