Predictor Feedback For Discrete-time Linear Systems With Multiple State And Input Delays

Time delay phenomenon exists widely in various practical engineering systems.Therefore,in the past decades,the analysis of time-delay systems has been the focus of attention in various fields.Time delay system is essentially a kind of infinite dimensional system,which brings great difficulties to system analysis and design.The problem of predictive feedback controller design for discrete-time linear systems with multiple states and input delays is considered in this paper.Different from general discrete time-delay systems,the main characteristic of this system is that the input and state delays of the system are not single.In this paper,the above system is divided into two cases to study.Firstly,the design of discrete-time linear systems with state delay greater than input delay is considered.By introducing new state variables,an equivalent first-order augmented state space model is constructed,and an effective predictor feedback is designed.Finally,the change of closed-loop poles is discussed.The problem of pole placement for discrete linear systems with multiple states and input delays is transformed into solving a linear matrix equation satisfying certain conditions.It is proved that the pole placement problem is solvable if and only if the solution of the corresponding linear matrix equation is nonsingular.In this paper,a parameterization method is adopted.The essence of this method is to use the right coprime decomposition of the original system to obtain the explicit solution of the linear matrix equation,and to ensure that the control law has sufficient degrees of freedom.Finally,the feasibility of this method is verified by numerical examples.Two different design methods are proposed for systems with input delay greater than state delay.The first method is based on generalized predictive control and selects new state variables to transform the original system into a system without time delay.The linear matrix equation similar to the original system can be solved by a parameterization method based on arbitrary matrix.The second method is to extend the time delay of the original system and extend the iterative method to this case.Two design methods are used to find the effective predictive feedback controller in the simplest case.According to the advantages and disadvantages of the two algorithms,the practical analysis is given.In this paper,a complete algorithm for pole placement problem of discrete linear systems with multiple states and input delays is summarized,which makes the solving process of this kind of problem simple and clear and easy to apply.Finally,different methods are used to simulate,and the results are compared to verify the effectiveness of the method.