Research On Nonlinear Control Technique For Spherical Mult-rod Inverted Pendulum System

In this paper, some control methods for mult-rod inverted pendulum system are studied and the physical structure of inverted pendulum system is optimized, which is aimed to implement the stabilization of the spherical quadruple inverted pendulum sys-tem. Firstly, the mathematical model of spherical n-rod inverted pendulum is derived; the variable gain linear quadratic regulator(VGLQR) control technique is proposed, and control method for filtering the white noise from the inverted pendulum control system is discussed. Secondly, the swing-up control of nonlinear double inverted pendulum un-der restricted rail length is realized based on VGLQR control technique. Thirdly, the swing problem of triple inverted pendulum is theoretically discussed, and the simulation for swing-up of a triple inverted pendulum is achieved. Finally, the qualitative analy-sis of spherical quadruple inverted pendulum is discussed; the structure of the spherical quadruple inverted pendulum system is designed and optimized; the spherical quadruple inverted pendulum is firstly stabilized in the world.The main works in the thesis are as follows:1. The detailed process of inferring the mathematical model of spherical single/ double/triple inverted pendulum is proposed, and the model of spherical n-rod inverted pendulum(SNIP) is conclude. In addition, the model of the spherical quadruple inverted pendulum is derived via the Lagrangian method, by which the correctness of the mathe-matical model of SNIP is Verified.2. For a class of autonomous nonlinear control systems, the VGLQR control tech-nique is proposed, and the controller of VGLQR is successfully applied to stabilizing a multi-rod inverted pendulum. This method is a kind of gain-scheduled feed-back control technique based on states variables. The main feature of VGLQR is to obtain the Jacobi matrix and resolve the algebraic Riccati equations at each sampling time online, and a more precise real-time feedback gain matrix, which is changing with respect to states, is obtained. Based on VGLQR technique, the controller of LQG is improved, and the results of simulation and experiments illustrate the new LQG controller is of significant performance.3. The VGLQR control technique is applied in control systems for swing-up inverted pendulum. At first, based on VGLQR technique, the swing-up of a double inverted pendulum under restricted rail length is realized. Then, the inversion-based controller is improved, and the simulation experiment for swinging-up of the triple inverted pendulum is achieved. 4. Physical experiment for the spherical quadruple inverted pendulum is imple-mented. Because the rod of ordinary inverted pendulum can only rotate in a plumb plane, and the rod of spherical inverted pendulum can free rotate in three-dimensional space, the degrees of freedom of the spherical inverted pendulum are as twice as that of ordinary inverted pendulum. Therefore, both in theory and practice, it is more difficult to stabilize a spherical inverted pendulum than to stabilize a ordinary inverted pendulum because not only the factors of state variables, nonlinearity and instability fold increase in spherical inverted pendulum systems but also great difficult is encountered for the design of physical construction and the component selection. In this paper, the qualitative anal-ysis of spherical quadruple inverted pendulum is discussed, and the physical construction, which is matched with the mathematical model, is designed. Finally, the physical exper-iment is very well implemented, and the result illustrates that it not only has quite good stability and robustness, but also is able to make the cart of the pendulum moving to the place where it is appointed in advance.