Linear discrete-time periodic systems are the simplest and most important time-varying systems.In the study of linear discrete systems,the design of state observer has always been a hot topic.Many scholars began to explore the design of state observer and continuously optimize it,and at the same time,more and more researchers have applied it in practice.However,for the study of the state observer of linear discrete periodic systems,the research results are very few at present,and the methods and algorithms used need to be further improved.In this paper,the design of controllers and state observers for linear discrete-time systems is studied,and the theory is applied to dc linear motors.This article mainly consists of the following parts:First,the design and robustness of state observer based on LDP system are discussed.Based on the well-known CG algorithm for matrix equation Ax = b as well as applying the lifting technique and algebraic operations,an iterative algorithm for both periodic observer gains and periodic state feedback gains can be generated simultaneously.By optimizing the free parameter matrix in the proposed algorithm,a robust stabilization algorithm based on periodic observer for LDP systems is presented.At the same time,a robust periodic controller and periodic state observer are designed.Second,the design of the reduced-order state observer for the LDP system is studied.In case that the linear discrete-time periodic system is observable,an algebraic equivalent system is obtained by non-singular linear transformation,and the partial states to be observed are separated simultaneously.Then the considered problem is transformed into the problem of solving a class of periodic Sylvester matrix equation and an iterative algorithm for periodic reduced-order state observers design is derived.In addition,robust consideration based on periodic reduced-order state observers for LDP systems is also conducted,and a robust periodic reduced-order state observer is designed.Based on the study of state observers for linear discrete periodic systems,a linear discrete periodic controller and a full-dimensional state observer are designed to ensure the dynamic tracking error and steady-state accuracy of DC linear motors.Firstly,a mathematical model is established for the DC Linear Motor,and the internal constant parameters are fixed to obtain a simplified mathematical model.Through a series of transformations,we obtain the state space expression of the DC Linear Motor.Then we discrete it and use the period controllaw to get a linear discrete period system.Using the iterative algorithm of Chapter 2,the feedback gain of the controller and the gain of the state observer can be obtained.Through simulation,it is proved that the designed controller and the full-dimensional state observer can ensure the dynamic tracking error and steady-state accuracy of the DC Linear Motor. |