The Control And Research Of Single-Rotational Inverted Pendulum Based On H-infinity Theory

Inverted pendulum is an ideal didactical experiment apparatus of study on automation control theory.It is clear when it is used to illustrate many abstract ideas in control such as stabilization of system,controllability,disturbance resistance etc. However,inverted pendulum is a control system,with the feature of high order, multivariable,non-liner,strong coupling,rapid movement and unstable by nature.So the control of inverted pendulum is a very complex,nonlinear and unstable control problem with the high request for veracity and celerity.Obviously the study of a typical,nonlinear and unstable system has the more important meaning on both the theory and methodology.In this paper,the theory of robust control is introduced at begin.The H_∞standard control problem and state-feedback control method are research and discussed in detail.The problem of sensitivity minimization,robust stabilization and mixed sensitivity and the designing method of H_∞robust controller of system with parameter uncertainty are systematically summarized.The H_∞robust controller design of inverted pendulum is summed up as mixed sensitivity problem.Based on this method,the H_∞state-feedback controller of single-rotational inverted pendulum system is designed.During the process of designing H_∞controller of inverted pendulum,the paper analyzes the movement of inverted pendulum form the movement of the rigid body first.The linear mathematical model of inverted pendulum is constructed by mechanical Equation.Then,the state space realization of general object is gained according to the data of actual plant and weight functions W₁(s),W₂(s),W₃(s) which are reasonably elected.The problem is that finding the K stabilize the closed-loop system and minimize the infinitely great norm of the objective functionΦ=[W₁S W₂R W₃T]^T.Last,the value of K can be got by resolving Riccati equation.Using this approach,the key is to determine the values of the C₁,D₁₂ matrixes which is relative with how to choose the weight functions W₁(s),W₂(s),W₃(s).So the performance of H_∞controller is directly influenced by weight functions.In this paper,a programming method in Matlab environment is introduced,which can resolve Riccati equation and work out the H_∞controller.Simulation is researched in this paper.The results of simulation illustrate that the stable time of system with the H_∞optimum controller is shorter than that with the LQG(Linear Quadratic Regulator)controller when there is no parameter perturbation, also amplitude is smaller.Although both H_∞state-feedback control and LQG control can stabilize the inverted pendulum system when less model parameter perturbation happens.But the performance of LQG control degrades,such as the stable time becoming long,the times of surge increasing and the amplitude aggravating.The LQG control system is no longer stable when the perturbations reach a certain quantity but the H_∞state-feedback control can still keep the system stabilized commendably.The output responses of inverted pendulum system whose mechanical parameters disturb in a certain spectrum are analyzed in four kind instances.The results of simulation illustrate that inverted pendulum system can reach stabilization finally under the H_∞state-feedback control.So the conclusion is that H_∞state-feedback control has stronger robust stability and anti-jamming capability than LQG control.