Modeling And Robust Control Of The Inverted Pendulum System

The inverted pendulum system is an inherently unstable complex nonlinear system with a simple physical structure and intuitive and clear experimental results,which can be used to evaluate the effectiveness of various control theories.Because the inverted pendulum system has complex characteristics such as nonlinearity and inherent instability,it is easily affected by uncertain factors such as model error,measurement error and external interference,which makes the control of the system very difficult.Therefore,the robust right coprime factorization method based on operator theory is comprehensively used in this paper,and the robust control and tracking control of the inverted pendulum system with uncertainty are studied.The main works of the paper are as follows:(1)An inverted pendulum dynamics model based on operator theory is established.Firstly,on the basis of the working principle of the linear double inverted pendulum,the dynamic differential equation of the inverted pendulum system is obtained by using the Lagrange equation method and linearization it at the equilibrium point.Then,aiming at the difficulty of algorithm design of complex multi-variable inverted pendulum system in traditional research,the dynamic differential equation is factorized by right coprime using operator theory.The inverted pendulum dynamic model with simple structure,which can be controlled and analyzed easily,is obtained.(2)A self-swing control scheme combining passive control method and robust right coprime factorization method based on operator theory is proposed.Aiming at the double inverted pendulum dynamics model based on operator theory modeling with limited guide rails,the passive control method and the robust right coprime factorization method are combined to design the switching control law for start-up and steady-swing.Thus,the robust stability of the two-stage inverted pendulum system from start-up and steady-swing is realized.Meanwhile,based on the scheme,the start-up and steady-swing control of the first-stage inverted pendulum is completed.The experimental results show that the scheme realizes rapid control of the self-starting pendulum of the inverted pendulum system,and its effectiveness has been well verified.(3)A robust tracking control scheme combining the robust right coprime factorization method based on operator theory,observer and tracking compensator is proposed.Aiming at the uncertain factors existing in the inverted pendulum system,firstly,the system is analyzed based on the operator theory,and the robust controller,the observer and the tracking compensator are designed to ensure the robustness and stability of the system.Secondly,according to the definition of Lipschitz norm and Unimodular operator,the sufficient conditions of the system tracking performance are deduced,and the accurate tracking of the system is realized.Finally,the feasibility and effectiveness of the control scheme are verified by MATLAB simulation.The simulation results show that,compared with the traditional linear quadratic optimal control method,the uncertainty of inverted pendulum system is effectively suppressed and the control accuracy is improved.While ensuring the robustness and stability of the inverted pendulum system,the precise tracking of the swinging process of the pendulum rod is realized.