SDRE Based Rotary Pendulum Control System

Inverted pendulum,which is wildly used in the research of control theory,is a typical under actuated system with characteristics of nonlinear and unstable.SDRE method,proposed recently,is like LQR which is a control schemes based on mathematical model,and its character is the system matrix after parameterization and the weight matrix relate to system state,therefore much more flexibilities in design. It is hard to get analytical formulations of close-loop system based on SDRE,rendering difficulties in analyzing stability of close-loop systems.A parameterization method is introduced after establishing nonlinear differential equations of one-level-rotary pendulum,thereby a pseudo-linearizing model.Methods of overvaluing system and Julia set for stabilities' analyzing are respectively represented for high order and low order systems,based on which attraction of region of close loop systems can be estimated.Research on system's point-wise controllability reveals the system is controllable except the points in which the pendulum is down vertical.Then a SDRE based controller is designed,and the controller's gain relates to the state,which results in complexity in computing but good effects of control.Comparative simulations between SDRE and LQR reflect that advantages of the SDRE technique such as small over-shooting,short settling time,are over the LQR design while a larger deflection from balancing point;and only a single controller is needed for both swing-up and balancing control of the pendulum.Since control gain relates to system states and it have to solve Riccati equations every time steps,which rendering complexity in computing,it's necessary to design a reasonable algorithm real-time implementation of SDRE.The paper presents a solving method called eigen-structure assignment.