Stabilization Of Distributed Parameter Systems With Disturbance Or Time Delay

In the practical engineering,the control problems of distributed parameter system with disturbance and delay have become the hotspot research in recent years,and this problem has an important role for practical engineering application.In this paper,we mainly studied the second type of thermal elastic system and Timoshenko beam,considered the control problems of the thermoelasticity of type ? with uncertain disturbances in the boundary input and the Timoshenko beam with internal delay,and gave the well-posedness and the stability of corresponding closed-loop system.Firstly,we considered the stability for a thermoelastic system of type ? with nonuniformly bounded external disturbances in the boundary control.Using the active disturbance rejection control,we designed extended state observers with the timevarying high gains to estimate the disturbances,which are different from the previous disturbance observer with the constant,and this estimator improved the accuracy of the disturbance estimation.In addition,the disturbances are canceled by the feedback controllers.Then,we proved that the closed-loop system is polynomially or exponentially stable in terms of Lyapunov method.This analysis improved the results of the present studies which are only bounded stable.Besides,we investigated the solvability of the closed-loop system by the dual theory and the theory of admissibility.Secondly,we analyzed the existence of the state observer of Timoshenko beam with internal delay,and discussed the interval of the system observer which are the conditions the system internal parameters should satisfied,while the system observer exists.In this paper,we transformed the problem of observer into the well-posedness and stability of the error system.The existence and uniqueness of the solution of the error system is analyzed by the linear operator semigroup theory.Using the Lyapunov function method,we constructed a Lyapunov functional for the error system,transformed the exponential stability issue of error system into the inequality equations,and obtained the region of the exponential stability of the system.Finally,we illustrated the stability range of the system through the numerical simulation which means the interval of system state observer.