Input-delay Compensation And Fixed-time Stabilization By Periodic Delayed Feedback For Linear Systems

Time-delay widely exists in practical engineering,such as network control system,communication system,process control system and so on.Because the time-delay system is essentially an infinite-dimensional system with infinitely many poles,and the control of the time-delay system with finite control parameters is a very complicated reduced-order control problem in nature.Therefore,the analysis and design of time-delay systems have received extensive attention from scholars at home and abroad in recent decades.On the one hand,time-delay compensation is a natural control strategy,which is to compensate the influence of time-delay by predicting the future state of the system through the predictor;on the other hand,the system can also have some ”unexpected” system performance by the introduction of additional delay,this is the delayed feedback strategy.These two methods complement each other and together form the main theoretical basis of this paper.In this paper,the time-delay compensation problem of linear systems with both input and state delays is studied firstly,and a time-delay system control strategy based on pseudo-predictor is proposed.The proposed controller does not involve memory terms of input signal,thus avoiding the security problem of numerical implementation theoretically.Furthermore,this paper proposes a periodic delayed feedback control strategy for the input time-delay system.By introducing an additional time-delay,the closed-loop system is transformed into a finite-dimensional periodic system on the basis of compensating the time-delay of the original system.The assignment of the monodromy matrix of the closed-loop system achieves the fixed-time stabilization the closed-loop system.The proposed controller is a linear controller,and the settling time is independent of the initial value of the system.At this point,there is an essential difference with the existing fixed-time stabilization control method.The main work contents of this paper include:Chapter 2 considers the stabilization problem of linear systems with both state and input delays.Under the assumption that the original system can be stabilized when there is no input time-delay,a time-delay compensation method based on an artificial closed-loop system is constructed by using the concept of the fundamental matrix,and the arbitrarily large bounded input time-delay is compensated.The obtained stabilizing controller does not contain the memory term of the input signal,thus avoiding the problem of numerical implementation theoretically.In this method,the stability of the original time-delay system is equivalently transformed into the stability of an integral delay system.When the original system has commensurate time-delay,an optimization-based controller gain design method is given.Chapter 3 considers the fixed-time stabilization problem of time-delay systems.A piecewise constant periodic delayed feedback control strategy is proposed.By switching the control gain to different values several times in one period,the closed-loop system is transformed into a finite-dimensional periodic system.Based on the controllability matrix,a method that can achieve the arbitrary assignment of the monodromy matrix of the closed-loop system is given,and a fixed-time stabilization strategy for the input timedelay system is given by the constructive method.Furthermore,combining the proposed periodic delayed feedback control method with the predictor,a predictor-based periodic delayed feedback control strategy is proposed,which can achieve more ideal system performance.On the basis of Chapter 3,Chapter 4 considers the problem of finite-time stabilization of linear time-delay systems with immeasurable states.Three different observer-based periodic delayed feedback controllers are proposed.The proposed observer-based periodic delayed feedback control method solves the open-loop problem faced by the standard periodic delayed feedback method on the one hand,and on the other hand,because the observer-predictor does not contain a memory term of the input signal,thus numerical implementation issues are avoided.Chapter 5 considers fixed-time functional observers based on periodic delayed feedback,including the design of reduced-order observer and dual observer-controller.In a fixed form of observer,the requirement of the system state is transformed into a problem of solving a set of algebraic matrix equations.The proposed method is more procedural and avoids the analysis and design of the system structure,which is more conducive to computer-aided design.On this basis,a dual observer design method based on model decomposition is given,revealing the design ideas and principles behind the dual observercontroller.Chapter 6 considers the fixed-time rendezvous control problem of an elliptical orbit spacecraft based on the idea of period delayed feedback control,and gives an explicit algorithm to implement the spacecraft rendezvous controller.The proposed controller is continuously differentiable and can even be smooth.Under the condition that the settling time independent of the initial value of the system,the proposed controller also avoids the singular value problem faced by general high-gain feedback controllers.