On The Modeling, Control And Experiment Of The Planar Inverted Pendulum

The inverted pendulum is characterized as a typical nonlinear, high order, unstable and underactuated system. Research on precise control of the inverted pendulum has great value for control problems of complicated plant in industrial production process, as a result, the inverted pendulum simulation and physical experiments are representative project for verifying a certain control theory or approach in the fields of control science and technology. At present, linear inverted pendulum has found wide use in instruction due to the advantages of simplicity, intuition, easy to change the components parameters and low price. Rotary and planar inverted pendulums remain the research stage so far.The base cart of a planar inverted pendulum moves in the 2D plane, i.e. the base motion has two degrees-of-freedom (DOF). As a result of the number of system parameters increase and coupling characteristic of motion and control, the control problem of the planar inverted pendulum is more challenging compared with linear or rotary inverted pendulums, which base has only one DOF.The GPIP2002 planar inverted pendulum is considered as a plant in this paper. The mathematical models of the planar single and double inverted pendulums are established by using the Lagrange equations. Taloy series expansion of the mathematical models around the balance postion and linearation result in the decoupled linear models in two orthogonal directions of the planar single and double inverted pendulums respectively.Based on optimal control theory, linear quadratic optimal regulator (LQR) is designed for linear model of the inverted pendulum. Balancing and positioning control goals are achived at the same time, furthermore, simulation results shows the validity of the linear quadratic optimal control strategy in the control of planar inverted pendulums.A typical intelligent variable structure control approach, fuzzy sliding mode control (FSMC) combines robustness of sliding mode (SMC) control and independence of system model of fuzzy logical control (FLC). Adaptive FSMC incorporates the adaption into SMC and improves the control performance by adjusting the slopes of the sliding mode functions adptively. An adaptive sliding mode fuzzy controller is designed in each direction of the planar inverted pendulum, and positing control is realized under the condition of pole balace.