Anti-windup Control Method For Linear Unstable Systems

According to the input saturation problem existing in Liu,Ne???i??? and mareels' research on the stabilization and tracking control of space inverted pendulum[13],and the cooperative foundation of Zaccarian,Teel,Ne???i??? and mareels et al for more than 20 years,the major breakthrough results on anti-windup control research are summarized[1,2,24].Therefore,this paper makes further research on the stabilization part of the input saturation problem in stabilization and tracking control of space inverted pendulum,following Zaccarian and Teel et al the founders of the anti-windup control theory.This topic mainly studies the stabilization problem of anti-windup control for linear unstable systems with input saturation.Under the condition of ignoring input saturation,a state feedback H_? controller is designed for a linear unstable systems to ensure the overall convergence rate of the closed-loop system,so that the closed-loop system meets the finite-gain L₂ stability and ensures the overall convergence rate of the closed-loop system.However,when the input saturation occurs,the transient response performance of the system will be affected,thus the stability of the system will be destroyed.Therefore,this paper proposes an analysis and design method of anti-windup controller,which can maintain good transient performance in the domain of attraction.However,through analysis,it is found that this method improves the transient performance by slightly sacrificing the domain of attraction,and then it is found that there is a compromise between expanding the domain of attraction and improving the transient performance in the synthesis of anti-windup controller.In addition,in order to objectively evaluate the compensation effect of anti-windup control,this paper proposes a more intuitive evaluation index of anti-windup control.The effectiveness of the anti-windup control method is verified by numerical simulation and analysis.Finally,the anti-windup control method for continuous systems is extended to discrete systems,and rigorous theoretical proof is given.Numerical simulation and analysis verify the effectiveness of the anti-windup control method for discrete systems.The main innovations of this paper are as follows:Firstly,this paper proposes a state feedback H_? controller design method to ensure the overall convergence rate for the stabilization control of linear systems with external disturbances,by using the theory of input-output stability and the concept of convergence rate,so that the closed-loop system is finite-gain L₂ stable.This design method not only guarantees the robustness of the closed-loop system but also guarantees the convergence rate of the system.Secondly,the anti-windup controller is designed to make the closed-loop system asymptotically stable for the stabilization of linear unstable systems with input saturation.Based on Lyapunov stability theory,it is proved that input saturation makes the closed-loop systemchange from the global asymptotic stability to the local asymptotic stability.In addition,this paper analyzes the influence of input saturation on the stability and performance of the system,and proposes an estimation method of the domain of attraction and domain of divergency,which intuitively reflects the relationship between the maximum allowable amplitude of saturation constraint and the size of the domain of attraction.The numerical simulation and analysis show that this anti-windup method improves the transient performance by sacrificing domain of attraction,and then the compromise problem of enlarging the domain of attraction and improving the transient performance is found in the synthesis of the anti-windup controller.Thirdly,in order to improve the transient performance of the input saturated linear unstable system with external disturbance in as possible as large domain of attraction,this paper proposes a new anti-windup controller design method and proves that the closed-loop system satisfies the finite-gain L₂ stability.In addition,an evaluation index of anti-windup control is proposed.The effectiveness of the anti-windup controller design method is verified by numerical simulation and analysis.Fourthly,this paper extends the proposed anti-windup controller design method to the discrete anti-windup controller design method for the stabilization of input saturated discrete linear unstable systems with external disturbances,and gives a rigorous proof process.Numerical simulation and analysis verify the effectiveness of the anti-windup controller design method for discrete systems.