Research On Stability Of The Parametric Inverted Pendulum

Inverted pendulum is a kind of important mechanical system, it is typically a nonlinear, strong coupling, non-autostability system, in practice, it is widely used in aerospace, mechanical design, artificial intelligence and many other fields. In the scientific research, it is an important model for testing various control method and mathematical methods. For such a system which is unstable in natural, to stabilize it, we could use control means such as changing external conditions and applied excited force, which is at last attributed to a parametric model, and we could study the relationship between the parameters of stability. In this process, many key problems can be effectively reflected, and there is intuitive effect on verifying the correctness and effectiveness of many mathematics, physics and control theory.In this paper, we focus on the following several aspects:Firstly, this paper introduces the developing history of the pendulum model and the present situation, and inverted pendulum model and its stability significance.Secondly, this paper introduces the concrete meaning of stability, lists the different stability concepts, and gives concrete examples to show the relationship and differences between them.Thirdly, this paper expounds the Liapunov methods, gives the specific theorem of judgment of stability and unstability. Then, this paper studys the stability conditions of periodic coefficient linear differential equations with stability theory and Floquet theory.Lastly, the stability theory of periodic coefficient linear differential equations are applied to a class of parametric inverted pendulum model. By using the method of numerical integration, the approximate stability boundary curves on the parameters of undamped inverted pendulum is obtained. In dealing with the damped case, we take the same method as we’ve used on undamped inverted pendulum and obtain the stability the boundary surface on the parameters of the inverted pendulum. Through numerical simulation, we compares the corresponding system trajectory of the points of the stability and unstable region. The results of the numerical test shows that our stability boundary has a good differentiation. Finally, in case that the length of the pendulum is long, by using variable subsystem to simplify the calculation, we also get a reliable stability boundary.