A fundamental matrix based approach to design predictor-based control laws for discrete-time systems with state and input delays

In this paper, the concept of fundamental matrices is generalized to discrete-time systems with state delays, and the commutativity-like property is derived. With the aid of fundamental matrices, an explicit expression of state responses is obtained for discrete-time systems with state delays. Then, a prediction scheme is constructed to predict the future states of discrete-time linear systems with delays in both states and inputs. Further, a predictor-based feedback law is presented to stabilize the considered delayed systems. It is revealed that the characteristic equation of the closed-loop system under the presented predictor-based feedback control law is the same as that of the closed-loop system of the corresponding system without input delays under the delayed state feedback. Two numerical examples show the effectiveness of the proposed approach. In this paper, it is illustrated that the proposed fundamental matrices of discrete-time systems with state delays are the basis of a prediction scheme for solving the problem in the presence of delays in both states and inputs.