This work studies the delayed output feedback (DOF) of discrete-time systems with input and output delays that are arbitrarily large, bounded, and exactly known. The significance of the DOF is that only the present and delayed inputs and outputs are used for feedback. For systems with a single input delay and multiple output delays, the idea of state prediction is utilised to construct the DOF controller. While for systems with multiple input and output delays, the DOF controller is designed by firstly reducing the original system to a delay-free one. It is shown in both cases that the closed-loop systems controlled by DOF are asymptotically stable and act like delay-free systems with the same dimensions as open-loop systems. Compared with the continuous-time case, the DOF controllers for discrete-time systems do not contain distributed delay terms and the implementation problems can be avoided. The DOF is used to stabilise the spacecraft rendezvous system with input and output delays and simulations show the effectiveness of the proposed approach.

In this study, the authors study the input delay compensation problem for discrete-time linear systems with both state and input delays. Under the assumption that the original time-delay system without input delay can be stabilised by state feedback, a nested predictor feedback controller is established to predict the future states such that the arbitrarily large yet exactly known input delay in the original system is completely compensated. Consequently, it is shown that the closed-loop system consisting of the original time-delay system and the nested prediction feedback controller is asymptotically stable. Under an additional assumption, an explicit nested predictor feedback controller without involving any nested summations is also established. Finally, two numerical examples are carried out to illustrate the obtained theoretical results.