Studies On Nonlinear Dynamics And Control Of Inverted Pendulum

Inverted pendulum is a typical model of multivariable, nonlinear, absolutely unsteady system, and research inverted system to theoretically have the profound meaning with methodology up. Researching on inverted pendulum can return the knot for nonlinear, multi-variable, absolutely unsteady system, method of its control is with the way of thinking right handle the general industry process, and also has the extensive use. There are abundant and complicated dynamical behaviors in the inverted pendulum system, such as the local and global bifurcations and the chaotic dynamics. In this dissertation, we investigate the nonlinear dynamics in the inverted pendulum system. The two-degree-of-freedom nonlinear system with cubic nonlinearities will be used to explore the bifurcations and chaotic dynamics in the inverted pendulum system. The results obtained by the dissertation show that there exist the chaotic motions in some parameter regions.The research contents and the major results obtained in this dissertation are as follows.(1) We give a review on the researches for the inverted pendulum system. The applications and developments of the inverted pendulum system in recent years. In particularly, we present the results for the studies of nonlinear dynamics in the inverted pendulum system. We also point out the trend of the inverted pendulum system in future and the necessity of the study on the nonlinear dynamics of the inverted pendulum system,and introduced the method to research invterted pendulum systems.(2) With Lagrange Equations, We establish two-degree-of-freedom nonlinear equations of motion for the invterted pendulum. By using feedback control theory, we simplify equations of motion for inverted pendulum, and then we obtain the dimensionless equations of notion for inverted pendulum.(3) We investigate the nonlinear dynamics of the inverted pendulum system in 1/1 internal resonance and 1/2 sub harmonic resonance. Using the multiple method of scale, the averaged equations of the inverted pendulum system are obtained. The amplitude-frequency response equations and the local bifurcations are respectively analyzed in the two resonant cases. The numerical simulations are given to obtain the amplitude-frequency response curve.(4) We discuss the control methods of inverted pendulum system based on modern control theories and intelligent control theories. Simulation research on the stability of inverted pendulum has been done using MATLAB software.Numerical simulation has been done to averaged equations and original system(5) equations using DYNAMICS software, we obtain the chaotic response of the averaged equations and original system equations. These mean that the chaotic motions can occur in the inverted pendulum system in some parameter regions, and it is also found from the numerical simulation that the chaotic responses are very sensitive to initial conditions.