Controller Design And Stabilization For One-Dimensional Distributed Parameter Systems With Time-delay Inputs

The phenomenon of time-delay exists widely in various engineering systems,which usually degrades the performance of the systems.Therefore,the anti-delay problem has become a hot and difficult point in modern control theory.In this paper,the backstepping method is applied to distributed parameter systems such as the string-payload system and the Schrodinger equation.The design of the controller against input delay and the stability of system are discussed.The research contents of this paper are summarized as follows.Via the Volterra integral theory,the backstepping method is applied to the distributed parameter system with a bounded control operator and a time-delay input,an anti-delay state-feedback law is proposed and the stability of the closed-loop system is analyzed.Based on duality theory,the realization and calculation method of controller are given.The idea of this design is to compensate for the effect of delay input by transferring the time-delay phenomenon to the dual system,so that the backstepping method is reinterpreted from the perspective of duality theory.The design principle and calculation implementation are applied to string system with a payload and delay input.Numerical simulation shows that the controller design is effective and feasible.By the way of the admissibility theory of operators and the regularity theory of system,the above method is extended to time-delayed systems with unbounded control operators.It is difficult to prove that the input signal is L2 and that the backstepping transformation is bounded and invertible.Thus,a state feedback control law is presented and the stability of the resulting closed loop system is proved.By constructing dual system,the realization and calculation method of controller are provided.Taking the Schrodinger system with a delay input as an example,the simulation results show that the proposed control law is effective.